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Number Sense Teacher’s Guide Workbook 3:2 1 Copyright © 2007, JUMP Math For sample use only – not for sale.

NS3-51 Ordinal Numbers Goal: Students will use ordinal numbers correctly.

Prior Knowledge Required: Number lines Counting

Vocabulary: ordinal numbers, position, ordinals from first to tenth.

Teach students that ordinal numbers are used to tell the position of objects.

Write the ordinal words from first to tenth on the board in order. Leave a lot of space between words. Ask

volunteers if they know what the words mean.

Hand out cards numbered 1 to 10 to ten volunteers. Ask the volunteers to line up in the order of their

number, underneath the appropriate word. The student with “1” should stand under the word “first” and so

on. Tell the class to pretend that these students are in line for tickets and ask for examples of what the

tickets might be for (a sporting event, a play, a movie theatre, and so on). Ask various volunteers to

demonstrate that they know where they are in line. EXAMPLES: Will the second person please clap their

hands? Will the 6th person please take one step forward and then one step back? Will the 8th person please

turn around in a complete circle?

Ask the class to tell you who is 3rd in line. Who is 4th? Who is 9th? Who is first? Who is last? How many

people are before the 4th person in line? How many are after the 8th person? Use your students’ names to

ask a question like “How many places before Mark is Sara?” Demonstrate counting back from Mark’s place

in line to Sara’s. SAY: “Tony is 1 place before Sara, Lisa is 2 places before Sara, Bilal is 3 places before

Sara” and so until you reach Mark.

ASK: How many are before the 8th person and after the 4th person? How many people are between the 2nd

and 4th people? Between the 2nd and 5th people? The 2nd and 7th? The 5th and 9th? Tell students that Sally

finds the number of people between the 2nd and 4th people by finding 4 – 2, so she says that there are 2

people between the 2nd and 4th people—is she right? How much is she off by? Then repeat for the other

examples: the 2nd and 5th people, the 2nd and 7th, the 5th and 9th. How much is Sally off by each time? How

can you change Sally’s answer to make it right? Then ask students to see if this strategy works for various

other pairs of small ordinal numbers. (EXAMPLES: 3rd and 8th, 2nd and 9th, 3rd and 6th, 6th and 9th. 7th and 9th,

8th and 9th.) Then challenge students to extend this pattern to larger ordinal numbers. For example, if there

are 30 people in line, have students find the number of people in between the 14th and the 28th by subtracting

and then subtracting 1, or between the 3rd and the 25th, and so on.

Review the words “vowel” and “consonant.” Then ASK: What is the 4th letter in the alphabet? What is the 4th

vowel in the alphabet? What is the 3rd letter in the word “Montreal”? What is the 3rd vowel in the word

“Montreal”? Which two letters are the same in the word “apple”? (Have students phrase their answer in terms

of ordinal numbers, i.e. 2nd and 3rd). Have students find two letters (and describe them by their ordinal

position) that are the same in each word:

a) moon b) sleep c) penny d) counting


Bonus: Make up a word that has 2 letters the same and describe the position of those letters. As an extra

challenge, make up a word with 3 letters the same and describe the position of those letters.

Have students count by 5s, starting at 5. ASK: What is the 3rd number you say? What is the 7th number you

say? Teach students to keep track by using their fingers.

ASK: What is the 4th letter you say if:

a) You say the alphabet starting from H?

b) You say the alphabet starting from V?

c) You say the alphabet backwards starting from T?

d) You say the vowels starting from A?

What is the 7th number you say if:

a) You count by 2s starting at 8?

b) You count by 2s starting at 17?

c) You count by 5s starting at 35?

d) You count by 5s starting at 49?

e) You count by 100 starting at 433?

Then draw a number line from 0 to 30, labelled with only the multiples of 10. Show students how to use the

number line to find the 7th number they say when counting by 3, starting from 3. Students can mark 3 with an

X, mark every third number with an X and then count to find the 7th X.

Activities:

1. Working in Pairs

For a random way to pair up students, give half the class cards with numbers (say 1 to 10 and 21 to 25

if there are 30 students in the class) and give the other half cards with ordinal number endings put st,

nd, rd, and th in quotation marks “st”, “nd”, “rd”, and “th” (st (2 cards), nd (2 cards), rd (2 cards) and th

(9 cards)). Have students find a partner that matches, for example, 2 matches with “nd” because the

corresponding ordinal is 2nd or second, 4 through 10 all match with “th”. Students will see quickly that

“th” is the easiest to match with.

2. Decoding Messages

Teach students how to use skip counting by 5 to find the position of letters in the alphabet quickly:

A B C D E = 5th

F G H I J = 10th

K L M N O = 15th

P Q R S T = 20th

U V W X Y = 25th

Z


ASK: What is the 22nd letter? The 14th? The 11th? The 19th? The 26th? And then, to make sure they’re paying

attention, ask: The 27th?

Have students decode messages using ordinal numbers. For example, the code for “recess” is:

18th 5th 3rd 5th 19th 19th

Some examples of messages you could encode for your students to decode include:

• To make glue, mix water and flour.

• No math homework today.

• Ordinal numbers are used to tell position.

• Ordinal number words are used for fractions too.

Students could make up a message for a partner to decode.

3. Money

The following is a sample problem from an Ontario Ministry of Education Guide for Teachers. Students

could solve the problem using play money:

Remove the 3rd coin.

Move the last coin into the second place.

Remove the 4th coin.

Move the 5th coin into 1st place.

Remove the second coin.

How much money do you have left?

Extension: Ask students to write the third letter of the word “fast,” the second letter of the word “puppy,”

the last letter of the word “mop,” the most common letter in the word “green” and then the fourth letter in the

word “fourth.” Ask students what word they spelled? Encourage them to make up their own similar puzzles,

either by using their classmates’ names as words or making up their own names.

25¢ 5¢ 5¢ 5¢ 5¢ 10¢ 10¢


NS3-52 Round to the Nearest Tens and NS3-53 Round to the Nearest Hundreds Goal: Students will round to the closest ten or hundred, except when the number is exactly half-way

between two multiples of ten or a hundred.

Prior Knowledge Required: Number lines

Concept of closer

Vocabulary: multiple

Show a number line from 0 to 10 on the board:

0 1 2 3 4 5 6 7 8 9 10

Circle the 2 and ask if the 2 is closer to the 0 or to the 10. When students answer 0, draw an arrow from the

2 to the 0 to show the distance. Repeat with several examples and then ask: Which numbers are closer to

0? Which numbers are closer to 10? Which number is a special case? Why is it a special case?

Then draw a number line from 10 to 30, with 10, 20 and 30 a different colour than the other numbers.

Circle various numbers (not 15 or 25) and ask volunteers to draw an arrow showing which number they

would round to if they had to round to the nearest ten.

Repeat with a number line from 50 to 70, again writing the multiples of 10 in a different colour. Then repeat

with number lines from 230 to 250 or 370 to 390, etc.

Ask students for a general rule to tell which ten a number is closest to. What digit should they look at? How

can they tell from the ones digit which multiple of ten a number is closest to?

Then give several examples where the number line is not given to them, but always giving them the two

choices. EXAMPLE: Is 24 closer to 20 or 30? Is 276 closer to 270 or 280?

Tell students that the multiples of 10 are the numbers they say when they start at 0 and skip counting by 10,

namely 0, 10, 20, 30, and so on. ASK: Is 70 a multiple of ten? Is 130 a multiple of 10? What about 37? How

can you tell whether or not a number is a multiple of 10?

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


ASK: Which two multiples of 10 is 37 between? (30 and 40) How can you tell? How many tens are in 37? (3)

What is one more ten? (4) So 37 is between 3 tens and 4 tens. How many tens are in 97? (9) What number

has exactly 9 tens? (90) What is one more ten than 9 tens? (10 tens) What number has 10 tens? (100) What

two multiples of 10 is 97 between? (90 and 100) How many tens are in 794? (79) Show students how they

can cover up the ones digit to find the number of tens. What is one more ten than that? (79 + 1 = 80 tens).

Which two multiples of ten is 794 between? (790 and 800)

Ensure that all students can tell you which number is another way of saying:

a) 54 tens b) 3 tens c) 10 tens d) 99 tens e) 100 tens f) 1430 tens

Have students find which two multiples of 10 the following numbers are between:

a) 53 b) 153 c) 671 d) 809 e) 998 Bonus: f) 789 432 g) 12 349 087

Then have students round each number to the nearest ten. Explain that to round a number to the nearest ten

means to find the multiple of ten that the number is closest to.

STEP 1: Decide which two multiples of ten the number is between.

STEP 2: Look at the ones digit to decide which multiple of ten the number is closest to.

a) 327 b) 411 c) 32 d) 48 e) 196 Bonus: 53 098 006

Tell students that when you round a 3-digit number to the nearest ten, you usually get a 3-digit number.

Challenge your students to find an exception. (The exceptions are 995, 996, 997, 998 and 999—995 is not

closer to either 990 or 1000; this case will be discussed in the next lesson)

Repeat the lesson with a number line from 0 to 100, that shows only the multiples of 10.

0 10 20 30 40 50 60 70 80 90 100

At first, only ask students whether numbers that are multiples of 10 (30, 70, 60 and so on) are closer to 0

or 100. (EXAMPLE: Is 40 closer to 0 or 100? Draw an arrow to show this.) ASK: Which multiples of 10

are closer to 0 and which multiples of 10 are closer to 100? Which number is a special case? Why is it a

special case.

Then include numbers that are not multiples of 10. First ask your students where they would place the

number 33 on the number line. Have a volunteer show this. Then ask the rest of the class if 33 is closer to 0

or to 100. Repeat with several numbers. Then repeat with a number line from 100 to 200 and another

number line from 700 to 800.

Review the word “multiple” and ASK: If a multiple of 10 means “the numbers you say when skip counting by

10s starting from 0,” what do you think a multiple of 100 is? Say various numbers and have students tell you

whether each number is a multiple of 100.

(EXAMPLES: 320; 1 500; 78 000; 341; 12 341; 12 300; 890)

ASK: How can you tell whether or not a number is a multiple of 100?

Remind students that to find the number of tens, we can cover up the ones digit and read the number we

see. ASK: How can we find the number of hundreds in a number? What digits should we cover up? (Cover

up the ones and tens digits.)


Have students find the number of hundreds in various numbers.

(EXAMPLES: 349; 890; 1 954; 39 876 421)

ASK: Once we’ve found the number of hundreds in a number, how can we find the two multiples of a

hundred the number is between?

Have students decide which two multiples of 100 the above examples are between.

Ask your students for a general rule to tell which multiple of a hundred a number is closest to.

What digit should they look at? How can they tell from the tens digit which multiple of a hundred a number is

closest to? When is there a special case? Emphasize that the number is closer to the higher multiple of

100 if its tens digit is 6, 7, 8 or 9 and it’s closer to the lower multiple of 100 if its tens digit is 1, 2, 3 or 4. If

the tens digit is 5, then any ones digit except 0 will make it closer to the higher multiple. Only when the tens

digit is 5 and the ones digit is 0 do we have a special case where the number is not closer to either.

Activity: Attach 11 cards to a rope so that there are 10 cm of rope between each pair of cards. Write the

numbers from 30 to 40 on the cards so that you have a rope number line. Make the numbers 30 and 40 more

vivid than the rest. Ensure that the numbers are stuck to the rope so that they cannot move. Take a ring that

can slide freely on the rope and pull the rope through it.

Ask two volunteers to hold the number line taught. Ask a volunteer to find the middle number between 30

and 40. How do you know that this number is in the middle? What do you have to check? (the distance to the

ends of the rope—make a volunteer do that). Let a volunteer stand behind the line holding the middle.

Explain to your students that the three students with a number line make a rounding machine. The machine

will automatically round the number to the nearest ten. Explain that the machine finds the closest ten. Put the

ring on 32. Ask the volunteer who is holding the middle of the line to pull it up, so that the ring slides to 30.

Try more numbers. Ask your students to explain why the machine works.


NS3-54 Rounding Goal: Students will round whole numbers to the nearest ten or hundred.

Prior Knowledge Required: Knowing which two multiples of ten or a hundred a number is between

Finding which multiple of ten or a hundred a given number is closest to

Vocabulary: rounding, multiple

Review rounding to the nearest ten when the ones digit is not 5.

Then tell your students that when the ones digit is 5, it is not closer to either the smaller or the larger ten,

but we always round up. Give them many examples to practice with: 25, 45, 95, 35, 15, 5, 85, 75, 55, 65. If

some students find this hard to remember, you could give the following analogy: “I am trying to cross the

street, but there is a big truck coming, so when I am part way across I have to decide whether to keep going

or to turn back. If I am less than half way across, it makes sense to turn back because I am less likely to get

hit. If I am more than half way across, it makes sense to keep going because I am again less likely to get

hit. But if I am exactly half way across, what should I do? Each choice gives me the same chance of getting

hit.” Have them discuss what they would do and why. Remind them that they are, after all, trying to cross

the street. So actually, it makes sense to keep going rather than to turn back. That will get them where they

want to be.

Another trick to help students remember the rounding rule is to look at all the numbers with tens digit 3 (i.e.

30-39) and have them write down all the numbers that we should round to 30 because they’re closer to 30

than to 40. Which numbers should we round to 40 because they’re closer to 40 than to 30? How many are

in each list? Where should we put 35 so that it’s fair?

Then move on to 3-digit numbers, still rounding to the nearest tens: 174, 895, 341, 936, etc.

Bonus: Include 4- and 5-digit numbers.

Then move on to rounding to the nearest hundreds. ASK: Which multiple of 100 is this number closest to?

What do we round to? Start with examples that are multiples of 10: 230, 640, 790, 60, 450 (it is not closest

to either, but we round up to 500). Then move on to examples that are not multiples of 10. (EXAMPLES:

236, 459, 871, 548)

SAY: When rounding to the nearest 100, what digit do we look at? (The tens digit). When do we round

down? When do we round up? Look at these numbers: 240, 241, 242, 243, 244, 245, 246, 247, 248, 249.

What do these numbers all have in common? (3 digits, hundreds digit 2, tens digit 4). Are they closer to 200

or 300? How can you tell without even looking at the ones digit?

Then tell your students to look at these numbers: 250, 251, 252, 253, 254, 255, 256, 257, 258, 259. ASK:

Which hundred are these numbers closest to? Are they all closest to 300 or is there one that’s different?

Why is that one a special case? If you saw that the tens digit was 5, but you didn’t know the ones digit, and

you had to guess if the number was closer to 200 or 300, what would your guess be?


Would the number ever be closer to 200? Tell your students that when you round a number to the nearest

hundred, mathematicians decided to make it easier and say that if the tens digit is a 5, you always round

up. You usually do anyway, and it doesn’t make any more sense to round 250 to 200 than to 300, so you

might as well round it up to 300 like you do all the other numbers that have tens digit 5.

Then ASK: When rounding a number to the nearest hundreds, what digit do we need to look at? (The tens

digit.) Then write on the board:

Round to the nearest hundred: 234 547 651 850 493

Have a volunteer underline the hundreds digit because that is what they are rounding to.

Have another volunteer write the two multiples of 100 the number is between, so the board

now looks like:

Round to the nearest hundred: 234 547 651 850 973

200 500 600 800 900

300 600 700 900 1000

Then ask another volunteer to point an arrow to the digit they need to look at to decide whether

to round up or round down. Ask where is that digit compared to the underlined digit? (It is the

next one). ASK: How do you know when to round down and when to round up? Have another volunteer

decide in each case whether to round up or down and circle the right answer.

Tell students that most 3-digit numbers, when rounded to the nearest hundred, will have 3 digits. Which

numbers will be exceptions? (any number from 950 to 999)

Extension: Ask students to round a 3-digit number to all possible places.

EXAMPLE: 1382

1000

thousands

1400

Hundreds

1380

tens


NS3-55 Estimating Sums and Differences Goal: Students will estimate sums and differences by rounding each addend to the nearest ten or

hundred.

Prior Knowledge Required: Rounding to the nearest ten or hundred

Adding and subtracting

Vocabulary: the “approximately equal to” sign ( ≈ ), estimating

Show students how to estimate 52 + 34 by rounding each number to the nearest ten: 50 + 30 = 80.

SAY: Since 52 is close to 50 and 34 is close to 30, 52 + 34 will be close to, or approximately, 50 + 30.

Mathematicians have invented a sign to mean “approximately equal to.” It’s a squiggly “equal to” sign: ≈.

So we can write 52 + 34 ≈ 80. It would not be right to put 52 + 34 = 80 because they are not actually equal;

they are just close to, or approximately, equal.

Tell students that when they round up or down before adding, they aren’t finding the exact answer, they are

just estimating. They are finding an answer that is close to the exact answer. ASK: When do you think it

might be useful to estimate answers?

Have students estimate the sums of 2-digit numbers by rounding each to the nearest ten. Remind them to

use the ≈ sign.

EXAMPLES:

41 + 38 52 + 11 73 + 19 84 + 13 92 + 37 83 + 24

Then ASK: How would you estimate 93 – 21? Write the estimated difference on the board with students:

93 – 21 ≈ 90 – 20

= 70

Ensure that students can add and subtract multiples of 10 (EXAMPLES: 30 + 20, 70 – 40, 130 – 50). Have

students estimate the differences of 2-digit numbers by again rounding each to the nearest ten.

EXAMPLES:

53 – 21 72 – 29 68 – 53 48 – 17 63 – 12 74 – 37

Then have students practise estimating the sums and differences of:

• 3-digit numbers by rounding to the nearest ten (EXAMPLES: 421 + 159, 904 – 219).

• 3- and 4-digit numbers by rounding to the nearest hundred (EXAMPLES: 498 + 123, 4 501 – 1 511).

Ensure that students can add and subtract multiples of 100 (EXAMPLES: 300 – 100, 600 + 300,

800 – 200)


Bonus: Students who finish quickly may add and subtract larger numbers, rounding to tens, hundreds, or

even thousands.

Teach students how they can use rounding to check if sums and differences are reasonable.

EXAMPLE:

Daniel added 273 and 385, and got the answer 958. Does this answer seem reasonable?

Students should see that even rounding both numbers up gives a sum less than 900, so the answer can’t

be correct. Make up several examples where students can see by estimating that the answer cannot be

correct.


NS3-56 Estimating Goal: Students will solve word problems by estimating rather than calculating.

Prior Knowledge Required: Estimating sums and differences by rounding

Reading word problems and knowing when to add or subtract

Vocabulary: estimating

Tell students that you want to estimate how many apples were sold altogether if 58 red apples were sold

and 21 green apples are sold. Tell students that it would be easier to add these numbers if they were

multiples of 10 and ASK: Are these numbers close to multiples of 10? What is the closest multiple of 10 to

58? (60) To 21? (20) What is 60 + 20? (80) Do you think that 80 is a good estimate for 58 + 21? Will

58 + 21 be close to 60 + 20? Why? What is the actual answer to 58 + 21? (79) Was 80 a good estimate?

Have students estimate the total number of apples sold in these situations:

a) 27 red apples were sold and 42 green apples were sold;

b) 46 red apples were sold and 78 green apples were sold;

c) Jenn sold 52 apples and Rita sold 31 apples;

d) Jenn sold 42 apples and Rita sold 29 apples.

Write out some of the answers on the board, using the “approximately equal to” sign

(EXAMPLE: 27 + 32 ≈ 30 + 30 = 60).

Write the following question on the board:

About how many more green apples than red apples were sold in part a)?

ASK: What word in that question tells you I only want an estimate? Does the question ask for the sum of the

numbers of green and red apples or the difference between them? How do you know? What operation

should I use to find the difference—addition or subtraction? Tell students that you would find it easier to

subtract if the numbers were multiples of 10. ASK: What multiples of 10 are closest to the number of red

and green apples? If there are about 30 red apples and about 40 green apples, about how many more

green apples were sold than red apples?

Have students estimate how many more green apples were sold than red apples in b), and then how many

more apples Jenn sold than Rita in c) and d).

Tell students (and write on the board): Greg collected 37 stamps, Ron collected 72 stamps, and Sara

collected 49 stamps. ASK: How many more stamps did Ron collect than Sara? How many more did Sara

collect than Greg? How many more stamps does Ron have than Greg?

Give students similar problems with 3-digit numbers, asking them to round to the nearest ten to estimate the

answer. Then give problems with 3- and 4-digit numbers and ask students to round to the nearest hundred.


Ask students to estimate this sum: 27 + 31. Then ask them to find the actual answer. What is the difference

between the estimate and the actual answer? Which number is larger, the estimate or the actual answer?

How much larger? What operation do they use to find how much more one number is than another? (They

should subtract the smaller number from the larger number.) Repeat with sums and differences of:

2-digit numbers, rounding to the nearest ten

(EXAMPLES: 39 + 41, 76 – 48).

3-digit numbers, rounding to nearest ten

(EXAMPLES: 987 – 321, 802 + 372).

3- and 4-digit numbers, rounding to nearest hundred

(EXAMPLES: 3 401 + 9 888, 459 – 121).

Draw on the board:

ASK: How many balls are in each box? If I want to know how many are in 4 boxes, what is an easier number

to multiply 4 by that is close to 12? (10) What makes that number easy to multiply by? About how many balls

are in 4 boxes? (40).

Repeat with the following pictures, having students estimate how many balls are in 7 boxes:

a) b) c)

NOTE: Encourage your students to use estimating to judge the reasonableness of their answers. Give them

the following questions and ask them to tell you what they would estimate the answer will be before they

perform the operation.

a) 382 + 217 b) 427 + 604 c) 923 – 422 d) 875 – 215


Activities:

1. Count the number of slices in an orange and estimate the number of slices in 6 oranges.

2. Take a handful of counters and drop them onto a sheet of paper folded into 4 equal regions. Count the

number of counters in one of the four regions and estimate the number of counters you dropped.

3. Fill up a clear plastic container with beans or small blocks. Ask students to estimate the number of beans

or blocks in the container. Ask them how they arrived at their estimate. Did they count out 10 blocks to

use 10 as the known quantity? Or 100 blocks? 10 beans or 100 beans?

Extensions:

1. Estimate the number of pages in your JUMP Math Part 2 workbook. Note that the page number on the

last page of the workbook shows the number of pages in both Part 1 and Part 2, so students will need to

round both this number and the number of pages in their Part 1 workbook and then subtract. To guide

students, ASK: What page does Part 2 end at? What page does it start at? So what page does Part 1

end at? How many pages are in both Part 1 and Part 2 together? How many pages are in just Part 1?

How can we find the number of pages in just Part 2? What operation should we use? To make the

numbers easier to subtract, what numbers close to these numbers are easier to work with?

2. Estimate the number of pages in all the JUMP Math Part 2 workbooks in the class. Hint: Round the

number of pages to the nearest hundred and the number of workbooks to the nearest ten.

The following three extensions are adapted from the Atlantic Curriculum Guide (A2):

3. Which estimate is closer to find 46 + 25? 50 + 20 OR 50 + 30 How do you know?

4. Is the following estimate for 82 – 47 too high or too low: 80 – 50? How do you know?

5. If you have a loonie, do you have enough money to buy:

• A pencil for 12¢

• An eraser for 25¢

• A notebook for 29¢

• A pen for 19¢

Explain your strategy.


NS3-57 Mental Math and Estimation Goal: Students will use doubles to add mentally.

Prior Knowledge Required: Counting by 2s

Doubling 2-digit numbers

Relationship between skip counting and multiplying

Arrays

Vocabulary: Double, double plus one, double minus one

Review doubling (see NS3-38) with your students.

Prepare 20 paper circles to use as counters and to tape to the board. Draw 2 circles and put 5 counters in

each of the two circles. ASK: What double have I shown? (Double 5)

Draw two new circles and put 3 counters in one circle and 5 in another. ASK: What addition statement does

this show? (3 + 5 = 8) Challenge students to think of a way to move one of the counters so that there is the

same number in each circle. What double have you shown? (4 + 4 = 8)

Repeat with larger numbers. EXAMPLES: 8 + 6, 7 + 9, 5 + 7, 11 + 9. Bonus: Move two counters to change

7 + 11 into a double.

Teach students to change addition statements into doubles without using counters to help them. Emphasize

that when you move a counter from one pile to the other, you are adding 1 counter to one pile and

subtracting 1 counter from the other pile, so 8 + 6 becomes 8 – 1 + 6 + 1 = 7 + 7.

Have students change 7 + 5 to a double, then 6 + 8, then 10 + 12, then 33 + 31, then 62 + 60.

Draw the following picture on the board:

Tell your students that the four counters are in front of a mirror. Ask a volunteer to draw what they would see

in a mirror on the other side of the dotted line. Have another volunteer write an addition sentence based on

the number of circles on one side of the mirror, and the total number of circles they see altogether. Do they

see a double anywhere?


Do more examples of this and then ask what happens if we put a circle over here outside the range of

the mirror:

Have a volunteer show where to draw the circles we would see in the mirror. Ask how many circles there

appear to be altogether. Suggest two different number sentences: 4 + 4 + 1 = 9 and 5 + 4 = 9 and ask your

students to tell you what you are thinking by each addition sentence. Ask what number is being doubled and

what they think you mean by a double plus one. Repeat with several other examples of doubles plus one.

Then write the sum: 3 + 4 and ask if that can be written as a double plus one. What number would be

doubled? Write on the board:

3 + 4 = + + 1 and tell students that you want to put the same number in each box. Ask them

how writing the sum this way can help them add 3 + 4. Show students how they can make a doubles chart if

they don’t have the doubles memorized:

0 1 2 3 4 5 6 7 8 9 10

0 2 4 20

Have students guide you in finishing this double’s chart. Then demonstrate how to use the chart to find

doubles:

Have students use the method above to find the following sums: 8 + 7; 4 + 5; 9 + 10; 9 + 8.

Bonus: 23 + 24; 35 + 36; 57 + 56;

NOTE: Students might also try finding the sums above by doubling the larger number and subtracting one:

7 + 6 = 7 + 7 – 1 = 14 – 1 = 13.

Teach students to subtract numbers from 100 using the following mental math strategy. Ensure that

students can …

1) Subtract single-digit numbers easily from 10 (EXAMPLE: 10 – 4 = 6) See the Modified Go Fish game in

the Mental Math section of this guide.

2) Subtract 2-digit multiples of 10 from 100 (EXAMPLE: 100 – 40 = 60)

3) Subtract single-digit numbers from multiples of 10 (EXAMPLE: 80 – 4 = 76)

4) Subtract two-digit numbers from 100 (EXAMPLE: 100 – 74 = 100 – 70 – 4 = 30 – 4 = 26)

Bonus: Subtract 2-digit numbers from multiples of 100 (EXAMPLE: 800 – 74 = 726)

Teach students to find sums by adding the digits separately. Ensure that students can …

1) Add 2 single-digit numbers (EXAMPLE: 7 + 8 = 15)

2) Add 3 single-digit numbers (EXAMPLE: 6 + 7 + 9 = 22)


3) Separate the digits (EXAMPLES: 48 = 40 + 8; 132 = 100 + 30 + 2)

4) Add a 2-digit number and a 1-digit number by separating the tens and ones

(EXAMPLE: 48 + 6 = 40 + 8 + 6 = 40 + 14 = 54)

5) Add three 2-digit numbers by separating the tens and ones

(EXAMPLE: 48 + 16 + 23 = 40 + 10 + 20 + 8 + 6 + 3 = 70 + 17 = 87)

6) Add 3 numbers that include a 3-digit number

(EXAMPLE: 532 + 54 + 7 = 500 + 30 + 50 + 2 + 4 + 7 = 500 + 80 + 13 = 593)

Activities:

1. If you have miras available, students can show different examples of doubles plus one, such as 3 + 3 + 1

using counters.

2. Compete (teacher against the class) to see who can come up with the most strategies to find 78 – 29.

Some strategies include:

• 78 – 28 = 50, so 78 – 29 = 49.

• 78 – 29 = 79 – 30 (because if I have two piles of counters, one with 78 counters and one with 29

counters and I add a counter to each pile, the difference stays the same) = 49.

• 78 – 29 = 80 – 31 (adding two counters to each pile instead) = 49.

• 29, 30, 70, 78 has differences 1, 40 and 8, so the total difference is 1 + 40 + 8 = 49.

• Separating the tens and ones and then regrouping:

70 + 8 60 + 18

– 20 + 9 – 20 + 9

40 + 9 = 49.

3. Repeat Activity 2, but with adding 27 + 49. Sample strategies include:

• 27 + 50 = 77, so 27 + 49 = 76

• 20 + 49 = 69, so 27 + 49 = 69 + 7 = 76

• 27 + 49 = 20 + 40 + 7 + 9 = 60 + 16 = 76

• 25 + 50 = 75, so 27 + 49 = 75 + 2 – 1 = 76

Extensions:

1. Teach students to subtract 2-digit numbers from 100 by adding: 100 – 73 = 100 – 80 + 80 – 73. = 20 + 7.

This can be represented as:

7 20 27

73 80 100

2. Find an easy way to add: 299 + 198 + 399.

3. Tell students that the numbers 40 and 60 are partners because they add to 100. Have students find a

number which is its own partner. (50). Then have students make a chart of various multiples of ten and

find their partners. What is an easy way to find the partner of a multiple of ten? Then move on to


examples that are not multiples of ten. Have students make a chart with headings: “Number” and

“Partner that adds to 100.”

Give students various numbers to put in the first column (24, 33, 19, 47, 85, 76, 93, 4, 11). Then have

students use various numbers of their choice to add more pairs to their chart. Challenge students to look

for a pattern so that they can find an easy way to find a number’s partner without using the method

taught in class. In particular, guide students’ attention to the ones digits and the tens digits. Look at the

ones digits of the pairs of numbers. What do they notice about them? (The ones digits add to ten.) Look

at the tens digits of the pairs of numbers. What do they notice about them? (The tens digits add to 9).

Why does this make sense? If the sum is 9 tens and 10 ones, what number is that? (90 + 10 = 100).

Have students use this method to find the partner of various numbers. Start by filling in the tens digit for

them and having the students only find the ones digit, then fill in the ones digit and have them only find

the tens digit. The mix up which digit you give them and which digit they need to find. Finally, have them

find both digits.

The following four extensions were taken from Atlantic Curriculum B6:

4. Why might someone find it easier to subtract 123 – 99 than 123 – 87?

5. Which sum is closest to 500? Explain how you know.

329 + 189 329 + 217 329 + 207

6. Which difference is closest to 50? Explain how you know.

125 – 30 168 – 115 103 – 82

7. You subtracted a number in the 3 hundreds from a number in the 5 hundreds. The answer was about

100. What might the numbers have been?

8. Teach students to estimate by clustering. For example: 23 + 24 + 34 is estimated, by rounding, as

20 + 20 + 30 = 70. But if students notice that 3 + 4 + 4 is about 10, a better estimate is 70 + 10 = 80.

Similarly, 232 + 244 + 322 is about 200 + 200 + 300 + 100.


NS3-58 Sharing – Knowing the Number of Sets ERRATA NOTE: This 2-sheet spread in the workbook is accidentally titled NS3-58: Multiplication and

Division (Review) and NS3-59: Knowing the Number of Sets

Goal: Given the total number of objects divided into a given number of sets, students will identify the

number of objects in each set.

Prior Knowledge Required: Dividing equally

Word problems

Vocabulary: set, divide, equally

Divide 12 volunteers into 4 teams, numbered 1-4, by assigning each volunteer a number in the following

order: 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 1. Separate the teams by their numbers, and then ask your students what

they thought of the way you divided the teams. Was it fair? How can you reassign each of the volunteers a

number and ensure that an equal amount of volunteers are on each team? An organized way of doing this

is to assign the numbers in order: 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4. Demonstrate this method and then

separate the teams by their numbers again. Does each team have the same amount of volunteers? Are the

teams fair now?

Present your students with 4 see-through containers and ask them to pretend that the containers represent

the 4 teams. Label the containers 1, 2, 3 and 4. To evenly divide 12 players (represented by a counter of

some sort) into 4 teams, one counter is placed into one container until all 12 counters are placed into the 4

containers. This is like assigning each player a number. Should we randomly distribute the 12 counters and

hope that each container is assigned an equal amount? Students should see that it makes more sense to

place the students’ counters (or name tags) into the container one at a time.

Now, suppose you want to share 12 cookies between yourself and 3 friends. How many people are sharing

the cookies? [4.] How many containers are needed? [4.] How many counters are needed? [12.] What do the

counters represent? [Cookies.] What do the containers represent? [People.] Instruct your students to draw

circles for the containers and dots inside the circles for the counters. How many circles will you need to

draw? [4.] How many dots will you need to draw inside the circles? [12.]

Draw 4 circles.

Counting the dots out loud as you place them in the circles, have your students yell “Stop” when you reach

12. Ask them how many dots are in each circle? If 4 people share 12 cookies, how many cookies does each

person get? If 12 people are divided among 4 teams, how many people are on each team? Now what do

the circles represent? Now what do the dots represent?


If 12 people ride in 4 cars, how many people are in each car? What do the circles and dots represent? Have

students suggest additional representations for circles and dots.

Assign your students practice exercises, drawing circles and the correct numbers of dots in each circle.

If it helps, allow students to first use counters and count them ahead of time so they know automatically

when to stop.

a) 12 cookies, 3 people b) 15 cookies, 5 people c) 10 cookies, 2 people

When students have mastered this, write the following word problem.

5 friends shared 20 strawberries. How many strawberries does each friend get?

Have a volunteer read the word problem out loud and then ask how many dots are needed. What is to be

divided into groups? [Strawberries.] How many circles are needed? What will the circles represent?

[Friends.] What will the amount of dots in each circle illustrate? [The number of strawberries that each friend

will receive.] Have another volunteer solve the problem for the rest of the class. Then assign your students

several word problems. Read all the word problems out loud, and remind students that they can use a

dictionary if they don’t understand a word.

EXAMPLES:

a) 3 friends picked 15 cherries. How many cherries did each friend pick?

b) Joanne shared 15 marbles among 5 people (4 friends and herself). How many marbles

did each person receive?

c) There are 18 plums on 6 trees. How many plums are on each tree?

d) There are 16 apples on 2 trees. How many apples are on each tree?

e) 20 children sit in 4 rows. How many children sit in each row?

f) Lauren’s weekly allowance is $21. What is Lauren’s daily allowance?

g) An egg carton has 12 eggs divided into 2 rows. How many eggs are in each row?

Bonus:

Have students use base ten materials for the following questions.

a) 3 friends picked 69 cherries. How many cherries did each friend pick?

b) Joanne shared 84 marbles among 4 people (3 friends and herself). How many marbles

did each person receive?

c) There are 63 plums on 3 trees. How many plums are on each tree?

d) There are 68 apples on 2 trees. How many apples are on each tree?

Activity: Students might act out their solutions to questions 5 and 6.


NS3-59 Sharing – Knowing How Many in Each Set

Goal: Given a number of objects to be divided equally and the number in each set, students will determine

the number of sets.

Prior Knowledge Required: Dividing objects into equal sets

Word problems

Vocabulary: sets, groups, divide, row, column

Distribute 12 counters to each student and have them divide the counters evenly into 3 piles, then 2 piles,

then 6 piles, and then 4 piles. Then have them divide the 12 counters first into piles of 2, then into piles of 3,

then 6, then 4. Tell your students that, last class, the question always told them how many piles (or sets) to

make. This class, the question will tell them how many to put in each pile (or in each set) and they will need

to determine the number of sets.

Tell your students that Saud has 30 apples. Count out 30 counters and set them aside. Saud wants to

share his apples so that each friend gets 5 apples. He wants to know how many people can get apples.

What can be used to represent Saud’s friends? [Containers.] Do you know how many containers we need?

[No, because we are not told how many friends Saud has.] How many counters are to be placed in each

container? [5.]

Put 5 counters in one container. Are more containers needed for the counters? [Yes.] How many counters

are to be placed in a second container? (5) How do they know? (Each container gets 5 counters because

each friend gets 5 apples.) Will another container be needed? (Yes, because there are at least 5 apples, or

counters, left) Continue until all the counters are evenly distributed. The 30 counters have been distributed

and each friend received 5 apples. How many friends does Saud have? How do they know? [6, because 6

containers were needed to evenly distribute the counters.]

Now draw dots and circles like in the last lesson. What will the circles represent? [Friends.] How many

friends does Saud have? How many dots are to be placed in each circle? [5.] Place 5 dots in each circle

and keep track of the amount used.

5 10 15 20 25 30

6 circles had to be drawn to evenly distribute 30 dots, meaning 6 friends can share the 30 apples. What is

the difference between this problem and the problems in the previous lesson? [The previous lesson

identified the number of sets, but not the number of objects in each set.]


Saud has 15 apples and wants to give each friend 3 apples. How many friends can he give apples to? How

can we model this problem? What will the dots represent? What will the circles represent? Does the

problem tell us how many circles to draw? (no) Or how many dots to draw in each circle? (yes, 3) Do we

know how many dots to draw altogether? (yes, 15) Draw 15 dots on the board and demonstrate distributing

3 dots in each circle:

There are 5 sets, so he can give apples to 5 people.

Have your students distribute 3 dots into each set, and then ask them to count the number of sets.

a)

b)

c)

Repeat with 2 dots into each set:

a)

b)

Have students draw the dots to determine the correct amount of sets.

a) 15 dots, 5 dots in each set b) 12 dots, 4 dots in each set c) 16 dots, 2 dots in each set

Bonus:

24 dots and

a) 2 dots in each set b) 3 dots in each set c) 4 dots in each set

d) 6 dots in each set e) 8 dots in each set f) 12 dots in each set

ASK: As the number of dots in each set gets bigger, what happens to the number of sets?

Activity: Students might act out their solutions to questions 5 and 6.


NS3-60 Sets Goal: Students will identify the objects to be divided into sets, the number of sets and the number of

objects in each set.

Prior Knowledge Required: Sharing equally

Vocabulary: groups or sets, divided or shared

Draw 12 children divided equally into 4 canoes, as illustrated in the worksheet. Ask your students to identify

the objects to be shared or divided into sets, the number of sets and the number of objects in each set.

Repeat with 15 apples divided into 3 baskets and 12 plates placed on 3 tables. Have students complete the

following examples in their notebook.

a) Draw 9 people divided into 3 teams (Team A, Team B, Team C)

b) Draw 15 flowers divided into 5 flowerpots.

c) Draw 10 fish divided into 2 fishbowls.

Note that, in division problems, the word that tells you what is being divided or shared will almost always

come right before the word “each” (“in each”, “on each”, “to each”, “for each”, “at each”). The word coming

after “each” is usually the set.

For instance, in the sentence: “There are 4 kids in each boat,” the word ‘”kids” comes right before the phrase:

“in each boat”. Boats are the sets and kids are being divided into sets.

Students should also think of the set as a kind of container that holds the things that are being divided

or shared.

On the board, write several phrases or sentences with the word “each” in them and ask your students to say

what is being divided or shared, and what are the sets.

a) 5 boxes, 4 pencils in each box (pencils are being divided, boxes are sets)

b) 3 classrooms, 20 students in each classroom (students are being divided, classrooms are sets)

c) 4 teams, 5 people on each team (people are being divided, teams are the sets)

d) 5 trees, 30 apples on each tree (apples are being divided, trees are the sets)

e) 3 friends, 6 stickers for each friend (stickers are being divided, friends are the sets)

f) There are 3 sides on each triangle.

g) There are 6 houses on each block.

h) There are 30 kids on each school bus.

i) There are 3 school buses for each school.

j) There are 6 schools in each neighbourhood.


To ensure contextual understanding, you might ask your students to draw some of the above situations.

Then have students use circles to represent sets and dots to represent objects to be divided into sets.

a) 5 sets, 2 dots in each set

b) 7 groups, 2 dots in each group

c) 3 sets, 4 dots in each set

d) 4 groups, 6 dots in each group

e) 5 children, 2 toys for each child

f) 6 friends, 3 pencils for each friend

g) 3 fishbowls, 4 fish in each bowl, 12 fish altogether

h) 20 oranges, 5 boxes, 4 oranges in each box

i) 4 boxes, 12 pens, 3 pens in each box

j) 10 dollars for each hour of work, 4 hours of work, 40 dollars

Bonus:

k) 12 objects altogether, 4 sets

l) 8 objects altogether, 2 objects in each set

m) 5 fish in each fishbowl, 3 fishbowls

n) 6 legs on each spider, 4 spiders

o) 3 sides on each triangle, 6 triangles

p) 3 sides on each triangle, 6 sides

q) 6 boxes, 2 oranges in each box

r) 6 oranges, 2 oranges in each box

s) 6 fish in each bowl, 2 fishbowls

t) 6 fish, 2 fishbowls

u) 8 boxes, 4 pencils in each box

v) 8 pencils, 4 pencils in each box

Activities:

1. The teacher starts by saying a sentence that describes some objects that are divided into groups.

EXAMPLE: There are 4 kids in each boat. The first student then says a sentence where the sets become

the objects being divided. For example, the first student might say: “There are 7 boats on each river.” Or

“There are 4 boats on each dock.” Or “There are 5 boats in each boathouse.” Students continue in this

way. For example, “there are 7 boathouses in each river” and then “there are 5 rivers in each province.”

2. A student makes up a division-related phrase, for instance, “There are 4 kids in each boat.” They then try

to make up a sentence where things that were previously divided are now the sets and the sets are the

things divided: ie, “Each kid has 4 boats.” (In some cases this will be impossible to do.)


NS3-61 Two Ways of Sharing Goal: Students will recognize which information is provided and which is absent among: how many sets,

how many in each set and how many altogether.

Prior Knowledge Required: Solving problems where either the number of sets or amount of

objects in each set is given

Samuel has 15 cookies. There are two ways that he can share or divide his cookies equally.

1. He can decide how many sets (or groups) of cookies he wants.

If he wants to share his cookies with two friends, he will have to divide

the cookies into 3 sets. He will draw 3 circles and place one cookie

into each circle until all 15 cookies are placed in the circles.

2. He can decide that he wants each person to receive 5 cookies.

He counts out sets of 5 cookies until he has counted all 15 cookies.

Show students how to divide 3 rows of 8 dots into 4 circles.

And so on. Or, instead of crossing out the dots, students might count the total number of dots (or multiply to

find the total number of dots) and then count as they place the dots in the circles.

Assign your students several similar problems.

a) 2 rows of 6 squares into 3 circles

b) 3 rows of 8 hearts into 6 circles

c) 1 row of 18 dots into 2 circles

d) 1 row of 12 vertical lines into 6 circles

NOTE: Some students may find it easier to draw dots instead of triangles, squares or hearts.

Then ask your students if they remember how to group the dots so that there are 4 dots in

each set, and to explain how this is different from the previous problem.

a)

b)

Have students draw 12 dots and group them so that there are…

a) 4 dots in each set

b) 2 dots in each set


c) 6 dots in each set

d) 3 dots in each set

Sometimes the problem provides the number of sets, and sometimes the problem provides the number of

objects in each set. Have your students tell you if they are given the number of sets or the number of

objects in each set for the following problems.

a) There are 15 children. There are 5 children in each canoe.

b) There are 15 children in 5 canoes.

c) Aza has 40 stickers. She gives 8 stickers to each of her friends.

Have your students draw and complete the following table for PROBLEMS a)–i). Insert a box for

information that is not provided in the problem.

What has been shared

or divided into sets?

How many sets? How many in each set?

EXAMPLE 18 6

a) 24 4

EXAMPLE: There are 6 strings on each guitar. There are 18 strings.

a) There are 24 strings on 4 guitars.

b) There are 3 hands on each clock. There are 15 hands altogether.

c) There are 18 holes in 6 sheets of paper.

Make sure your students understand that words such as “book shelves” or “tables” or “containers” or

“vehicles” might refer to the sets or the objects being divided into sets.

Ask students to decide in each statement below whether the words “book shelves” represent sets or objects

being divided into sets.

a) There are 5 books on each bookshelf.

b) There are 6 bookshelves in each room.

c) Each bookshelf has several books on it.

d) Each library has several bookshelves.

NOTE: In question 8 all of the examples tell you the number of containers (or sets). For variety assign your

students several questions which give the number of items in each set. For example:

1. Paul has 15 stamps. He put 5 on each page.

2. 12 kids sit down to dinner. There are 3 kids at each table.


NS3-62 Division and NS3-63 Dividing by Skip Counting

Goal: Students will learn the division symbol through repeated addition and skip counting.

Prior Knowledge Required: Skip counting using fingers or a number line

Sharing or dividing into sets or groups

Vocabulary: division (and its symbol ÷), divided by, dividend, divisor

Ensure that students can tell you, for various pictures: a) how many objects altogether, b) how many sets

and c) how many objects in each set.

Then write two division statements for each picture: 8 ÷ 4 = 2 and 8 ÷ 2 = 4 for the first picture and

15 ÷ 5 = 3 and 15 ÷ 3 = 5 for the second picture.

Explain that 15 objects divided into sets of 5 equals 3 sets. This is written as 15 ÷ 5 = 3 or as 15 ÷ 3 = 5 and

read as “fifteen divided by 5 equals 3” or “fifteen divided by 3 equals 5.”

Distribute 12 counters to each of your students and then ask them to divide the counters into sets of 3. How

many sets do they have? What two division statements can they write?

The following website provides good worksheets for those having trouble with this basic definition of

division:

http://math.about.com/library/divisiongroups.pdf.

Using various symbols, have your students find 12 ÷ 2, 12 ÷ 3, 12 ÷ 4 and 12 ÷ 6. For example:

So 12 ÷ 2 = 6.

Have your students illustrate each of the following division statements with two pictures.

6 ÷ 3 = 2

12 ÷ 3 = 4 9 ÷ 3 = 3 8 ÷ 2 = 4 8 ÷ 4 = 2


Then have your students write two division statements for each of the following illustrations.

A B C

D E F G H

I J K

Challenge students to find another way to group the objects in each illustration that will show the same

division statement. For example, instead of 3 groups of 4 in diagram A, students can draw 4 groups of 3;

instead of 4 groups of 1 in diagram I, students can draw 1 group of 4. You might also draw several

pictures and have students pair up the pictures that show the same division statements. (In the pictures

above, G and I both show 4 ÷ 1 = 4 and 4 ÷ 4 = 1; B and C both show 12 ÷ 2 = 6 and 12 ÷ 6 = 2).

WRITE: 15 ÷ 3 = 5 (15 divided into sets of size 3 equals 5 sets).

3 + 3 + 3 + 3 + 3 = 15

Explain that every division statement implies an addition statement. Ask your students to write the addition

statements implied by each of the following division statements. Allow them to illustrate the statement first,

if it helps.

15 ÷ 5 = 3 12 ÷ 2 = 6 12 ÷ 6 = 2 10 ÷ 5 = 2

10 ÷ 2 = 5 6 ÷ 3 = 2 6 ÷ 2 = 3

Add this number

WRITE: 15 ÷ 3 = 5 This many times.

Ask your students to write the following division statements as addition statements, without

illustrating the statement this time.

12 ÷ 4 = 3 12 ÷ 3 = 4 18 ÷ 6 = 3 18 ÷ 3 = 6

18 ÷ 2 = 9 18 ÷ 9 = 2 25 ÷ 5 = 5

Bonus:

132 ÷ 43 = 3 1700 ÷ 425 = 4 90 ÷ 30 = 3 1325 ÷ 265 = 5

Then have your students illustrate and write a division statement for each of the following

addition statements.

4 + 4 + 4 = 12 2 + 2 + 2 + 2 + 2 = 10 6 + 6 + 6 + 6 = 24

3 + 3 + 3 = 9 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21 5 + 5 + 5 + 5 + 5 + 5 = 30


Then have your students write division statements for each of the following addition statements, without

illustrating the statement.

17 + 17 + 17 + 17 + 17 = 85

21 + 21 + 21 = 63

101 + 101 + 101 + 101 = 404

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 36

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 21

Draw:

What division statement does this illustrate? What addition statement does this illustrate? Is the addition

statement similar to skip counting? Which number could be used to skip count the statement?

Explain that the division statement 18 ÷ 3 = ? can be solved by skip counting on a number line.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

How many skips of 3 does it take to reach 18? (6.) So what does 18 ÷ 3 equal? How does the number line

illustrate this? (Count the number of arrows.) NOTE: Some students might find it helpful if you start with 18

counters and count out 3 at a time, drawing an arrow for each set of 3 counters that you set aside. You

know to stop when all 18 counters are used up, or when you reach 18 on the number line.

Explain that the division statement expresses a solution to 18 ÷ 3 by skip counting by 3 to 18 and then

counting the arrows.

Ask volunteers to find, using the number line: 12 ÷ 2; 12 ÷ 3; 12 ÷ 4; 12 ÷ 6;

0 1 2 3 4 5 6 7 8 9 10 11 12

ASK: If I want to find 10 ÷ 2, how could I use this number line?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

What do I skip count by? How do I know when to stop? Have a volunteer demonstrate the solution.

Ask your students to solve the following division statements with a number line to 18.

8 ÷ 2, 12 ÷ 3, 15 ÷ 3, 15÷ 5, 14 ÷ 2, 16 ÷ 4, 16 ÷ 2, 18 ÷ 3, 18 ÷ 2

Then have your students solve the following division problems with number lines to 20 (see the BLM

“Number Lines to Twenty”). Have them use the top and bottom of each number line so that each number

line can be used to solve two problems. For example, the solutions for 6 ÷ 2 and 8 ÷ 4 might look like this:


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

6 ÷ 2 8 ÷ 4 6 ÷ 3 14 ÷ 2 9 ÷ 3 15 ÷ 3

10 ÷ 5 20 ÷ 4 20 ÷ 5 18 ÷ 2 20 ÷ 2 16 ÷ 4

Provide various solutions of division problems and have your students express the corresponding division

and addition statements. For example, students should give the statements “20 ÷ 4 = 5”

and “4 + 4 + 4 + 4 + 4 = 20” for the following number line.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Draw number lines for 16 ÷ 8, 10 ÷ 2, 12 ÷ 4, and 12 ÷ 3.

Explain that skip counting can be performed on the fingers, as well as on a number line. If your students

need practice skip counting without a number line, see the MENTAL MATH section of this teacher’s guide.

They might also enjoy the following interactive website:

http://members.learningplanet.com/act/count/free.asp

Draw 12 dots on the board. Ask your students if they can figure out how to divide a set of 12 dots into 3

equal sets by skip counting by 3s. As students count up by 3s ask them to imagine putting one dot into each

set (thus placing 3 dots altogether) every time they say a multiple of 3. The skip counting helps them keep

track of the number of dots placed altogether and the number of fingers raised helps them keep track of the

number of dots placed in each set.

“3” I’ve placed one dot in each

set (3 altogether).

“6” I’ve placed 2 dots in each set

(6 altogether).


(9 altogether).


(12 altogether).


To solve 12 ÷ 3, skip count by 3 to 12. The number of fingers it requires to count to 12 is the answer. [4.]

Perform several examples of this together as a class, ensuring that students know when to stop counting

(and in turn, what the answer is) for any given division problem. Have your students complete the following

problems by skip counting with their fingers.

a) 12 ÷ 3 b) 6 ÷ 2 c) 10 ÷ 2 d) 10 ÷ 5 e) 12 ÷ 4 f) 9 ÷ 3

g) 16 ÷ 4 h) 50 ÷ 10 i) 25 ÷ 5 j) 15 ÷ 3 k) 15 ÷ 5 l) 30 ÷ 10

Bonus:

a) 200 ÷ 50 b) 125 ÷ 25

Two hands will be needed to keep track of the count for the following questions.

a) 12 ÷ 2 b) 30 ÷ 5 c) 18 ÷ 2 d) 28 ÷ 4 e) 30 ÷ 3 f) 40 ÷ 5

Bonus:

a) 450 ÷ 50 b) 175 ÷ 25

As an extra challenge, provide problems that require counting beyond the fingers on two hands.

Bonus:

a) 22 ÷ 2 b) 48 ÷ 4 c) 65 ÷ 5 d) 120 ÷ 10 e) 26 ÷ 2

Then have your students express the division statement for each of the following word problems and

determine the answers by skip counting. Ask questions like: What is to be divided into sets? How many sets

are there and what are they?

a) 5 friends share 30 tickets to a sports game. How many tickets does each friend receive?

b) 20 friends sit in 2 rows at the movie theatre. How many friends sit in each row?

c) $50 is divided among 10 friends. How much money does each friend receive?

Have your students illustrate each of the following division statements and skip count to

determine the answers.

3 ÷ 3 5 ÷ 5 8 ÷ 8 11 ÷ 11

Without illustrating or skip counting, have your students predict the answers for the following division

statements.

23 ÷ 23 180 ÷ 180 244 ÷ 244 1 896 ÷ 1 896

Then have your students illustrate each of the following division statements and skip count to determine the

answers.

1 ÷ 1 2 ÷ 1 5 ÷ 1 12 ÷ 1

Then without illustrating, have your students predict the answers for the following

division statements.

18 ÷ 1 27 ÷ 1 803 ÷ 1 6 692 ÷ 1


Extension: Teach students that division is similar to repeated subtraction. Start with 20 counters and

take away 4 each time until there is none left. Symbolize this on the board by writing:

20 – 4 – 4 – 4 – 4 – 4 = 0

ASK: How many times did you subtract 4? What is 20 ÷ 4?

Compare this method to dividing on a number line or by skip counting backwards by 4s:

20 16 12 8 4 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


NS3-64 Division and Multiplication Goal: Students will understand the relationship between division and multiplication

Prior Knowledge Required: Relationship between multiplication and skip counting

Relationship between division and skip counting

Vocabulary: divided by

Write “10 divided into sets of 2 results in 5 sets.” Have one volunteer read it out loud, another illustrate it,

and another write its addition statement. Does the addition statement remind your students of

multiplication? What is the multiplication statement?

10 ÷ 2 = 5

2 + 2 + 2 + 2 + 2 = 10

5 × 2 = 10

Another way to express “10 divided into sets of 2 results in 5 sets” is to write “5 sets of 2 equals 10.”

Have volunteers illustrate the following division statements, write the division statements, and then rewrite

them as multiplication statements.

a) 12 divided into sets of 4 results in 3 sets. (12 ÷ 4 = 3 and 3 × 4 = 12)

b) 10 divided into sets of 5 results in 2 sets.

c) 9 divided into sets of 3 results in 3 sets.

Assign the remaining questions to all students.

d) 15 divided into 5 sets results in sets of 3.

e) 18 divided into 9 sets results in sets of 2.

f) 6 people divided into teams of 3 results in 2 teams.

g) 8 fish divided so that each fishbowl has 4 fish results in 2 fishbowls.

h) 12 people divided into 4 teams results in 3 people on each team.

i) 6 fish divided into 3 fishbowls results in 2 fish in each fishbowl.

Then ask students if there is another multiplication statement that can be obtained from the same picture as

3 × 4 = 12? How can the dots be grouped to express that 3 sets of 4 is equivalent to 4 sets of 3?

3 sets of 4 is equivalent to 4 sets of 3.

ANSWER: The second array of dots should be circled in columns of 3.

ASK: If 12 ÷ 4 = 3 and 3 × 4 = 12 are obtained from the same picture, what division statement comes from

the same picture as 4 × 3 = 12? (12 ÷ 3 = 4)


Draw similar illustrations and have your students write two multiplication statements and

two division statements for each. Have a volunteer demonstrate the exercise for the class

with the first illustration.

Demonstrate how multiplication can be used to help with division. For example, the division statement

20 ÷ 4 = _____ can be written as the multiplication statement 4 × _____ = 20. To solve the problem, skip

count by 4 to 20 and count the number of fingers it requires. Demonstrate this solution on a number line,

as well.

20 divided into skips of 4 results in 5 skips. 20 ÷ 4 = 5

5 skips of 4 results in 20. 5 × 4 = 20, SO: 4 × 5 = 20

Assign students the following problems.

a) 9 × 3 = 27, SO: 27 ÷ 9 = _____

b) 2 × 6 = 12, SO: 12 ÷ 2 = _____

c) 8 × _____ = 40, SO: 40 ÷ 8 = _____

d) 10 × _____ = 30, SO: 30 ÷ 10 = _____

e) 5 × _____ = 30, SO: 30 ÷ 5 = _____

f) 4 × _____ = 28, SO: 28 ÷ 4 = _____

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NS3-65 Knowing When to Multiply or Divide Goal: Students will understand when to use multiplication or division to solve a word problem.

Prior Knowledge Required: Relationship between multiplication and division

Vocabulary: each, per, altogether, in total

Have your students fill in the blanks.

sets 2 sets

objects per set 5 objects per set

objects altogether 10 objects altogether

When you are confident that your students are completely familiar with the terms “set,” “group,” “for every,”

and “in each,” and you’re certain that they understand the difference between the phrases “objects in each

set” and “objects” (or “objects altogether,” or “objects in total”), have them write descriptions of the

diagrams.

2 groups

4 objects in each group

8 objects

Explain that a set or group expresses three pieces of information: the number of sets, the number of objects

in each set, and the number of objects altogether. For the problems, have your students explain which piece

of information isn’t expressed and what the values are for the information that is expressed.

a) There are 8 pencils in each box. There are 5 boxes. How many pencils are there altogether? (5 groups,

8 objects in each group, how many altogether?)

b) Each dog has 4 legs. There are 3 dogs. How many legs are there altogether?

c) Each cat has 2 eyes. There are 10 eyes. How many cats are there?

d) Each boat can fit 4 people. There are 20 people. How many

boats are needed?

e) 30 people fit into 10 cars. How many people fit into each car?

f) Each apple costs 20¢. How many apples can be bought for 80¢?

g) There are 8 triangles divided into 2 sets. How many triangles are

there in each set?

h) 4 polygons have a total of 12 sides. How many sides are on

each polygon?

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Introduce the following three problem types:

Type 1: You know the number of sets and the number of objects in each set.

Example: You have 4 sets of objects and 2 objects in each set. How many objects do you have in total?

STEP 1: Draw 4 boxes to represent the 4 sets:

STEP 2: Fill each box with 2 objects:

STEP 3: Count the number of objects: 8 objects (or “8 objects altogether” or

“8 objects in total”)

Your student can then write a multiplication statement to represent the solution: 4 � 2 = 8

ASK: If you know the number of objects in each set and the number of sets, how can you

find the total number of objects? What operation should you use—multiplication or division?

Write on the board:

Number of sets × Number of objects in each set = Total number of objects

Other examples you could use:

a) 3 sets b) 3 groups

5 objects in each set 7 objects in each group

How many objects? How many objects in total?

Type 2: You know how many objects there are altogether and how many objects there are in each set.

Example: You have 6 objects altogether and 3 objects in each set. How many sets do you have?

STEP 1: Draw the total number of objects:

STEP 2: Draw a box around three objects

at a time until you’ve put all the

objects in boxes:

STEP 3: Count the number of boxes: 2 boxes (or "sets")

Your student should then write a division statement to represent the solution: 6 ÷ 3 = 2

Again use the same multiplication statement as before.

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If there are 4 objects in each set and 12 objects in total, how many sets are there?

_____ × 4 = 12, SO: 12 ÷ 4 = 3. There are 3 sets.Other examples you could use:

a) 12 objects altogether b) 16 objects

4 objects in each group 2 objects per set

How many groups? How many sets?

Type 3: You know how many objects there are and how many sets there are.

Example: You have 10 objects and 5 sets. How many objects are there in each set?

STEP 1: Draw the total number of sets:

STEP 2: First, put one object in each set.

STEP 3: Check to see if you have placed all

the objects. If not, put one more object

in each set and continue until you have

placed all the objects:

STEP 4: Count the number of objects in each set: 2 objects in each set

Your student should then write a division statement to represent the solution: 10 ÷ 5 = 2


If there are 6 sets and 12 objects in total, how many objects are in each set?

6 × _____ = 12, SO: 12 ÷ 6 = 2. There are 2 objects in each set.

Other examples you could use:

a) 15 objects altogether b) 3 sets

5 sets 12 objects altogether

How many objects in each set? How many objects in each set?

Have students write multiplication statements for the following problems with the blank in the correct place.

a) 2 objects in each set. b) 2 objects in each set. c) 2 sets.

6 objects in total. 6 sets. 6 objects in total.

How many sets? How many objects in total? How many objects in

each set?

[ _____ × 2 = 6 ] [ 6 × 2 = _____ ] [ 2 × _____ = 6 ]

Which of these problems are division problems? [Multiplication is used to find the total number

of objects, and division is used if the total number of objects is known.]

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Assign several types of problems.

a) 5 sets, 4 objects in each set. How many objects altogether?

b) 8 objects in total, 2 sets. How many objects in each set?

c) 3 sets, 6 objects in total. How many objects in each set?

d) 3 sets, 6 objects in each set. How many objects in total?

e) 3 objects in each set, 9 objects altogether. How many sets?

When your students are comfortable with this lesson’s goal, introduce alternative contexts for objects and

sets. Start with point-form problems (EXAMPLE: 5 tennis courts, 3 tennis balls on each court. How many

tennis balls altogether?), and then move to complete sentence problems (EXAMPLE: If there are 5 tennis

courts, and 3 balls on each court, how many tennis balls are on the tennis courts altogether?).

Have your students solve these problems:

a) 20 apples; 4 baskets.

How many apples in each

basket?

b) 3 birds; 5 cages.

How many birds?

c) 18 bottles of water, 3 cases.

How many bottles in each

case?

When your student is able to distinguish between (and solve) problems of Type 1, 2, and 3 readily, you can

teach them how to solve more general word problems involving multiplication and division. Tell them to think

of a container (like a box or pot) or a carrier (like a car or a boat) as a set, and the things contained or carried

as objects in the set.

EXAMPLE: Ten people need to cross a river. A boat can hold two people. How many boats are needed to

take everyone across?

Hint: Think of the boats as sets (or boxes) and the people as objects placed in the sets. This is a problem of

Type 2, as discussed in above, ie. you know the total number of objects and the number of objects in

each set.

Draw 10 lines to represent 10 people:

Put boxes around every 2 lines (each

box represents a boat):

Count the number of boats: 5 boats

This approach also works for things that have parts (think of the things that have parts as sets, and the parts

as objects in the set).

Example: A cat has 2 eyes. How many eyes are there on 5 cats?

Hint: Note that the first statement is really saying that “Each cat has 2 eyes,” so cats are the sets and eyes

are being divided into sets. Use boxes to represent each cat and lines in each box to represent the eyes.

This is a problem of Type 1, as discussed above, i.e. you know the number of sets and the number of

objects in each set.

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Draw 5 boxes to represent the 5 cats:

Draw 2 lines in each box

(representing the eyes):

The number of lines gives you the answer: 10 eyes

This approach also works for things that have a value or a price (think of the thing with value as a set or box,

and the price or value as the objects in the set, i.e. you can think of dollars or cents as lines which you can

place inside the box representing the thing you are buying).

Example: A piece of gum costs 5 cents. You have 15 cents. How many pieces of gum can you buy?

Hint: Again, this is a problem of Type 2.

Draw 15 lines to represent 15 cents:

Put boxes around every 5 lines (each

box represents a piece of gum):

Count the number of boxes: 3 boxes (or pieces of gum)

Once you give your students enough practice with this type of problem, they should eventually see that they

simply have to divide 15 by 5 to find the answer.

Activity: Students could model their solutions to questions 5 a) and b) with counters. It is important,

however, that students also be able to solve the problems by drawing a sketch with dots or lines.

Extension: Tell your students that you met someone from Mars last weekend, and they told you that

there are 3 dulgs on each flut. If you count 15 dulgs, how many fluts are there? Explain the problem-solving

strategy of replacing unknown words with words that are commonly used. For example, replace the object in

the problem (dulgs) with students, and replace the set in the problem (flut) with bench. So, if there are 3

students on each bench and you count 15 students, how many benches are there? It wouldn’t make sense to

replace the object (dulgs) with benches and the set (flut) with students, would it? [If there are 3 benches on

each student and you count 15 benches, how many students are there?] A good strategy for replacing words

is to replace the object and the set and then invert the replacement with the same two words. Only one of the

two versions of the problem with the replacement words should make sense.

Students may wish to create their own science fiction word problems for their classmates. Encourage them

to use words from another language, if they speak another language.

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NS3-66 Remainders Goal: Students will divide with remainders using pictures.

Prior Knowledge Required: Relationships between division and multiplication, addition, skip

counting, number lines

Vocabulary: remainder R, quotient, divisor

Draw:

6 ÷ 3 = 2

7 ÷ 3 = 2 Remainder 1


9 ÷ 3 = 3


Ask your students if they know what the word “remainder” means. Instead of responding with a definition,

encourage them to only say the answers for the following problems. This will allow those students who don’t

immediately see the pattern a chance to detect it.

7 ÷ 2 = 3 Remainder _____

11 ÷ 3 = 3 Remainder _____

12 ÷ 5 = 2 Remainder _____

14 ÷ 5 = 2 Remainder _____

Challenge volunteers to find the remainder by drawing a picture on the board. This way, students who do

not yet see the pattern can see more and more examples of the rule being applied.


SAMPLE PROBLEMS:

9 ÷ 2 7 ÷ 3 11 ÷ 3 15 ÷ 4

15 ÷ 6 12 ÷ 4 11 ÷ 2 18 ÷ 5

What does “remainder” mean? Why are some dots left over? Why aren’t they included in the circles? What

rule is being followed in the illustrations? [The same number of dots is placed in each circle, the remaining

dots are left uncircled]. If there are fewer uncircled dots than circles then we can’t put one more in each

circle and still have the same number in each circle, so we have to leave them uncircled. If there are no

dots left over, what does the remainder equal? [Zero.]

Introduce your students to the word “quotient”: Remind your students that when subtracting two numbers,

the answer is called the difference. ASK: When you add two numbers, what is the answer called? In

7 + 4 = 11, what is 11 called? (The sum). When you multiply two numbers, what is the answer called? In

2 × 5 = 10, what is 10 called? (The product). When you divide two numbers, does anyone know what the

answer is called? There is a special word for it. If no-one suggests it, tell them that when you write

10 ÷ 2 = 5, the 5 is called the quotient.

Have your students determine the quotient and the remainder for the following statements.

a) 17 ÷ 3 = _____ Remainder _____ b) 23 ÷ 4 = _____ Remainder _____

c) 11 ÷ 3 = _____ Remainder _____

Write “2 friends want to share 7 apples.” What are the sets? [Friends.] What are the objects being divided?

[Apples.] How many circles need to be drawn to model this problem? How many dots need to be drawn?

Draw 2 circles and 7 dots.

To divide 7 apples between 2 friends,

place 1 dot (apple) in each circle.

Can another dot be placed in each

circle? Are there at least 2 dots

left over? So is there enough to put

one more in each circle? Repeat this

line of instruction until the diagram

looks like this:


How many apples will each friend receive? Explain. [There are 3 dots in each circle.] How many apples will

be left over? Explain. [Placing 1 more dot in either of the circles will make the compared amount of dots in

both circles unequal.]

Repeat this exercise with “5 friends want to share 18 apples.” Emphasize that the process of division and

placing apples (dots) into sets (circles) continues as long as there are at least 5 apples left to share. Count

the number of apples remaining after each round of division to ensure that at least 5 apples remain.

Have your students illustrate each of the following division statements with a picture, and then determine

the quotients and remainders.

Number in each circle

a) 11 ÷ 5 = _____ Remainder _____ Number left over

b) 18 ÷ 4 = _____ Remainder _____

c) 20 ÷ 3 = _____ Remainder

d) 22 ÷ 5 = _____ Remainder

e) 11 ÷ 2 = _____ Remainder

f) 8 ÷ 5 = _____ Remainder

g) 19 ÷ 4 = _____ Remainder

Explain to students that the word “Remainder” is sometimes written just as “R.” For example,

11 ÷ 5 = 2 R 1.

Teach students that there are many ways to think about division. To find the answer to 14 ÷ 3, students might use

any of the following methods.

a) Forming equal groups (of size 3) using a picture of 14 objects: b) Sharing 14 things (candies for instance), three apiece, among friends: c) Adding threes repeatedly. Stop before you reach 14:

There are 4 groups of 3 with 2 objects left over

3 6 9 12

You stop here, one more three would take you beyond 14 to 15. (You could show this with a number line).

You can share with 4 friends. Two are left over.


So 3 goes into 12 (and 14!) four times.

The remainder of 14 ÷ 3 is the difference between 14 and the number you reached when you were

counting up (12). So the remainder is 2.

d) Guessing and multiplying by 3 until you get close to 14:

e) Subtracting 3 from 14 repeatedly until you get close to 0 (this is not the most practical method). You can

subtract 3 four times from 14 before you reach 2 (the remainder).

Activity: Students could model their solutions to all of the questions on this worksheet with counters.

Extensions:

1. 22 kids go on a picnic. Hot dogs come in packs of 8. Buns come in packs of 12. How many packs of

buns and hot dogs should the children take if each kid wants one hot dog and bun? Will there be any

buns or hotdogs left over?

2. Which number is greater, the divisor (the number by which another is to be divided) or the

remainder? Will this always be true? Have your students examine their illustrations to help explain.

Emphasize that the divisor is equal to the number of circles (sets), and the remainder is equal to the

number of dots left over. We stop putting dots in circles only when the number left over is smaller

than the number of circles; otherwise, we would continue putting the dots in the circles. See the

journal section below.

Which of the following division statements is correctly illustrated? Can one more dot be placed

into each circle or not? Correct the two wrong statements.

15 ÷ 3 = 4 Remainder 3 17 ÷ 4 = 3 Remainder 5 19 ÷ 4 = 4 Remainder 3

Without illustration, identify the incorrect division statements and correct them.

a) 16 ÷ 5 = 2 Remainder 6 b) 11 ÷ 2 = 4 Remainder 3 c) 19 ÷ 6 = 3 Remainder 1

3. Explain how a diagram can illustrate a division statement with a remainder and a multiplication

statement with addition.


3 × 4 + 2 = 14

4 × 3 = 12 5 × 3 = 15

This is too high, so 3 divides into 14 four times (with 2 remainder).


Ask students to write a division statement with a remainder and a multiplication statement

with addition for each of the following illustrations.

4. Compare mathematical division to normal sharing. Often if we share 5 things (say, marbles) among 2

people as equally as possible, we give 3 to one person and 2 to the other person. But in mathematics, if

we divide 5 objects between 2 sets, 2 objects are placed in each set and the leftover object is

designated as a remainder. Teach them that we can still use division to solve this type of real-life

problem; we just have to be careful in how we interpret the remainder. Have students compare the

answers to the real-life problem and to the mathematical problem:

a) 2 people share 5 marbles (groups of 2 and 3; 5 ÷ 2 = 2 R 1)

b) 2 people share 7 marbles (groups of 3 and 4; 7 ÷ 2 = 3 R 1)

c) 2 people share 9 marbles (groups of 4 and 5; 9 ÷ 2 = 4 R 1)

ASK: If 19 ÷ 2 = 9 R 1, how many marbles would each person get if 2 people shared 19 marbles?

Emphasize that we can use the mathematical definition of sharing as equally as possible even when the

answer isn’t exactly what we’re looking for. We just have to know how to adapt it to what we need.

5. Find the mystery number. I am between 22 and 38. I am a multiple of 5. When I am divided by 7 the

remainder is 2.

6. Have your students demonstrate two different ways of dividing…

a) 7 counters so that the remainder equals 1.

b) 17 counters so that the remainder equals 1.

Journal

The remainder is always smaller than the divisor because…


NS3-67 Multiplication and Division and NS3-68 Multiplication and Division (Review)

Goal: Students will consolidate their learning on multiplication and division done so far.

Vocabulary: fact family, multiple, division, remainder, twice as many

Tell students that you have 13 apples and you have divided them into equal groups so that there is

one left over:

ASK: What division statement does this picture show? Then challenge students to find another way to divide

the apples so that there is only 1 left over. (Students might divide the apples into 2, 3 or 4 circles.) Take up

the various solutions.

Draw a 2 × 3 array on the board:

ASK: What multiplication sentence does this show? Challenge students to find another array that shows a

similar multiplication sentence but uses twice as many dots. (Review the phrase “twice as many” if needed.)

ASK: What addition sentence does this show:

ANSWER: 3 + 4 = 7. To guide students, ASK: How many dots are on the left side of the vertical line? How

many are on the right side? How many are there altogether?

Repeat with several similar examples. Then tell students that you are going to show them a more

complicated picture. This time the left side and the right side both show multiplication statements. Can they

tell which ones?


The left side shows 3 × 2 and the right side shows 4 × 2. ASK: What does the whole picture show? What

multiplication statement is shown by all 7 circles? (7 × 2) Emphasize that this picture shows that seven sets

of 2 equals three sets of 2 plus four sets of 2. Do they think that 7 sets of 3 will equal three sets of 3 plus four

sets of three? Have students draw the picture to show this. Then have students change their picture, one

step at a time to show that:

a) seven sets of 3 equals two sets of 3 plus five sets of 3 (move the vertical line one place left – erase the

vertical line that exists and add a new one)

b) eight sets of 3 equals two sets of 3 plus six sets of 3 (add another circle to the right of the line)

c) ten sets of 3 equals four sets of 3 plus six sets of 3 (add two more circles to the left of the line)

d) ten sets of 4 equals four sets of 4 plus six sets of 4 (add one more dot in each circle)

e) ten sets of four equal three sets of 4 plus one set of 4 plus six sets of 4 (add another vertical line after the

third circle; do not erase the vertical line that exists)

Have students practise doing puzzles similar to question 10 on the worksheet.

Use the numbers 3, 4 and 6 once each to make the sentences true:

× + = 18 × + = 22

× + = 27 × – = 21

÷ + = 6 × – = 6

× ÷ = 2 + – = 7

Bonus: Have students make up a similar puzzle for a partner to solve. They can use any operation, but if

they use multiplication or division, it must only be the first operation used; addition and subtraction can be

either first or second. (This condition avoids potential problems with the order of operations, which they are

not required to know at this point.)

Bonus: Make 0 using the numbers 2, 3 and 6 once each in as many ways as you can.

(EXAMPLES: 2 × 3 – 6 = 0, 3 × 2 – 6 = 0, 6 ÷ 3 – 2, 6 ÷ 2 – 3) Note: Students should not be taught the order

of operations at this point. That 6 – 2 × 3 is also 0 should not be discussed.

The activity and extensions 5-8 below are designed to satisfy the Atlantic Curriculum.

Activity: This activity is best done after completing the extensions below. Play a game with your students

to see who (you or your class) can come up with the most multiplication or division questions that you could

solve in one or two steps just from knowing 5 × 6 = 30. Sample questions:

• 6 × 5 = 30

• 5 × 7 = 5 × 6 + 5 = 30 + 5 = 35

• 6 × 6 = 5 × 6 + 6 = 30 + 6 = 36

• 30 ÷ 5 = 6

• 30 ÷ 6 = 5


• 60 ÷ 10 = 6

• 90 ÷ 15= 6

• 120 ÷ 20 = 6

• 60 ÷ 5= 12

• 5 × 12 = 60

• 10 × 6 = 60

• 4 × 6 = 30 – 6 = 24

Extensions:

1. Ask students to write a full fact family for a division problem (including 2 addition statements).

For example: The fact family for 14 ÷ 2 = 7 is 14 ÷ 7 = 2, 7 × 2 = 14, 2 × 7 = 14,

2 + 2 + 2 + 2 + 2 + 2 + 2 = 14, and 7 + 7 = 14.

2. Ask your students to complete the following story problems with their own numbers and solve the

problems.

a) Janice had _____ apples. She shared them equally with _____ friends. How many did each

person get?

b) Tim had _____ boxes. He placed _____ watermelons in each box. How many watermelons did he

have altogether?

3. The BLMs “Always, Sometimes, or Never True (Numbers)” and “Define a Number” will help students

sharpen their understanding of numbers.

4. (Adapted from the Atlantic Curriculum) Show students how 5 sets of 3 can be broken down into subsets

in various ways:

• 4 sets of 3 and 1 set of 3

• 3 sets of 3 and 2 sets of 3

• 5 sets of 2 and 5 sets of 1


Challenge students to write multiplication and division statements that relate to these pictures. For example,

we might write:

• 4 × 3 + 1 × 3 = 5 × 3 or 12 ÷ 3 + 3 ÷ 3 = 15 ÷ 3

• 3 × 3 + 2 × 3 = 5 × 3 or 9 ÷ 3 + 6 ÷ 3 = 15 ÷ 3

• 5 × 2 + 5 × 1 = 5 × 3 or 10 ÷ 5 + 5 ÷ 5 = 15 ÷ 5

5. Ask students what happens to the product when you double one of the numbers in a

multiplication statement:

2 × 3 4 × 3 or 2 × 3 2 × 6

ANSWER: The product doubles.

What happens to the product when you multiply one of the factors by 3? (The product is multiplied by 3.)

Have students investigate what this means for division. For example, 2 × 3 = 6, so 2 × 6 = 12 becomes, in

terms of division: 6 ÷ 3 = 2, so 12 ÷ 6 = 2. Note that doubling both terms of the division statement will keep

the answer the same. Ask students how they could use doubling both terms to find 45 ÷ 5.

6. Remind students that, from Extension 5, the quotient remains the same when both the dividend and the

divisor are multiplied by 2 (or when both are multiplied by 3, or both multiplied by 4, and so on). Ask

students to investigate what happens to the quotient when the dividend is multiplied by 2 and the divisor

remains the same. (The quotient doubles.) What happens to the quotient when the divisor is multiplied

by 2 and the dividend remains the same? (The quotient is divided in half.) Tell students that 21 ÷ 7 = 3.

ASK: What is 42 ÷ 7? 84 ÷ 7? 168 ÷ 3? How does knowing 60 ÷ 6 help you to know 30 ÷ 6?

7. Teach students how to estimate products. Draw a number line as follows:

3 × 0 3 × 10

0 3 6 9 12 15 18 21 24 27 30

ASK: Is 3 × 7 closer to 3 × 0 or 3 × 10? Have a volunteer circle 3 × 7. Have volunteers circle 3 × 1,

3 × 9, 3 × 5. Then draw on the board the following number line:

4 × 0 4 × 10

0 40


ASK: Is 4 × 8 closer to 4 × 0 or 4 × 10? Is 4 × 8 closer to 0 or 40? How much closer? Have a volunteer mark

where 4 × 8 occurs with an X. Then have another volunteer mark where they think 10, 20 and 30 occur.

Which two multiples of ten is 4 × 8 between? Which multiple of ten is 4 × 8 closest to? If you were to

estimate 4 × 8 to the nearest ten, which multiple of ten would you pick?

4 × 0 4 × 10

0 10 20 30 40

Have students use the same method and the following number line to estimate 3 × 17:

3 × 10 3 × 20

30 60

Where are 40 and 50?

8. Teach students how to estimate quotients. For example, draw the following number line:

20 40 60 80

5 10 15 20

Tell your students that the bottom numbers were all obtained from the top numbers in the same way.

Can they see how? (divide by 4) Bring to the students’ attention that the numbers on the bottom, just like

the numbers on the top, go up by a fixed amount. When the numbers on the top skip count by 20, and

you divide by 4 to get the numbers on the bottom, then the numbers on the bottom skip count by

20 ÷ 4 = 5. ASK: What do they think 35 ÷ 4 will be close to? Have students mark where 35 is on the top

and then use that to mark where they think 35 ÷ 4 will be on the bottom. What is the closest whole

number to 35 ÷ 4?


NS3-69 Patterns Made with Repeated Addition Goal: Students will use their calculator to discover patterns made from repeated addition.

Prior Knowledge Required: Addition, patterns

Give each student a calculator. If available, photocopy a calculator onto an overhead transparency, so that

students can see all the buttons. Tell students to use their calculator to find 7 + 2. Have a volunteer show on

the overhead the buttons they pressed to find the answer (7, +, 2, =). Then have all students press the “=”

button again. What number showed up? Why? What operation did the calculator do to get from 9 to 11?

Again, have students press the “=” button. What number does the calculator show? Challenge students to

predict the next number the calculator will show when they press the “=” button, and then the next number.

Explain to students that pressing 7, +, 2, =, =, =, =, =, results in: 7 + 2 + 2 + 2 + 2 + 2.

Challenge students to find, by pressing “=” the correct number of times:

a) 3 + 2 + 2

b) 5 + 3 + 3 + 3

c) 3 + 5 + 5 + 5

d) 8 + 2 + 2 + 2 + 2 + 2 + 2

e) 7 + 3 + 3 + 3 + 3

f) 8 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9

Then show students how to record each step of the process:

8 8 + 9 = 17 8 + 9 + 9 = 26 8 + 9 + 9 + 9 = 35, and so on.

Write the sequence of results, all in a row:

8 17 26 35 44 53 62 71 80 89 98 107

Have students record the patterns of ones digits only:

8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7.

Can your students predict what the next ones digit will be? Have them explain their answer.

Have students record the patterns in the number of tens:

0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10.

Tell your students that you showed this pattern to someone who said that the numbers always increase by 1.

Tell them to look closely at the sequence and to tell you if that is correct. (No, the 8 is repeated) Have

students extend the pattern on their calculators to find out when the next repetition is. ASK: How many

numbers occur before the first repetition? How many numbers come before the second repetition? How

many numbers do they think will occur before the next repetition? Have students write what they think the

next ten terms of the sequence (of the number of tens) will be and then to check their answer using the

calculator.

Activity: Review the Skip Counting Machine activity from section NS3-13: Counting by 5s and 25s.

Have students build a skip counting by 9s machine, starting from any number. Students may also enjoy

building a skip counting by 11s machine.


NS3-70 Counting by Dollars and Coins Goal: Students will use dollar and cent notation for money amounts that involve either only cents or only

dollars. Students will skip count to add money amounts that consist of a single type of coin or dollar bill.

Review the Canadian coins: pennies, nickels, dimes and quarters, and then introduce loonies and toonies.

Give students play coins. Ask them if the amount of money is shown on the coins. Does every coin have the

amount written on it? Is the value of the coin printed on both sides or just one? Ask them if there is a word

that shows the number is talking about money and not how many trees, cats, or houses. Tell them there are

two different words used to show money and challenge them to find both of them (they are written on the

coins). Ask them how cents and dollars are like units in measurement. Remind them that if something is 5

paper clips long, it is shorter than if it is 5 notebooks long. Tell them that if something is worth 3 cents, it is

worth less than if it is worth 3 dollars. A dollar is worth a hundred cents, so if Isobel has 2 dollars and Soren

has 30 cents, Isobel has more money even though 2 is less than 30. If Isobel is 2 m tall and Soren is 30 cm

tall, who is taller?

Write on the board:

“1 dollar = 100 cents and 2 dollars = 200 cents.” Have students look at the amount on their coins and

arrange them in order from least value to most value, keeping in mind that a dollar is worth a hundred cents.

Then have them arrange the coins in order from smallest to largest in size. Which coin is out of place?

To introduce the symbols $ and ¢, tell students that in math we use the words plus, minus, and equals a lot

and ask if there is a symbol we use instead of writing out the words all the time. Write on the board the words

“plus,” “minus,” and “equals” and have volunteers write the symbols used to show those words. Ask them if

there are words we use a lot when talking about money. Ask if they think those words should have a symbol

for them. Ask if anyone knows what the symbols are. Then write on the board, “$1 = 100¢ and $2 = 200¢.”

Teach students that when we count money in cents, we write the cent sign after the amount of cents, but

when we count money in dollars, we write the dollar sign before the amount of dollars. Have students

practice by using the correct notation for the following money amounts (do not include money amounts that

involve both dollars and cents).

a) five cents b) twelve dollars c) 9 cents d) 9 dollars e) 83 dollars f) 46 cents

Draw a circle and write “5¢” inside. ASK: Which coin does this represent? Repeat for several Canadian coin

values (EXAMPLES: 1¢, $1, 10¢, $2, 25¢)

Review skip counting. Then, have students skip count to find the total amount of money:

a) 5¢ 5¢ 5¢ 5¢

b) 10¢ 10¢ 10¢ 10¢


c) 25¢ 25¢ 25¢ 25¢

d) 10¢ 10¢ 10¢ 10¢ 10¢ 10¢ 10¢ 10¢

e) $1 $1 $1 $1 $1 $1 $1

f) $2 $2 $2 $2 $2

To guide students, have them write the subtotal over each coin. For example:

5¢ 10¢ 15¢ 20¢

5¢ 5¢ 5¢ 5¢

Then teach students to skip count without writing the subtotals. Students should count how many coins there

are and then skip count on their fingers to find the total amount of money.

Present a problem: Jane emptied her piggy bank. She asked her elder brother John to help her count the

coins. He suggested she stack the coins of the same value in different stacks and count each stack

separately. Jane has:

19 pennies 8 nickels 7 dimes

3 quarters 6 loonies 3 toonies

How much money is in each stack? Ask students to write the amount in dollar notation when counting

loonies and toonies and to write the amount in cent notation when counting pennies, nickels, dimes and

quarters. Give them more practice questions, such as:

9 pennies 14 nickels 13 dimes

6 quarters 5 loonies 9 toonies

Then ask students how much money they have if they have 6 nickels. ASK: What do I skip count by? How do

I know when to stop? Repeat with several examples of a single type of coin. (EXAMPLES: 3 quarters, 5

dimes, 7 loonies, 4 nickels, 3 toonies).

Then tell students that they are only allowed to use one type of coin. ASK: How can I make 8 cents? (Use 8

pennies.) How can I make 10 cents? (Use 10 pennies or 2 nickels or 1 dime.) How many I make 20 cents?

25 cents? 50 cents? 3 dollars? 4 dollars? Students need only find 1 solution. Bonus: Faster students should

be challenged to find several solutions or even all solutions.

Then write the amount in words that you want the students to make, again using only 1 type of coin.

(EXAMPLES: eight cents, eighty cents, five dollars, forty-five cents, fifty cents) Bonus: Students can try to

make these amounts using more than one denomination. For example, five dollars can be made with a

toonie and three loonies or two toonies and a loonie, or twelve quarters and a toonie, and so on.


Activities:

Heads and Tails Give students play money coins – one of each: penny, nickel, dime, quarter, loonie, and

toonie. Ask them to turn all the coins so that the heads side is up. Then tell them to take a white sheet of

paper and fold it in half so that the fold line separates the top to the bottom. They then unfold the paper and

place all the coins under the top half of the page and they rub a pencil over the paper so that they can see

the coin images. When this is done, they turn the coins around so that the tails side is showing, rearrange

the coins, place the coins under the bottom half of their sheet, and rub the pencil over the paper again. They

should then match each heads side with the tail of the same coin.

Pick the Right Coin Give students a bag of coins including pennies, nickels, dimes, and quarters. Ask

students to try to pick out a dime without looking. Repeat with picking a penny, a nickel and a quarter. What

characteristics are they looking for? What is the easiest coin to pick out? Why is it the easiest? What is the

hardest coin to pick out?


NS3-71 Dollars and Cent Notation Goal: Students will express monetary values less than a dollar in dollar and cent notation.

Prior Knowledge Required: Place value to tenths and hundredths

Canadian coins

Vocabulary: penny, loonie, nickel, toonie, dime, cent, quarter, dollar, tenths, notation, hundredths

Have students show how to make various amounts of money using dimes and pennies (EXAMPLES:

35¢ = dimes and pennies, 22¢, 25¢, 30¢, 36¢, 41¢)

Explain that there are two standard ways of representing money. The first is cent notation: you simply write

the number of cents or pennies that you have, followed by the ¢ sign. In dollar notation, “twenty-five cents”

is written $0.25. The first number to the right of the decimal represents the number of dimes and the next

digit to the right represents the number of pennies.

25¢ = $0.25

2 dimes 5 pennies

Have students write both the cent notation and the dollar (decimal) notation for each amount shown above.

Have students write the total value of a collection of coins both in dollar and cent notation (begin with only

one type of coin, then two types, then three types and finally four types of coins).

a) 25¢, 25¢ b) 10¢, 10¢, 10¢ c) 10¢, 10¢, 10¢, 5¢, 5¢, 5¢

d) 10¢, 10¢, 1¢, 1¢, 1¢, 1¢, 1¢, 1¢ e) 10¢, 10¢, 5¢, 1¢, 1¢, 1¢ f) 25¢, 25¢, 10¢, 10¢, 1¢

g) 25¢, 5¢, 5¢, 5¢, 1¢ h) 25¢, 25¢, 25¢, 10¢, 1¢, 1¢ i) 25¢, 10¢, 10¢, 5¢, 1¢, 1¢

j) 25¢, 10¢, 10¢, 10¢, 5¢, 1¢, 1¢, 1¢.

Then tell students that you have 7 nickels and 4 pennies. Show this on the board in a random arrangement:

5¢, 1¢, 1¢, 5¢, 5¢, 5¢, 5¢, 1¢, 5¢, 5¢, 1¢

Ask your students to write an addition sentence to show the total amount of money. Remind them that when

they have a long sequence of numbers to add, they should keep track as they go along by putting the total at

each stage in squares above the numbers:

6 7 12 17 22 27 28 33 38 39

5 + 1 + 1 + 5 + 5 + 5 + 5 + 1 + 5 + 5 + 1


Then show the coins on the board, putting the nickels first. Tell them you want to add the nickels first and ask

what you should skip count by. Demonstrate doing this until you get to the pennies (5, 10, 15, 20, 25, 30, 35)

by crossing out the fives as you count them and then ask: Should I still skip count by 5s? Now what do I

count by? Demonstrate continuing the count (36, 37, 38, 39) by crossing out the pennies as you count them.

ASK: Did we get the same total both ways?

Repeat with pennies and dimes, then dimes, nickels and pennies and then quarters, dimes, nickels and

pennies, all arranged randomly and then in order. ASK: What if they didn’t have nickels or dimes or quarters,

and they only had pennies. How would that affect their counting? (It would make it a lot slower to count how

much money they have and it would make carrying the money around a lot heavier.)

Then introduce dollar and cent notation for money amounts that are at least a dollar:

100¢ = $1.00, 200¢ = $2.00.

Ask students how they would show, in dollar notation, the following various amounts: 300¢, 700¢, 900¢,

500¢. Bonus: 1200¢, 1000¢, 3000¢.

Then have students total the following amounts and then write the dollar notation for the total:

a) $1, $1, $1, $1, $1 b) $2, $2, $2, $2 c) $2, $2, $1, $1, $1

d) $5, $1, $1 e) $5, $5, $2, $2, $1 f) $10, $5, $5, $2, $2, $2, $1

Then tell students that 300¢ is written as $3.00 in dollar notation and 47¢ is written as $0.47 in dollar

notation. Challenge students to guess how 347¢ is written in dollar notation. (ANSWER: $3.47)

$3.47

dollars dimes pennies

Have students (volunteers at first and then individually) show how to write the following cent amounts in

dollar notation: 321¢, 21¢, 320¢, 301¢, 478¢, 408¢, 78¢, 470¢, 603¢, 57¢, 430¢, 541¢)

Activities:

1. Ask students to pretend that there is a vending machine which only takes loonies, dimes and pennies.

Have them make amounts using only these coins (EXAMPLE: 453¢, 278¢, 102¢, etc).

2. A Game for Two: The Change Machine

One player makes an amount using nickels and quarters. The other (“the machine”) has to change the

amount into loonies, dimes and pennies.

Extensions:

1. Discuss the difference and similarities between the dollar and cent notation and the metre and

centimetre notation. Tell students that 134 cm can equally be written as 1m 34 cm or as 1.34 m. But

134¢ can be written as $1.34 (not 1.34 $) but not as 1 $ 34 ¢.


2. Have students write in symbols to make the following sentences correct (do the first one for them):

135¢ = $1 + 35¢

246 = 2 + 46

887 = 8 + 87

432 = 32 + 4

Remind students that the dollar sign goes to the left and the cents sign to the right of the number, but

you always say 3 dollars, not dollars 3. If students are comfortable with cm and m, have them fill in the

same number sentences with those units to make the sentences true and then discuss the comparison.

Then give only the number of cents (a 3-digit number) and have students break it up into dollars and

cents.


NS3-72 Counting and Changing Units and NS3-73 Converting Between Dollar and Cent

Notation

Goal: Students will convert between dollar and cent notation.

Prior Knowledge Required: Understanding of dollar and cent notation

Knowledge of place value

Vocabulary: penny, loonie, nickel, toonie, dime, cent, quarter, dollar, notation, convert

Changing from dollar to cent notation is easy—all you need to do is to remove the decimal point and dollar

sign, and then add the ¢ sign to the right of the number.

Changing from cent notation to dollar notation is a little trickier. Make sure students know that, when an

amount in cent notation has no tens digit (or a zero in the tens place), the corresponding amount in dollar

notation must have a zero in the dimes place.

EXAMPLE: 6¢ is written $.06 or $0.06, not $0.60.

0 dimes 6 pennies 6 dimes 0 pennies

Make a chart like the one shown below on the board and ask volunteers to help you fill it in.

Amount Amount in Cents Whole Dollars Dimes Pennies Amount in Dollars

3 toonies 800 8 0 0 $8.00

2 loonies

2 quarters

5 dimes

2 nickels

4 quarters

2 pennies

Students could also practice skip counting by quarters, toonies and other coins (both in dollar and in cent

notation).

$2.00, $4.00, ____, ____ $0.25, $0.50, ____, ____, ____, ____

200¢, 400¢, ____, ____ 25¢, 50¢, ____, ____, ____, ____


Bonus: What coin is being used for skip counting?

$1.00, ____, ____, ____, $1.20

$2.00, ____, ____, ____, $3.00

You may wish to give your students some play money and to let them practice in pairs—each player picks 2

coins and then both students individually write the total amount of the 4 coins in dollar and cent notation.

Partners then compare answers.

Assessment

1. Write the total amount in dollar and cent notation:

a)

Total amount = _____ ¢ = $ ______

b)

Total amount = _____ ¢ = $ ______

2. Convert between dollar and cent notations:

$ .24 = ___¢, _____=82¢, _____ = 6¢, $12.03 = _____, $120.30 = ______

Bonus:

3. Change these numbers to dollar notation: 273258¢, 1234567890¢.

4. Change $245.56 to cent notation. Try these amounts: $76.34, $12.03, $120.30, $123.52, $3789.49.

Challenge students to make various money amounts using exactly two coins (EXAMPLES: 6¢, 10¢, 11¢,

15¢, 20¢, 30¢, 35¢, 50¢, Bonus: $2, $3, $1.05, $2.25)

Then challenge students to make various money amounts using exactly three coins (EXAMPLES: 7¢, 12¢,

15¢, 27¢, 30¢, 31¢, 40¢, 51¢, 75¢, Bonus: $3, $4, $5, $6, $2.10, $1.10, $1.15, $1.11)

Challenge students to make various money amounts using exactly four coins (EXAMPLES: 8¢, 13¢, 22¢,

30¢, 40¢, 45¢, 60¢, 65¢, 70¢, 76¢, 80¢, 81¢, 85¢, Bonus: $4, $5, $6, $7, $8, $1.40, $2.10, $3.25, $4.30,

$3.50, $5.10)

25¢ 10¢ $2 5¢ 1¢ 1¢

25¢ $1 10¢ 5¢ 1¢


Activities:

1. Guess the hidden coins Player 1 hides several coins: for example, a loonie, a toonie and a dime or

three loonies and a dime or a loonie, a toonie and two nickels for $3.10. Player 1 tells her partner how

many coins she has and the value of the coins in dollar notation. The partner has to guess which coins

she has. There may be more than one possibility for the answer. The player should try to use money and

coin amounts that have more than one possibility to increase the chances of an incorrect guess.

Students might discuss whether it is a good strategy for this game to use the least number of coins. (It is

not, because the least number of coins is always unique and quite easy to guess.) Challenge students to

change the rules of the game so that it is better to use the least number of coins. Then play the game.

2. Money Matching Memory Game. (See the BLM “Money Matching Memory Game”) Students play this

game in pairs. Each student takes a turn flipping over two cards. If the cards match, that player gets to

keep the pair and takes another turn. Students will have to remember which cards are placed where and

also match up identical amounts written in dollar and cent notation.

3. Each pair should have the BLM “Adding or Trading Game” as a game board and each player should

have a different token to use as their playing piece. They will also need a die to know how many pieces

to move forward. When they roll, they move forward the correct number of squares and receive the coin

shown on the board. When both players are at the end of the board (not necessarily by the exact amount

shown on the die), they count up their money – the player with the most amount of money wins.

Variation: The player with the fewest coins wins; players may trade from a shared bank for equal value

coins at the end of the game to try to have fewer coins.

4. Trading Game Have students work in pairs with different goals. Give each player 10 pennies, 4 nickels,

and 1 dime. Player One’s goal is to get 10 coins and Player Two’s goal is to get 20 coins. They must only

trade for equivalent values. To ensure this is always the case, have students add their money at the

beginning and periodically. They should always have 40¢. Note that both players will achieve their goals

at the same time, so they should cooperate.

Variation: Player One’s goal: 18 coins; Player Two’s goal: 12 coins.

Variation: Give each player 5 pennies, 5 dimes, 4 nickels, and 1 quarter. Player 1 tries to get 20 coins –

the other person will then try to get 10 coins. Students should repeatedly check that they have a total of

$1 or 100¢.

Variation: Give each player 2 loonies, 3 quarters, 10 dimes and 7 nickels, so that each player has 22

coins. Player 1 aims for 11 coins while Player 2 aims for 33 coins. Students should repeatedly check that

they have a total of $4.10.

Extensions:

1. Provide the BLM “Smallest Number of Coins Chart.”

2. Give students the BLM “Dimes, Pennies, and Base Ten Materials” to show them the relationship

between dimes and base ten blocks, and pennies and ones blocks.

3. (Atlantic Curriculum A5.11) Mary won $5000 in a contest. If she wants all her prize money in $10 bills,

how many would she receive?

4. (Atlantic Curriculum A4.4) Pretend that you won three thousand dollars. How many hundred dollar bills

would that be?

5. (Atlantic Curriculum A4.5) Martin said the car cost thirty-four hundred dollars, while Sam said he thought

it cost over three thousand dollars. Are they disagreeing? Explain.


NS3-74 Canadian Bills and Coins Goal: Students will identify Canadian coins by name and denomination, and express their values in dollar

and cent notation. They will identify correct forms of writing amounts of Canadian currency.

Prior Knowledge Required: Dollar and cent notation

Vocabulary: penny, toonie, nickel, dollar notation, dime, cent notation, quarter, currency, loonie

Remind students that when writing in dollar notation, the number of full dollars is written to the left of the

decimal point. There is no limit to how many place value columns this can extend to. Demonstrate by writing

$2.00, $22.00, $222.00 and $2222222222222.00 on the board.

However, there are only two place value columns to the right of the decimal place. Remind students that, if

you have a number of cents which is only a single digit (say 3), in dollar notation a zero is placed in the

dimes column.

Ask the class to invent some ways to write money amounts in incorrect notation. Allow them to come up

with a wide variety of ideas and welcome silly answers. Something that looks like this: 54$, is incorrect. Add

several examples yourself, for instance:

2.89$, $26.989, $67¢, ¢45, ¢576, 37.58¢, ¢67.89, $12.34¢, $1 35

Review the names and values of Canadian coins and bills. Discuss the images depicted on the coins. Point

out that the animal on the quarter is a caribou (not a moose) and that the dime shows the Bluenose, a type

of sailing ship called a schooner.

Discuss with students the relationships between various coins and bills.

Ask students how many…

a) dimes are in a loonie, toonie, or a 5 dollar bill.

b) nickels are in a loonie or toonie.

c) quarters are in a loonie, toonie, 5 or 10 dollar bill.

d) toonies are in a 10 or 20 dollar bill.

Students could use play money as manipulatives for the above problems.


Activities:

1. As a class or in small groups, have students visit the Canadian Currency Museum website and their

exhibit on Canada’s Coins. This has great information about the history and symbolism of Canada’s

coins. You could have students write a short report about one of the symbols on a Canadian coin and

why they are important enough to appear on a Canadian coin.

http://www.bankofcanada.ca/currencymuseum/eng/learning/canadascoins.php

2. Cross-curricular connection: Students could design coins that represent important symbols of the

First Nation peoples in Upper Canada in the years around 1800. Ask them to explain what image they

chose and why they think the symbol could be important.

3. (Atlantic Curriculum A4) Have students play “Race For a Loonie.” Ask each student to repeatedly

toss a die and count out pennies on a mat. Ten pennies are exchanged for a dime, and ten dimes for

a loonie. You may wish to have the students toss the die onto a plate to prevent the die from going

too far.

4. (Atlantic Curriculum A4.2) Have the students use a mat with sections marked off for $1, 10¢ and 1¢.

Ask them to toss two dice, find the sum, and place the total on the mat. Have them exchange 10

pennies (or 10 counters in the 1¢ section) for one dime (or one counter in the 10¢ section) and

continue until they have reached a dollar.

Extensions:

1. The Royal Canadian Mint has great resources available on their website. Their Currency Timeline might

be of great interest to the students. This outlines the history of Canadian settlement and development

and all the varieties of currency that have been used from the early 16th century to the present. Use this

as a resource and have students research other kinds of coins (denominations, forms, images, etc.),

that have existed in Canada’s past.

www.mint.ca/teach

Project Ideas

Choose a coin and find the following information on the mint.ca/teach web-site.

• What are the different images that have appeared on this coin and what did they

commemorate?

• Who drew the design for the coin?

• When was the coin introduced?

• Were there changes in the metal used to make the coin? When and why were these changes

made?

2. If you have any students who have lived in or travelled to other countries, have them bring in samples

of the other currency as a ‘show and tell’ for the class. Discuss the different images and shapes of the

coins and bills.


NS3-75 Adding Money Goal: Students will be able to add money amounts using traditional algorithms.

Prior Knowledge Required: Adding three-digit numbers

Adding using regrouping

Converting between dollar and cent notation

Familiarity with Canadian currency

Vocabulary: penny, quarter, nickel, loonie, dime, toonie, cent, adding, dollar , regrouping

Review adding 2-digit numbers. Demonstrate the steps: line up the numbers correctly, then add the digits in

each column starting from the right. Start with some examples that would not require regrouping, and then

move on to some which do (EXAMPLES: 15 + 23, 62 + 23, 38 + 46).

Use volunteers to pass to 3-digit addition questions, first without regrouping, then with regrouping.

SAMPLE QUESTIONS:

545 + 123, 123 + 345, then 132 + 259, 234 + 556, 578 + 789, 346 + 397.

Tell your students that you want to add $5.45 + $1.23. ASK: How many cents are in $5.45? (545¢) How

many cents are in $1.23? (123¢) How many cents is that altogether? (668¢) What is that in dollar notation?

($6.68) Show on the board:

$5.45 545¢

+ $1.23 123¢

$6.68 668¢

When adding money, the difference is in the lining up—the decimal point is lined up over the decimal point.

ASK: Are the one dollars lined up over the one dollars? The ten dollars over the ten dollars? The dimes

over the dimes? The pennies over the pennies? Tell them that if the decimal point is lined up, all the other

digits must be lined up correctly too, since the decimal point is between the ones and the dimes. Students

can model regrouping of terms using play money: for instance, in $2.33 + $2.74 they will have to group 10

dimes as a dollar.

Students should complete a number of problems in their notebooks. Some SAMPLE PROBLEMS:

a) $5.08 + $1.51 b) $3.13 + $2.98 c) $1.74 + $5.22

d) $3.95 + $4.28 e) $1.79 + $2.83

Ask volunteers to help you solve several word problems, such as: Julie spent $4.98 for a T-shirt and $3.78

for a sandwich. How much did she spend in total?


Students should also practice finding the value of sets of coins and bills and writing the amount in dollar

notation, such as:

There are 19 pennies, 23 nickels and 7 quarters in Jane’s piggybank. How much money does

she have?

Helen has a five-dollar bill, 1 toonie and 9 quarters in her pocket. How much money does she have?

Randy paid 2 toonies, a loonie, 5 quarters and 7 dimes for a parrot. How much did his parrot cost?

A mango fruit costs 69¢. I have a toonie. How many mangos can I buy?

If I add a dime, will it suffice for another one?

Have students do the following questions individually.

1. Add:

a) $7.25 b) $3.89 c) $5.08 d) $5.37 e) $2.86 + $5.17

+$1.64 +$6.23 +$3.87 +$2.79

2. Sheila saved 2 toonies, 4 dimes and 8 pennies from babysitting. Her brother Noah saved 4 loonies and

6 quarters from mowing a lawn.

a) Who has saved more money?

b) They want to share money to buy a present for their mother. How much money do

they have together?

c) They’ve chosen a teapot for $9.99. Do they have enough money?

Bring in fliers from local businesses. Ask students to select gifts to buy for a friend or relative as a

birthday gift. They must choose at least two items. They have a $10.00 budget. What is the total

cost of their gifts?

Extension: Fill in the missing information in the story problem and then solve the problem.

a) Betty bought _____ pairs of shoes for _____ each. How much did she spend?

b) Una bought ____ apples for _____ each. How much did she spend?

c) Bertrand bought _____ brooms for _____ each. How much did he spend?

d) Blake bought _____ comic books for _____ each. How much did he spend?

Bonus: Have students make up their own story problem and have a partner solve it.


NS3-76 Subtracting Money Goal: Students will subtract money amounts using traditional algorithms

Prior Knowledge Required: Subtraction of 3-digit numbers

Subtracting using regrouping

Converting between dollar and cent notation

Familiarity with Canadian currency

Vocabulary: penny, toonie, nickel, cent, dime, dollar, quarter, subtracting, loonie, regrouping

Review 2-digit and 3-digit subtraction. Start with some examples that would not require regrouping: 45 − 23,

78 − 67, 234 − 123, 678 − 354.

Show some examples on the board of numbers lined up correctly or incorrectly and have students decide

which ones are done correctly.

Demonstrate the steps: line up the numbers correctly, subtract the digits in each column starting from the

right. Move onto questions that require regrouping, EXAMPLE: 86 − 27, 567 − 38, 782 − 127, 673 − 185,

467 − 369.

Tell your students that you want to subtract $0.38 from $5.67. ASK: How many cents are in $5.67? (567)

How many cents are in $0.38? (38) What is 567 – 38?

567¢

– 38¢

529¢

ASK: What is 529¢ in dollar notation? ($5.29) Show on the board:

567¢ $5.67

– 38¢ – $0.38

529¢ $5.29

When subtracting money, the difference is in the lining up—the decimal point is lined up over the decimal

point. ASK: Are the one dollars lined up over the one dollars? The dimes over the dimes? The pennies over

the pennies? Remind them that if the decimal point is lined up, all the other digits must be lined up correctly

too, since the decimal point is between the ones and the dimes. Students can model regrouping of terms

using play money: for instance, in $2.74 – $2.36 they will have to group a dime as ten pennies.


Students should complete a number of problems in their notebooks. For EXAMPLE:

Subtract:

a) $8.89 b) $9.00 c) $4.00 d) $8.37 e) $8.47 - $0.62

−$1.64 −$7.23 −$3.87 −$5.79

Bonus: $28.14 – 17.43

ASK: Alyson went to a grocery store with $10.00. She would buy buns for $1.69, ice cream for $3.99 and

tomatoes for $2.50. Does she have enough money? If yes, how much change will she get?

Next, teach your students the following fast way of subtracting from powers of 10 (numbers such as 10,

100, 1 000 and so on) to help them avoid regrouping:

For example, you can subtract any money amount from a dollar by taking the amount away from 99¢ and

then adding one cent to the result.

1.00 .99 + one cent .99 + .01

− .57 = −.57 = −.57

.42 + .01 = .43 = 43¢

As another example, you can subtract any money amount from $10.00 by taking the amount away from

$9.99 and adding one cent to the result.

10.00 9.99 + .01

− 8.63 = − 8.63

1.36 + .01 = 1.37

NOTE: If students know how to subtract any one-digit number from 9, then they can easily perform the

subtractions shown above mentally. To reinforce this skill have students play the Modified Go Fish game

(in the MENTAL MATH section) using 9 as the target number.

Literature Connection

Alexander, Who Used to Be Rich Last Sunday. By Judith Viorst.

Last Sunday, Alexander’s grandparents gave him a dollar—and he was rich. There were so many things

that he could do with all of that money!

He could buy as much gum as he wanted, or even a walkie-talkie, if he saved enough. But somehow the

money began to disappear…

A great activity would be stopping to calculate how much money Alexander is left with every time he ends

up spending money.

Students could even write their own stories about Alexander creating a subtraction problem of their own.


Extensions:

1. a) Ella paid for a bottle of water with $2 and received 35¢ in change. How much did the water cost?

b) Jordan paid for some spring rolls with $5 and received 47¢ in change. How much did his meal cost?

c) Paige paid for a shirt with $10 and received 68¢ in change. How much did the shirt cost?

d) Geoff paid for a hockey stick with $10 and received 54¢ in change. How much did the hockey

stick cost?

2. Fill in the missing information in the story problem and then solve the problem.

a) Kyle spent $4.90 for a notebook and pencils. He bought 5 pencils for ______. How much did the

notebook cost?

b) Sally spent $6.50 for a bottle of juice and 3 apples. The apples cost ______How much did the

juice cost?

c) Clarke spent $9.70 for 2 novels and a dictionary. The ______ cost ______. How much did the

dictionary cost?

d) Mary spent $16.40 for 2 movie tickets and a small bag of popcorn. ___________________. How

much did her popcorn cost?

3. (Atlantic Curriculum B4.1) Count back the change for $5.00, if the bill totalled $3.59.

4. (Atlantic Curriculum B4.3) Write a subtraction problem that includes $1.40 and 16¢. Solve the problem.


NS3-77 Estimating Goal: Students will use rounding to estimate money amounts.

Prior Knowledge Required: Rounding

Canadian bills and coins

adding and subtracting money

dollars and cents

word problems

Vocabulary: rounding, estimating, estimate, about

Begin with a demonstration. Bring in a handful of change. Put it down on a table at the front of the class.

Tell your students that you would like to buy a magazine that costs $2.50. Do they think that you have

enough money? How could they find out?

Review rounding. Remind the class that rounding involves changing numbers to an amount that is close to

the original, but which is easier to use in mental calculations. As an example, ask what is easier to add:

8 + 9 or 10 + 10.

Explain to the class that they will be doing two kinds of rounding—rounding to the nearest 10¢ and rounding

to the nearest dollar. To round to the nearest 10¢ look at the digit in the pennies column. If the pennies digit

is less than 5, you round down. ASK: For which pennies digits would you round down? 7? 0? 3? 4? 9?

(round down for 0, 1, 2, 3 or 4) If the pennies digit is 5 or more, you round up. For which ones digits would

you round up? (5, 6, 7, 8 and 9) For example 33¢ would round down to 30¢, but 38¢ would round up to 40¢.

Use volunteers to round: 39¢, 56¢, 52¢, 75¢, 60¢, 44¢.

Model rounding to the nearest dollar. In this case, the number to look at is in the tenths place (the dimes

place). If an amount has 50¢ or more, round up. If it has less than 50¢, round down. So, $1.54 would round

up to $2.00. $1.45 would round down to $1.00. Use volunteers to round: $1.39, $2.56, $3.50, $4.75, $0.60,

$0.49.

Give several problems to show how rounding can be used for estimation:

Make an estimate and then find the exact amount:

a) Dana has $5.27. Tor has $2.38. How much more money does Dana have than Tor?

b) Mary has $3.74. Sheryl has $5.33. How much money do they have altogether?

a) Jason has saved $9.95. Does he have enough money to buy a book for $4.96 and a binder for $5.99?

Why is rounding not helpful here?

Ask students to explain why rounding to the nearest dollar isn’t helpful for the following question:

“Millicent has $2.15. Richard has $1.97. About how much more money does Millicent have than Richard?”

(Both round to $2. Rounding to the nearest 10 cents helps.)


Make sure students understand that they can use rounding to check the reasonableness of answers. Ask

students to explain how they know that $2.52 + $3.95 = $9.47 can’t be correct.

Have students solve the following problems individually.

1. Make an estimate and then find the exact amount: Molina has $7.89. Vinijaa has $5.77. How much

more money does Molina have?

2. Benjamin spent $2.94 on pop, $4.85 on vegetables and dip and $2.15 on bagels. About how much did

he spend altogether?

Activities:

1. Ask students to estimate the total value of a particular denomination (EXAMPLE: dimes, loonies, etc.)

that would be needed to cover their desk or a book. Students could use play money to test their

predictions. Students could also trace an outline of their hand and predict the value of dimes that would

cover their hand and then test their prediction.

2. Place a handful of play money bills and coins on a table and cover up the amount after students have

had a chance to look at it 5 or 10 seconds. Ask students to estimate the total amount. Alternatively, you

might ask students to estimate how much extra money would be needed to make a particular amount

(say $5 or $10).

1. Pairs of students practice estimating by using play money coins. Player 1 places ten to fifteen play

money coins on a table. Player 2 estimates the amount of money before counting the coins.

Extensions:

1. What is the best way to round when you are adding two numbers: to round both to the nearest dollar, or

to round one up and one down?

Explore which method gives the best answer for the following amounts:

$2.56 + $3.68 $4.55 + $4.57 $6.61 + $1.05

Students might notice that, when two numbers have cent values that are both close to 50¢ and that are

both greater than 50¢ or both less than 50¢, rounding one number up and one down gives a better

result than the standard rounding technique. For instance, rounding $2.57 and $3.54 to the nearest

dollar gives an estimated sum of $7.00 ($2.57 + $3.54 ≈ $3.00 + $4.00), whereas rounding one

number up and the other down gives an estimated sum of $6.00, which is closer to the actual total.

2. (Adapted from Atlantic Curriculum B10)

a) Popsicles cost 10¢ each. How many popsicles could you buy with a loonie? (ANSWER: 10)

b) Popsicles cost 20¢ each. How many popsicles could you buy with a loonie? (ANSWER: 5)

c) Popsicles cost 14¢ each. About how many popsicles could you buy with a loonie? Will the answer

be closer to 5 or 10? If I want to buy a popsicle for myself and five others, will a loonie be enough?

d) If erasers are on sale for 19¢, how many would you estimate you could buy with a loonie?

e) If I have a loonie, can I buy 3 party favours that cost 29¢ each?

f) If I have a loonie, can I buy 5 packages of stickers that cost 21¢ each?


NS3-78 Equal Parts Goal: Students will verbally express fractions given as diagrams or numerical notation.

Prior Knowledge Required: The ability to count

Fraction of an area

Ordinal numbers

Vocabulary: part, numerator, whole, denominator, fraction, area

Draw:

Ask your students how many circles are shaded.

Draw then ask them again how many circles are shaded.

Explain that the whole circle is no longer shaded. Ask your students if they know the word for a number that

is not a whole number, but is only part of a whole number. [Fraction.]

Explain that a fraction has a top and bottom number, then ask your students if they know what the numbers

represent. Explain that the shaded fraction of the circle is written as 34 (pronounce this as “three over four” for

now). What does the 3 represent? What does the 4 represent? Draw more fractions and ask students who

understand the significance of the numbers to identify the fractions without explaining it to the rest of the

class.

14

34

24

24

Draw examples with different denominators. Ask: What does the top number of the fraction represent? [The

number of shaded parts.] What does the bottom number of the fraction represent? [The number of parts in a

whole.]

If some students say that the bottom number is counting the number of parts altogether, tell them that from

what they’ve seen so far, that’s a good answer, but later we will see improper fractions where we have more

than one whole pie, so if each pie has 4 pieces and we have 2 pies, there are 8 pieces altogether, but we still

write the bottom number as 4. For example, if you bought 2 pies and 3 pieces were eaten from each pie, you

could say that 64 of a pie was eaten. (You could also say that

68 of what you bought was eaten, but that would

change the whole to 2 pies instead of 1 pie). Explain that fractions aren’t generally pronounced as they are.

Ask your students if they know the expressions for “three over four.” [Three-fourths or three-quarters.] Then

ask if they know the expression for 12 . Draw a half-shaded circle to encourage answers.


Draw several diagrams and have students express the fraction for the shaded parts of each whole by

writing the top and the bottom numbers.

Gradually increase the number of parts in each whole and shaded parts.

Ask them to explain their methods (skip counting or multiplication, for example) for determining the total

number of squares. At first, shade parts in an orderly way to facilitate a count. Then shade parts randomly,

but never exceed more than 20 shaded parts, and start with a small number of parts.

Ask students if they know which number—the top or bottom—is called the numerator. [Top.]

ASK: Does anyone know what the bottom number is called? [The denominator.] Which number—the

numerator or denominator—expresses the amount of equal parts in the whole?

Have students shade the correct number of parts to illustrate the following fractions.

16

35

47

Extension: A sport played by witches and wizards on brooms regulates that the players must fly higher

than 5 m above the ground over certain parts of the field (shown as shaded). Over what fraction of the field

must the players fly higher than 5 m?


NS3-79 Models of Fractions Goal: Students will name fractions given as diagrams or in numerical notation. Students will understand

that parts of a whole must be equal to determine a fraction that one or more parts represent.

Prior Knowledge Required: Expressing fractions

Fraction of an area

Fraction of a length

Vocabulary: numerator, denominator, fraction, part, whole

Illustrate these fractions — 12 ,

13 ,

14 ,

15 ,

16 ,

17 ,

18 — with circles.

Ask students if they know the proper way to pronounce these fractions. Remind them of ordinal numbers.

Say: If Sally is first in line, Tom is second in line and Rita is in line behind Tom, in what place is Rita? If Bilal

is in line behind Rita, in what place is Bilal? Continue to the eighth position. Explain that most of the ordinal

numbers—except for first and second—are also used for fractions. No one refers to half of a pie as one-

second of a pie, but we do say one-third, one-fourth, one-fifth, and so on.

Then ask your students to pronounce these fractions: 1

11 , 124 ,

113 ,

119 ,

1100 ,

192 .

If your students are comfortable with ordinal numbers up to a hundred, 1

92 could lead to some confusion,

since “first” and “second” are usually unused when dealing with fractions. In this case, the fraction is

expressed as “one ninety-second of a whole.” Explain that fractions with a numerator larger than one are

expressed the same way, with the numerator followed by the ordinal number. For example, 311 is expressed

as “three elevenths”. The ordinal number is pluralized when the numerator is greater than one, i.e., one

eleventh, two elevenths, three elevenths, and so on. Some ESL students might find it helpful to contrast this

with how we say 200 = two hundred, not two hundreds.

Ask your students to pronounce these fractions: 314 ,

295 ,

17100 ,

9495 ,

6183 ,

4151 ,

3052 .

Include fractions on spelling tests by writing the numeric fraction on the board.

Ask your students if they have ever been given a fraction of something (like food) instead of the whole, and

gather their responses. Bring a banana (or some easily broken piece of food) to class. Break it in two very

unequal pieces. Say: This is one of two pieces. Is this half the banana? Why not? Emphasize that the parts

have to be equal for either of the two pieces to be a half. Then draw the following rectangle.


Ask your students if they think the rectangle is divided in half. Explain that the fraction 12 not only expresses

one of two parts, but it more specifically expresses one of two equal parts. Draw numerous examples of this

fraction, some that are equal and some that are not, and ask volunteers to mark the diagrams as correct or

incorrect.

�� x x �

Ask: Which diagram illustrates one fourth? What’s wrong with the other diagram? Isn’t one of the four pieces

still shaded?

Then draw the following diagram on the board:

Ask volunteers to come to the board to show 34 in different ways. Then challenge a volunteer to draw the

circle themselves and to show 34 in yet a different way. Have students practice individually drawing a circle

divided into fourths in their notebooks and to find as many ways of showing 24 as they can.

Then show students the four ways of showing 14 :

Challenge students to find another way of showing 14 in a circle. Draw the following pictures on the board to

help them:

Have volunteers draw their own circle divided into fourths to show yet another way of modelling 1/4. You

could help them get started by drawing one line for them:

Then challenge students to individually show in their notebooks many ways of dividing a pie into halves, and

then into thirds (they should find a pie drawn in the workbook that is already divided into thirds to help them)

and then into eighths.


Bonus: Cut the following pies into thirds:

Activity: After students have completed exercise 6 and are familiar with the words “half,” “third,” “fourth,”

and so on, ask them to identify fractions (i.e. of pizzas, of objects in the classroom such as the blackboard,

the carpet, their desk, their pencil, and of diagrams) using phrases such as “one half,” “two thirds,” etc.

Extension: Have students ask their French teacher if ordinal numbers are used for fractions in French,

and have them tell you the answer next class.


NS3-80 Fractions of a Region or Length Goal: Students will understand that parts of a whole must be equal to determine an entire measurement,

when given only a fraction of the entire measurement.

Prior Knowledge Required: Expressing fractions

Fraction of an area

Fraction of a length

Vocabulary: numerator, denominator, fraction, part, whole

Explain that it’s not just shapes like circles and squares and triangles that can be divided into fractions, but

anything that can be divided into equal parts. Draw a line and ask if a line can be divided into equal parts.

Ask a volunteer to guess where the line would be divided in half. Then ask the class to suggest a way of

checking how close the volunteer’s guess is. Have a volunteer measure the length of each part. Is one part

longer? How much longer? Challenge students to discover a way to check that the two halves are equal

without using a ruler, only a pencil and paper. [On a separate sheet of paper, mark the length of one side of

the divided line. Compare that length with the other side of the divided line by sliding the paper over. Are

they the same length?]

Have students draw lines in their notebooks and then ask a partner to guess where the line would be

divided in half. They can then check their partner’s work.

ASK: What fraction of this line is double?

Say: The double line is one part of the line. How many parts, equal to that one, are in the whole linne,

including the double line? [5, so the double line is 15 of the whole line.]

Mark the length of the double line on a separate sheet of paper. Compare that length to the entire line to

determine how many of those lengths make up the whole line. Repeat with more examples.

Then ask students to express the fraction of shaded squares in each of the following rectangles.


Have them compare the top and bottom rows of rectangles.

ASK: Are the same fraction of the rectangles shaded in both rows? Explain. If you were given the

rectangles without square divisions, how would you determine the shaded fraction? What could be used to

mark the parts of the rectangle? What if you didn’t have a ruler? Have them work as partners to solve the

problem. Suggest that they mark the length of one square unit on a separate sheet of paper, and then use

that length to mark additional square units.

Prepare several strips of paper with one unit shaded, and have students determine the shaded fraction

without using a pencil or ruler. Only allow them to fold the paper. For instance they could fold the following

pieces of paper into 3, 4 and 5 equal parts respectively.

Draw a shaded square and ask students to extend it so the shaded part becomes half the size

of the extended rectangle.

Repeat this exercise for squares that are one-third and one-quarter the size of extended rectangles. ASK:

How many equal parts are needed? [Three for one-third, four for one-quarter.]

How many parts do you already have? [1.] So how many more equal parts are needed?

[Two for one-third, three for one-quarter.]

Activities:

After students have completed Question 3 from worksheet NS3-80, ask them to use a ruler to draw

rectangles in which:

a) 13 of the area is shaded b)

34 of the area is shaded

Extensions

1. On grid paper draw a rectangle with a width of 2 boxes and a length of 3 boxes. Shade 13 of the boxes.

2. On grid paper draw a rectangle with a width of 2 boxes and a length of 5 boxes. Shade 15 of the boxes.

3. a) Sketch a pie and cut it into fourths. How can it be cut into eighths?

b) Sketch a pie and cut it into thirds. How can it be cut into sixths?

4. Ask students to identify several fractions they see in the classroom and name them (e.g. about one third

of the chalk board is covered with writing, the room is about 4 fifths wide as it is long).


NS3-81 Equals Parts of a Set Goal: Students will understand that fractions can represent equal parts of a set.

Prior Knowledge Required: Fractions as equal parts of a whole

Review equal parts of a whole. Tell your students that the whole for a fraction might not be a shape like a

circle or square. Tell them that the whole can be anything that can be divided into equal parts. Brainstorm

with the class other things that the whole might be: a line, an angle, a container, apples, oranges, amounts

of flour for a recipe. Tell them that the whole could even be a group of people. For example, the grade 3

students in this class is a whole set and I can ask questions like: what fraction of students in this class are

girls? What fraction of students in this class are eight years old? What fraction of students wear glasses?

What do I need to know to find the fraction of students who are girls? (The total number of students and the

number of girls). Which number do I put on top: the total number of students or the number of girls? (the

number of girls). Does anyone know what the top number is called? (the numerator) Does anyone know

what the bottom number is called? (the denominator) What number is the denominator? (the total number of

students). What fraction of students in this class are girls? (Ensure that they say the correct name for the

fraction, for example: “eleven twentieths” instead of “eleven over twenty”) Tell them that the girls and boys

don’t have to be the same size; they are still equal parts of a set. Ask students to answer: What fraction of

their family is older than 8? Younger than 8? Female? Male? Some of these fractions, for some students,

will have numerator 0, and this should be pointed out. Avoid asking questions that

will lead them to fractions with a denominator of 0 (For example, the question “What fraction of your siblings

are male?” will lead some students to say 0/0).

Then draw pictures of shapes with two attributes changing:

a)

ASK: What fraction of these shapes are shaded?

What fraction are circles?

What fraction of the circles are shaded?

b)

ASK: What fraction of these shapes are shaded? What fraction are unshaded? What fraction are

squares? What fraction are triangles? Bonus: What fraction of the triangles are shaded? What fraction

of the squares are shaded? What fraction of the squares are not shaded?

Have students write fraction statements in their notebooks for similar pictures.

Then tell your students that you have five squares and circles. Some are shaded and some are not. Have

students draw shapes that fit the puzzles:


a) 25 of the shapes are squares.

25 of the shapes are shaded. One circle is shaded.

SOLUTION:

b) 35 of the shapes are squares.

25 of the shapes are shaded. No circle is shaded.

c) 35 of the shapes are squares.

35 of the shapes are shaded.

13 of the squares are shaded.

Ask some word problems:

A basketball team played 5 games and won 2 of them. What fraction of the games

did the team win?

A basketball team won 3 games and lost 1 game. How many games did they play altogether? What

fraction of their games did they win?

Bonus: A basketball team won 4 games, lost 1 game and tied 2 games. How many games did

they play? What fraction of their games did they win?

Activities:

1. On a geoboard, have students enclose a given

fraction of the pegs with an elastic.

For instance, 1025 .

2. (Adapted from Atlantic Curriculum Grade 4) Have the students “shake and spill” a number of two-

coloured counters and ask them to name the fraction that represents the red counters.

Extensions:

1. There are 8 figures in total. 12 of the figures are squares. The rest of the shapes are triangles.

58 of the figures are shaded. One triangle is shaded. How many squares are shaded?

2. a) Complete the following sentences (by writing a fraction in the first blank):

ii) _____ of the children in my family are _____ ii) _____ of the children in my class are _____

b) Make up your own sentence like the ones above in a).

3. Then draw the following picture.

ASK: How many pieces are shaded? (1) How many pieces are there altogether? (4)

Are one quarter of the pieces shaded? (yes) Is 14 of the shape shaded? (No, because

the four pieces do not have equal area.) Note that, just like boys and girls don’t have

to be the same size to be equal parts of a set, the pieces don’t have to be the same

size to be equal parts of a set.


NS3-82 Parts and Wholes Goal: Students will compare fractions with the same denominator. Students will compare fractions with

the same numerator. Students will divide shapes into equal parts to determine what fraction of the shapes

are shaded.

Prior Knowledge Required: Fractions as area

Parts of a whole must be equal to determine a fraction that one or

more parts represent.

Comparing fractions with the same denominator.

Draw on the board:

Have students name the fractions shaded and then have them say which circle has more shaded. ASK: Which is

more: one fourth of the circle or three fourths of the circle? Then draw different shapes on the board:

14

34

14

34

Have volunteers show the fractions on the board by shading and ASK: Is three quarters of something

always more than one quarter of the same thing? Is three quarters of a metre longer or shorter than a

quarter of a metre? Is three quarters of a dollar more money or less money than a quarter of a dollar? Is

three fourths of an orange more or less than one fourth of the orange? If possible, bring in an actual orange,

or use a paper circle, cut into quarters. Put one quarter aside and count the remainder: one quarter, two

quarters, three quarters. ASK: Which is more: one quarter or three quarters? Is three fourths of the class

more or less people than one fourth of the class? If three fourths of the class have brown eyes and one

quarter of the class have blue eyes, do more people have brown eyes or blue eyes?

Tell students that if you consider fractions of the same whole—no matter what whole you’re referring to—

three quarters of the whole is always more than one quarter of that whole, so mathematicians say that the

fraction 34 is greater than the fraction

14 . Ask students if they remember what symbol goes in between:

34

14 (< or >).

Remind them that the inequality sign is like the mouth of a hungry person who wants to eat more of the

pasta but has to choose between three quarters of it or one quarter of it. The sign opens toward the bigger

number:

34

14 or

14

34 .


Ensure that students understand that three of anything is always more than one of them. ASK: Are three

fifths more than one fifth?

Count as you colour in the fifths: one fifth, two fifths, three fifths.

ASK: Are three eighths more than one eighth? (Draw a picture of two pies, one with one eighth shaded and

the other with three eighths shaded). Which is more: five eighths or seven eighths? Five apples or seven

apples? $5 or $7? If Sally gets one sixth and Tony gets two sixths, who gets more? If Sally gets three sixths

and Tony gets two sixths, who gets more?

Have volunteers circle the largest fraction: 35 ,

25 ;

17 ,

47 ;

36 ,

56 ;

58 ,

38 .

Have students solve the following problems individually.

19 or

29 ?

112 or

212 ?

153 or

253 ?

1100 or

2100 ?

1807 or

2807 ?

29 or

59 ?

311 or

411 ?

911 or

811 ?

3587 or

4387 ?

91102 or

52102 ?

Bonus: 7 432

25 401 or 869

25 401 ? 52 645

4 567 341 or 54 154

4 567 341 ?

Comparing fractions with the same numerator.

Draw on the board:

12

13

14

15

Have a volunteer colour the first part of each strip of paper and then ask students which fraction shows the

most: 12 ,

13 ,

14 or

15 . Ask: Do you think one sixth of this fraction strip will be more or less than one fifth of it?

Will one eighth be more or less than one tenth? Then colour the second fifth and ASK: Which is more: two

fifths or one half. ASK: Two is more than one; why aren’t two fifths more than one half? (The fifth-sized

pieces are smaller than the half-sized pieces. It’s like saying two pencils are longer than one desk because

2 is more than 1 – show the two pencils next to your desk).


Then draw two circles the same size on the board:

Have volunteers shade one part of each circle. ASK: What fractions are represented? Which fraction

represents more? Repeat with different shapes:

12

14

12

14

12

14

Ask: Is one half of something always more than one quarter of the same thing? Is half a metre longer or

shorter than a quarter of a metre? Is half an hour more or less time than a quarter of an hour? Is half a dollar

more money or less money than a quarter of a dollar? Is half an orange more or less than a fourth of the

orange? Is half the class more or less than a quarter of the class? If half the class has brown eyes and a

quarter of the class has green eyes, do more people have brown eyes or green eyes?

Tell students that no matter what quantity you have, half of the quantity is always more than a fourth of it, so

mathematicians say that the fraction 12 is greater than the fraction

14 . Ask students if they remember what

symbol goes in between: 12

14 (< or >). What symbol goes in between now?

14

12

Tell your students that you are going to try to trick them with this next question so they will have to listen

carefully. Then ASK: Is half a minute longer or shorter than a quarter of an hour? Is half a centimetre longer

or shorter than a quarter of a metre? Is half of Stick A longer or shorter than a quarter of Stick B?

Stick A:

Stick B:

Ask: Is a half always bigger than a quarter?

Allow everyone who wishes to attempt to articulate an answer. Summarize by saying: A half of something is

always more than a quarter of the same thing. But if we compare different things, a half of something might

very well be less than a quarter of something else. When mathematicians say that 12 >

14 , they mean that a

half of something is always more than a quarter of the same thing; it doesn’t matter what you take as your

whole, as long as it’s the same whole for both fractions.

Draw the following strips on the board:

Ask students to name the fractions and then to tell you which is more.


Have students draw the same fractions in their notebooks but with circles instead of strips.

Is 34 still more than

38 ? (yes, as long as the circles are the same size)

Bonus:

Show the same fractions using a line of length 8 cm.

Ask students: If you cut the same strip into more and more pieces of the same size, what happens to the

size of each piece?

Draw the following picture on the board to help them:

1 big piece

2 pieces in one whole



Many pieces in one whole

Ask: Do you think that 1 third of a pie is more or less pie than 1 fifth of the same pie? Would you rather have

one piece when it’s cut into 3 pieces or 5 pieces? Which way will you get more? Ask a volunteer to show how

we write that mathematically (13 >

15 ).

Do you think 2 thirds of a pie is more or less than 2 fifths of the same pie? Would you rather have two pieces

when the pie is cut into 3 pieces or 5 pieces? Which way will get you more? Ask a volunteer to show how we

write that mathematically (23 >

25 ).

If you get 7 pieces, would you rather the pie be cut into 20 pieces or 30? Which way will get you more pie?

How do we write that mathematically? (720 >

730 ).

Ask the students to answer individually: Which is greater?

a) 12 or

17 b)

13 or

14 ? c)

19 or

16 ? d) Bonus:

1132 or

1147

e) 37 or

38 f)

419 or

415 g)

822 or

825 h) Bonus:

132234 or

132198

SAY: Two fractions have the same numerator and different denominators. How can you tell which fraction is

bigger? Why? Summarize by saying that the same number of pieces gives more when the pieces are

bigger. The numerator tells you the number of pieces, so when the numerator is the same, you just look at

the denominator.

The bigger the denominator, the more pieces you have to share between and the smaller the portion you

get. So bigger denominators give smaller fractions when the numerators are the same.


SAY: If two fractions have the same denominator and different numerators, how can you tell which fraction

is bigger? Why? Summarize by saying that if the denominators are the same, the size of the pieces are the

same. So just as 2 pieces of the same size are more than 1 piece of that size, 84 pieces of the same size

are more than 76 pieces of that size.

Emphasize that students can’t do this sort of comparison if the denominators are not the same. ASK: Would

you rather 2 fifths of a pie or 1 half? Draw the following picture to help them:

Tell your students that when the denominators and numerators of the fractions are different, they will have

to compare the fractions by drawing a picture or by using other methods that they will learn in later grades.

The same fraction of the same thing are always equal.

Draw:

ASK: What fraction of the first square is shaded? What fraction of the second square is shaded? Are the

squares the same size? Are the shaded parts the same size?

Repeat for various pairs of shapes:

Challenge students to find various ways of dividing these shapes into quarters.

EXAMPLES:

Dividing shapes into equal parts.

Draw on the board the shaded strips from before:


ASK: Is the same amount shaded on each strip? Is the same fraction of the whole strip shaded in each

case? How do you know? Then draw two hexagons as follows:

ASK: Is the same amount shaded on each hexagon? What fraction of each hexagon is shaded? Then

challenge students to find the fraction shaded by drawing their own lines to divide the shapes into

equal parts:

Give your students pattern blocks. Ask them to make a rhombus from the triangles. How many triangles do

they need? What fraction of a rhombus is a triangle? Challenge them to find:

a) What fraction of a hexagon is the rhombus? The trapezoid? The triangle?

b) What fraction of a trapezoid is the triangle?

Activities:

1. Give each group of 3 students 3 large congruent shapes, but cut differently into the same number of

equal parts. EXAMPLE:

Have students shade one part of each shape and then cover, in a single layer, the shaded part with as

many small counters as they can. Did each quarter of the shape require the same number of counters, at

least approximately?

2. Have students toss several coins (or two-colour counters). What fraction of the coins came up heads?

What fraction of the coins came up tails? What do the two numerators add to? (the denominator) Why do

they think that happened? Will it always happen?

3. Ask students to enclose a fraction of the pegs on a geoboard and then to write the fraction of pegs that

are not enclosed. Have students investigate with many examples, so that they see that the two

numerators always add up to the denominator. For extra practice with fractions that add to 1, use the

2-page BLM “Fractions That Add to 1.”


Extensions:

1. a) What fraction of a tens block is a ones block?

b) What fraction of a tens block is 3 ones blocks?

c) What fraction of a hundreds block is a tens block?

d) What fraction of a hundreds block is 4 tens blocks?

e) What fraction of a hundreds block is 32 ones blocks?

f) What fraction of a hundreds block is 3 tens blocks and 2 ones blocks?

2. On a geoboard, show 3 different ways to divide the area of the board into 2 equal parts.

EXAMPLES:

3. Give each student a set of pattern blocks. Ask them to identify the whole of a figure

given a part.

a) If the pattern block triangle is 16 of a pattern block, what is the whole?

Answer: The hexagon.

b) If the pattern block triangle is 13 of a pattern block, what is the whole?

Answer: The trapezoid.

c) If the pattern block triangle is 12 of a pattern block, what is the whole?

Answer: The rhombus.

d) If the rhombus is 16 of a set of same-shaped pattern blocks, what is the whole?

Answer: 2 hexagons or 6 rhombuses or 12 triangles or 4 trapezoids.

2. Draw each shape below onto cm grid paper so that each square takes up a 4 cm by 4 cm square. Have

students find the area of each shaded piece:

Bonus:

To help students count half squares and whole squares, see ME3-30.

5. The pattern block triangle represents 14 . What might the whole be?


6. (From Atlantic Curriculum A3.9) This shape is 12 of a larger one. What could the larger one look like?

How many different possibilities can you find?

a) b) c) d)

NOTE: If you prefer, you could assign these shapes on grid paper (rather than dot paper or a

geoboard).

7. Students can construct a figure using the pattern block shapes and then determine what fraction of the

shapes is covered by the pattern block triangle.

8. What fraction of the figure is covered by…

a) The shaded triangle

b) The small square

9. This extension is best done after Activity 2 above. Compare the following fractions by comparing how

much of a whole pie is left if the following amounts are eaten: 34 or

45 . Emphasize that the fraction with

a bigger piece left-over is the smaller fraction.

10. Write the following fractions in order from least to greatest. Explain how you found the order. 13 ,

12 ,

23 ,

34 ,

18

HINT: Use the ideas from Extension 9 above.

11. Fold a strip of paper 3 times to create eighths. Write the following fractions over top of the

corresponding folds:

18

38

12

78 Bonus:

14

12. Give each student three strips of paper. Ask them to fold the strips to divide one strip into halves, one

into quarters, one into eighths. Use the strips to find a fraction between

a) 38 and

58 (one answer is

12 ) b)

14 and

24 (one answer is

38 ) c)

58 and

78 (one answer is

38 )


13. Have students fold a strip of paper (the same length as they folded in EXTENSION 12 into thirds by

guessing and checking. Students should number their guesses.

EXAMPLE:

Try folding here too short, so try a little further

1st guess

2nd

guess

Is 13 a good answer for any part of Extension 12? How about

23 ?

14. Why is 23 greater than

25 ? Explain.

15. Why is it easy to compare 25 and

212 ? Explain.


NS3-83 Sharing and Fractions Goal: Students will find fractions of numbers. Students will see examples of equivalent fractions.

Prior Knowledge Required: Fractions as area

Fractions of a set

Finding fractions of whole numbers.

Brainstorm the types of things students can find fractions of (circles, squares, pies, pizzas, groups of

people, angles, hours, minutes, years, lengths, areas, capacities, apples).

Brainstorm some types of situations in which it wouldn’t make sense to talk about fractions. For example:

Can you say 3 12 people went skiing? I folded the sheet of paper 4

14 times?

Explain to your students that it makes sense to talk about fractions of almost anything, even people and

folds of paper, if the context is right: EXAMPLE: Half of her is covered in blue paint; half the fold is covered

in ink. Then teach them that they can take fractions of numbers as well. ASK: If I have 6 hats and keep half

for myself and give half to a friend, how many do I keep? If I have 6 apples and half of them are red, how

many are red? If I have a pie cut into 6 pieces and half the pieces are eaten, how many are eaten? If I have

a rope 6 metres long and I cut it in half, how long is each piece? Tell your students that no matter what you

have 6 of, half is always 3. Tell them that mathematicians express this by saying that the number 3 is half of

the number 6.

Tell your students that they can find 12 of 6 by drawing two circles. Put one dot in each circle until you have

placed 6 dots.

2 dots 4 dots 6 dots

Now, one half of the dots are in the first circle and one half of the dots are in the second circle. So 12 of 6

is 3. Have students use this method to find 12 of a) 10 b) 8 c) 14

Then have students draw 3 circles to find 13 of a) 6 b) 12 c) 9 d) 15 e) 3 f) 18.

Then have students write a division statement for each picture a) to f) above. (6 ÷ 3 = 2 and 12 ÷ 3 = 4 and

so on)


Then have students draw a picture and write a division statement for each fraction of a number:

a) 12 of 8 b)

13 of 12 c)

14 of 8 d)

12 of 10

e) 15 of 10 f)

15 of 20 g)

14 of 12 h)

15 of 15

Write a division statement to find the fraction of the number without using pictures.

a) 12 of 22 b)

14 of 84 c)

15 of 100 d)

110 of 70

Teach your students to see the connection between the fact that 6 is 3 twos and the fact that 13 of 6 is 2. The

exercise below will help with this:

Complete the number statement using the words “twos”, “threes”, “fours” or “fives”. Then draw a picture

and complete the fraction statements. (The first one is done for you.)

Number Statement Picture Fraction Statement

a) 6 = 3 twos 13 of 6 = 2

23 of 6 = 4

b) 12 = 4 ________ 14 of 12 =

24 of 12 =

34 of 12 =

c) 15 = 3 ________ 13 of 15 =

23 of 15 =

Equivalent fractions

Tell your students that sometimes two fractions that look different can mean the same thing. Show the

following pictures:


ASK: What fraction does each picture show? (12 ,

24 ,

48 and

36 ) ASK: Is the same amount shaded in each

picture? Ask students to find other fractions that mean the same as 12 . ASK: If a pie had 10 pieces, how

many would be half? What fraction with denominator 10 means the same thing as 12 ?

Have students find fractions that mean the same as 12 and have given denominators:

12 = 6

12 = 12

12 = 8

12 = 10

12 = 20

12 = 14

Then have students find fractions that mean the same as 12 and have given numerators:

12 =

2

12 =

5

12 =

3

12 =

4

12 =

8

12 =

50

Finally, mix up questions of both types:

12 =

6

12 = 6

12 =

10

12 = 10

12 = 16

12 =

16

Activities:

1. Have students compare using fraction strips 12 and

24 ,

34 and

68 ,

14 and

28 ,

12 and

48 ,

24 and

48 .

2. Students can find fractions of a whole number using the following method: Find 12 of 12.

STEP 1 – Make a model of 12 using 12 yellow counters

STEP 2 – Replace yellow counters one at a time with red counters until an equal number of the

counters are red and yellow.

Ask students to use this method to find 12 of 6, 8, 10, 14, etc.

3. Students can find 12 of 6 by drawing rows of dots. Put 2 dots in each row until you have placed 6 dots.

STEP 1 STEP 2 STEP 3

There are 3 dots in each column, so 3 is 12 of 6. Have students find

12 of 10, 12, 8 and 16 using this

method. Students might also find 13 of various numbers by drawing rows of dots with 3 dots in each row.


Extensions:

1. How many months are in:

a) 12 year? b)

13 year? c)

14 years?

2. How many minutes are in:

a) 13 of an hour? b)

12 of an hour? c)

14 of an hour?

3. How many hours are in:

a) 12 of a day? b)

13 of a day? c)

14 of a day?

4. Give your students counters to model the following problems.

a) 2 is 13 of a set. How many are in the set? b) 3 is

14 of a set. How many are in the set?

5. Show students how to find 23 of 6 dots.

STEP 1. Find 13 of 6 dots.

3 dots 6 dots

So 13 of 6 is 2.

STEP 2. Multiply by 2.

Each circle has 13 of 6 dots, so 2 circles have

23 of 6 dots.

23 of 6 is 4. Have students use this method

to find:

a) 23 of 9 b)

34 of 12 c)

23 of 12 d)

25 of 10 e)

35 of 15

6. Have students investigate.

a) 34 of 4 b)

45 of 5 c)

37 of 7 Bonus: What is

13215 of 215?


7. (Atlantic Curriculum A3.4) Tell the student that Lee and Teddy bought their mother a gift for Christmas

which cost $20. Lee paid 34 of the cost, and Teddy paid the balance. Ask: How much did each pay?

Provide coloured counters to help him/her solve the problem.

8. (Adapted from Atlantic Curriculum A3.5)

a) Pair each student with a partner to solve this problem: Eight-year-old Samantha, whose birthday is

January 25th, said, “I can’t wait until I’m 8 and 1112 .” Ask: Why was she excited?

b) Natalia said that she will turn 8 and 13 on Christmas day. When is her birthday?

9. (Atlantic Curriculum A3.6) Ask the student to tell why, whenever you see a model of 13 , there is always a

model of 23 associated with it.

10. (From Atlantic Curriculum A3.8) You have 8 coins. Half of them are pennies. More than 18 of them

are quarters. The others are nickels. Use coins to represent the situation. How much money might

you have?


NS3-84 Comparing Fractions Goal: Students will decide when a picture shows more than half or less than half, about a half, or

about a third or about a fourth.

Prior Knowledge Required: Comparing fractions

Using models to represent fractions

Vocabulary: less than, more than, half, close to

Ask students to tell you whether the shaded fraction is more than half or less than half and how they can tell:

Then repeat the exercise, but this time have a volunteer extend one of the border lines of the shaded region

to show half. Then give several similar problems for students to do individually, and tell them to imagine

where the line would be extended.

Ask students to tell you whether the shaded fraction is less than 13 , more than

13 but less than

23 or more than

23 and have students explain how they can tell:

Then repeat the exercise, but this time have a volunteer use one of the border lines of the shaded region to

show the pie cut into thirds. Then give several similar problems for students to do individually, and tell them

to imagine where the lines would be to divide the pies into thirds.

Repeat with fractions that show less than a quarter, between one and two quarters, between two and three

quarters and more than three quarters.

Draw some fraction strips on the board:

12

13

14


Give students fraction strips with a part shaded. EXAMPLES:

Have students estimate what fraction of the strips are shaded. Challenge the students to find a way to check

their answer by folding the fraction strips.

Then ask: Which figure shows about 13 ?

Then show on a strip of paper the following circles:

Ask students to estimate what fraction of the circles are shaded. ASK: How can we check by folding the

paper? Demonstrate folding the paper and then circling the 4 groups of 3 circles. ASK: How many circles are

shaded? (3) How many groups of 3 circles are there altogether? What fraction of the circles are shaded?

(one group out of four groups are shaded, so one fourth or one quarter of the circles are shaded. Show

students how they can do this without folding.

ASK: How many circles are shaded? (2) Demonstrate circling groups of 2 circles until you have circled all the

circles. ASK: How many groups of 2 circles are there? (5) What fraction of the circles is shaded? (one group

out of five groups are shaded, so one fifth of the circles is shaded. Give students several such problems to

practise with.

Activities:

1. Oranges or Apples

Bring oranges or apples into the class and cut some in fourths and some in half and some in eighths. Ask

students to decide which is more between different fractions: for example, one half or three eighths.

Encourage them to put three eighths together so that they can see that it is not quite half.

2. Mark off 13 and

23 of a clean plastic cylinder with scotch tape. Partially fill the cylinder with water and turn

the cylinder so students can’t see the tape marks. Ask them whether the cylinder is closer to 13 or

23 full.

Then turn the cylinder so that students can see the tape marks. Did they guess correctly? Repeat with

different amounts of liquid


Extension:

Ordering Fraction Strips

Use the BLM “Shaded Fraction Strips” and give each student three fraction strips to compare. They can tape

the fraction strips to a coloured piece of paper in order from smallest fraction to largest fraction and write

their conclusions on the same paper. If they are familiar with the < and > symbols for “less than” and “greater

than” students can write their answers in the form 38 <

12 <

23 . Otherwise, they can write sentences: “

38 is less

than 12 ” and “

12 is less than

23 .”

Literature/Cross-Curricular Connections:

Apple Fractions by J. Pallotta Different apples are used to teach kids all about fractions. Students will learn

to divide apples into halves, thirds, fourths, and more. See the above activity.


NS3-85 Fractions Greater than One Goal: Students will model fractions greater than 1 by using pictures.

Prior Knowledge Required: comparing fractions, using models to represent fractions

Vocabulary: less than, more than

Tell students that beads come in packs of 4. Ask volunteers to show, by modelling beads with circles:

a) one whole pack, b) one half of a pack, c) 2 packs, d) one pack and another half of a pack.

Tell students that, rather than saying one pack and then another half of a pack, we describe this as one and

a half packs. Ask a volunteer to draw what they think two and a half packs will look like.

Then tell students that beads come in packs of 6. Ask students to show individually in their notebooks: a) one

half of a pack, b) one and a half packs, c) two packs, d) two and a half packs, e) three and a half packs.

Then tell students that beads come in packs of 3. Ask students to show individually in their notebooks: a) one

third of a pack, b) two thirds of a pack, c) one and a third packs, d) two and a third packs, e) four and two

thirds packs.

NOTE: You might also demonstrate what it means to have “one and a half” or “two and a half” of something

using an area model: for instance, one and a half pizzas.

Extension: Teach students that one whole can be written as a fraction in many different ways.

Have students name the fractions shaded. Tell them that they are all one whole and write 1 = 44 and 1 =

66 .

Then have a student volunteer to fill in the blanks: 9 7

Then repeat with larger numbers and have students fill in the denominator (give them only the numerator).

Repeat the above exercises with fractions that show two wholes. Ensure that students understand that to

find the numerator, they double the denominator, or to find the denominator, they take half of the numerator.

Gradually increase the denominators to make them more difficult to double: 3, 7, 23, 34, 36, 52, 47, 74, 78.

1 = 1 =


NS3-86 Puzzles and Problems Goal: students will consolidate and apply their learning about fractions.

Prior Knowledge Required: fractions as equal parts of an area or set

fractions that look different but mean the same amount

comparing fractions

fractions greater than one

fractions of numbers

Vocabulary: less than, more than

This worksheet is review. It can be used as an assessment.


NS3-87 Decimal Tenths Goal: Students will learn the notation for decimal tenths. Students will add decimal tenths with sum up

to 1.0. Students will recognize and be able to write 10 tenths as 1.0.

Prior Knowledge Required: Fractions Thinking of different items as a whole (EXAMPLE: a pie, a hundreds block)

Place value

0 as a place holder

Vocabulary: decimal, decimal tenth, decimal point

Draw the following pictures on the board and ask students to show the fraction

110 in each picture:

Tell students that mathematicians invented decimals as another way to write tenths: One tenth ( 1

10 ) is

written as 0.1 or just .1. Two tenths ( 2

10 ) is written as 0.2 or just .2. Ask a volunteer to write 310 in decimal

notation. (.3 or 0.3) Ask if there is another way to write it. (0.3 or .3) Then have students write the following

fractions as decimals:

a) 710 b)

810 c)

910 d)

510 e)

610 f)

410

In their notebooks, have students rewrite each addition statement using decimal notation:

a) 310 +

110 =

410 b)

210 +

510 =

710 c)

210 +

310 =

510 d)

410 +

210 =

610

Bonus:

Include subtraction problems such as:

a) 710 –

310 =

410 b)

910 –

410 =

510 c)

310 –

110 =

210 d)

610 –

310 =

310


Draw on the board:

ASK: What fraction does this show? (410 ) What decimal does this show? (0.4 or .4)

Repeat with the following pictures:

Have students write the fractions and decimals for similar pictures independently, in their notebooks.

Then ask students to convert the following decimals to fractions, and to draw models in

their notebooks:

a) 0.3 b) .8 c) .9 d) 0.2

Demonstrate the first one for them:

0.3 = 3

10

Have students write addition statements, using fractions and decimals, for each picture:

+ = + = + =

Draw on the board:

0 110

210

310

410

510

610

710

810

910 1

Have students count out loud with you from 0 to 1 by tenths: zero, one tenth, two tenths, … nine

tenths, one.

Then have a volunteer write the equivalent decimal for 1

10 on top of the number line:

0.1

0 1

10 210

310

410

510

610

810

910 1

Continue in random order until all the equivalent decimals have been added to the number line.


Then have students write, in their notebooks, the equivalent decimals and fractions for the spots marked on

these number lines:

a) b)

0 1 0 1

c)

0 1

d)

0 A 1 B C 2 D 3

Have volunteers mark the location of the following numbers on the number line with an X and the

corresponding letter.

A. 0.7 B. 2 710 C. 1.40 D.

810 E. 1

910

0 1 2 3

Invite any students who don’t volunteer to participate. Help them with prompts and questions such as: Is

the number more than 1 or less than 1? How do you know? Is the number between 1 and 2 or between

2 and 3? How do you know?

Tell your students that there are 2 different ways of saying the number 1.4. We can say “one decimal four” or

“one and four tenths”. Both are correct.

Have students write the following numbers as decimals:

a) four tenths b) one and six tenths c) three and one tenth d) two and five tenths

Have students write the following decimals as words:

a) 1.2 b) 2.1 c) 3.4 d) 7.3 e) 9.1 f) 2.9

Have students place the following fractions on the number line from 0 to 3:

A. three tenths B. two and five tenths C. one and seven tenths

D. one decimal two E. two decimal eight

Which of the numbers (from A, B, C, D and E above) are less than 1? Which are more than 1 and less

than 2? Which are more than 2? Is 2.3 more or less than 1.8? How do you know? Which two whole numbers

is 2.3 between? Which two whole numbers is 1.8 between?


Teach students to count forwards and backwards by decimal tenths using dimes. ASK: How many dimes

make up a dollar? (10) What fraction of a dollar is a dime? (one tenth) Tell students that we can write the

dime as .1 dollars, since .1 is just another way of writing 110 . ASK: What fraction of a dollar are 2 dimes?

How would we write that using decimal notation? What fraction of a dollar are…

a) 7 dimes? b) 3 dimes? c) 9 dimes? d) 6 dimes? e) 10 dimes?

Have students write all the fractions as decimals.

ASK: How many dimes are in … a) $0.70 b) $0.20 c) $0.60 d) $0.30

ASK: How many tenths of a dollar are in … a) $0.70? b) $0.20 c) $0.60 d) $0.80 e) $0.90

Activity: (From Atlantic Curriculum A8)

Decimal War. Prepare a deck of cards with numbers such as 0.1, 0.2, …, 0.9, 1.0, 1.1, …,

1.9, 2.0, 2.1, …, 2.9 for each pair of students. Each student gets half the deck. They both turn over one

card at a time. The student with the card showing the greater number keeps both cards. Play continues

until someone has all the cards. Variation: Give each pair two identical sets of cards so that ties are

possible.

You might choose to have students play the same game with cards numbered 1 through 29 instead of 0.1

through 2.9 and have them compare the two games. Notice that 1.1 is greater than 0.9 precisely because

11 (tenths) is greater than 9 (tenths).

Extensions:

1. Put the following sequence on the board: .1, .3, .5, _____ and have students extend the pattern.

2. If your students are familiar with equivalent fractions, have them convert each fraction to an equivalent

fraction with denominator 10 and then to a decimal:

a) 25 b)

12 c)

45

3. If your students are familiar with equivalent fractions, have them rewrite each addition or subtraction

statement using decimal notation by first changing all fractions to an equivalent fraction with

denominator 10:

a) 12 +

15 =

710 b)

12 +

25 =

910 c)

35 –

12 =

110 d)

12 –

15 =

310

4. Teach students numbers with two decimal places and where to find the tenths.

ASK: How many dimes are in … a) $0.78 b) $0.93 c) $0.21 d) $0.35

ASK: How many tenths of a dollar are in … a) $0.78 b) $0.93 c) $0.21 d) $0.35

ASK: Where do you see the number of tenths in each number:

a) 0.78 b) 0.93 c) 0.21 d) 0.35


5. (From Atlantic Curriculum A8)

a) Identify the decimals and then put them in order:

A.

B.

C.

D.

b) Model the decimals and then write and model a decimal that is between the two decimals:

3.4 and 3.8 2.8 and 3.1 1.4 and 3.9 0.5 and 1.2 1.9 and 3.2

c) Complete the pattern: 2.9, _____, 3.1, 3.2, _____, 3.4

6. (From Atlantic Curriculum A7)

a) Which number is larger: 2.9 or 4.2? How do you know?

b) Which number is larger: 6.2 or 40? How do you know?

c) Which number is 0.2 more than 0.4? Use a number line or ten frame to help you.

d) Which number is 0.2 less than 1? Use a number line or ten frame to help you.

e) Teach students that decimals are equivalent to fractions with denominator 10 and just as they can

take a fraction of a set, they can take a decimal of a set. Ask students to circle about 0.4 (or 410 ) of

the dots.


NS3-88 Word Problems (Warm Up) Goal: Students will decide when to use addition, subtraction, multiplication or division in word problems

given in point-form notation.

Prior Knowledge Required: addition, subtraction, multiplication, division

Ask students whether the underlined words make them think of adding or subtracting:

1) Three more joined.

2) Two children left to go skipping.

3) There are five altogether.

4) She took five away.

5) There are seven in total.

6) How many are leftover?

7) How many cookies are left?

8) How many altogether?

9) How many are not red?

10) How many more apples than oranges are there?

11) How many fewer days are in a week than in a month?

12) How many days are in a month and a week altogether?

13) How much longer is a school bus than a car?

Have students write down the important words in each question and then the symbol (+ or –) that it makes

you think of:

a) How many apples were sold altogether?

b) How many more red apples than green apples were sold?

c) How many apples were not red?

d) How many stickers were collected altogether?

e) How many stickers were not from Canada?

f) How much longer is a ruler than my pencil?

g) How long are my ruler and pencil when placed end to end?

Tell students that you went to a zoo and saw 14 spotted animals and 9 striped animals and 6 plain animals.

ASK:

a) How many animals are there altogether? 14 + 9 + 6 = 29

b) How many animals are not plain? (29 – 6 = 23)


c) How many spotted and striped animals are there? (Have students discuss two different solutions:

14 + 9 = 23, or 29 – 6 = 23)

d) How many animals are not striped? Discuss the different solutions.

e) How many more spotted animals are there than plain animals?

f) How many fewer plain animals are there than striped animals?

g) Make up your own question and have a partner solve it, then check the partner’s solution.

Tell students that you went to a different zoo and saw 12 spotted animals and 15 striped animals and 34

animals altogether. ASK:

a) How many spotted and striped animals are there altogether?

b) How many more striped animals are there than spotted animals?

c) How many plain animals are there? (I.e. How many animals are there that are NOT spotted or striped?)

d) How many animals are not spotted? (Discuss the two solutions here.)

e) How many animals are striped or plain?

Then ask students to decide between addition and multiplication and how they know:

a) There are 4 pages in each chapter. There are 5 chapters. How many pages are there altogether?

b) There are 7 pages in Chapter 1, 4 pages in Chapter 2 and 8 pages in Chapter 3. How many pages are

there altogether?

c) There are three bookshelves. The shelves have 9, 4, and 6 books each. How many books are there

altogether?

d) There are three bookshelves. There are 6 books on each shelf. How many books are there altogether?

Ask students to tell you how they know whether to add or multiply. Emphasize that when the same number is

on each shelf or in each page, or on each whatever, then they know to multiply (they could add too, but

multiplication is less work).

Have students solve the following multiplication word problems:

a) There are 5 bookshelves and 3 books on each shelf. How many books are there altogether?

b) There are 100 cm in each metre. How many cm are in 3 metres?

c) There are 3 medals given in each event. How many medals are given in 7 events?

d) Bonus: There are 52 weeks in each year. If Jenny turned 8 today, how many weeks until she turns 10?

Remind students how to decide between multiplication and division (see NS3-65):


Have students decide which two pieces of information they are given (between the number of sets, the

number of objects in each set and the total number of objects) and which piece they need to determine.


a) 30 boxes. 6 shelves. How many boxes on each shelf?

6 × _________ = 30

number of sets × number of objects in each set = total number of objects

b) 30 books. 6 books on each shelf. How many shelves?

_______ × 6 = 30


c) 3 books on each shelf. 5 shelves. How many books?

5 × 3 = _______


When they are given the total number, they should divide. When they have to find the total number, they

should multiply. Give students more practice with this skill.


NS3-89 Word Problems and NS3-90 Planning a Party and NS3-91 Additional Problems

Goal: Students will solve word problems involving addition, subtraction, multiplication and division.

Prior Knowledge Required: addition, subtraction, multiplication and division, word problems

Vocabulary: sum, difference

Most of these worksheets provide review and extra practice.

For Question 5 on NS3-89, remind students how to list in an organized way the pairs of numbers that sum to

a given number. They could then look at all their pairs to see if there is one pair with the correct difference.

Extension: After all students have completed Question 6 on NS3-89, discuss the various solutions. For

example, some students might multiply 79¢ by 5 to find the total and then subtract the total from $5.00. Other

students might calculate the change from $1 for each pack (21¢) and then multiply by 5. This works because

the total number of dollars is the same as the number of packs.


NS3-92 Charts Goal: Students will use charts to find possibilities.

Prior Knowledge Required: addition, subtraction, multiplication, division

Tell students that a restaurant has orange juice or apple juice to drink and eggs or pancakes or French toast

for breakfast to eat. If we want to choose one of each, what are the possible choices? Have students

volunteer possibilities. Then tell them that you want an organized way to make sure that you don’t miss any

choices. Write down orange juice on the board and ask how many choices to eat you can have if you pick

orange juice to drink. What are those choices? Write on the board:

Orange juice, eggs

Orange juice, pancakes

Orange juice, French toast

Apple juice,

Apple juice,

Apple juice,

ASK: Why did I write down apple juice three times? Have a volunteer finish the chart. ASK: How many

choices are there altogether? Did we find all of them? How do you know? Could I have organized my chart

differently? Write on the board:

Eggs,

Eggs,

Pancakes,

Pancakes,

French toast,

French toast,

ASK: How am I organizing my chart now? Why did I write each choice twice instead of 3 times? Then have a

volunteer finish the chart.

Repeat with several similar examples, always keeping two choices for one option and three choices for

the other.

Then show students a dart board where they get 2 points for hitting the board but 5 points for hitting the

centre of the board:


Ask students to make a chart to show all the scores they could get by throwing the dart twice. Assume that

they will never miss the board, so they never get 0 points (if they miss the board, they get another throw).

Help get them started:

1st

dart 2nd

dart Total Score

2

2

ASK: Why did I write 2 twice?

Have students individually finish the chart in their notebooks?

Bonus:

Assume that they can get a score of 0 by missing the board. Help them get started:

ASK: Why did I write each first score three times?

1st

dart 2nd

dart Total Score

0

0

0

2

2

2

5

5

5


NS3-93 Arrangements and Combinations and NS3-94 Arrangements and Combinations

(Advanced)

Goal: Students will learn problem-solving.

Ask: I want to make a 2-digit number that uses the digits 1 and 2 each once. How many different numbers

can I make? (2) What are they? 12 and 21. What 2-digit numbers can I make using the digits 3 and 5 each

once? 4 and 7? 2 and 9?

Ask students to make 2-digit numbers:

a) an even number using the digits 3 and 4.

b) An odd number using the digits 5 and 6.

c) A number greater than 40 using the digits 1 and 7.

d) A number less than 30 whose tens digit is four times smaller than its ones digit.

e) A number divisible by 5 using the digits 3 and 5.

f) A number divisible by 5 using the digits 0 and 6.

Ask students to find a 3-digit number that uses each of the digits 1, 2 and 3 once. Write down all their

answers and do not stop until they found all of them. Encourage them to write the 6 different numbers in an

organized list. Repeat with the digits 2, 5 and 7 and then with the digits 3, 4, and 8 and then with the digits 4,

6 and 7. Ask them how they are organizing their list (for example, to write all the numbers using digits 1,2

and 3 each once, they can start with numbers having hundreds digit 1, then numbers with hundreds digit 2

and then numbers with hundreds digit 3; there will be 2 of each) Ask how does the organization make it easy

to know whether they have all of them or not?

Ask students to make 3-digit numbers:

a) An even number using the digits 3, 4 and 5

b) An odd number using the digits 4, 5 and 6.

c) A number greater than 400 using the digits 2, 3, and 6.

d) An even number greater than 400 using the digits 3, 4 and 5.

e) An odd number less than 200 using the digits 1, 3 and 4.

f) An odd number divisible by 5 using the digits 0, 3 and 5.

Draw on the board:


Show students how to arrange these circles in a row in 2 different ways: or

Then draw a third colour and have them find different ways to arrange the circles in a row. Have

volunteers come up to show the different ways. After all 6 ways are shown, tell your students that it is a good

idea to reflect back and ask if there is an organized way to find all 6 ways. If we want to make sure we found

all of them and didn’t leave out any, how could we start? (start by deciding what colour the first one is) Let’s

say the first one is white. What can the second one be? (solid or striped) Have a volunteer show the 2

possibilities that start with the white circle. Have another volunteer show the 2 possibilities that start with the

solid circle and another volunteer show the 2 possibilities that start with the striped circle.


NS3-95 Guess and Check Goal: Students will learn how to improve previous guesses to get closer to an answer.

Bring in a large strip of paper and ask a volunteer to guess where half is. Ask the class: Should the next

guess be further or not as far? How can they tell? Repeat until half is found. (NOTE: One way of deciding if

the guess is too far or not far enough is to directly compare the two halves either by folding the paper or

tracing one part of the line and placing it over the other part.

Then write on the board: 23 + __ __ = 58. Have students guess an answer that they think is close to the

correct answer. Students might say 30 or 35 or 40, for example. Then ASK: Is the guess too high or too low?

Should I guess a larger number or a smaller number next? Continue until students have found the correct

answer.

Then write on the board: 47 + __ __ = 74. Repeat the process. ASK: How am I improving my guess each

time? Would guessing and checking be a good strategy if I didn’t improve my guess each time? What if I just

randomly tried a bunch of numbers and hoped to eventually get it right—is that a good strategy?

Ask students if they know of any games that use the guess and check strategy by improving the guess each

time? (warm and cold, the person tells you whether you are looking close or far away from the object and

you adjust your search accordingly; guessing a number between 1 and 20 that the other person is thinking of

and they tell you whether you are too high or too low) Which games do they know that use guessing without

improving their guess each time? (Hide and seek unless you are given clues such as noise) Which game

usually takes longer to find the correct answer? Games like guess a number between 1 and 20 or games like

hide and seek? Encourage students to share times when hide and seek took a long time precisely because

guessing the possibilities didn’t help with eliminating any option except the one they just checked.

Then write on the board: 2 ___ + ___ 4 = 65. ASK: How is this problem different from: 47 + __ __ = 74?

Emphasize that now they are guessing digits of different numbers instead of guessing the whole number.

Tell students that it might seem a bit overwhelming to try to guess two different numbers at the same time.

One way they can do this is to guess something for one of the numbers and then change the other number

repeatedly until you get close to the answer. For example: 20 + ____ 4 is about 65. We can try 1 and see

that 20 + 14 is 34, which is too low, so our next guess should be higher. Continue in this way until students

are confident they found the closest number they can: 20 + 44 = 64, is about 65.

Then have students adjust the first number until they get 65: 21 + 44 = 65. Repeat this with several examples

where no regrouping is required in the adding of 2-digit numbers. (EXAMPLES: 3 __ + __ 4 = 89,

4 __ + __2 = 59) Bonus: 2 __ 4 + __ 3 ___ = 796.

Then write on the board: 3 __ + __ 7 = 85 Again have students guess a number to fill in: 30 + __7 is about

85. (30 + 57 = 87 and 30 + 47 = 77) Since 87 is closer to 85, students might think that 5 is the best digit to

use. However, they are not allowed to change the 7 to a 5, so they would have to reduce the 30. This would

require changing the 3 to a 2, which they are also not allowed to do. Emphasize that 30 is the smallest the

first number can be, so whatever digit we put in front of the 7 cannot make the answer become more than


85. Since 87 is too high, the second number must be 47. Once we found the second number, the problem

becomes 3 __ + 47 = 85. This can again be done by guessing and checking. Give students practice with

more examples of this sort (EXAMPLES: 1__ + __8 = 81, 3__ + __ 9 = 56.)

Then tell students that you are looking for two numbers that add to 10 and have a difference of 2. Ask

students for some numbers they know that have a difference of 2. Is there a way of listing numbers with

difference 2 so that their sums always increase? (1 and 3, 2 and 4, 3 and 5, and so on). So we can guess

a pair and check to see if the sum is too high or too low. We then know whether to go further in the list

or earlier in the list. For example, if we try 5 and 7, we know that the sum is too high, so try 4 and 6.

We’re done.

Have students find two numbers with difference two that add to:

a) 16 b) 24 c) 22 d) 48 e) 56

Bonus: Find two numbers that add to 234 and have a difference of 2. Students might notice the pattern that

the answer is always very near to half of the given number. For example 7 and 9 are close to 8, which is half

of 16.

Ask the class to list in order the first ten pairs of numbers with difference 3: (1 and 4, 2 and 5, 3 and 6, … ,

10 and 13). Tell students that you are looking for a pair of numbers with difference 3 that add to 37. Have a

volunteer guess a number they think will be close to the right number. For example 15 + 18 is 33, which is

too low. Then try 16 + 19 = 35, again too low. Then try 17 + 20 = 37; done.

Repeat for various such examples.

Cross-Curricular Connection: If you have a piano available and a tape recorder, record yourself playing

several piano notes with enough time in between so that rewinding between notes is easy. Take your

students to a room with a piano. Play your first recorded note once and tell your students that you want to

guess which note this was. Demonstrate guessing a note on the piano right in the middle. ASK: Should my

next guess be higher or lower? Where should my next guess be – to the right or to the left? Rewind and play

the note again. If the next guess should be higher, ASK: A lot higher or a little higher? How about here?

Repeat several times until they guess it correctly. If your students have trouble listening to musical notes, put

tape on certain keys (say 10 keys apart) and tell your students that the note you played is one of the taped

keys. This will make it easier to guess and check. As your students become more comfortable, you can

gradually move the tape pieces closer together.

Extensions:

1. Find two numbers that:

a) Have a difference of 9 and add to 9;

b) Have a difference of 9 and multiply to 10

2. Refer back to finding pairs of numbers that add to 10 and have a difference of 2. Ask students if, instead

of starting with numbers that differ by 2, could they have started by listing numbers that add to 10. What

is an ordered way of listing numbers that add to 10? (Start with the first number 1 and increase the first

number by 1 each time: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5) Then notice that the differences

decrease so if you start by guessing 2 and 8, your next guess needs to be further in the list.


NS3-96 Puzzles Goal: students will improve their problem-solving ability.

Prior Knowledge Required: adding two or three 1-digit numbers, trading for coins of equal value.

Write the following numbers on the board: 1, 2, 3, 4, 5, 6. Tell your students that they are only allowed to use

each digit once for each question below. There may be more than one solution for each part.

a) + = 4 (ANSWER: 1, 3)

b) + = 6 (ANSWER: (1, 5 or 2, 4)

Ask students to place numbers from 1 to 6, at most once each, so that any pair of numbers joined by a

straight line add to 6:

c) + = 8 (ANSWER: 2, 6 or 3, 5)

Repeat the puzzle above with numbers joined by a straight line adding to 8 instead of 6.

d) + = 7 (ANSWER: 1, 2, 4)

e) + = 8 (ANSWER: 1, 2, 5 or 1, 3, 4)

Take up both answers for part e) with the class. Then tell the students to solve the following problem

using the numbers from 1 to 5 each once so that both of the diagonal sums equal 8:


ASK: Which number is in both sums? Where does that number need to be placed? How do you know? Can

any other number be at the top? Why not?

Then challenge your students to use the numbers from 1 to 5 each once so that both lines of 3 numbers

add to 8:

f) + + = 9 (ANSWER: 1, 2, 6 or 1, 3, 5 or 2, 3, 4)

Challenge students to find all possible solutions, then have them solve the following problem using the

numbers from 1 to 6 each once so that all of the edge sums equal 9:

ASK: Which numbers need to be in the corners? Why?

e) Repeat part d) with + + = 10 (ANSWER: 1, 3, 6 or 1, 4, 5 or 2, 3, 5)

Now tell students that you have 3 dimes, 5 nickels and 5 pennies. ASK: How can this be evenly split

among 2 people. Draw the coins or write their letters (D, D, D, N, N, N, N, N, P, P P, P, P) and 2 circles

to divide them into. Divide the dimes first, using nickels if necessary:

D, N, N D, D D, N, N D, D

N, N N, P, P

P, P, P

N, N, N, P, P, P, P, P

ns3-51 ordinal numbers - pbworkscommondrive.pbworks.com/f/tg3_ns2.pdf · ns3-51 ordinal numbers ......

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