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NUMBERS AND OPERATIONSGrade K

BIG IDEA (1): Understand numbers, ways of representing numbers, relationships among numbers and number systems

CONCEPT EXPECTATION EXAMPLE

A Read, write and compare numbers

*Rote count to 100 and recognize numbers up to 31

Problem:Have students stand in a circle and count by ones from 1 to 10 in unison. Then, call on students to do this in pairs or individually. Follow the same procedure for counting by ones from 1 to 20, then to 30, to 40, to 50…and eventually to 100.

Or, give the students a starting number between 2 and 20 and have them count by ones from that number to a designated number. Repeat the process using a starting number between 11 and 20.

Have the students count in unison before asking them to count in pairs or individually.

Problem:Give each student a copy of a blank calendar for the month. Begin each school day by having students write in their calendar the number that represents the day of the month as you record it on the classroom calendar.

Ask students to use the classroom calendar to answer questions you ask themabout the calendar that require specific number answers

Problem:Provide students with daily experiences with the calendar. Refer to numbers on the calendar in connection with special events, field trips, birthdays, parties, Fridays, Mondays, assemblies, etc. Always draw attention to the calendar numeral by writing it on the board, counting it out with concrete objects, writing it in the air, and having the students write it.

NUMBERS AND OPERATIONS – Kindergarten DRAFT 1

TEACHER NOTES: As students go beyond rote counting, they should be learning to count small collections of objects, keeping track of what they have already counted. Gradually, they should learn to count larger groups of objects by correctly keeping track of what they have already counted. Opportunities to count things should be natural situations within the classroom, such as counting how many snacks are at a table, how many snacks are needed at a table, etc.

Students should also begin to establish a system of tagging (one number to one object) as they move or touch objects when counting a collection.

Students learn to rote count through repeated experiences with counting and listening to the counting sequence. One way to give this practice meaning is to offer students one row at a time of the 100s chart. As the students master counting 1 to 10 (in various venues), the next row can be added so that they learn to count 1 to 20, but they also focus on the bridge numbers (9–10, 19–20, 29–30, etc.) and visually see how the tens continue to grow: 10–20–30–40…

Evidence of mastery is the student counting 1 to 100 without hesitation at the bridge numbers.Teachers may want to highlight the bridge numbers on the 100s chart.Differentiation: They may want to challenge some students by asking them to begin counting at a number other than “1” to check for understanding.

Students should be able to recognize the numerals when dealing with the calendar.

In order for children to work with and understand the function of the calendar as a tool that we use to measure or keep track of time, students need to be able to recognize numerals up to 31 at least.

Students learn to recognize numerals through a number-rich classroom.



B Represent and use rational numbers

*Recognize ½ a shape

Problem:Prepare several sheets of paper each with a line drawn that separate the sheet into equal or unequal regions. Ask students to respond with a “thumbs up” for those that show equal (same size) regions and “thumbs down” for those that show unequal regions.

Problem:Prepare several sheets of paper that have been cut into two equal parts. Show ½ of one of the sheets and ask the students to identify the other halve of the sheet. Another way to do this activity would be to give each student ½ of a sheet of paper that has been cut apart and ask them to find the person who have the other half of their sheet of paper.



C Compose and decompose numbers

*Use concrete objects to compose and decompose values up to 10

Problem:Use various objects arranged different ways to represent the same number.For example:

1. 5 buttons2. 5 triangles3. plastic numbers 4 and 14. 5 pieces of candy


BIG IDEA (3): Compute fluently and make reasonable estimates


B Develop and demonstrate fluency

*Connect number words (orally) and quantities they represent

Problem:Have students sit in a semicircle facing you. Each of you has a mat (blank sheet of paper) and 10 counting pieces. Read the story “Mr. Vet’s Dogs.” Say to the students, “For every dog named in the story, we will place a counting piece on our mat to represent that dog.”

“Mr. Vet has a black dog with one brown spot.”(Pause. Put one counting piece on your mat to stand for the black dog with one brown spot.)“Mr. Vet has one white dog.”(Pause. Put one counting piece on your mat to stand for the white dog.)“Mr. Vet has one dog with long, brown hair.”(Pause. Put one counting piece on your mat to stand for the dog with long, brown hair.)“Mr. Vet has one gray dog with short hair.”(Pause. Put one counting piece on your mat to stand for the gray dog with short “Mr. Vet has one gray dog with short hair.”(Pause. Put one counting piece on your mat to stand for the gray dog with short hair.)“Mr. Vet has one dog that is four different colors.”(Pause. Put one counting piece on your mat to stand for the dog that is four different colors.)“How many dogs does Mr. Vet have altogether?”(Pause. Let the students counts the chips on their mat to tell how many dogs Mr. Vet has.)

Answer: 5


Problem:Students count the crayons in their supply box on their desk (6–10 items). Have them start counting with a different color each time. They count the same set of crayons several times, getting the same answer, no matter what the sequence of the colors.Check to see that each student understands the ordinal sequence of numbers, as well as the cardinal meaning of the numbers. Being able to count accurately using the ordinal sequence is not the same as knowing that the ending number of a sequence tells the quantity of the things we have counted.

Problem:Display different numbers of items and have student count them. Ask questions such as:

“How many Unifix cubes are on the table?” “How many blue color tiles are in your pattern?” “How many students are here today?” “How many snacks or milk do you need at your table?” “Can you show me five fingers?”

TEACHER NOTES: Students can count from 1 to 100 and still not have the skills and the understanding of one-to-one correspondence necessary to connect number to quantity. Some students can connect number to quantity but still not truly understand the difference between ordinal and cardinal numbers.Students should be learning to count small collections of objects, keeping track of what’s been counted, and gradually learning to count larger groups of objects by correctly keeping track of what they have already counted. Opportunities to count things should be natural situations within the classroom, such as situations identified in the problem above.Students should establish a system of tagging (one number to one object) as they touch or move objects while they are counting. This counting and tagging should follow the rote number sequence of counting.


NUMBERS AND OPERATIONSGrade 1




*Read, write and compare whole numbers less than 100.

Problem:Provide each student with a card with a number less than 100 written on it. Ask each one to find someone with a number on their card that has something in common with the number on their card. Ask each pair to report what their numbers have in common and it can’t be simply that they both have a 6 or 2, etc.. Example responses might include: they both are greater than 10, they both have a 4 in the tens place, they both have 8 ones, they are both less than 50, they have the same digits but a different value, etc.

Problem:Using the same number cards described in the previous activity, ask different questions for students to respond to including but not limited to the following:

Stand up if your number is greater than 35. (Ask each student standing to read their number as they hold it for others to see.)

OR Ask the students standing to come to the front of the room and without

talking arrange their numbers in order from least to greatest, greatest to least, etc.

OR Ask the students not standing to identify the greatest number of those

standing

Problem:Using base ten blocks, display different values and have students write the numbers that are being represented by the blocks.

NUMBERS AND OPERATIONS – Grade 1 DRAFT 7



*Recognize ½ and ¼ of a shape

Problem:Provide different sizes of paper and have students fold some of them in half vertically and some horizontally, pointing out each half is the same size as the other side. Also discuss that ½ is one of 2 total pieces.

Problem:Provide students with pictures of various shapes divided into equal parts. Ask them to color ¼ of each of the shapes divided into fourths and ½ of the shapes that are divided into halves.




*Compose or decompose whole numbers up to 20 using multiple strategies such as known facts, doubles and close to doubles, tens, and one place value

Using quick images or flashing (showing very quickly so that all the images cannot be counted), show a collection of dots. Ask students, “How many did you see?” and “How did you see them?” or “How did you know that there are that many?”

Problem:Ask students to identify the sums of the doubles of numbers 1 through 10 (e.g.,1 + 1 = 2, 2 + 2 = 4, 3 + 3 = 6, etc.). Then present an addition example such as5 + 6. Discuss with students the different ways to find the sum.

“Double plus”—adding 1. For example, in 5 + 6, identify the smaller of the two addends. (5). Rename the larger number as the smaller number plus 1. (5 + 1). Double the smaller number. (5 + 5). Add the 1. (11). Repeat the process with 6 + 7, 8 + 7, etc.

1. “Double minus”—subtracting 1. Give students the same example (5 + 6) that you used for “double plus.” Identify the greater of the two addends. (6) Rename the other number as the greater number minus the difference between the two numbers. (6 – 1). Double the 6 (6 + 6), subtract the 1(11). Repeat the process with 6 + 7, 8 + 7, etc.

After students have explored both strategies, give them several sums to determine. They can select the “double plus” or “double minus” strategy. Help students recognize the relationship between the two strategies. For example, to find the sum of 6 + 8, you can either double 6 and add 2 or double 8 and subtract 2.


TEACHER NOTES:The use of flashing or quick images encourages students to determine the quantity through seeing or visualizing smaller quantities within a larger set, e.g., seeing a collection of four or a collection of 5 and 2 or 4 and 3, etc., within a set of seven. Students should have numerous opportunities to do quick images to be able to compose and decompose numbers. This skill of composing and decomposing numbers will be used by students as they begin to make sense of combinations that are not already known. The skill also helps students make sense of number and perform mental math.

DEFINITION:compose or decompose numbers—flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. For example knowing that in 4 there is a “3” and a “1” allows a student to think about 7 + 4 as being 7 + 3 + 11

close to doubles—number combinations such as 3 + 4, 6 + 7, etc. that are 1 apart.

1 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 82). Reston, VA: Author.



D Classify and describe numeric relationships

*Skip count by 2s, 5s, and 10s

Problem:Provide a 100 number chart to each student. Have them select a color and color all the squares that are multiples of two the same color. Repeat for number patterns for counting by fives and tens by selecting a different color. The chart becomes a great visual for skip counting by 2's, 5's, and 10's.

Problem:Use various rhymes, such as the following, and trade books for various skip counting activities.

Substitute the name of each child in the class when saying the following rhyme to make them feel important and to encourage repeating the sequence over and over.

Two, four, six, eight,Who do we appreciate?Martin, Martin, Yeah!


BIG IDEA (2): Understand meanings of operations and how they relate to one another


A Represent operations

*Represent/model a given situation involving addition and subtraction of whole numbers using pictures, objects or symbols

Problem:There were nine children playing on the swings. Then eight more children came to play on the swings. How many children were playing on the swings? Show how you solved the problem. Use pictures, numbers or words.

Answer: = 17or,9 + 9 = 18 and 18 – 1 = 17

Problem:Give each student six cards with a different number of stars on each (as shown below). Ask students to

choose cards and write an addition/subtraction problem for the cards they have chosen, along with the sum/difference of the stars on their cards, or

use specific cards to write an addition/subtraction problem, or find sums/differences of specific cards (the cards they use may vary).


TEACHER NOTES: “As students in the early grades work with complex tasks in a variety of contexts, they also build an understanding of operations on numbers. Appropriate contexts can arise through student-initiated activities, teacher-created stories, and in many other ways. As students explain their written work, solutions, and mental processes, teachers gain insight into their students’ thinking.”2

“An understanding of addition and subtraction can be generated when young students solve ‘joining’ and take-away problems by directly modeling the situation or by using counting strategies, such as counting on or counting back. Students develop further understandings of addition when solving missing-addend problems that arise from stories or real situations.”3

BIG IDEA (3): Compute fluently and make reasonable estimates2 Carpenter, T. P., & Moser, J. M. The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education,15, 179–202. 3 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 83). Reston, VA: Author.



A Describe or represent mental strategies

*Describe or represent the mental strategy used to compute addition and subtraction problems

Problem:Roll a die four times, recording the number rolled each time. Have students find the sum/difference of the numbers and share their strategies of how they know they have the correct answer.

Problem:Find the sum/difference of the numbers below. Show with pictures, numbers or words how you found your answer.

4 + 5 + 2 + 6Answer: Answers may vary. Examples: + + + = 174 + 5 = 9 and 2 + 6 = 8I used doubles 9 + 9 to add 8 and 9 then subtract 1 to get 17.

Problem:Give Me Two GameWrite the following numbers on the board in the arrangement shown.

12 9 15

6 1 10

4 7 3


Ask students to give you two numbers that have a sum less than 20(e.g., 15 and 4; 10 and 6; 7 and 9; 12 and 3, etc.) and to describe the mental strategy they used to solve the problem.Additional questions to ask might include “Which two numbers have a difference of 3?” Which two numbers have a difference of 1? Which two numbers have a sum greater than 10?

TEACHER NOTES:“Teachers can help students increase their understanding and skill in single-digit addition and subtraction by providing tasks that (a) help them develop relationships with subtraction and addition combinations, (b) elicit counting on for addition and counting up for subtraction and unknown-addend situations.

Teachers should also encourage students to share the strategies they develop in class discussions. Students can develop and refine strategies as they hear other students’ descriptions of their thinking about number combinations. For example, a student might compute 8 + 7 by counting on from 8: ‘…., 9, 10 , 11, 12, 13, 14, 15.’ But during a class discussion of solutions for this problem, she might hear another student’s strategy, in which he uses knowledge about 10: namely, 8 and 2 make ten, and 5 more is 15. She may then be able to adapt and apply this strategy later when she computes 28 + 7 by saying 28 + 2 make 30, 5 more make 35.”4





*Use strategies to develop fluency with basic number relationships of addition and subtraction for sums up to 20

Students should have plenty of experiences working with number combinations that are selected to provide opportunities to see or discuss strategies of plus-one and minus-one combinations, doubles, and facts to ten. This does not necessarily mean “teaching” these strategies in direct instruction, but rather, capitalizing on and labeling the strategies that students will naturally use when presented with the task of finding an answer. This also means asking students to find the sum of more than two single-digit numbers. Let’s say you roll a die or number cube three or four times, and ask students to find the sum of the numbers rolled (such as 3 + 5 + 3 + 5). Rather than instructing or expecting students to combine the numbers in order from left to right, ask them if there are other ways in which the numbers can be combined. Some students will choose to combine the two 5s to get 10 and the two 3s to get 6 for a sum of 16. Some students may choose to separate the problem into 3 + 5 and 3+ 5 thus leading to 8 + 8.

Problem:John drew numbers cards with 2, 5 and 6 on them. 1. Find the sum of the two largest numbers.2. Find the difference between the smallest and largest number. Use numbers, pictures, or words to show how you got your answer.

Answer:1. 5 + 6 = 112. 6 – 2 = 4


TEACHER NOTES: Throughout first grade, students should rely less and less on counting strategies to compute basic number relationships. Students should begin to use strategies such as doubles, doubles plus 1, sums of 10s. By the end of first grade, most students should “know” the doubles combinations, plus 1 and minus one combinations, and sums and differences of 10s.

“Students should develop strategies for knowing basic number combinations (the singe-digit addition pairs and their counterparts for subtraction) that build on their thinking about, and understanding of, numbers.

Teachers can help students increase their understanding and skill in single-digit addition and subtraction by providing tasks that (a) help them develop the relationships within subtraction and addition combinations and (b) elicit counting on for addition and counting up for subtraction and unknown-addend situations.”5

DEFINITION:develop fluency—developing fluency means the process of memorizing some combinations or—having command of some combinations—not having to count, use manipulatives or draw pictures to find the sum or difference; fluency means that students are able to compute efficiently and accurately with single digit numbers.6

5 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 9). Virginia: Author.6 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 84). Reston, VA: Author.



C Read, write and compare numbers

*Apply and describe the strategy used to solve addition or subtraction problems.

Problem:Have a student choose a two-digit number between 1 and 20. Write the number on the board and have all students write down all the whole number pairs that will add to the given number. Record all the solutions on the board. (Note: If none of the students use zero in their pairs, make sure to include them in your solutions.)

Problem:Have students identify all the sums of the doubles of the numbers 1 through 10 and record them on the board. Then ask students to find the sum of 5 + 6 and talk about the different strategies they may use including “doubles plus” and “doubles minus”.

For example, double the smaller number 5, which results in 10 then add 10 + 1 since 6 is one more than 5, resulting in an answer of 11 OR double the larger number 6, which results in 12 then subtract 1 since 5 is one less than 6, resulting in and answer of 11.






*Read, write and compare whole numbers less than 1000

Problem: Have students identify which of two sets of objects (each less than 100)

has the greatest number or the least number of items. Given two or three numbers, students should be able to identify which

number is the greatest or which number is the least. Students should be able to read random numbers—e.g., 57, 38, 29. Given two sets of objects (each less than 100), students should be able

to record (write) the number of the set that has the greatest amount (or the

least).

Problem:Place students in groups of four, and give each group a pair of number cubes (with numbers 1 through 6) and three copies of the “Order! Order!” recording sheet.


Order! Order! Recording Sheet

Round __Predict Correc

t Highest Number

Third Highest Number

Second Highest Number

Smallest Number

Score __

Have the first student in each group roll the cubes, use the results to form a two-digit number, and record that number in one of the four lines for round 1 on the game sheet. Explain to students how they might decide on which line to put their number. “If you think that your number is likely to be larger than the number that the other three students will form during their turns, you should write it on the first line. If you think it is likely to be smaller than the numbers that the other students will form, you should write it on the last line,” etc.

Have the second student roll the cubes, form a two-digit number, and record the number on one of the remaining three lines.

Have the third and fourth students play the same way.

Next, have the students in each group record the four numbers in the “correct” column, and compare the two lists (in the “predict” and “correct” columns). If the order of all their numbers in the “predict” column is correct,

They get a group score of 4. If the order of three of their numbers in the “predict” column is correct, they get a group score of 3. A game consists of three rounds. The group’s total score for the game is the sum of the scores for the three rounds.


TEACHER NOTES:During second grade, students should have numerous opportunities to read and record whole numbers. These experiences should be in a natural context as well as during math instruction time.

It is essential that by the end of second grade, students develop a solid understanding of the base-ten number system. This involves more than just being able to identify a number in a particular place. It also involves understanding how numbers are written and the idea that one ten is the same as ten ones. Students need to understand that when we represent a number, the order of the digits matter. However, when we illustrate a number (as with cubes), the order of the digits does not matter. For instance, we can represent the number 43 by putting out 43 Unifix or multilink cubes. We can also link 10 of the multilink or Unifix cubes together to make a stick of “ten.” This would give us 1 ten stick and 33 single cubes to represent 43. We can also represent 43 by making four sticks of 10 and 3 individual cubes. It would not matter whether those 3 units were to the left or the right of the four sticks of 10.




*Recognize unit fractions of a shape

Problem:

Which picture shows of the hexagon shaded?

A.

B.

Answer: A

Problem:

In which of the figures of the figure shaded?

A.

B.

Answer: A


Problem:

Shade in of the hearts below.

Answer: Answers may vary.

or or

TEACHER NOTES:Students at the second-grade level should have experiences with simple fractions as they relate to everyday life experiences and expressed in the language the students bring to the classroom. The focus at this level should be on students recognizing when things are divided into equal parts rather than on fractional notation. Second graders should be able to identify three parts out of four parts, or three-fourth of a folded sheet of paper that has been shaded, and to understand that fourths mean four equal parts of a whole.1

1 National Council of Teachers of Mathematics. (2000) Principles and standards for school mathematics (p. 82). Reston, VA: Author.




*Compose or decompose numbers using a variety of strategies, such as using known facts, tens or landmark numbers to solve problems

Using quick images or flashing (showing very quickly so that all the images cannot be counted), show a collection of dots. Ask students, “How many did you see?” and “How did you see them?” or “How did you know that there are that many?” This activity should be expanded to include flashing completed or partially completed ten frames or base-ten collections.

Problem:Find the missing numbers and sums in the grid below. The numbers to the right and at the bottom of the grid are the sums of the rows and columns.


Answer:

TEACHER NOTES:Students should have numerous opportunities to do quick images to be able to compose and decompose numbers. These experiences should include numbers between 10 and 99. Students who are accustomed to decomposing and composing numbers will use these strategies naturally as they make sense of addition and computation of larger numbers. For instance, a student may mentally combine 37 and 28 using several of the following strategies:

combining the 30 and 20 to get 50 then combining the 7 and 8 to get 15 and finally combining the 50 and 15 to get 65,

combining 40 and 28 (40 being 3 more than 37) to get 68 then subtracting the extra three to get 65,

combining 40 and 25 (taking 3 from the 28 to combine with the 37) to get 65.2

2 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (pp. 83–84). Reston, VA: Author.




*Skip count by 2s, 5s, and 10s

Problem:Count by 2s. Shade in all numbers on the chart that would be included in your count. Count by 5s. Put an X on all numbers that would be included in your count. Count by 10s. Circle all numbers that would be included in your count.

Answer:

Have students practice rote counting by 2s, 5s, or 10s as they line up for different activities during the school day.

Counting nickels is an application of skip counting by 5s, and counting dimes is an application of skip counting by 10s.

When students are counting collections of things, encourage them to count by


something other than ones.

Recording data with grouped tally marks encourages and provides opportunity to practice counting by 5s and 10s.

TEACHER NOTES:“In pre-kindergarten through grade two, students should begin to develop an understanding of the concepts of multiplication and division. Through work in situations involving equal subgroups within a collection, students can associate multiplication with the repeated joining (addition) of groups of equal size. Similarly, they can investigate division with real objects and through story problems, usually one involving the distribution of equal shares.”3

Second graders will most likely be able to rote count by 2s, 5s and 10s. This rote counting should lead eventually to recognizing that when we count by 2s, 5s or 10s, we are actually counting a quantity, not just repeating a sequence of numbers. This is similar to how young children learn the rote sequence of our number system. Later, they will be able to determine the quantity of a set. By the end of second grade, students should know that the amount in a set does not change, no matter how you count them.

Of course, the ability to skip count comes into play when students are counting money as well as leading into multiples in multiplication.






*Represent/model a given situation involving two-digit whole number addition or subtraction

Problem:Write an addition or subtraction problem to show why these numbers belong together.

Answer:12 – 7 = 5 or 12 – 5 = 7 or 7 + 5 = 12 or 5 + 7 = 12

Problem:There were 9 children playing on the swings. Then 8 more children came to play on the swings. How many children were playing on the swings? Show how you solved the problem. Use pictures, numbers or words.

Answer:9 + 8 = 17

Problem:Sam was playing marbles with his friends. He started with 16 marbles. At the end of the game, he had 48 marbles. How many marbles did Sam win? Show how you solved the problem. Use pictures, numbers or words.

Answer:48 – 16 = 32


TEACHER NOTES:“As students in the early grades work with complex tasks in a variety of contexts, they also build an understanding of operations on numbers. Appropriate contexts can arise through student-initiated activities, teacher created stories, and in many other ways. As students explain their written work, solutions, and mental processes, teachers gain insight into their students’ thinking.

An understanding of addition and subtraction can be generated when young students solve joining and take-away problems by directly modeling the situation or by using counting strategies, such as counting on or counting back (Carpenter and Moser, 1984). Students develop further understandings of addition when solving missing-addend problems that arise from stories or real situations. Further understandings of subtraction are conveyed by situations in which two collections need to be made equal or one collection needs to be made a desired size.”4






*Describe or notate the mental strategy used to compute addition or subtraction of whole numbers, including 2-digit numbers

Problem:1. Pick two numbers from below that give a sum greater than 50. Do not use

pencil and paper. 12, 40, 27, 28, 11, 21

2. Explain how you knew the sum would be greater than 50.

Answer:1. 12 and 40; 40 and 27; 40 and 28; 40 and 11; 27 and 282. Answers may vary. Examples: To get a sum greater than 50, you only need

to add 11 more to 40. 12 is more than 11, so 12 plus 40 would be greater than 50; or, I know that half of 50 is 25, so if I add 27 (which is more than half of 50) and 28 (which is more than half of 50), I will get a sum greater than 50.

Problem:Mentally solve 25 + 37. Then tell and show how you solved the problem.

Answers: Answers may vary. Examples:


TEACHER NOTES:“Students learn basic number combinations and develop computing strategies that make sense to them when they solve problems with interesting and challenging contexts. Through class discussions, they can compare the ease of use and ease of explanation of various strategies. In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different. Many times, students’ invented approaches are based on a sound understanding of numbers and operations, and they can often be used efficiently and accurately. Some sense of the diversity of approaches students use can be in the example above, which shows the ways several students in the same second-grade classroom computed 25 + 37.”5

5 National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp. 84–85). Reston, VA: Author.




*Demonstrate fluency including quick recall with basic number relationships or addition of subtraction for sums up to 20

Problem:How could you use “double minus” to help you solve the following problem: 18 + 19?

Answer:I would take the largest number, 19, and double it to get 38. But then I need to minus one because 18 is one less than 19, so that gives me 37.

Problem:Show two ways to find the answer to the following subtraction problem: 15 – 9.

Answer:I could borrow to solve it the regular way and get 6 or I could add “1” to the 9 to get 10 and 1 to the 15 to get 16 and have 16 – 10 which gives me 6 also.

Students should have plenty of experiences working with number combinations that are selected to provide opportunities to see or discuss strategies of plus-one and minus-one combinations, doubles, and facts to ten. This does not necessarily mean “teaching” these strategies in direct instruction but rather, capitalizing on and labeling the strategies that students will naturally use when presented with the task of finding an answer. This also means asking students to find the sum of more than two single-digit numbers.

Students should also be given plenty of opportunities to generate combinations of numbers, such as by using 20 two-color counters. Students can shake and spill the two-color counters on their desk, then record the number of yellow (or white) counters and the number of red counters that turn up.

Students can also be given a number and asked to generate as many equations as they can think of that equal that number.


TEACHER NOTES:“Students should develop strategies for knowing basic number combinations (the singe-digit addition pairs, and their counterparts for subtraction) that build on their thinking about, and understanding of, numbers.

Teachers can help students increase their understanding and skill in single-digit addition and subtraction by providing tasks that (a) help them develop the relationships within subtraction and addition combinations and (b) elicit counting on for addition and counting up for subtraction and unknown-addend situations.”6

DEFINITION:demonstrate fluency—demonstrate fluency means that students are able to compute efficiently and accurately with single-digit numbers.7

6 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 84). Reston, VA: Author.7 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 84). Reston, VA: Author.



C Compute problems

*Apply and describe the strategy used to compute 2-digit addition or subtraction problems with regrouping

Problem:Solve the following problems, then describe the strategy/strategies you used to solve them.1. 37 + 18 =2. 35 – 18 =

Answers: 1. 552. 17

Strategies may vary. For 37 + 18, some students might add the 10s first, then the ones, either in horizontal or vertical notations. Other students might add the ones then the tens. Still others might use a mental strategy of making 37 40 and making 18 15, then adding 40 + 15 to get 55.

For the subtraction problem, students may use the traditional algorithm, indicating a need to regroup the 35 to a 20 + 15 and then performing the subtraction…15 – 8 = 7, 20 – 10 = 10, answer = 17. Some might say that 30 – 10 = 20,5 – 8 = –3, so 20 – 3 = 17.

TEACHER NOTES:


“As students work with larger numbers, their strategies for computing play an important role in linking less formal knowledge with more sophisticated mathematical thinking. Research provides evidence that students will rely on their own computational strategies (Cobb et. al., 1991). Such inventions contribute to their mathematical development (Gravemeijer, 1994; Steffe, 1994). Moreover, students who used invented strategies before they learned standard algorithms demonstrated a better knowledge of base-ten concepts and could better extend their knowledge to a better knowledge of base-ten concepts and could better extend their knowledge to new situations, such as finding how much of $4.00 would be left after a purchase of $1.86 (Carpenter et al., 1998, p. 9). Thus, when students compute with strategies they invent or choose because they are meaningful, their learning tends to be robust—they are able to remember and apply their knowledge.”8

8 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (pp. 85–86). Reston, VA: Author.



D Estimate and justify solutions

*Estimate and justify sums and differences of whole numbers

Problem:Write three estimation problems that would have an answer of 30.

Answer:Examples include:1. 61 – 32 = 302. 19 + 10 = 303. 8 + 24 = 304. 56 – 29 = 30






Read, write and compare whole numbers up to 10,000

Problem:Using the digits below, make three different 4-digit numbers. Write them in order from least to greatest.

5, 2, 7, 6 Answer: Answers may vary. For example— 2765, 5276, 7652—written in order from least to greatest.

Problem:Bob and Barbara are looking for a mystery number using the clues below.

It is greater than 600. There is a 2 in the ones place. There is a 6 in the tens place.

Write a three-digit number using the clues.

Answer: Answers may vary. Any of the following: 662, 762, 862, 962.

Students should be able to read numbers such as 157, 638, 929, 3246, 8041

Given two or three numbers less than 10,000, students should be able to identify the biggest and/or the smallest.


TEACHER NOTES:During third grade, students should have numerous opportunities to read and record whole numbers. These experiences should be in natural contexts as well as during math instruction time.

Teachers may want to begin introducing the inequality symbols of greater than and less than and explain them as being taken from the ends of the number line. The number line presents an authentic model for the symbols.< is less than and > is greater than.




*Represents, halves, thirds and fourths

Problem:

of the word rectangle is a color. Shade in the letters of the color below.

Answer:

Problem:

Draw 12 shapes and circle of the shapes.

Answer: Answers may vary. Acceptable answers include 12 drawn shapes with any three of them circled. Example:


Problem:1. Which picture below represents ½ shaded in?

A.

B.

2. Explain how you know that is shaded in.

Answers:1. B

2. There are 6 pieces in the shape and 3 of them are shaded. 3 is of 6 so

3 shapes would be of the 6 shapes;

or,The shape has 6 pieces, and the same amount are shaded and not shaded.

Problem:

1. Shade of the picture below.

2. Color in of the objects below.


Answers:1. Answers may vary. Any of the following:

2. Answers may vary. Any of the following:

TEACHER NOTES:During third grade, students should begin developing their understanding offractions as parts of a whole and as division (fair shares) and parts of a set.




Recognize equivalent representations for the same number and generate them by decomposing and composing numbers including expanded notation

Problem:Which of the following show another way to write 37?A. 27 + 10B. 10 + 10 + 10 + 3 + 4C. 18 + 18D. 50 – 13

Answer: A, B, D

Problem:Using the numbers below, write a subtraction problem.

17 25 8 9 3 16

Answer: Answers may vary. Examples:25 – 8 = 17 or 25 – 17 = 825 – 9 = 16 or 25 – 16 = 917 – 8 = 9 or 17 – 9 = 8


TEACHER NOTES:Students who have developed number sense are able to think about numbers flexibly. They recognize that in order to work with numbers mentally, they can split numbers apart to make combinations that they already know.

This flexibility builds an understanding of number as a specific quantity as well as a relationship between and among numbers, which allows students to compute with numbers mentally.

Being able to break a number into a multiple of 10 or 25 allows students morestrategies to think about the number.

DEFINITIONS:decomposing and composing numbers—flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. For example, 36 can be thought of as 32 + 4, 20 + 16, 40 – 4, 12 3, etc.1





Classify numbers by their characteristics, including odd and even.

Problem:John is playing a game and has drawn five numbers. His five numbers are greater than 20 but less than 50. What numbers did he draw?

Answer: Answers may vary. Any five of the following: 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48.

Problem:Which of the problems below will have an odd answer?A. 23 + 16B. 14 + 12 + 18C. 16 + 17D. 9 + 9 + 9

Answer: A, C, D

Problem:Place the following numbers in the Venn Diagram below:

5, 18, 23, 25, 8, 55, 26


Answer:

TEACHER NOTES:“Throughout their study of numbers, students in grades 3–5 should identify classes

of numbers and examine their properties.”2

DEFINITIONS:classify numbers—to group a set of numbers together by an attribute, such as even or odd, less than 20, more than 20, etc., recognizing that different types of numbers have particular characteristics.3

2 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 151). Reston, VA: Author.3 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 151). Reston, VA: Author.





*Represent/model a given situation involving multiplication and related division using various models including sets, arrays, areas, repeated addition/subtraction sharing and partitioning

Problem:Use numbers to write a multiplication sentence for the following array.

Answer:3 x 4 = 12

Problem:Use pictures to create an array for 2 8.Answer:

Problem:Mrs. Jones has nine chickens. She has placed eight eggs under each chicken to try to hatch some chicks. If all the eggs hatch, how many chicks will she have? Write a multiplication problem to help you find the answer.

Answer:9 8 = 72 chicks or 8 9 = 72 chicks


Example:Draw an array for 6 X 7. XX X X X X X X X X X X XX X X X X X XX X X X X X X X X X X X X X X X X X X X X X

TEACHER NOTES: Students should have numerous opportunities to use manipulatives to model problem situations. Modeling multiplication problems with pictures, diagrams, or concrete materials helps students learn what the factors and their products represent in various contexts.4 Arrays are created Rows X Columns

Partitioning-determining out how many are in the group when the number of groupsis known. Example: How would you divide 24 cookies equally among 6 children?(Think of dividing the cookies into 6 equivalent subsets.)

4 National Council of Teachers of Mathematics. (2000) Principles and standards for school mathematics (p. 149). Reston, VA: Author.



B Describe effects of operations

*Describe the effects of adding and subtracting whole numbers as well as the relationship between the two operations

Problem:What happens to the value of 25 when you subtract 12?

Answer:It becomes smaller because subtraction means taking away 12 of the 25.

Problem:If I have only 16 oranges, and I want a total of 20 oranges, how would I find out how many more I need?

Answer: Student answers may vary. Examples: Add 4 to 16, or, Take away 16 from 20.

Problem:Which subtraction problem is related to 5 + 12 = 17?A. 12 + 5 = 17B. 17 – 5 = 12C. 17 – 17 = 0D. 5 + 12 = 17

Answer: B. 17 – 5 = 12





*Represent a mental strategy used to compute a given multiplication problem up to 9 x 9

Problem:Describe how you would solve 8 x 9 mentally.

Answer:I rounded 9 up to 10 and took 8 x 10 for 80. Then I subtracted 80 – 8 = 72.

Problem:Joann used “doubling” and “halving” mental math to solve 4 x 12 and used 8 x 6 to get 48. Explain what she did.

Answer: She doubled 4 to get 8 and took half of 12 to get 6 then multiplied 8 x 6 to get 48 which is the same as 4 x 12.




Use strategies to develop fluency with basic number relationships (9X9)of multiplication and division

Problem:Sue and Sid are discussing how to solve the following problem:

“Allen wants to know how many quarters he can save in 9 weeks if he saves 8 quarters per week.”

Sue says that Allen can just do 9 8 to find out. Sid says Allen can add 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 +.

Who is right? Explain your answer.

Answer:They are both right. You get the same answer when you multiply 9 8 as you do when you add 8nine times. The answer is 72.



TEACHER NOTES: Students should have plenty of experiences working with number combinations that provide students with opportunities to see or discuss strategies that involve using facts to find unknown facts. These opportunities do more than help students memorize facts; they help students build number sense. This does not necessarily involve teaching these strategies in direct instruction, but rather, capitalizing on and labeling the strategies that students will naturally use when presented with the task of finding an answer.

Throughout third grade, students should rely less and less on counting strategies to compute the basic multiplication and division relationships. Students should begin to use strategies such as skip counting, arrays and using known facts to solve unknown facts. By the end of third grade, most students should “know” the multiples for 2, 5, and any other multiple where any number that is 5 or less is one of the factors.

Teachers can help students increase their understanding and skill in single-digit multiplication and division by providing tasks that (a) help them develop the relationships within multiplication and division combinations, (b) help students build models for the relationships, such as arrays, and (c) elicit the use of known multiples to find unknown multiples.

DEFINITION:develop fluency—developing the ability for efficient and accurate methods of computing and being able to demonstrate flexibility in computational methods chosen which result in students being able to explain their methods and produce accurate answers.5




C Compute problems

Apply and describe the strategy used to compute up to 3-digit addition or subtraction problems

Problem:How would you solve 290 + 310?

Answer:Strategies may vary. Example: I would take the 10 from 310 and add it to the 290 and have 300 + 300 to get 600.

Problem:How would you solve 550 – 270?

Answer:Strategies may vary. Example: I would change 270 to 300 and subtract it from 550 to get 250. Then I would take 30 away from 250 for the 30 I added on to 270. This would leave 220.

Problem:Alicia says she can solve 160 + 161 by using doubles. Explain how she could use doubles to help solve this problem.

Answer:Strategies may vary. Example: Alicia may know that 16 + 16 is 32 so she would know that 160 + 160 is 320. So to get 160 + 161 Alicia could just add 1 to get a total of 321.

TEACHER NOTES:Having students discuss their solution strategies helps them solidify their understanding and see likeness and differences between their strategies and their classmates.’ Having students verbalize or write about their thinking gives the teacher a wealth of information regarding a student’s understanding about the concept and about number in general.




Estimate and justify sums and differences of whole numbers

Problem:Which amount is the best estimate for the sum of 29 and 19? Explain why.A. 40B. 50C. 100D. 20

Answer: B. 50 29 is almost 30, and 19 is almost 20, so 20 + 30 gives a sum of 50.

Problem:Raymond had 125 rabbits. He gave 36 away. About how many rabbits does he have left? Show how you got your answer.

Answer: Answers may vary. Example: 130 – 40 = 90

TEACHER NOTES: Estimation serves as an important companion to computation. It provides the tool for judging the reasonableness of calculator, mental, and paper-and-pencil computations. Being able to compute exact answers does not automatically lead to the ability to estimate or judge the reasonableness of an answer. Students in grades3–5 will need to be encouraged to routinely reflect on the size of anticipated solutions.6







Read, write and compare whole numbers less than 100,000

Problem:Which of the following is the correct way to write twenty-five thousand six?A. 6,256B. 250,600C. 25,006D. 2,506

Answer:C. 25,006

Problem:Write the numbers in order from least to greatest.81,140; 81,143; 81,104

Answer:81,104; 81,140; 81,143

Problem:Use >, <, or = to compare the numbers below.1. 371 3172. 2,304 2,3403. 60,532 60,032


Answers:1. >2. <3. >

TEACHER NOTES: During fourth grade, students should have numerous opportunities to read and record whole numbers. These experiences should be within natural contexts as well as during math instruction time.

Teachers should also use the inequality symbols of greater than or less than (< or >). Teachers may want to introduce them as part of the number line, which presents an authentic model for the symbols.




*Use models,

benchmarks (0,

and 1) and equivalent forms to judge the size of fractions

Problem:

Using models, Rex is trying to decide whether is greater or less than .

Shade in the values on each model below.

Shade

Shade

Answer: Answers may vary. Examples:

Shade

Shade

Problem:

Which is greater, or ?

Answer:


Problem:Jan is sorting fractions into two categories using the following rules:

Box A—closer or equal to

Box B—closer or equal to 1

Which box should she put each of the following fractions in: , , , ,

, , ?

Answer:

Box A— , , , ,

Box B— ,

TEACHER NOTES: Students working with fractions at the fourth-grade level should focus on making sense of fractions and on being able to determine the size of a fraction by relating the given fraction to common fractions they already know.

During fourth grade, students should also explore the idea that fractional representations can be equal even if they are not necessarily the same shape. Students should also become familiar with commonly used equivalent fractions for halves, fourths and tenths.




Recognize equivalent representations for the same number and generate them by decomposing and composing numbers

Problem:Which of the following does not have the same value as 7,523?A. 7,000 + 500 + 23B. 7,800 – 277C. 8,000 – 473D. 6,000 + 1,000 + 250 + 250 + 23

Answer:C. 8,000 – 473

Problem:What is another way to write 76?

Answer:Answers may vary. Examples: 80 – 4; 70 + 6; 75 + 1; 100 – 24

TEACHER NOTES: By fourth grade, students should be able to think of 36 as 30 + 6 or 20 + 16 or 9 4. This flexibility in number sense builds on strategies that students should be developing in the primary grades.

DEFINITION:


decomposing and composing numbers—flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. For example, 36 can be thought of as 32 + 4, 20 + 16, 40 – 4, 12 3, 72/2 etc.1



Classify and describe numbers by their characteristics, including odd, even, multiples and factors

Problem:Tess is playing a game where she scores a point for every even number she rolls on the number cube. She rolls the following numbers: 6, 3, 2, 1, 3, 5, 4, 2, 4, 2, 1, 2 , 2, 3. How many points does she score? Show how you got your answer.

Answer:The even numbers that Tess rolled were 6, 2, 2, 2, 2, and 2. She scored 1 point for every even number, so she scored 6 points since she rolled six even numbers.

Problem:1. Mrs. Mathforu put three lists of multiples of numbers on the board and

asked her students to label the lists. How should the student label each list?

LIST A LIST B LIST C1 5 32 10 63 15 9



4 20 12

2. What would be the next three numbers in list C?

DEFINITIONS:even—a whole number that is divisible by 2.2

multiples—products of a whole number and any other whole number.3

odd—a whole number that is not divisible by 2.4

factor – an integer that will divide evenly into another number5




*Represent and recognize multiplication and related division using various models, including equal intervals on the number line, equal size groups, distributive property, etc.

Problem:Which of the following is another way to write 6 x 12.

a. 6 x 2 x 2b. 6 x (10 + 2)c. 10 x 2 x 6d. 2 x (6 + 10)

Answer: b. 6 x (10 + 2)

Problem:Write a related division problem for 8 x 4 = 32.

Answer:32 4 = 8; 32 4 = 8

Problem: Alice used shapes to make the following arrangement to illustrate7 x 6 = 42.

2 Math at hand: A mathematics handbook (p. 523). (1999). Wilmington, MA: Great Source Education Group, Inc.3 Math at hand: A mathematics handbook (p. 528). (1999). Wilmington, MA: Great Source Education Group, Inc.4 Math at hand: A mathematics handbook (p. 529). (1999). Wilmington, MA: Great Source Education Group, Inc.


Using the same number of shapes, make another arrangement and write the multiplication problem to illustrate your shape.

Answer:

6 x 7 = 42 Note: 6 rows of 7 columns




Describe the effects of multiplying and dividing whole numbers as well as the relationship between the two operations

Problem:John's teacher told him that when he's stumped on a division problem, he can think of a related multiplication fact to help him solve it. What multiplication fact could he use to help him find the answer to this division problem?

45 ÷ 9 =

Answer: 9 5 = 45

Problem:Give an example to show how multiplication and division are related.

Answer:Answers may vary. Examples: 5 6 = 30, 30 ÷ 6 = 5, 30 ÷ 5 = 6





*Represent a mental strategy used to compute a given multiplication problem (up to 2-digit by 2-digit multiple of)

Problem:Describe how you would solve 14 10 without pencil and paper.

Answer: You could borrow the 0 from 10 and “add” it on to 14 to get 140, or you could remember that 14 tens is the same as 14.

Problem:Explain how to use halving and doubling to solve 14 5.

Answer: 7 10 = 70To use halving and doubling, I would take half of the first factor, which is 14, and make it a 7. I would double the second factor, which is 5, and make it a 10. Then I would multiply 7 10 to get 70.

TEACHER NOTES: To use halving and doubling, take half of the first factor and multiply it by double the second factor (e.g., change 14 5 to 7 10).

Note: Additional mental math strategies and activities may be found in NumberSense: Simple Effective Number Sense Experiences.5

5 McIntosh, A., Reys, B., Reys, R., & Hope, J. (1997). Number sense: Simple effective number sense experiences. Palo Alto, CA: Dale Seymour Publications.




Demonstrate fluency with basic number relationships (12 12) of multiplication and related division facts

Problem:Mr. Brown has some candies that he wants to share with his class. There are 24 students in the class, and he knows that he has enough for each student to have three pieces of candy. Write a multiplication expression to help you find out how many pieces of candy he has.

Answer:24 3 = 72

Problem:Three classes in a school participated in a contest for selling the greatest number of raffle tickets. One class has 15 students, one has 12, and one has 10. The principal baked 24 cookies for the winning class so that all the students in the class would get the same number of whole cookies. Which is the winning class? Explain how you know.

Answer:The class that has 12 students is the winner because 24 can be divided equally by 12, which means that every student would get 2 cookies each. 24 cannot be evenly divided by 15 or 10.


TEACHER NOTES:

Students should have plenty of experiences working with number combinations that provide opportunities to see or discuss strategies for quickly figuring out multiplication and division “facts.” This does not necessarily involve teaching these strategies in direct instruction, but rather, capitalizing on and labeling the strategies that students will naturally use when presented with the task of finding an answer.

By the end of fourth grade, this grade-level expectation is about having a command of the multiplication and division relationships through 12 12. This means that students should be able to use the “multiplication and division facts” without counting, tallying, or drawing pictures. Some students might be using known facts to arrive at some of the “facts,” e.g., to compute 4 8, a student might think to himself, “Two 8s are 16, and 16 + 16 = 32.” This would need to be done as quickly as the student who has memorized the fact 4 8 = 32.

Having command of these relationships will aid students in their mental arithmetic skills as well as lay the foundation for them to work with fractional number relationships. It is important that students learn these “facts” as relationships rather than in isolation. It is more difficult for students to retrieve or use the “facts” when they are learned in isolation.

DEFINITION:demonstrate fluency—demonstrating the ability for efficient and accurate methods of computing and being able to demonstrate flexibility in computational methods chosen which result in students being able to explain their methods and produce accurate answers.6




C Compute problems

Apply and describe the strategy used to compute a given 2-digit by 2-digit numbers and related division facts

Problem:Use multiplication and another strategy to solve 35 11.

Answer: Answers may vary. Example: You could take 35 10 = 350 and 35 1 = 35 then add 350 + 35 to get 385 for your answer.

TEACHER NOTES: Having students discuss their solution strategies helps students solidify their understanding and see likenesses and differences between their own strategies and their classmates’. Having students verbalize or write about their thinking gives the teacher a wealth of information regarding a student’s understanding about the concept and about numbers in general.




Estimate and justify products of whole numbers

Problem:Rooms at a hotel cost $99 a night. About how much money would a travel club need if 23 people each need a room for one night in the hotel?

Answer: Answers may vary. Example: $99 is near $100. $100 23 is $2,300, so the club would need about $2,300.

Problem:Use estimation and the number wheel to help answer the questions below.

1. What two numbers on the number wheel have a product between 3200 and 3500?

2. What two numbers on the number wheel have a product near 800?3. What two numbers on the number wheel have a product less than 500?

Answers:


1. 76 and 432. 18 and 433. 18 and 25

TEACHER NOTES:Estimation serves as an important companion to computation. It provides the tool for judging the reasonableness of calculator, mental, and paper-and-pencil computations. Being able to compute exact answers does not automatically lead to an ability to estimate or judge the reasonableness of an answer. Students in grades 3–5 will need to be encouraged to routinely reflect on the size of anticipated solutions.7







*Read, write, and compare whole numbers less than 1,000,000, unit fractions and decimals to hundredth (including location on the number line)

Problem:

Identify which fraction is greater— or —and explain why.

Answer:

Answers may vary. Examples: is greater because it is one out of 3 pieces, and

would be 1 out of five pieces, or is greater because 1 is closer to half of 3

than 1 is to half of 5, or students could draw a picture to explain their thinking.

Problem:Order the following decimal numbers from least to greatest: 0.3, 0.004, 0.105

Answer:



0.004, 0.105, 0.3

Problem:

Write each of the following fractions or decimals in words: , , 1.18 and 0.124

Answer:One eighth, three sixths, one and eighteen hundredths, one hundred twenty-four thousandths

TEACHER NOTES: During fifth grade, students should have numerous opportunities to read and record whole, fraction and decimal numbers. These experiences should be within natural contexts as well as during math instruction time.

Students should use models and other strategies to represent and study decimal numbers, including number lines and money.

Students can investigate decimal numbers on a calculator by counting by tenths, adding one tenth each time and watching the display to see how and where the quantity display changes.

DEFINITION:

unit fractions—a fraction with a numerator of 1, for example, , , , .1

1 Math at hand: A mathematics handbook (p. 537). (1999). Wilmington, MA: Great Source Education Group, Inc.




Recognize and generate equivalent forms of commonly used fractions, decimals, and percents

Problem:Solve each of the fraction riddles below.

a. ½ of matrix is what you sleep or exercise onb. ¼ of mountain is the opposite of outc. 2/3 of review means to look at somethingd. 5/8 of examples means you have enoughe. 2/5 of calculates means tardyf. 4/6 of number means unable to feelg. 3/10 of conference is someone that is dishonest

Answer:a. mat b. in c. view d. ample e. late f. numb g. con

Problem:What fraction and decimal are represented by the shaded region of the figure.



Answer:10/25 or 2/5 or 0.4

TEACHER NOTES: During fifth grade, students should recognize fractions, decimals and percents as being different ways of representing the same quantity or number. Through a variety

of activities, students should understand that a fraction such as is equivalent to

and that the decimal representation is 0.5 and the percent representation is

50%. They should become flexible switching between fractions and decimals.

During fifth grade, students should also be broadening their knowledge of fractions

to include the meaning of a fraction as a quotient of two whole numbers, e.g., is

3 divided by 4 or 0.75. To help see this relationship, a calculator may be used to perform the division yielding the decimal equivalent.

DEFINITION:commonly used fractions—halves, thirds, fourths, fifths, sixths, eighths, and tenths.2





*Recognize equivalent representations for the same number and generate them by decomposing and composing numbers

Problem:Compute 36 25 mentally.

Answer:Student changes 36 to 9 4, so the problem becomes 9 4 25 which is then equal to 9 100.

Problem:Rita said that 52, 675 could also be written as 50,000 + 2,000 + 675. What’s another way to represent 52,675?

Answer:Answers may vary. Examples: 50,000 + 600 + 70 + 5; 52,000 + 675; 52,600 + 75

TEACHER NOTES By fifth grade, students should be able to think of 36 as either 30 + 6 or 20 + 16 or9 4. Having students discuss their solution strategies helps them solidify their understanding and see likenesses and differences between their solution strategies and their classmates.’ Having students verbalize or write about their thinking gives the teacher a wealth of information regarding a student’s understanding about the concept and about number in general.3



DEFINITION:decomposing and composing numbers—flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. For example, 36 as 32 + 4, 20 + 16, 40 – 4, 12 3, 72 / 2, etc.4



*Describe numbers according to their characteristics, including whole number common factors and multiples, prime or composite, and square numbers

Problem:Marsha is playing a game with some number cubes. She scores one point for every composite number she scores and two points for every prime number she scores. How many points does she score if she has the following numbers?

12, 31, 39, 24, 7, 23

Answer:She scores two points each for 31, 7, and 23. She scores one point each for 12, 39, and 24. She scored a total of 6 + 3 or 9 points.

Problem:What square number is represented in each picture below?

1. 2.

Answers:1. 25

3 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (pp. 82–85). Reston, VA: Author. 4 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 149). Reston, VA: Author.



2. 9

Problem:Fill in the blank with O (odd) and E (even).1. E + E + O = __2. O + O + O + __ = E

Answers:1. O2. OTEACHER NOTES: “Throughout their study of numbers, students in grades 3–5 should identify classes of numbers and examine their properties.”5

DEFINITIONS:composite number—a number that has more than two factors.6

even—a whole number that is divisible by 2.7

factor—an integer that will divide evenly into another number.8

odd—a whole number that is not divisible by 2.9

prime number—a number that has exactly two different positive factors, itself and 1.10

square number—the number of dots in a square array; the product of an integer multiplied by itself.11

5 National Council of Teachers of Mathematics. (2000). Principals and standards for school mathematics (p. 151). Reston, VA: Author.6 Math at hand: A mathematics handbook (p. 520). (1999). Wilmington, MA: Great Source Education Group, Inc.7 Math at hand: A mathematics handbook (p. 523). (1999). Wilmington, MA: Great Source Education Group, Inc.8 Math at hand: A mathematics handbook (p. 524). (1999). Wilmington, MA: Great Source Education Group, Inc.9 Math at hand: A mathematics handbook (p. 529). (1999). Wilmington, MA: Great Source Education Group, Inc.10 Math at hand: A mathematics handbook (p. 531). (1999). Wilmington, MA: Great Source Education Group, Inc.11 Math at hand: A mathematics handbook (pp. 531, 535). (1999). Wilmington, MA: Great Source Education Group, Inc.





Represent and recognize division using various models, including quotative and partitive

Problem:I have $12. If a pair of socks cost $3, how many pairs of socks can I buy?

Answer: 4 pair of socks. (Note: This is an example of a quotative division problem—How many groups of $3 will fit into $12? Skip counting by 3s to 12 would be another way of solving this problem.)

Problem:Socks are on sale at $12 for three pairs. How much is each pair?

Answer: $4.00 a pair. (Note: This is an example of a partitive or distribution problem—How is $12 distributed across three pairs? It is a dealing-out type of problem—I have three pairs, and I know that altogether they cost $12. So, I can deal out $1 at a time to


each pair until I have used all 12 dollars.)

TEACHER NOTES: Students need to consider both forms of division problems. Students who believe or are lead to believe that division is just a quick way of doing subtraction or repeated subtraction are often hindered in developing the part/whole relationship between multiplication and division.12

DEFINITIONS:partitive—distribution division that involves figuring out how many are in the group when the number of groups is known. Example: How would you divide 24 cookies equally among six children? (Think of dividing or partitioning the cookies into six equivalent subsets.)13

quotative—measurement division that involves seeing how many groups will fit into a number. Example: If a serving consists of 4 cookies and you have 24 cookies, to how many children can you give a serving of cookies? (Think of making one pile of 4 cookies, then a second pile of 4 cookies, etc.)14

Note: The quotative model is also known as the “repeated-subtraction” model.

12 Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Heinemann Press.13 Fosnot, C., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division (pp. 53–57). Portsmouth, NH: Heinemann.14 Fosnot, C., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division (pp. 53–57). Portsmouth, NH: Heinemann.




*Describe the effects of addition and subtraction on fractions and decimals

Problem:John had ¾ of a candy bar. How much could he eat and still have ½ of a candy bar left? Explain your answer.

Answer:He could 3 eat ¼ of the candy bar and still have ½ candy bar left because ¾ - ¼ leaves 2/4 or ½ of a candy bar.

Problem:Jill had $35 to spend on when she began the day. She spent $25.87 on a purchase. Explain how you know whether or not she has $10.00 left to pay for the movie.

Answer: $25.00 + $10.00= $35.00 and Jill spent over $25 so she does not have enough left to pay for her movie, because $35.00 - $25.87= $9.13 which is less than $10.





*Describe a mental strategy used to compute a given division problem, where the quotient is a multiple of 10 and the divisor is a 1 digit number (e.g., 350 / 7)

Problem:Describe a mental strategy you would use to solve 320 ÷ 4.

Answer: I would take 32 and divide by 4, which gives me 8, and then multiply by 10 to get an answer of 80. Or I could also divide 32 by 4 to get 8, then put a 0 after the 8 to get 80.

TEACHER NOTES: These are only two of many mental math strategies than can be used to solve problems. Additional mental strategies may be found in Mental Math in the Middle Grades. 15

15 Hope, J., Reys, B., & Reys, R. (1987). Mental math in the middle grades. Palo Alto, CA: Dale Seymour Publications.




*Demonstrate fluency with efficient procedures for adding and subtracting decimals and fractions (with unlike denominators) and division of whole numbers.

Problem:Anna ran the a quarter of a mile in 84.5 seconds. Sue ran the same distance in 79.44 seconds. How much faster was Anna than Sue?

Problem.5.06 Before having students solve the problem, asked them to give an estimate of the difference in the two numbers or 85 – 80.. This should help them in knowing how to line the decimal points up.

Problem:John’s teacher told him when he’s stumped on a division problem, he can think of a related multiplication fact to help him solve it. What multiplication fact could he use to help him find the answer to the division problem?

45 9

Answer:9 x 5 =45

TEACHER NOTES:A common mistake that students make when adding and subtracting decimals is the placement of the decimal point and not knowing how to line them up if one decimal has more places than the other. Estimating the sum or difference prior to doing the actual computation can be beneficial in helping students understand addition and subtraction of decimals. The estimation practice will help focus their attention on the meaning of the numbers.



C Compute problems

Apply and describe the strategy used to compute a given division problem up to a 3-digit by 2-digit and addition and subtraction of fractions and decimals

Problem:Mary's mother has agreed to buy a new piano, which costs $3,240. The music store offers a payment plan of 24 equal monthly payments. How much will Mary's mother pay per month for the piano? Describe the strategy you would use.

Answer:$135 for each monthly payment Strategies may vary. Example: You could write a number sentence. Since the amount is divided into 24 equal payments, you would divide the total cost of the piano by 24. $3,240 ÷ 24 = 135

Problem:Tasha knows that 580 pieces of candy come in a bag. If she wants to give each person 20 pieces of candy. How many people can she give 20 pieces of candy?

Answer:I wrote 580 20 as a fraction and reduced it. 580/20 = 29

Problem:Allen studied ½ hour on Monday, 1/4 of an hour on Tuesday, and 1/6 of an hour on Wednesday for the test he was having on Thursday. How much time did he study for his test?

Answer:½ +1/4 + 2/6 = 6/12 + 3/12 + 2/12 = 11/12 (55 minutes)Another strategy would be to convert to minutes before adding: 30 min + 15 min + 10 min = 55 min



TEACHER NOTES:In addition to the traditional algorithms, there are a number of various alternative methods for solving problems of this nature. Having students discuss their solution strategies helps them solidify their understanding and see likenesses and differences between their own strategies and their classmates’. Having students verbalize or write about their thinking gives the teacher a wealth of information regarding a student’s understanding about the concept and about number in general.




Estimate and justify products, and quotients of whole numbers and sums differences of decimals and fractions

Problem:Ben earned $355.68 in six weeks mowing lawns and cleaning windows. About how much did Ben earn each week?

Answer:$355.68 is about $360. $360 ÷ 6 is $60. So Ben earned about $60 per week.

Problem:John was asked to give an estimate for the sum of ½ and 5/8. He responded that 2 would be a good estimate. Explain why you agree or disagree with John.

Answer:I disagree with John because 5/8 is just a little over ½ and if you add one half to it, you get a little over 1. So a reasonable estimate would be 1 not 2.

Problem:Write a decimal subtraction problem that would have an estimated difference of 25.

Answer:Examples include 44.995 – 19.5; 168.1 -

Problem:In each example below what numbers would you use to find the estimate?

a. Tickets sell for $5 each. If the arena has 3,569 seats, about how much money would tickets sales be if the tickets for every seat were sold?

b. Lance drove 185.5 miles on Monday and 123.75 miles on Tuesday. About how many more miles did Lance drive on Monday than Tuesday?

c. Nancy spent $49.86 on pizzas for the class party. If each of the 5 room mothers agree to pay an equal amount for the pizzas, about how much would each room mother need to pay?



Answers:Example responses:

a. $3,600 x 5 = $18,000; $4,000 x 5 = $20,000b. 190 – 120 = 70; 200 – 100 = 100; 200 – 120 = 80c. $50 5 = $10

TEACHER NOTES: Estimation serves as an important companion to computation. It provides the tool for judging the reasonableness of calculator, mental and paper-and-pencil computations. Being able to compute exact answers does not automatically lead to an ability to estimate and judge the reasonableness of an answer. Students in grades 3–5 will need to be encouraged to routinely reflect on the size of anticipated solutions.16







Apply and understand whole numbers to millions, fractions and decimals to the thousandths (including location on the number line)

Problem:1. Place the appropriate sign < , = , or > to make the following statement true:

0.87 __

2. Order the following numbers from the least to the greatest.

60%, 0.06, 60,

Answer: 1. 0.87 < 7/8

2. 0.06, 60%, , 60

Problem:Place the following numbers in their appropriate location on the number line by matching them to the correct letter on the number line.

50%, 0.07, , 0.42

Answer:


(Note: These are just some suggested numbers. You can use others as you see fit.)



Recognize and generate equivalent forms of fractions, decimals and benchmark percents

Problem:Complete the chart below.

FRACTION DECIMAL PERCENT

0.5

75%

Answer:

FRACTION DECIMAL PERCENT

0.33... 33 %

2 0.5 50%

0.75 75%

Problem:Tim and Lana were discussing their answers to a problem. Tim said that he got

0.25. Lana said she got . Explain how you know that their answers are the same


value.

Answer:

They are both the same because 0.25 is equal to which simplifies to ; or

0.25 is the same as 25 cents out of a dollar, and 25 cents is the same as of a

dollar.

Problem:Represent four-tenths as a fraction, decimal and percent.

Answer:

, 0.4, 40%

Problem:What fraction, decimal and percent is represented by the shaded region of the figure below?

Answer:

or or ; 0.4; 40%




*Recognize equivalent representations for the same number and generate them by decomposing and composing numbers

Problem:Write in standard form the number that is represented by the following expansion:

6 10,000 + 5 1,000 + 3 10 + 4 + 6 0.1 + 5 0.001

Answer: 60,000 + 5,000+ 30 + 4 + 0.6 + 0.005 = 65,034.605

Problem:Write the number that is the equivalent representation of the following:

300 + 50 + 50 + 10 + 8

Answer:418

Problem:Write another representation for 472.


Answer: Answers may vary. Examples: 400 + 70 + 2; 400 + 72; 500 – 28

Problem:Write 45,739 in expanded notation.

Answer:(4 10,000) + (5 1,000) + (7 100) + (3 10) + (9 1)

Problem:

Luther said that he could use to explain the percentage for th. Explain how.

Answer:

is equal to 25%. is half of , so half of 25% is 12 %.

DEFINITION:decomposing and composing numbers—flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. For example, 36 as 32 + 4, 20 + 16, 40 – 4, 12 3, 72 / 2, etc.1






Describe the effects of multiplication and division on fractions and decimals

Problem:Omar was perplexed when he did the following division problem on his calculator, 2 ¼ ¾, and it resulted in an answer of 3 which was bigger than the number he started with. Explain how to show Omar that this answer is correct.

Answer:If you think of 2 ¼ as 9/4, then you have 9/4 ¾. So Omar is really trying to find out how many sets of 3 fourths are in a set of 9 fourths.. His answer should be 3 sets of three fourths.

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ 1 set of ¾ 1 set of ¾ 1 set of ¾ = 3 sets of ¾

Problem:Provide an example and explain why multiplying a number by 0.25 results in a number less than the original number.

Answer:Example:12 x 0.25 = 3 , 0.25 is the same as 25/100 or ¼. 12 x 1/4 is the same as dividing 12 by four which results in 3.



C Apply properties of operations

*Apply properties of operations (including order of operations) to positive rational numbers

Problem:Melissa worked the following problem incorrectly. Explain to Melissa where she made her mistake and how you would work the problem correctly.

Melissa’s ProblemProblem: 5 + 2 x (3+4)Step 1: 7 x (3 + 4)Step 2: 7 x 7Wrong Answer: 49

Answer:Melissa started by adding. You should always do what is inside the parenthesis first, then multiplication, then addition.

Rework Problem Correctly5 + 2 x (3 + 4) 5 + 2 x 75 + 14Correct Answer: 19

Problem:Insert parentheses to make the following statement true: . Show work to justify that your changes will result in the correct answer.

Answer:5 + 8 must be enclosed in some type of grouping symbols.

3 ( 5 + 8) – 2 3 (13) – 2 39 – 2 37


Problem:Simplify the following and show work to justify your answer:1. 2. 3.

Answer:1. 3 – 5 (4 +7) 2. (2 + 7)2 3. (4 + 6) - 32

3 – 5 (11) (9)2 (10) - 32

3 – 55 81 10-9 -52 1

Problem:Identify the property illustrated with the following statements:1. 2. 7 8 = 8 73.

Answer:1. Distributive2. Commutative3. Associative for addition

Problem:Replace with the correct operation to make a true statement.

Answer:a. (5 ÷ ½) + 3 = 13

(4 + 6) x (5 – 2) = 30



D Apply operations on real and complex numbers

Identify square and cubic numbers and determine whole number roots and cubes

Problem:Place an S by each square and a C by each cubic number below.

a. 25b. 27c. 64d. 100

Answer:a. Sb. Cc. S and Cd. S

Problem:Write a cubic number greater than 100, find the cubic root, and explain how you know it’s a cubic number.

Answer:125 is a cubic number. The cubic root of 125 is 5 because 5 x 5 x 5 = 125. You could build a 3-D cube that is 5 cubes long by 5 cubes wide by 5 cubes long using 125 cubes.


Problem:What is the square root of each number?

a. 81b. 121c. 144

Answers:a. 9b. 11c. 12

DEFINITION:properties of operations: associative and commutative properties of addition and multiplication, the distributive property of multiplication over addition to simplify computations; order of operations should be followed.

Example: 3(3+52)÷7+1 = 3(3+25) ÷7+1 = 3(28)÷7+1=84÷7+1=12+1 =132

Mathematicians have agreed that when there is more than one operation within a problem, the operations should be done in the following order:

1. Perform all operations that are in grouping symbols2. Do all work with exponents3. Multiply and divide in order from left to right4. Add and subtract in order from left to right3

2 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 214). Reston, VA: Author3 Chaplin, Suzanne. (1995), Middle Grade Mathematics: An Interactive Approach. (p. 524). Needham, MA: Prentice Hall Publishers.




C Compute problems

Multiply and divide positive rational numbers

Problem:Hamburger is on sale at $1.50 per pound. You order 6 pounds.

1. Show work or explain in words how you could find the cost using mental math methods.

2. Use the algorithm to check your work.

Answer:1. Students should be encouraged to use benchmarks as they do mental math.

In this problem, the distributive property can serve as an aid in getting the

solution. $1.50 can be broken down as . and of 6 is 3.

So I know that we are going to add $6 + $3 = $9 for the total cost of the hamburger.

2. 1.5 x 6 = 9

Problem:

of students in your middle school love to go to the movies. of the students

who love to go to the movies also love to shop. What fractional part of the students love to go to the movies and shop? Show work on how you arrived at your answer.

Answer:



Problem:Simplify.

1.

2.

3.

Answer:1. 7/2 or 3 1/22. 103. 3/10

Problem:

Bill has of a yard of rope. He wishes to divide the rope into pieces that are

of a yard. How many pieces will he have? Show all work on how you arrived at your answer.

Answer:Use the invert and multiply method:4/5 1/10 4/5 x 10/1 40/5 = 8

OR

Using the common denominator method:4/5 1/10 8/10 1/10 8 1



10 10 = 8/1 = 8

Problem:Explain how to find an estimate for 48% of 450.

Answer:48% is close to ½. Half of 450 is 225.

Problem:

If the radius of a circle cm, what is the area? Show work on how you arrived at

your answer. Use 3.14 for .

Answer:A= r2

A= (1/2)2 x 3.14A= 0.785

Problem:Find the area of the rectangle below. Show all work on how you arrived at your answer.

(Object is not to scale.)



Answer:8/9 x 3/8 = 3/9 = 1/3 cm2

TEACHER NOTES:Students should be aware that answers are best given in lowest terms.

In NCTM’s Principles and Standards for School Math, students are asked to develop a deeper understanding of rational numbers (numbers that can be expressed as the ratio of two integers) by using a variety of models such as fraction strips, number lines, 10 10 grids, fraction circles, and other concrete representations.4

4 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.




*Estimate and justify the results of multiplication and division of positive rational numbers

Problem:Janet had to buy 6 tickets for $4.00 each so that she and some friends could go to the talent show. She had $32.00 in her pocket. On her way to buy the tickets, she decided to stop and buy 4 birthday cards for $2.79 each. Does she have enough money to buy the birthday cards and the tickets? Explain in detail how estimation could be used to determine your answer.

Sample Answer:6 tickets at $4.00 each will total $24.00. If the cards were only $2.00 each, four of them would be $8.00. When you add the $8.00 to the $24.00, the total would be exactly $32.00. Since the cards were more than $2.00 each, Janet will not have enough money to buy the birthday cards and the tickets.

TEACHER NOTES: In grades 6–8, students should acquire computational fluency—the ability to compute efficiently and accurately—with fractions, decimals, and integers. Teachers should help students learn how to decide when an exact answer or an estimate would be more appropriate, how to choose the best computational method to use, and how to evaluate the reasonableness of answers to computations.5




E Use proportional reasoning

Solve problems using ratios and rates

Problem:

In a commercial, the Cookie Crumbles Company said that of the cookies in a bag

were loaded with extra chocolate chips. If there are 48 cookies in the bag, how many of them are loaded with extra chocolate chips?

Answer:

means that 2 out of every 3 cookies are loaded with extra chips. The total number

of cookies in the bag is 48. If the commercial is correct, it should be possible to put those 48 cookies into 16 groups of 3 so that each group had 2 cookies that are loaded with extra chocolate chips. 2 16 = 32 cookies that are loaded with chocolate chips

Problem:Jessie was told that inside a box, the ratio of blue tiles to total tiles was 7 to 12. After further investigation and questioning, he was able to determine that there was a total of 48 tiles in the box. However, no one in class would tell him how many blue tiles were in the box. Help Jessie determine how many of the tiles are blue.

Answer:

= .

Note: Many methods can help determine this solution. One such method would be

with equivalent fractions. = 4 so the scale is 4. We can take the scale 4 and

multiply by 7 to get an answer of 28. Encourage students to look at other methods to solve proportion problems rather than just cross multiplying and dividing.






Compare and order all positive rational numbers and find their approximate location on a number line

Problem:Draw a number line marked –6 to 6 on the board. Write each of the following numbers on a separate Post-it note:

8, 7 , , , 20%, 50%, 150%, 4.5, 5 , 5%.

Have students place those numbers in the proper order on the number line.

Answer:

5%, 20%, , 50%, , 150%, 2, 4.5, 5 , 7 , 8

Problem:List the following rational numbers in order from least to greatest. Describe how you determined the order in which the numbers should be placed.

0.7 3.5 2.4 12%

Answer:

12%, , 0.7, , 2.4, 3.5

Many methods may be used in determining the order of the numbers. Converting all the numbers to all to fractions or decimals is likely.



Problem:List the following decimals from least to greatest.

0.2 0.2374 0.2472 0.2158 0.2111 0.02889

Answer: 0.02889, 0.2, 0.2111, 0.2158, 0.2374, 0.2472

Problem:

What number is of the way between and ? Explain your reasoning.

Answer:1/3 is equal to 3/9. Halfway between 3/9 and 7/9 is 5/9.

Problem:Complete the Venn diagram to sort the fractions listed below into appropriate groups

based upon their relationship to .



Answer:

Problem:Fill in the boxes to create positive fractions to make a true statement.



Is there more than one answer? How many possibilities could there be? Where on the number line would these fractions be located in reference to 1?

Answer:½ + ¼ < 1. There is an infinite number of possible answers where the sum of the two fractions is less than one. Both numbers would need to be to the left (smaller) than one. One of the two numbers would have to be less than or equal to ½.

Problem:

1. What happens to the value of the fraction if the numerator gets bigger

and the denominator stays the same?

2. What happens to the value of the fraction if the denominator gets bigger

and the numerator stays the same?

Answer:1. The value of the fraction gets larger.2. The value of the fraction gets smaller.



B Use proportional reasoning

Solve problems using ratios and rates

Problem:Your dinner bill came to $27.83. You decide to leave a 15% tip.

1. You do not have a calculator with you. Estimate what would be an appropriate amount to leave as a tip. Show your work or explain how to get your answer.

2. You have a calculator with you. What exact amount would be left as a tip. Show your work or explain how to get your answer.

Answer:1. Answers will vary. A quick method for figuring tip would be to change the

amount to 28 and find 10%, which is 2.8, and then 5%, which is 1.4. Then, add the two to get $4.20 as the tip. Another interesting method is to double the amount of tax that is charged, since the tax would be a little over 7%.

2. $27.83 x .15 = $4.17 Problem:You need to build a rectangular fence for your dog. The dimensions of the rectangle


need to be feet long and feet wide. How many linear feet of fencing

material do you need? Show all work to support your answer.

Answer:

linear feet of fencing material would be needed.

Add the two fractions first to get 8 and 13/24; then multiply that answer by two.

Problem:

A parking garage contains 34 blue cars, which is of the total cars in the garage.

What is the total number of the cars in the garage? Explain your reasoning.

Answer:34 x 5 = 170

Problem:Estimate the percent of the following. Explain your thinking.

1. 2. 3.

Answer:Answers will vary.

1. 25 times 4 is 100, so I would take 4 times 4 which is 16%. Another method would be to round to 5/25 which is 1/5 or 20%.

2. Round to tenths and multiply 2 times 10 which is 20%. Another method would be to rould to 2/8 which is ¼ or 25%.

3. Round to twentieths and multiply 19 x 5 which is 95%. Another method is to assume that since the pieces for eighteenths would be small, it is close to 100%.

Problem:Three stores in town had the same type of t-shirt on sale. The original price for one


shirt was $6.48. The following sale signs were seen at the stores. Determine which store had the best price per t-shirt. Explain your reasoning.

Answer:The first shop offers 60% off, the second 33% off, and the third 50% off. Therefore, the first store has the best price per t-shirt, because it has the greatest discount.

TEACHER NOTES:Some students will multiply by the original price by the different sale reductions. Students should realize that since the price is constant, the best buy is obtained by the sale that has the greatest markdown.


C Compose and Decompose Numbers

*Recognize equivalent representations for the same number and generate them by decomposing and composing numbers, including exponential notation

Problem:A student’s answer for the equivalent of is 15. Explain the student’s error. Then show what you would do to get the correct answer.

Answer:The answers will vary, but may say something similar to this: The student took 5 times 3 to the answer of 15. What they should have done is take (5)(5)(5), which would have given them the correct answer of 125.

Problem:1. What number does represent?2. Express 72 in exponential notation using its prime factorization.


60% Off the Original Price!

All T-Shirts Reduced by 1/3!!!

Get 2 shirts for the price

of 1


3. Express as the sum of two different fractions.

4. Express 3.421 as the sum of two different decimals.

Answer:1. 6002.

3. Answers will vary. Sample answer:

Answers will vary. Sample answer: 2.321 + 1.100

Problem:The following are written in expanded form. Rewrite each number in standard form.

1.2. 100

3.

Answer:1. 3,4502. 703,0853. 5,020,009

Problem:Use exponential notation to write the number 64.

Answers:43, 82, 26, 4222, 24x4, etc.

Problem:



Jill wants to use her calculator to determine the value of 163. Unfortunately, the 3 key is broken. Explain how Jill could still use her calculator to find this value.

Answer: Jill could take 16 x 16 x 16 which equals 4096.

Problem:One number in each of the following sets is not equivalent to the others. Determine the number and explain why. _1. a)1/3 b) 0.3 c) 3/9 d) 30%

2.

Answer:1. 30%; 30% = .30 and all others are equivalent to 1/3.

2. = 5.75 and all the others are equivalent to 5 and ¼.


DEFINITIONS:compose or decompose numbers: flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. Grade 7 should include exponential notation. For example, 36 can be thought of as 32 + 4, 20 + 16, 40 – 4, 12 X 3, 22 x 32 , etc.1 Added Note: Decomposing numbers means rewriting a number as the sum, product or difference of other numbers. Composing numbers means putting together numbers that are represented as a sum or product in order to obtain a single number.factor: an integer that will divide evenly into another number (the quotient has a remainder of 0). 1, 2, 3, 4, 6, 12 are all factors of 12, since 12 is divisible by each.2

multiple: the product of a whole number and any other whole number. Multiples of 5 include 0, 5, 10, 15, … etc.3




*Describe the effects of all operations on rational numbers including integers

Problem:Lucy told Max a negative minus a negative always gives you a negative answer. Max said that was only sometimes true. Help Max prove his point to Lucy by writing two number sentences. One sentence should show a negative minus a negative can give you a negative answer. The other number sentence should show that a negative minus a negative can also give you a positive answer.

Possible Answer:-3 - (-1) = -2-3 - (-8) = 5Sometimes a negative minus a negative gives you a negative answer sometimes it gives you a positive answer: -3 + (1) = -2 and -3 + 8= 5

1 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 149).2 Math at hand: A mathematics handbook (p. 524). (1999). Wilmington, MA: Great Source Education Group, Inc.3 Math on call: A mathematics handbook (p. 584). (1998). Wilmington, MA: Great Education Source Group, Inc.



Problem:The temperature reading on a thermometer is –5° F. Tell what the new temperature will be if the temperature

1. falls 15° F.2. rises 4° F. 3. rises 7° F.

Write an equation (number sentence) that shows how you arrived at your answer.

Answer:1. –5° F - 15° F = -20° F2. –5° F + 4° F = -1° F3. –5° F + 7° F = 2° F

Problem:Write an example of an addition problem where the sum is smaller than either of the numbers in the problem.

Answer:Example: (-4) + (-6) = -10-10 is smaller than either –4 and –6.

Problem:Fill in the blanks with positive rational numbers that will make a true statement.

1.2.3.



Answer:Answers will vary. A sample answer for each is given.

1. (Any proper fraction will work.)

2. or (Any number greater than 5 will work.) With problem number two,

students may realize that it is possible to make any whole number a fraction by making one the denominator. This would only reinforce the concept that whole numbers can also be fractions.3. 1/6 (Any proper fraction will work.)

Problem:

1. is closest to which integer?

a. 0 b. 1 c. 7 d. 81

2. is closest to which integer?

a. 0 b. 1 c. 3 d. 7

Answer:1. a



2. b

Problem:Explain what types of numbers can be placed in the to make a true statement.

Answer: must be a rational number greater than 3.

Problem:Jody believes that when you multiply, your answer is always larger than the number you started with. Is this always true? Explain your reasoning.

Answer:This is not always true. When a number is multiplied by a fraction or if you multiply by a negative number, your answer will be smaller than the number you started with.

For example, 6 x ½ = 3 or 6 x -2= -12

Problem:2. Determine the quotient in the following problems.

3. Using the problems above, what patterns do you see regarding the quotient as the divisors get smaller?

4. Suppose the divisor is a negative number, what is true about the quotient?



Answer:

1.

2. The quotient gets larger.3. All the answers would be the same but negative, therefore the quotient gets

smaller.

Problem:Illustrate pictorially (using circles, rectangles, or other models) the following division problems:

1.

2.

Answer:1. 2 one fourth pieces fit into one half


¼ ¼


2.

1/61/6

1/61/61/6

1/61/6

1/61/61/65/6

Two and one half 1/3’s fit into 5/6



Apply properties of operations (including order of operations) to positive rational

Problem:Insert parentheses to make the following statement true: . Show work to justify that your changes will result in the correct answer.

Answer:



numbers and integers

5 + 8 must be enclosed in some type of grouping symbols.

3 ( 5 + 8) – 2 3 (13) – 2 39 – 2 37

Problem:Simplify the following and show work to justify your answer:1. 2. 3.

Answer:1. 3 – 5 (4 +7) 2. (2 + 7)2 3. (4 + 6) - 32

3 – 5 (11) (9)2 (10) - 32

3 – 55 81 10-9 -52 1

Problem:Identify the property illustrated with the following statements:1. 2. 7 8 = 8 73.

Answer:1. Distributive2. Commutative3. Associative for addition

Problem:



Replace with the correct operation to make a true statement.

Answer:a. (5 ÷ ½) + 3 = 13

(4 + 6) x (5 – 2) = 30

DEFINITION:properties of operations: associative and commutative properties of addition and multiplication, the distributive property of multiplication over addition to simplify computations; order of operations should be followed.

Example: 3(3+52)÷7+1 = 3(3+25) ÷7+1 = 3(28)÷7+1=84÷7+1=12+1 =134

Mathematicians have agreed that when there is more than one operation within a problem, the operations should be done in the following order:

1. Perform all operations that are in grouping symbols2. Do all work with exponents3. Multiply and divide in order from left to right4. Add and subtract in order from left to right5


D Apply operations on real and complex

*Approximate the value of square roots to the nearest whole number

Problem:How do you know the is closer to 6 than it is to 7?

Possible Answer:

4 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 214). Reston, VA: Author5 Chaplin, Suzanne. (1995), Middle Grade Mathematics: An Interactive Approach. (p. 524). Needham, MA: Prentice Hall Publishers.



numbers and the . Even though 40 is between 36 and 49, it is only 4 units from 36, but it is 9 units from 49. Therefore will be closer to than it is to

.

Problem:What is the approximate value of the square root of 74 to the nearest whole number? Show work or explain your reasoning to justify your answer.

Answer:74 is between two perfect square numbers 64 and 81. The square roots of these perfect square numbers are 8 and 9. The number 74 is closer to 81 than 64; so the nearest whole number is 9.

Problem:The value of each of the problems below would be between what two whole numbers on the number line?

1. 2. 3.

Answer:1. between 9 and 10, but closer to 9 since 9 x 9 = 81.2. between 7 and 8, but closer to 8 since 8 x 8 = 64.3. exactly 11. 11 x 11 = 121

Problem:Trish solved the following problems correctly.



Use the solutions to Trish’s problems to help you complete the following.1.2.

Answers:1.

2.



C Compute problems

Apply all operations on rational numbers including integers

Problem:Show your work to find the quotient when 2 ½ is divided by ¾.

Answer:Invert and multiply algorithm



2 ½ ¾5/2 x 4/3 20/6 = 3 1/3

Or Common denominator method10/4 ¾ 10 3 = 10/3 = 10/3 = 3 1/3 4 4 = 1

Problem: -3 + 8 (6 x 5) + 3

Answer:240

Problem:Louis bought 46 small note pads for $34.50. How much did he pay for each note pad?

Answer: $34.50

Problem:22 32 7 is the prime factorization for what composite number?Answer: 252

Problem:John made a profit of $28 on Monday, had a loss of $13.25 on Tuesday, a profit of



$65.87 on Wednesday, and a loss of $15.25 on Thursday. How much was his overall profit or loss in the four days?

Answer:$65.37 or a profit of $65.37

Problem:A store had an item on sale for 50% off the original price of $68. Today they are offering an additional 20% off the already reduced price. What is the sale price of the item if it’s purchased today? Explain how you got your answer.

Answer:I divided 68 by 2 since 50% is the same as ½. $ 68 2 =$ 34. Then I divided $34 by 5 since 20% is the same as 1/5. $34 5 = $6.80. Then I subtracted $6.80 from $34 to get the final sale price of $27.20

OR½ of $68 = $34

$34 x 4/5 = $27.20



*Estimate and justify the results of all operations on rational numbers

Problem:About 5/8 of the 483 students who attended Friday’s football game wore the home team colors. About how many students wore the home team colors? Explain how you got your answer.



Answer:5/8 is just a little over half and 483 is about 480. So I took ½ of 480 to get 240.

OR

476 is about 480 so I took 5/8 of 480 to get 300.

Problem:Suppose you have $80 dollars to spend on CDs that range in price from $8 to $12 each. What’s a good estimate for the number of CDs you could buy? Explain your thinking.

Answer:If you purchase the $12 CDs, you could only get 6 but if you purchase the $8 CDs you could get 9. Students can use any combination as long as they stay within the $80 limit.

Problem:Find an estimate for the quotient when you 68,495 is divided by 37.

Answer:70,000 35 = 2,000


E Use Solve problems Problem:



proportional reasoning

involving proportions, such as scaling and finding equivalent ratios

A map scale is given as 2cm: 25km. Use a proportion to find the actual distance between two cities if the map measured distance is 14 cm. Show all work.

Answer:

The distance traveled is 175 kilometers.

Problem:The polygons below are similar:(drawing is not to scale)

Determine x.



1. What must be true for the ratio w:z?2. If w = 9, what is z?3. Using the values determined for x, w, and z, what is the ratio of the perimeters? 4. How does this compare to the ratio of corresponding sides?

Answer:1. w:z = 3:42. z = 123. perimeter=33, perimeter=444. the ratio of the perimeter is the same as the ratio of corresponding sides

TEACHER NOTES:This would be an excellent time to develop map skills and scaling skills. Take the opportunity to look for classroom activities that require students to measure and then to either draw to scale or to determine actual amounts.






*Compare and order all rational numbers including percents, and find their approximate location on a number line

Problem:Arrange the following rational numbers in order from smallest to largest:

Answer:

Problem:Place the following values in their approximate location on the number line below:

Answer:



Problem:Find two numbers between the following pairs of numbers:

1. 0 and 12. -1 and 03. -4 and -34. -1 and 1

Answer:1. any fraction or decimal between 0 and 1 ex. ½ , 0.5, ¼ …2. any fraction or decimal between 0 and -1 ex. -½ , -0.5, -¼ …3. any mixed number, improper fraction, or decimal between -3 and -4

ex. -3½ , -3.5, 4. any fraction or decimal between -1 and 1 ex. ½ , -0.5, ¼, 0 …

Problem:Replace with a single digit value that will make a true statement.

Answer:1. any negative number or any positive number greater than 7 2. any one digit whole number greater than 3 3. any one digit whole number less than 54. any number greater than 65. 8

Problem:



Place the numbers 3, 5, 7, and 9 in the boxes to get a value that is less than 1.

Answer: Answers will vary, examples include:

Ex. , , , , ,….

Problem:Find two rational numbers which have a product between 12 and 13.

Answer:Answers will vary, examples include:

1. 4.1 x 3, , …




Use fractions, decimals and percents to solve problems

Problem:Mary receives $7.50 per hour at her after school job. Last week she worked 14 hours. If 12% is deducted for taxes, determine Mary’s take-home pay. Show all work on how you arrived at your answer.

Answer:7.50 x 14 = 105 105 x .12 = 12.60 105 – 12.60 = $92.40

OR

($7.50 x 14) x .88= $92.40

Problem:Dan completed 60% of his homework. If he has completed 12 problems, how many problems does he have left to complete? Show all work on how you arrived at your answer.



Answer:

12 40 = 60 x 480 = 60 x 480/60 = X 8= xOR

.6x = 12 40 .6x= 480 x= 480 .6 x= 8

OR

12/x = 60/10012 100 = 60x1200 = 60 x1200/60 = x 20 = x

20 – 12 = 8

Problem:If 8 cartons of milk cost $12.50, how much will 12 cartons cost? Show all work on how you arrived at your answer.



Answer:

8x = 12 .50 x 128x = 150 x= $18.75

OR$12.50 8 = 1.5625 1.5625 x 12= $18.75

Problem:Last year a pair of jeans cost $36.00. This year they sell for $42.00. What percent increase does this represent? Show all work on how you arrived at your answer.

Answer: 100[($42-$36)/36] = 16.7% or other valid response

Problem:The length and width of the rectangle below, are each increased by 20%.

1. Determine the dimensions (length and width) of the new rectangle. Show all of your work.

2. Find the area of the new rectangle. Show all of your work.3. Find the perimeter of the new rectangle. Show all of your work.



Answer: Numerous procedures can be used to solve each problem. Accept any valid response.

1. Length = 6 x 1.2 = 7.2 ftWidth = 2 x 1.2 = 2.4 ft

2. New area = 7.2 x 2.4 = 17.28 ft2 3. New Perimeter = 2 ( 7.2 + 2.4) = 19.2 ft

Problem:If 2% of a number is 0.52, what is the number?

a. 104b. 26c. 2.52d. 2.6

Answer: B 26

Problem:The price for airline tickets has increased by 130% from last year. If the price to fly to a certain city was $145.00 last year, what is the new price?

Answer:$333.50




*Recognize equivalent representations for the same number and generate them by decomposing and composing numbers, including scientific notation

Problem:Which of the following is equivalent to 33%?

A.

B. 3.3C. 0.033

D.

Answer: A. 33/100

Problem:Which of the following correctly expresses 203,000 in scientific notation?

A. 2.03 x 105

B. 2.03 x 106

C. 203 x 103

D. 20.3 x 104

E. 20.3 x 105

Answers:A. 2.03 x 105 C. 203 x 103 D. 20.3 x 104

Problem:Express each of the following in standard form.

1. 3.12 x 103

2. 5.68 x 102

3. 1.01 x 104

Answer:1. 3,120 2. 568 3. 10,100



Problem:Using exponents, write the prime factorization of 48.

Answer:2 x 2 x 2 x 2 x 3 = 24 x 3

Problem:Using four 4s, compose an expression that will result in 1.

Answer:(4 – 4) + 4/4

DEFINITION:compose or decompose numbers–flexibly using or knowing numbers through creating and breaking numbers apart to form equivalent representations. Grade 8 should include scientific notation. For example, 36 can be thought of as 32 + 4, 20 + 16, 40 – 4, 12 X 3, 22 x 32 , 3.6 x 101 etc.1 Decomposing numbers means rewriting a number as the sum, product or difference of other numbers. Composing numbers means putting together numbers that are represented as a sum or product in order to obtain a single number.

1 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 149).


BIG IDEA (2): Understand meanings of operations and how they relate to one another.



Apply properties of operations to rational numbers, including order of operations and inverse operations

Problem:Explain how Barbara could use two different methods to mentally calculate the exact answer to the following problem:

Answer:One method Barbara could use is to add 4 and 24, which is 28, then multiply by 1/4. Her answer would be 7. Another method Barbara could use is to use the distributive property. She could take ¼ of 4, which is 1, plus ¼ of 24, which is 6. 1 plus 6 is also 7.

Problem:Find the value for n in the following. (2/3 x 11) + (2/3 x 4) = 2/3 (11 + n)

A. 11B. 2/3C. 15D. 4

Answer:D. 4

Problem:Solve the following:

1.

2.

3. .12(-0.7+0.7)+1



Answer:1. 1 2. 03. 1

Note: Students need to be aware to look for ways of using basic rules of multiplication and addition that “cancel out” numbers, such as multiplying by 1 or 0.

Problem:Insert parentheses to make the following statements true.

1.

2.

Answer:

1.

2.

DEFINITION:properties of operations-associative and commutative properties of addition and multiplication, the distributive property of multiplication over addition to simplify computations; order of operations should be followed.

Example: 3(3+52)÷7+1 = 3(3+25) ÷7+1 = 3(28)÷7+1=84÷7+1=12+1 =132

Mathematicians have agreed that when there is more than one operation within a problem, the operations should be done in the following order: 1. Perform all operations that are in grouping symbols

2. Do all work with exponents3. Multiply and divide in order from left to right

2 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (p. 214). Reston, VA: Author


4. Add and subtract in order from left to right3



C Compute problems

Apply all operations on rational numbers including integers

Problem:In preparing for a barbecue, the Cook’s bought the following at the grocery store: 6½ pounds of hamburger at $1.89 a pound, 12 pounds of chicken at $.59 a pound, and 4 pounds of hotdogs at $2.10 a pound. Find the total amount paid by the Cook’s for the barbecue. Show all work to support your answer.

Answer:(6.5 x 1.89) + (12 x .59) + (4 x 2.1) = $33.59

Problem:I’m thinking of a number. If I subtract 1 and 1/3, add 5 and /12, and then divide by

, the answer is 7 and 2/5. What was the original number? Show all work on how

you arrived at your answer.

Answer:By working backwards from the 7 and 2/5 x 5/6= 37/6. 37/6 – 5 and ½ =4/6 or 2/3.2/3 plus 1 and 1/3 = 2.

Problem:Solve the following problems.

1.

2. 3.

4.

5.

3 Chaplin, Suzanne. (1995), Middle Grade Mathematics: An Interactive Approach. (p. 524). Needham, MA: Prentice Hall Publishers.



Answer:

1.

2. –1.28163. 7.64. 30

5. 2


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