black 4 step problem solving 1. 2. - stutz family · a venn diagram is used to organize information...

Unit 1 – Number Sense and Algebraic Thinking Lesson 6 – 4 Step Problem Solving

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Black – 4 Step Problem Solving

1. Suppose an imaginary tree was planted at the time of the earth's origin and has since

grown continuously at the rate of 3 inches per year. It would now just be reaching

the moon. How old is the earth?

2. You see lightning flash in the North and 7.5 seconds later you hear thunder. Twelve

minutes later you see lightning in the South and hear thunder 12.5 seconds after

that. How fast is the storm moving?

3. Jupiter is the largest of the nine planets with a diameter

of 88,000 miles. In fact, it is larger than all the other

planets combined. Jupiter's rotation is so fast that it

takes only 10 hours to complete one rotation. How fast is

Jupiter turning? (Speed, at the equator)

4. At what speed is the earth traveling as it circles the sun? (The earth orbits the sun

in an elliptical orbit, but for this problem, assume that the orbit is circular).

5. Pluto is approximately 3,348,000,000 miles away from earth. Suppose there are

Plutonians with advanced technology that enable them to look through a telescope

and read a clock on your wall. You look at the clock and it reads exactly 1:35. At

exactly the same time, a Plutonian is looking through his telescope at your clock.

What time would the Plutonian be seeing?

6. A Greek mathematician named Erotosthenes used mathematics to find the

circumference of the earth. He knew that the world was round and he also knew

something very interesting about the placement of the sun at two cities he visited.

The sun was directly overhead in the city of Syene and at the same time it was 7½°

away from directly overhead in, the city of Alexandria. Erotosthenes also knew

that Alexandria was 500 miles north of Syene. With this information, Erotosthenes

determined the circumference of the earth. How did he do it?

VENN DIAGRAMS

In a group of 22 students, 7 children like math, 12 children like science, 6 children like

English, 3 like math and science, 2 like English and math, 2 like science and English, and 1

student likes all three subjects. How many students don't like any subject?

A Venn diagram is used to organize

information so that the brain can

more easily understand and analyze it.


2

Make three circles, each one representing a different subject.

Math Science EnglishMath Science English

MathScience

English

MathScience

English

MathScience

English

Math and Science

Science andEnglish

Math and English

2

11 1

MathScience

English

Math and Science

Science andEnglish

Math and English

2

11 1

Step 1 Since 1 student likes all three subjects, we put the number 1 in the area where

all three circles intersect.

Step.2 Since 3 children like math and science, we need a total of 3 in that area. Since

there is already 1 in the center, we need to add 2 more for a total of 3.

Step 3 Since 2 students like math and English, we'll put a 1 where those two circles

intersect because we already have 1 in the center.

Step 4 Because 2 students like science and English, well also put a 1 in that, space so

there is a total of 2 students.

Now make the

here circles

intersect.


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Step 5 We know that the math circle should have 7 children in it, and since there are I

already 4 children within the math circle, the remaining space gets 3 children.

MathScience

English

Math and Science

Science andEnglish

Math and English

2

11 1

3 8

3

MathScience

English

Math and Science

Science andEnglish

Math and English

2

11 1

3 8

3

Now we are ready to solve the problem. We have a total of 19 children in our circles. Since

we started with 22 children, it is easy to see that we have 3 missing children who didn't

like any subject. (22 minus 19)


4

In a classroom of 29 children, 7 have been to Mexico, 11 have been to Canada, and 4

children have been to both countries. How many children have not been to either country?

Mexico Canada

43 7

Mexico Canada

43 7

Step 1 The 4 is placed in the intersection of the 2 circles because 4 children have been

to both countries.

Step.2 Since Mexico's circle must add up to 7, a 3 is placed inside the circle.

Step 3 Because Canada's circle must add up to 11, a 7 is placed inside that circle.

VENN DIAGRAMS

7. In a 4th grade classroom there are 14 children

who have a dog as a pet. There are 11 who have

cats and 3 who have both. If there are 24

children in this classroom who have neither a cat

or a dog, how many children are in the classroom?

8. In a group of 30 high school students, 8 take French, 12 take Spanish, and 3 take

both languages. How many students in the group take neither French nor Spanish?

9. A baseball team has 20 players. Seven players have a brother, while 6 players have a

sister. If 3 players have a brother and a sister, how many members of the team have

no brothers or sisters.

10. Look at the following information about the favorite colors of a group of students.

How many students are there in the group?

* 19 like the color green * 4 like green and red

* 11 like blue * 5 like green and blue

* 16 like red * 4 like all three colors

* 7 like red and blue * 5 don‟t like any color


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11. Elementary students at a small school were asked about their favorite fruits. This is

the information that was gathered:

How many students are in the group?

12. In a classroom of 45 children, 20 like the Green Bay Packers, 15 like the Vikings, 19

like the Chicago Bears, 8 children like the Packers and the Bears, 9 like the Vikings

and Bears, 6 like the Vikings and Packers and 5 children like all three teams. How

many children don't like any of the three teams?

13. Twenty-three children in an English class had an assignment over Christmas vacation

to read either The Call of the Wild or a book of poetry by Edgar Allen Poe. When the

teacher asked about what book the children decided Q to read, she learned the

following:

17 read The Call of The Wild How many children didn‟t read

9 read Edgar Allen Poe any book over Christmas vacation?

7 read both

The following information is to be used to answer question 14 – 16.

All people who ate a certain restaurant were surveyed as to what they ordered on a

particular Sunday morning.

40 ate eggs

42 ate sausage

32 ate pancakes

13 ate sausage and eggs

15 ate pancakes and sausage

11 ate eggs and pancakes

6 ate all three items

3 people didn‟t eat any of the item

14. How many people ate only pancakes?

* 23 like bananas

* 19 like apples

* 26 like raisins

* 6 like apples and bananas

* 8 like bananas and raisins

* 2 like all three fruit

* 3 don‟t like any fruit


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15. How many people ate at the restaurant that morning?

16. How many people didn't eat eggs?

17. Nikki and Jing collect action figures. Together, they have 48 figures. Jing has six

fewer figures than twice the number Nikki has. How many action figures does each

girl have?

18. Amelia, Brian, and Cody have three bicycles. One bike is blue, one is green, and one is

red. Amelia‟s brother rides a red bike. Brian does not have a blue bike. Cody‟s bike is

either red or green. Cody is not related to Amelia. Who owns the red bike?

19. Number sense. Mr. Lopez is packing the muffins he made for the school bake sale.

He finds that whether he puts four, five, or six muffins in each bag, he has two

muffins left over. What is the least number of Mr. Lopez could have?

20. Roger has band practice every other day, meets with a math tutor every fourth day,

and volunteers at a children‟s hospital once a week. Today, Roger had practice, met

with his tutor, and worked at the hospital. In how many days will Roger have all three

activities on the same day again?

21. Patterns. If the pattern below continues, how many squares will be in Stage 23?

Stage 1 Stage 2 Stage 3 Stage 4

22. Fund-raising. The basketball team is selling calendars to raise money for new

uniforms. Wall calendars to raise money for new uniforms. Wall calendars sell for $7

each and desk calendars sell for $5 each. On Saturday, Maya sold 11 calendars for a

total of $67 dollars. How many calendars of each type did she sell?

23. Maps. Rama and Dan are using a trail map to plan a backpacking trip. On the first

day, they plan to hike from the trail head to Miller‟s Pond – a map distance of about 4

in. The key on the map says that 1.25 in. represents 2.5 mi. About how many miles will

Rama and Dan hike the first day?

24. Geometry. How can you arrange 24 identical square tiles, without stacking, to create

a shape with the least possible perimeter?

25. Number Sense. Rose has forgotten the combination to her gym locker. She knows it

has the digits 1, 3, 5, 7, and 9, but she doesn‟t remember the order of the digits. She

decides to try every possible order until she finds the right one. How many five-digit

numbers does Rose have to try?


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26. How many letters are in either the rectangle or the

square at the right, but not in both?

27. A piece of pastry is rolled out to make a large square with an area of 2025 cm2. If

this piece of pastry is to be cut into 9 identical squares, what is the length of the

smaller squares?

28. A bin holding 100 apples weighs 20.6 kg. After 50 apples were sold, the bin and its

content weigh 11.8 kg Find the weight of the empty bin and the weight of 1 apple.

29. Nicola and Matthew are comparing how much pocket money they each have. Nicola

says, “If you give me $1, I will have as much as you.” Matthew says, “If you give me

$1, I will have twice as much as you.” How much did they each have?

30. A father opens a bank account when his daughter is born and deposits $5. Each year

on her birthday he deposits twice the amount that he did in the previous year. (i.e. on

her first birthday he deposits $10) He continues this until she is 16 years old.

Assuming that no interest is paid, how much is in the account after this time?

31. Trick question: Jason came to an old bridge. The sign in front of the bridge read

“Weight limit 100 kg”. Jason weighed 90 kg, but he was also carrying 3 watermelons

weighing 5 kg each. Jason could not throw the watermelons across the bridge

because they would be ruined. How did Jason cross the bridge with the melons in one

trip?

32. The Broken Window. The Principal of a school is questioning five boys who are

suspected of breaking a window. The Principal knows that at her school, guilty

students lie while innocent students tell the truth.

These were the answers when she questioned the boys:

Pete said “Just one of us did it”.

Quintin said “Two of us are guilty”.

Roy said “Three of us broke the window”.

Sam said “Four of us did it”.

Tom said “All five of us smashed it”.

Assume Pete tells the truth, then Pete is innocent and just one of the others did it.

Is that possible? ____ because ____.

If this is not possible then Pete is lying, he is guilty and his statement is false.

Continue with Quintin. Use a pencil to complete the table with ticks and crosses, it

will help you find the guilty one (s).

D

I

K

J

A

H

CF

B

E G

D

I

K

J

A

H

CF

B

E G


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So, who is/are guilty of breaking the window?

True Statement

Innocent (V)

False Statement

Guilty (X)

Pete

Quintin

Roy

Sam

Tom

33. Bribery at the Border. Gina, a photo journalist, is stuck in a country at war. To get

herself and her photos out of the country she must bribe five officials, one by one,

to give her a passport and leaving papers. Each official demands half of the money

she owns at that moment. Once the bribes are paid she must pay for her plane ticket

costing $1500.

What is the least amount of money Gina must have to pay for the bribes and the

plane ticket?

34. Squares & Rectangles. Three identical squares are pushed together to make a

rectangle. The perimeter of the rectangle is 60 cm.

xx

Question: What is the area of the rectangle?

35. Lovely Lamingtons. „Lotza Fun‟ play centre is running a cake stall. Eight lamingtons

cost the same as five muffins. Four fruit mince pies cost the same as six lamingtons.

How many fruit mince pies can you buy for the same cost as 15 muffins?

Solutions

1. 5,280,000,000 years old

The first piece of information that is needed is the distance to the moon in inches.

One mile has 12 x 5280 = 63,360 inches. The Earth is 250,000 miles away from the

moon so there are 250,000 x 63,360 = 15,840,000,000 inches to the moon. If the

tree grows at a rate of 3 inches per year, it will take 15,840,000,000 3 =

5,280,000,000 years to grow to the moon.

2. 20 mph

At the-first flash, the storm is 1.5 miles away. (The bound took 7.5 seconds to reach

your ears) 12 minutes later the storm is 2.5 miles away because it took the sound

12.5 seconds to reach your ears. The storm traveled 4 miles in 12 minutes, which is a

speed of 20 miles in one hour.


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3. 27,632 mph

The circumference of Jupiter is 88,000 x 3.14 = 276,320 miles. It takes the planet

10 hours to rotate once. It travels 276,320 = 10 = 27,632 miles in one hour.

4. 66,958 mph

The Earth is approximately 93,000,000 miles from the sun. If we pretend the orbit

is circular, then the diameter of the circle is 93,000,000 miles plus the diameter of

the sun (miles) plus 93,000,006 miles. This equals 186,800,000 miles.

186,800,000 x 3.14 =586,552,000 miles in one orbit.

There are 365 x 24 =8760 hours in a year.

The speed is 586,552,000 8760 = 66,958 miles per hour

5. 8:35

Light takes one second to travel 186,000 miles. The light leaving the clock on Earth

takes 3,348,000,000 186,000 =18,000 seconds to reach Pluto. 18,000 seconds is

equal to 5 hours. Because it takes 5 hours for the light to reach Pluto, the Plutonian

is looking at the clock as it appeared 5 hours ago ------ 8:35.

6. 7.5° is 1/48 of the entire circumference of the Earth (3600° 75° is equal to 48)

If 7.5° is equal to 500 miles, then 360° is equal to 48 x 500 = 24,000 miles.

7. 46 children

Dog Cat

311 814 total in dog circle 11 total in cat circle

Dog Cat

311 8

Dog Cat

311 814 total in dog circle 11 total in cat circle

8. 13 students

French Spanish

35 98 total in French circle 12 total in Spanish circle

French Spanish

35 9

French Spanish

35 98 total in French circle 12 total in Spanish circle

9. 10 members

34 3 6 players total in sister circle7 players total in brother circle 34 3 6 players total in sister circle34 334 3 6 players total in sister circle7 players total in brother circle


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10. 39 students

1

4

14 3

Green Blue

Red

0 3

9

4 + 1 or 5 total in green and blue

19 total in green circle 11 total in blue circle

4 + 3 or 7 total in red and blue

16 total in red circle

4 + 0 in green and red

55 don‟t like any color

1

4

14 3

Green Blue

Red

0 3

9

4 + 1 or 5 total in green and blue

19 total in green circle 11 total in blue circle

4 + 3 or 7 total in red and blue

16 total in red circle

4 + 0 in green and red

55 don‟t like any color

11. 57 students

4

2

11 13

Bananas

Raisins

6

18

6 total like apples and bananas

23 total like bananas 19 total like apples

26 total like raisin

8 total like bananas and raisins

3

Apples

3 don‟t like any fruit

4

2

11 13

Bananas

Raisins

6

18

6 total like apples and bananas

23 total like bananas 19 total like apples

26 total like raisin

8 total like bananas and raisins

3

Apples

3 don‟t like any fruit

12. 9 children

1

5

11 5

Packers

Bears

3 4

7

6 total like the Packers and Vikings

20 total like the Packers 15 total like the Vikings

19 total like the Bears

8 total like the Packers and Bears

Vikings

9 total like the Vikings and Bears

1

5

11 5

Packers

Bears

3 4

7

6 total like the Packers and Vikings

20 total like the Packers 15 total like the Vikings

19 total like the Bears

8 total like the Packers and Bears

Vikings

9 total like the Vikings and Bears

13. 4 children

710 217 total in Call of the Wild Circle

9 total in the Edgar Allen Poe circle

710 217 total in Call of the Wild Circle

9 total in the Edgar Allen Poe circle


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14. 12 people

7

6

22 20

Egg

Pancake

5 9

12

13 total ate sausage and eggs

40 are in the egg circle 42 are in the sausage circle

32 are in the pancake circle

11 total ate eggs and pancakes

3

Sausage

3 didn‟t eat any of the three foods

15 total ate sausage and pancakes

7

6

22 20

Egg

Pancake

5 9

12

13 total ate sausage and eggs

40 are in the egg circle 42 are in the sausage circle

32 are in the pancake circle

11 total ate eggs and pancakes

3

Sausage

3 didn‟t eat any of the three foods

15 total ate sausage and pancakes

15. 84 people

If you combine the people in the diagram and the 3 that didn't eat either eggs,

pancakes or sausage, you will get an answer of 84 people.

16. 44 people

The people outside the egg circle total 41 + the 3 who didn't eat any of the three

items.

17. Nikki 18

Jing 30

18. Brian

19. 62

20. 28 days

21. 67 squares

22. 6 wall calendars and 5 desk calendars.

23. 8 mi.

24. In a 6 x 4 array.

25. 120 numbers

26. 6 letters

27. 15 cm

28. Bin = 3gs; 1 apple = 0.176 gs

29. Nicola = $2 ; Matthew = $3


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30. 138240

31. Explanation vary – he could juggle them across the bridge

32. True Statement

Innocent (V)

False Statement

Guilty (X)

Pete X

Quintin X

Roy X

Sam V

Tom V

No, because all others say multiple did it.

Same innocent.

33. Question: What is the least amount of money Gina must have to pay for the bribes

and the plane ticket?

Working : To end up with $1500 she needs to have $3000 before she meets with the

last (5th) official.

3000 x 2 = 6000 (4th)

6000 x 2 = 12000 (3rd)

12000 x 2 = 24000 (2nd)

24000 x 2 = 48000 (1st)

Answer : Gina needs at least $48000 to get out of the country safely.

34. Squares & rectangles.

Question: What is the area of the rectangle?

Working : Let the side of the squares be x cm long, then an expression for the

perimeter is that there are 8 xs or simplified 8 x = 60.

The perimeter is 60 cm, so x x 8 = 60, x = 7.5

Then width of rectangle = 7.5 cm and length = 22.5 cm

Answer : Therefore area of rectangle = 168.75 cm2

35. Lovely Lamingtons.

Question 1: How many fruit mince pies can you buy for the same cost as 15 muffins?

Working 1 : Using the letters L, M, and F for the cost of each cake, you can write the

information as 8L = 5M 6L = 4F

Then 15M is the same as 24L, and this is the same as 16F

Answer 1 : 24 lamingtons


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Bibliography Information Teachers attempted to cite the sources for the problems included in this problem set. In some cases, sources may not have been known.

Problems Bibliography Information

1 - 16 Zaccaro, Edward. Challenge Math (Second Edition): Hickory Grove Press, 2005.

32 – 35

Geldof, W. Homework Book 3 /4, 2nd edition. NZ Mathematics Curriculum, Sigma Publications, 1999.

27, 28

Lakeland, Robert, Claire Norton, and Carl Nugent. Maths Homework Book Levels 5/6. Wanganui: Nulake Texts, 1998.

black 4 step problem solving 1. 2. - stutz family · a venn diagram is used to organize information...

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