Elementary Word Problem Bank

Tanya DaSilva, Annamaria Frasca, Alison Goddard, Christina Glover, Colleen McCarthy and Amy Shea for the Balanced Math Project

York Catholic District School Board

2017

This document was created as part of a Provincial Knowledge Exchange Grant funded by the Ministry of Education.

The intention of this Problem Bank was to provide elementary teachers with a “go to” document of words problems. Three levels of word problems are provided for use to increase the challenge of problem solving with students, or; to use for Guided Learning, then Shared Learning, and finally Independent Learning. We have used the Nelson Math Textbooks for our formatting and referenced where problems were directly quoted. The second section of this document has taken problems from Problem Solving in Mathematics, Grades 3­6: Powerful Strategies to Deepen Understanding 1st Edition by Alfred S. (Steven) Posamentier (Editor), Stephen Krulik (Editor)Copyright © 2009 by Corwin Copyright Guidelines have been respected by copying less than 10% of the documents: and not allowing public access. Our intention is to introduce you to these excellent resources and to encourage you to purchase the resources in full.

2

Primary ­ Grade 1

Curriculum Connection → compose and decompose numbers up to 20 in a variety of ways, using concrete materials (e.g., 7 can be decomposed using connecting cubes into 6 and 1, or 5 and 2, or 4 and 3) → relate numbers to the anchors of 5 and 10 (e.g., 7 is 2 more than 5 and 3 less than 10); → count forward by 1’s, 2’s, 5’s, and 10’s to 100, using a variety of tools and strategies

Nelson Chapter 2: Exploring Number

Low Level Problem Choose a number less than 10. Tell as many things are you can. (Source: Marian Small open questions for three part lessons page. 31)

Middle Level Problem Choose a number closer to 40 than to 20. How would you arrange objects to show that the number is closer to 40 than 20? (Source: Marian Small Open questions for three part lessons page. 31) Choose five numbers less than 50 and put them in order from least to greatest. Tell why the least one is least. (Source: Marian small Open questions for three part lessons page 38)

High Level Problem Choose two numbers that you think are alike in some ways and different in some ways. Represent each of your numbers in three ways. Tell how you representations show what makes the numbers alike. Tell how the representations show what makes them different. (Source: Marian Small open questions for three part lesson page. 31)

3

Primary ­ Grade 1

Curriculum Connection → solve a variety of problems involving the addition and subtraction of whole numbers to 20, using concrete materials and drawings (e.g., pictures, number lines) → solve problems involving the addition and subtraction of single­digit whole numbers, using a variety of mental strategies (e.g., one more than, one less than, counting on, counting back, doubles); → count backwards by 1’s from 20 and any number less than 20 (e.g., count backwards from 18 to 11), with and without the use of concrete materials and number lines;

Nelson Chapter 4: Introduction to Addition and Subtraction

Low Level Problem There are 7 frogs on a log, 3 frogs left the log. How many are frogs are left of the log? There are 6 frogs on the log, 3 more frogs come on the log. How many frogs are on the log? (Source Nelson Math teacher resource page. 16)

Middle Level Problem Joe caught 16 fish. 5 slipped out, how many fish does Joe have left? Answer: 16 ­ 5 = 11, Joe has 11 fish left 13 bunnies were playing in the grass. 5 more came in to join them. How many bunnies are playing? Answer: 13 + 5 = 18, 18 bunnies are playing in the grass

High Level Problem Sarah has 6 dolls. Her mom gave her some more and now she has 12. How many dolls did her mom give her? Answer: 12 ­ 6 = 6 , Her mom gave her 6 dolls Amy put 6 beads on her bracelet. She put 4 more beads on. How many beads are on her bracelet? Answer: 6 +4 = 10 , There are 10 beads on her bracelet.

4

Primary ­ Grade 1

Curriculum Connection → solve a variety of problems involving the addition and subtraction of whole numbers to 20, using concrete materials and drawings (e.g., pictures, number lines) → solve problems involving the addition and subtraction of single­digit whole numbers, using a variety of mental strategies (e.g., one more than, one less than, counting on, counting back, doubles); → count backwards by 1’s from 20 and any number less than 20 (e.g., count backwards from 18 to 11), with and without the use of concrete materials and number lines; → compose and decompose numbers up to 20 in a variety of ways, using concrete materials (e.g., 7 can be decomposed using connecting cubes into 6 and 1, or 5 and 2, or 4 and 3.

Nelson Chapter 6: Addition and Subtraction

Low Level Problem Making 10’s A baker has 10 candies to decorate 2 cupcakes. How many different ways can she decorate the 2 cupcakes? (Source: Nelson math teacher resource page.12) There are 3 apples, 2 oranges and 1 pear in the fruit basket. How many pieces of of fruit are all in the basket? Answer (3 + 2 + 1 = 6)

Middle Level Problem I have 5 cookies, my friend gave me some more, now I have 10 cookies. How many did my friend give me? (Source: Nelson math teacher resource page. 19) I had 12 muffins, my family ate some, now I have 8. How many muffins did my family eat? Answer: 12 ­ 8 = 4 , My family ate 4 Are there more ways to subtract to get 3 than ways to add to get three? Explain your answer. (Source: Marian Small Open questions for three part lessons pg. 50)

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High Level Problem Grandpa caught 10 fish. 3 are trout, 2 are bass. The rest are sunfish. How many sunfish did he catch? (3 + 2 = 5 , 10 ­ 5 = 5 , There are 5 sunfish) Sally has 10 beads. Some are blue, 2 are green and 1 is yellow. How many blue beads does she have? (2 +1 = 3, 10 ­ 3 = 7, There are 7 blue beads) Dan baked 16 loaves of bread. 4 were banana bread, and 7 were chocolate bread, and the rest were lemon bread. How many loaves of lemon bread did Dan bake? (4 + 7 = 10 , 16 ­ 10 = 6 , 6 loaves were lemon bread)

6

Primary Grade 1

Curriculum Connection → represent, compare, and order whole numbers to 50, using a variety of tools (e.g., connecting cubes, ten frames, base ten materials, number lines, hundreds charts) and contexts → identify and describe various coins → represent money amounts to 20¢, through investigation using coin manipulatives

Nelson Chapter 8: Exploring Greater Numbers

Low Level Problem You have five coins. When you count the value of your coins out loud, what do you say? How much might your coins be worth? (Source: Marian Small Open Question for the three­part lesson page 35) Choose a number that is near the top of a 100­chart and another number that is farther down. How does the chart help you tell that the second number is greater? (Source: Marian Small Open Questions for three­part lesson page 36)

Middle Level Problem There are two different amounts of money that can be shown the same number of ways with coins. For example 15 ¢ and 20 ¢ can be shown two ways (without pennies). What might the two amounts be? (Source: Marian Small Open Questions for three­part lesson page 30)

High Level Problem How could you break the number 12 into parts t o show each of these things?

a) 12 is 2 equal groups of something. b) 12 is between 10 and 20. c) 12 can be broken up into 3 equal groups. d) 12 is a lot less than 50. e) 12 is 10 plus something.

(Source: Marian Small Open Questions for three­part lesson page 34)

7

Primary Grade 1

Curriculum Connection → represent money amounts to 20¢, through investigation using coin manipulatives → add and subtract money amounts to 10¢, using coin manipulatives and drawings.

Nelson Chapter 10 Time and Money

Low Level Problem What different coins can to make 10 ₵? (Source: Nelson Math Teacher resource grade 1 page 21) You can show an amount of money with coins in two ways. What might the amount be?

Middle Level Problem You have 10 You want to buy at toy for 7 ₵ and a candy for 2 ₵. Do you have.₵ enough money to purchase those items? Answer: 7₵ + 2 ₵ = 9 ₵, Yes you have enough money.

High Level Problem You bought two things and spent a total of 8 ¢. What could the more expensive item have cost? (Source: Marian Small Open Questions for three­part lessons page 48)

8

Primary Grade 1

Curriculum Connection → divide whole objects into parts and iden­ tify and describe, through investigation, equal­sized parts of the whole, using fractional names (e.g., halves; fourths or quarters).

Nelson Chapter 9 Mass and Capacity Fractions

Low Level Problem Is one­half big or small? (Source: Marian Small Open Questions for three­part lessons page 40.) Choose a book. Cover half of the book with yellow pattern blocks. How many blocks did you use? How many do you think it would take to cover the entire book? (Source: Marian Small Open Questions for three­part lessons page 41.) Take each pattern block and trace it. Show how to divide each block into equal parts and tell what fractions each part would be. Try not to use the same number of parts each time. (Source: Marian Small Open Questions for three­part lessons page 41.)

Middle Level Problem What does one­fourth look like? (Source: Marian Small Open Questions for three­part lessons page 40.) Using square ties to show that a single square tile could be one­half, one­third, or one­fourth or something. (Source: Marian Small Open Questions for three­part lessons page 41.)

High Level Problem You cut a shape into fourths. Do you think the fourths have to look the same? (Source: Marian Small Open Question for three­part lessons page 42.) Explain how you can fold a piece of paper to make eight equal parts. (Source: Marian Small Open Questions for three­part lessons page 41.)

9

Primary ­ Grade 2

Curriculum Connection → represent, compare, and order whole numbers to 100 → locate whole numbers to 100 on a number line and on a partial number line

Nelson Chapter 2 Numbers to 50

Low Level Problem How can you tell that 42 is closer to 40 than to 50? (Source: Nelson Teacher Resource page 17) On a 100­ chart , choose one number that is far from 1 another number that is close to 1. How can you tell that the first number is far from 1? How can you tell that the second number is close to 1? (Source: Marian Small Open Questions for the three part lesson page 59)

Middle Level Problem How much would four coins be worth? (Source: Marian Small Open Questions for the three part lesson page 56) Choose 5 two­digit numbers and put them in order. Tell how you know which number is least. (Source: Marian Small Open Questions for the three part lesson page 61)

High Level Problem Someone said Charlotte had about 40 counters. How many do you think “about 40” is? (Source: Marian Small Open Questions for the three part lesson 61)

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Primary Grade 2

Curriculum Connection → solve problems involving the addition and subtraction of whole numbers to 18, using a variety of mental strategies.

Nelson Chapter 4: Addition and Subtraction Stories

Low Level Problem There are 4 red leaves, 5 brown leaves and 9 yellow leaves. How many leaves are in the pile? (Source: Nelson Teacher's manual pg 28.)

Middle Level Problem Your friend just gave you 12 sparkly markers, now you have 19. How many did you have before your friend gave you 12 more?

High Level Problem 22 students and their teacher are going on a trip to a pet store. Three parent helpers will drive the students. One parent can take 4 people at a time; one parent can take 5 people at a time; and one parent can take 6 people at a time. How many total trips will it take to get all the children and their teachers to the pet store? (Source: Nelson Teacher Manual page 26.)

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Primary Grade 2

Curriculum Connection → represent, compare, and order whole numbers to 100 using concrete materials and drawings → solve number problems involving addition and subtraction and describe and explain the strategies used

Nelson Chapter 6: Place Value

Low Level Problem Jenna built a number with 9 base ten blocks. Some were tens and some were ones. What could the number have been? (Source: Nelson Mathematics Teacher’s Resource page 33)

Middle Level Problem Choose a number between 20 and 100. Show your number 4 different ways. What number would you have to add to your number to get to the next 10? Show how you know. (Source: Nelson Mathematics Teacher’s Resource page 50)

High Level Problem The Red Team has 45 points and Blue has 36 points. How many more points does Blue need to score to tie the game? Show how you got your answer. (Source: Nelson Mathematics 2 Teacher’s Resource page 49)

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Primary Grade 2

Curriculum Connection → develop proficiency in adding and subtracting 1 digit and 2 digit whole numbers. → add and subtract 2­digit numbers with and without regrouping, with the sum less than 101, using concrete materials. → solve number problems involving addition and subtraction, and describe and explain the strategies used.

Nelson Chapter 8: Two­Digit Addition and Subtraction

Low Level Problem Addition→ Mary has 10 stuffed animals. Brandy has 15 stuffed animals. How many do they have altogether? Subtraction→ Bob has 20 cars. Bob gave 5 cars to his friend Scott. How many cars does Bob have now?

Middle Level Problem Addition → Tanya is selling cookies for the Girl Guides. She sold 47 boxes on Monday and 38 boxes on Tuesday. How many cookies did Tanya sell? Subtraction→ Josh and Jake planted a tulip garden in their backyard. Josh planted 27 tulips and Jake planted 49 tulips. How many more bulbs did Jake plant than Josh?

High Level Problem Addition → The sum of two numbers that are pretty close to each other is a little less than 60. What could the two numbers be? Subtraction → The movie theatre sold sixty­seven tickets to see ‘Finding Dory’. Forty­three tickets were for children up to 12 years old. How many were for people older than 12?

13

Primary Grade 2

Curriculum Connection → understand and explain basic operations of multiplication and division of whole numbers → represent multiplication as repeated addition using concrete materials → demonstrate division as sharing

Nelson Chapter 9: Multiplication and Division

Low Level Problem Multiplication → A spider has 8 legs, how many legs does 3 spiders have altogether? Division → Mike has 15 stickers for 3 friends to share. How many does each friend get.

Middle Level Problem Multiplication → A tablet takes 3 hours to fully charge. How many hours would it take to charge 6 tablets one after the other. Division → Bradley saved 9 dollar in a piggy bank in 3 days. If he saved equal amount of money in each day, find the number of dollars he saved in one day.

High Level Problem Multiplication → Use Modelling clay to create a monster with 3 antennae, 4 eyes, and 5 legs. Decide on how many monsters you would like. How many antennae, eyes, and legs would there be altogether? Division → If six students shared cookies, would it be easier to share 10 cookies or 12 cookies? Explain.

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Primary Grade 2

Curriculum Connection → Estimate and count money amounts to $1 and record money amounts using the ȼsymbol.

Nelson Chapter 10: Measuring Time and Mone y

Low Level Problem I have 2 nickels, 3 dimes and 1 quarter. How much money do I have?

Middle Level Problem Candice had 75¢ until she spent 5 dimes. How much money does Candice have now?

High Level Problem Alison bought a gum ball for a quarter, a sour key for a dime and a bouncy ball for using 2 nickels. If she gave the cashier $1, how much change will she get back?

15

Primary ­ Grade 3

Curriculum Connection → represent, compare, and order whole numbers to 1000, using a variety of tools (e.g.,base ten materials or drawings of them, number lines with increments of 100 or other appropriate amounts); → compose and decompose three­digit numbers into hundreds, tens, and ones in a variety of ways, using concrete materials

Nelson Chapter 2 Numeration

Low Level Problem Write a number that fits each of these rules:

1) The tens digit is greater than the ones digit. 2) The hundreds digit is greater than the tens and the tens digit is less than the

ones digit. 3) The hundreds digit is less than the tens digit and the tens digit is less than the

ones digit.

Middle Level Problem Alana has 5 base 10 blocks. She has at least one of each type of block. The value of her blocks is greater than 140. What blocks could she have? (Nelson page 39 textbook. #5) Use the digits 0­9, using each digit only once to fill in the blanks. Then, put the numbers in order from least to greatest. Repeat two or more times. Do you notice a pattern? __ 9 , __ __ 4, 11__ , __ __ __ , 3__ __ , 40 __ Sample answer: 93, 104, 117, 236, 384, 405 (Marian Small open questions NSN Primary Page 94)

High Level Problem Mystery number­ Using these clues, find the mystery number:

1. a) I am a 3 digit number greater than 400 b) The sum of my 10’s column and

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1’s column is 10. c) all my digits are odd d) no digits can be repeated ** Find a least 4 possibilities ** (573, 591, 973) ** Does finding one solution help you find more?

2. a) I am a 4 digit number greater than 6000 b) the sum of my 100’s digit and and 1’s digit is 10. c) The sum of all my digits is 20. d) no digits can be repeated.

** Find at least possibilities** ( 6842, 6248, 8426, 8624) Does finding one solution help you find more? 3. a) The house number is less than 871. b) The number of hundreds is the same as the number of tens. c) The sum of the 3 digits is 22. (Marian Small open questions NSN Primary Page 94) You write down a three digit number. You switch some of the digits around, and the value of your number increases by 54. What could your original number have been? Does finding one way help you find another way? Sample answer: 384 and 438 or 495 and 549 or 783 and 837 I realize that the two numbers I made up using those digits has to be less than 100 apart, so the two digits used in the hundreds place had to be 1 apart. For example, if I switched 384 and 834 the the two numbers would be more than 100 apart, but if I switched 374 and 437 the 2 numbers are less than 100 apart so they might work out..

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Primary ­ Grade 3

Curriculum Connection → solve problems involving the addition and subtraction of two­digit numbers, using a variety of mental strategies. → use estimation when solving problems involving addition and subtraction, to help judge the reasonableness of a solution;

Nelson Chapter 4 addition and subtraction

Low Level Problem Liam started with 5 skateboard stickers. He collected more stickers in 3 equal groups. He ended up with between 25 and 30 stickers. How many stickers could have been in each group Liam collected? Find 2 possible answers. Explain how you know. (Nelson Page 89 textbook #6)

Middle Level Problem In games 4 and 5, the sharks scored 36 points and 42 points. The Jets scored 85 points altogether. Which team scored more in 2 games? How do you know? (Nelson Page 95 textbook #9)

High Level Problem Kara takes 1 step forward, then 2 steps backward. She repeats this again and again. She takes 20 steps altogether. How many steps is she from where she started? (Nelson teacher resource) Carla wants to buy a CD player that costs $48.70 including tax. She has saved $10.50. Carla earns $10 each weekend babysitting. How many weekends does she need to babysit to earn enough money to buy the CD player?

18

Primary ­ Grade 3

Curriculum Connection → add and subtract three­digit numbers, using concrete materials, student­generated algorithms, and standard algorithms; → use estimation when solving problems involving addition and subtraction, to help judge the reasonableness of a solution;

Nelson Chapter 6 adding and subtracting with greater numbers.

Low Level Problem Two numbers have a sum of 845. The first number is one less than the second number. What are the numbers? Solve and show your work. Twenty coins are worth about $8. What might the coins be? Give 3 or more possibilities. (Nelson Math test question #5)

Middle Level Problem There are 750 Children's books are in the library. 139 are about animals. 287 have been checked out. How many are left in the library? Teddy bear $4.50 Dart game $2.75 Book $1.25

Blocks $3.25 Car $2.25 Doll $3.75 Which 2 toys cost exactly $5 without going over? How much more is the teddy bear than the book? What is the cost of the dart game and the doll?

High Level Problem Teddy bear $4.50 Dart game $2.75 Book $1.25

Blocks $3.25 Car $2.25 Doll $3.75

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Which two toys cost the same as one doll? Mary bought 2 toys. She got back $2.25 in change. Which 2 toys did she buy? When you add 2 numbers they give you a sum of 976. When you subtract one of the numbers from 976 you get 634. What is the missing number?

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Primary ­ Grade 3

Curriculum Connection → relate multiplication of one­digit numbers and division by one­digit divisors to real life situations, using a variety of tools and strategies

Nelson Chapter 9 Multiplication

Low Level Problem There are 7 bicycles. Each bicycles has 5 streamers. How many streamers are there on 7 bicycles?

Middle Level Problem Mary has less than $0.50. 2 coins are pennies. She has more nickels than pennies. She has more dimes than nickels. What is the greatest number of dimes she could have? For every year a dog lives, it ages about 7 years. Martys dog is 8 years old? How old is Marty’s dog in human years?

High Level Problem You are hired for a job for 15 days. The boss asks if you would like to be paid for $1.00 each day or if you would like to be paid 1 cent on the first day, two cents on the second day,4 cents on the third day and so on, doubling your pay each day. Which way would you choose? Why?

21

Primary ­ Grade 3

Curriculum Connection → relate multiplication of one­digit numbers and division by one­digit divisors to real life situations, using a variety of tools and strategies

Nelson Chapter 10 Division

Low Level Problem There are 24 students who want to play soccer. There must be 6 students on each team. How many teams can be made? (Nelson problem of the week. Teacher's guide. Page 3)

Middle Level Problem 3 tennis balls can be stacked in a container. The tennis club collected 85 loose balls. How many containers would 85 tennis balls fill? (Nelson textbook page 253 #5) Flowers cost $7 a bunch. How many bunches can you buy for $30?

High Level Problem My house number is between 19 ­ 29. It is divisible by 2 and 5. What is my house number? Explain how you know.

22

Primary ­ Grade 3

Curriculum Connection → divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g., one half; three thirds; two fourths or two quarters), without using numbers in standard fractional notation;

Nelson Chapter 12 Fraction

Low Level Problem

Colour the quilt squares so that ½ are green, ⅖ are red. Colour the rest yellow. What fraction is yellow?

Middle Level Problem

Describe the pictures using fractions in two ways (Ex: 6/10 are shaded. 3/10 are triangles, ¾ of the circles are shaded)

a) What fraction of the pictures are not shaded? b) What fractions of the triangles are shaded?

High Level Problem ⅘ of the basketball team members are very tall. ⅖ of the team members are excellent dribblers. How many of the tall members of the team could be excellent dribblers? Is there more than one answer? (Nelson textbook page 303 #1)

23

Junior ­ Grade 4

Curriculum Connection → solve problems involving whole numbers (and decimals) and describe and explain the variety of strategies used → recognize and read numbers (from 0.01) to 10 000 represent the place value of whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols → compare and order whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols

Nelson Chapter Two Numeration Place Value p. 13 ­ 16

Low Level Problem A number is represented by 10 hundreds blocks and some tens blocks. What could the number be? (Nelson Grade 4, p.51)

Middle Level Problem Patrick wants to build a model of 2343 He has these blocks. Can Ravi build the model? Use pictures, numbers, and words to explain your answer.

High Level Problem A Girl Guide is selling cookies, her sign that reads 1255 boxes sold. Each week for the next 5 weeks, the she sells 100 more boxes. a) Model 1255 using the least number of blocks. b) Add blocks to your model to include the additional 100 boxes sold each week. Regroup blocks so the model uses the least number of blocks. c) What should the sign read at the end of 5 weeks? d) Which blocks changed in your model? Why?

24

Junior ­ Grade 4

Curriculum Connection → solve problems involving whole numbers (and decimals) and describe and explain the variety of strategies used → recognize and read numbers (from 0.01) to 10 000 represent the place value of whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols → compare and order whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols

Nelson Chapter Two Numeration Expanded Form P. 17 ­ 20

Low Level Problem When I write the number as 3562, it is in standard form. Show this number 3562 in expanded form using words and using numerals.

Middle Level Problem Using the numbers 7 8 4 0, create as many numbers as possible and show them in a place value chart.

High Level Problem Austin Matthews of Toronto Maple Leafs will score 1325 goals as a professional hockey player. a) Write 1325 in a place value chart. b) Write 1325 as thousands, hundreds, tens, ones. c) How would you read this number? d) Write 1325 in expanded form using numbers.

25

Junior ­ Grade 4

Curriculum Connection → compare and order whole numbers (and decimals) using concrete materials and drawings → recognize and read numbers (from 0.01) to 10 000 → represent the place value of whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols → compare and order whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols → identify and appreciate the use of numbers in the media

Nelson Chapter Two Numeration Comparing and Ordering Numbers p. 21 ­ 24

Low Level Problem Write a number between 1453 and 1860. What digits can be in the thousands place and hundreds place?

Middle Level Problem The total RBI (Runs Batted In) for the Blue Jays is 1376; Boston Red Sox have 1297; Chicago Cubs have 2036. Which number is greater? What number is least? Display these in a Place Value Chart.

High Level Problem The Raptors are on a road trip. These are the places they have to travel to: Boston: 885 km Chicago: 835 km New York: 790 km Detroit: 372 km Los Angeles: 4053 km Colorado: 2026 km Place these numbers in order from greatest to least. What do you think the best route for the Raptors to travel so they spent the least amount of time on the road? You must explain your thinking. Which games do you think the Raptors will play better in because of the amount of travel they must do? You must explain your thinking.

26

Junior ­ Grade 4

Curriculum Connection → compare and order whole numbers using concrete materials and drawings → recognize and read numbers (from 0.01) to 10 000 → represent the place value of whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols → represent and explain number concepts and procedures

Nelson Chapter Two Numeration Exploring 10,000 p. 25 ­ 27

Low Level Problem How many people would be at a Raptors games if the stadium holds 30,000 people and 20,000 tickets were sold?

Middle Level Problem The number of people at the Toronto Rock Game was 10,438, more than any other game. How many people might have been at the game before? You must explain your thinking. How many people do you think go to the Toronto Maple Leaf games? Is this more or less than Toronto Rock games? You must explain your thinking.

High Level Problem You would eat 10,000 apples in about 27 years if you ate one apple a day. How many years would it take you to brush your teeth 10,000 times? Our school has 300 students and we have to buy pizza for the entire school. Each student will receive two slices. Each Pizza comes cut into 10 slices. How many Pizzas need to be ordered?

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Junior ­ Grade 4

Curriculum Connection → represent and explain number concepts and procedures → identify and appreciate the use of numbers in the media

Nelson Chapter 2 Numeration Rounding to the Nearest 10, 100, or 1000 p. 37 ­ 40

Low Level Problem Allison, Calvin, and Pedro report on school sports for the local newspaper. 2943 spectators came to a track and field meet. They decide to round the number of spectators for the headline in the newspaper. (Nelson p.42)

Middle Level Problem You are having a Blue Jays Party for 88 of your friends; and you need to order pizza. You can only order by full Pizzas. Pizza comes in 10 slices to a Pizza. How many slices do you need to order?

High Level Problem The Ministry of Education has decided that schools with student populations over 500 will get free pencils; and schools over 1000 will get free books; and school over 10,000 will have a day off. You will have to round these numbers to determine what each school will receive. School Number of Students Rounded Number Will receive

St Mary 495 500 Pencils

St Paul 702

St Michael 992

St Kateri 1675

St Brendan 1222

St Brigid 9250

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Junior ­ Grade 4

Curriculum Connection → compare and order whole numbers (and decimals) using concrete materials and drawings → compare and order whole numbers (and decimals from 0.01) to 10 000 using concrete materials, drawings, and symbols → represent and explain number concepts and procedures → explain their thinking when solving problems involving whole numbers

Nelson Chapter 4 Addition and Subtraction

Low Level Problem The Disney Cruise line uses 9284 L of hand sanitizer. The shipment comes in 3 boxes. The first box has 2499 L and the second has 4892 L. How much hand sanitizer is in the third box?

Middle Level Problem Rey estimated a sum by rounding 2 numbers to the nearest thousand. His estimate was 7000. What could the numbers have been? Find at least 2 answers. (Nelson Chapter 4 textbook page 117)

High Level Problem Change 1 digit in each price so the total cost is exactly $40.00. Find at least 2 ways. $18.37 and $22.73

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Junior ­ Grade 4

Curriculum Connection → select the appropriate operation and solve one step problems involving whole numbers with and without a calculator → pose problems involving whole numbers and solve them using the appropriate calculation method: pencil and paper, or calculator, or computer → explain their thinking when solving problems involving whole numbers → recognize situations in problem solving that call for multiplication and division and interpret the answer correctly

Nelson Chapter 6 Multiplication and Division Facts

Low Level Problem Mackenzie is arranging 3 rows of 6 flowers for the school garden a) How many flowers are there? b) How many flowers will there be if she adds another row? Show your work.

Middle Level Problem For Spencer’s birthday his mother has reserved a Box for a Blue Jay Game. Spencer has 18 friends he wants to invite. In the box there are 4 rows of 6 seats. How many friends can he invite? How can he divide his friends so that no one friend feels left out or not included?

High Level Problem Parent Council has donated lego kits for the junior division of the school. Each box contains 8 kits. There are 7 boxes. Will this be enough for each child, if there are 90 children in the junior division. How many more boxes will Parent Council need to donate? Will you have any leftover lego kits?

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Junior ­ Grade 4

Curriculum Connection → solve problems involving the multiplication of one­digit whole numbers, using a variety of mental strategies (e.g., 6 x 8 can be thought of as 5 x 8 + 1 x 8); → multiply two­digit whole numbers by one­digit whole numbers, using a variety of tools (e.g., base ten materials or drawings of them, arrays), student­generated algorithms, and standard algorithms;

Nelson Chapter 9 Multiplying Greater Numbers

Low Level Problem You arrange some flowers into groups of equal sizes but there are 2 flowers left over. How many flowers might there have been in total? How many flowers were in each group? Write the division

Middle Level Problem What if the 9 key is not working on your calculator. How else could you solve the the problem 7 x 59 if you HAD to use the calculator? Find 2 different ways. Check that your strategy worked using both a calculator and pencil and paper. (possible answer: 7 x 60 = 420 and then subtract 7) (Source: Marian Small Making Math Meaningful page 239)

High Level Problem Use the digits 0 ­ 9 once each in the blanks to make these equations all true: __3 x 4 = 92 __0 x 9 = 45__ __ __ x 3 = 2 __ __ 5__ x 6 = 30__ Sample Answer: 23 x 4 = 92, 50 x 9 = 450, 78 x 3 = 234, 51 x 6 = 306 (Source: Marian Small Open Questions Junior Page 53)

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Junior ­ Grade 4

Curriculum Connection → divide numbers using concrete materials, drawings, and symbols → interpret division problems using concrete materials, drawings, and symbols → recall division facts to 81 → recognize situations in problem solving that call for division and interpret the answer correctly

Nelson Chapter Chapter 10 Dividing greater numbers

Low Level Problem You arrange some flowers into groups of equal sizes but there are 2 flowers left over. How many flowers might there have been in total? How many flowers were in each group? Write the division sentence that describes it. Think of three more possibilities. (Source: Mairan Small Open Questions Junior Page 47.) a) Find a divisor for 81 that has no remainder. b) Find a divisor for 81 that has a remainder of 3. 4. F (Source: Nelson Text p.287)

Middle Level Problem Place the digits 1, 3, 5, 7, and 9 to create the greatest and least quotients

[ ] [ ] [ ] [ ] ÷ [ ] (Source: p.239 Making Math Meaningful Marian Small)

High Level Problem Carol had 810 apples to pack in bags of 6. a) How many bags could she fill? Show your work. b) How many apples will be left over? Explain how you know. c) If she found another 15 apples, would she be able to pack another 4 bags of 6 apples? Explain.

32

33

Junior ­ Grade 4

Curriculum Connection → demonstrate and explain the relationship between equivalent fractions, using concrete materials (e.g., fraction circles, fraction strips, pattern blocks) and drawings represent, compare, and order decimal → represent fractions using concrete materials, words, and standard fractional notation, and explain the meaning of the denominator as the number of the fractional parts of a whole or a set, and the numerator as the number of fractional parts being considered; → compare and order fractions (i.e., halves, thirds, fourths, fifths, tenths) by considering the size and the number of fractional parts. → Compare fractions to the benchmarks of 0, half, and 1.

Nelson Chapter 12 Fractions and Decimals

Low Level Problem Show that the same amount might be ½ of one thing but ⅓ of another thing. Marcus delivers 56 papers every day. He delivered ½ of the papers before 5pm. How many papers did he deliver before 5pm?

Middle Level Problem I have 63 in the denominator. I am equivalent to . What fraction am I ?7

2 Pens come in pack of 6. Sam game 1 pen to each friend. She gave out 2 and ½ packs. How many friends did she give out?

High Level Problem

Using the picture above, find fractions that describe

a) _______ of all the shapes are shaded b) _______ of all the shapes are triangles c) _______ of the triangles are shaded. d) _______ of the shapes have straight sides. e) 3/10 of the shapes __________________

34

Junior ­ Grade 4

Curriculum Connection → represent, compare, and order decimal → numbers to tenths, using a variety of tools and using standard decimal notation (Sample problem: Draw a partial number line that extends from 4.2 to 6.7, and mark the location of 5.6.);

Nelson Chapter 12 Fractions and Decimals

Low Level Problem Mary ran 3.5 km, Josh ran 5.3 km, Amy ran 2.1 km and Colleen ran 5.5 km. Who ran the farthest? How much farther did Josh run than Amy? What’s the total distance they ran?

Middle Level Problem Mr. T is lifting weights. He wants to lift exactly 5.0 kg. He has the following weights. 1.5 4.2 0.8 1.0 2.2 1.7 0.3 0.6 Which 2 weights can he use? Which 4 can he use?

High Level Problem Carey says that 0.23 of the whole numbers between 1 and 100 have the digit 8 in them. Is she correct?

35

Junior ­ Grade 5

Curriculum Connection → represent, compare, and order whole numbers and decimal numbers from 0.01 to 100 000, using a variety of tools → demonstrate an understanding of place value in whole numbers and decimal numbers from 0.01 to 100 000, using a variety of tools and strategies → read and print in words whole numbers to ten thousand, using meaningful contexts

Nelson Chapter 2 Numeration

Low Level Problem Use three words to write out a four digit number on a check. List four or more numbers that you could be writing. Then, write the number out in words. What do you notice about the numbers you wrote. (Source: Open Questions for three­part lesson, by Marian Small, page 63)

Middle Level Problem Attendance at a Montreal Canadians Hockey game was posted using number cards. The cards fell down. a.) The arena holds 21 273 people. What might have been the attendance? List five numbers made up of the following digits. 1­ 0­ 2­ 6­ 8 b.) Order your numbers from least to greatest. c.) How did you decide which number was greatest? (Source: Nelson Mathematics Textbook, page 35)

High Level Problem You read two numbers, greater than 1000 that are written in words. Everything is exactly the same when you read them except for two words. What could the numbers be? How are their standard forms alike? How are their standard forms alike? (Source: Open Questions for three­part lesson, by Marian Small, page 64)

36

Junior ­ Grade 5

Curriculum Connection → add and subtract decimal numbers to hundredths, including money amounts, using concrete materials, estimation,and algorithms (e.g., use 10 x 10 grids to add 2.45 and 3.25)

Nelson Chapter 4 ­ Addition and Subtraction

Low Level Problem Amy added 2.78 and 5.49. She also added 278 and 549. She compared his answers. Explain how the answers are the same and how are they different? (Source: Nelson Mathematics workbook page 36)

Middle Level Problem Mark bought a set of novels for $29.95 and a dictionary for $30.50. He paid with five bills and two coins. He received $1.50 amount of change. What bills and coins did he use to pay for the books? (Source: Nelson Mathematics Textbook page 121)

High Level Problem You bought something and paid the clerk $50. You got two bills and four coins in change. How much money might you have spend? Think of two or more possibilities. Explain your answer. (Source: Open Questions for three­part lesson, by Marian Small, page 94)

37

Junior ­ Grade 5

Curriculum Connection → multiply two­digit whole numbers by two­digit whole numbers, using estimation, student­generated algorithms,and standard algorithms; → divide three­digit whole numbers by one­digit whole numbers, using concrete materials, estimation, student­generated algorithms,and standard algorithms;

Nelson Chapter 6 ­ Multiplication and Division

Low Level Problem Multiplication → There are 48 seats in each row in the auditorium. There are 72 rows. About how many people can sit in the auditorium? Division → There are 5 skiers in each class at the Ski school. How many classes can be made with 90 skiers?

Middle Level Problem Multiplication→ Choose three pairs of two­digit numbers. Create problems that would involve multiplying these numbers. Then, solve the problems. (Source: Open Questions for three­part lesson, by Marian Small, page 99) Multi­step → Pierre uses 3 eggs to make a cake. There are 12 eggs in a carton. Pierre has 7 cartons of eggs. How many cakes can Pierre make?

High Level Problem Multiplication→ Three Children, Frank, Katie and Sherrie are entered in the charity Bike­a­Thon. Their pledges total $2.00 per mile for each of them. Katie, the oldest, rode twice as far as Sherrie. Frank, the youngest child, rode 8 miles. Sherrie rode 5 miles more than Frank. How much money did each child earn for the charity? (Source: Problem Solving in Mathematics, Grade 3­6, by Alfred S. Posamentier, Page 94) Multi­step→ The four children in the Barnes family decided to have a surprise party for their dad. Lucy paid $12 for paper plates and cups. Mark spent $10 for the decorations. Neville spent $8 on apple juice. Olivia bought the cake for $10. To share the costs equally, how much money should each child give to the others? (Source: Problem Solving in Mathematics gr 3­6. By Alfred S. Posamentier pg 70)

38

Junior ­ Grade 5

Curriculum Connection → multiply decimal numbers by 10, 100, 1000,and 10 000,and divide decimal numbers by 10 and 100, using mental strategies → multiply and divide decimal numbers by whole numbers

Nelson Chapter 9 & 10 ­ Multiplication and Division of Decimals

Low Level Problem Multiplying → The school bus travels 37.6 km each school day. How far does it travel in 21 school days. Dividing → Frank is going to make 12 loaves of bread. He has 4.08 kg of flour. How much flour is there for each loaf?

Middle Level Problem Multiplication → A lumber company produced 2000 wooden boards. Unfortunately, each board was 2.2 mm too long and that amount had to be cut off from each board! In all, how many metres of wood were cut off? Division → Katherine spends $1,089.72 each month for rent and supplies to run her hair salon. If she charges $18 for a haircut, how many haircuts must Katherine do to cover her monthly expenses? Round to the nearest whole number. (Source: Math Connects, Course 1, Page 30)

High Level Problem Multi­Step→ The local health club is advertising a special for new members: no initiation fee to join and only $34.50 every 3 months. They also have a 2 year membership for $290. If Andy joins the health club what is the better deal?

39

Junior ­ Grade 5

Curriculum Connection → represent, compare, and order fractional amounts with like denominators, including proper and improper fractions and mixed numbers → demonstrate and explain the concept of equivalent fractions,

Nelson Chapter 12 ­ Fractions

Low Level Problem There are 30 Grade 5 students at Central School. If ⅔ ride a bus, how many pupils ride the bus?

Middle Level Problem In a recent survey, ⅖ of the people surveyed said their favorite food was pizza, ¼ said it was hot dogs, and said it was popcorn. Which food was favored by the3

10 greatest number of people? Explain. (Source: Math Connects Course 1, Pg. 36)

High Level Problem On a 12 slice pizza, ¼ has mushrooms, ¾ has extra cheese, and ⅙ has pepperoni and ⅕ has eggplant. What is the least number of slices that have more than one topping? Show your work. (Source: Nelson Textbook Grade 5, page 354)

40

Junior ­ Grade 6

Curriculum Connection → represent, compare, and order whole numbers and decimal numbers from 0.001 to 1 000 000, using a variety of tools → demonstrate an understanding of place value in whole numbers and decimal numbers from 0.001 to 1 000 000, using a variety of tools and strategies → read and print in words whole numbers to one hundred thousand, using meaningful contexts

Nelson Chapter 2 ­ Numeration

Low Level Problem The population of the Toronto area increased from about 4.3 million in 1996 to 4.7 million in 2001. About how many additional people were there each day in the Toronto area? Explain your thinking.

Middle Level Problem I am a number between 7 000 000 and 8 000 000. All my digits are odd. All the digits in my thousands period are the same. The sum of my digits is 31. What number am I? Give as many answers as you can. What strategies did you use to find the mystery number? (Source: Math Makes Sense Textbook page 38)

High Level Problem Think of at least three numbers that make each of these statements occur. Explain why they occur. a.) You end up with the same number when you round to the nearest hundred thousand or the nearest ten thousand. b.) You end up with a smaller number when you round to the nearest hundred thousand than when you round to the nearest thousand. c.) You end up with a greater number when you round to the nearest hundred thousand than when you round to the nearest thousand. d.) You end up with a greater number when you round to the nearest hundred thousand when you round to the nearest ten thousand. (Source: Open Questions for three­part lesson, by Marian Small, page 118)

41

Junior ­ Grade 6

Curriculum Connection → use a variety of mental strategies to solve addition & subtraction problems involving whole numbers and decimals → use estimation when solving problems involving the addition and subtraction of whole numbers and decimals, to help judge the reasonableness of a solution

Nelson Chapter 4 ­ Addition and Subtraction

Low Level Problem You add two decimal numbers and show the sum using 1 thousand block, 5 hundred blocks, 9 ten blocks, and 7 one blocks. What two decimals might you be adding? (Source: Open Questions for three­part lesson, by Marian Small, page 148)

Middle Level Problem Anna sells freezies to sell money for her school. From Monday to Friday, Anna sold $873 worth of freezies. On Monday she sold $117.36, on Tuesday she sold $131.24, on Wednesday she sole $143.02, and on Friday she sold $156.12. Calculate how much was sold on Thursday. Show your work.

High Level Problem Mark is painting the walls in two rooms in a Boys and Girls club. He has two 0.90L cans of paint and one 3.79L can of paint. One litre of paint can cover between 23 and 37. Two walls in each room measure 6m by 2.44m. The other two walls in each room measure 4.00 m by 2.44m. Will he have enough to paint all the walls with two coats of paint. (Source: Nelson Mathematics Textbook page 129)

42

Junior ­ Grade 6

Curriculum Connection → solve problems involving the multiplication and division of whole numbers (four­digit by two­digit), using a variety of tools (e.g., concrete materials, drawings, calculators) and strategies (e.g., estimation,algorithms);

Nelson Chapter 6 Multiplication and Division

Low Level Problem Multiplication→ Michael needs 375g of chocolate chips for one batch of cookies. He has 2­kg bags of chocolate chips. Does Michael have enough chocolate chips to make 12 batches of cookies? Explain. (Source: Math Makes Sense, Pg. 66) Division → Estimate to solve each problem. Explain your reasoning. The total attendance at a powwow in July was 5208 people in 7 days. The total attendance at a powwow in August was 2934 in 3 days. (Source: Nelson Grade 5 Textbook Page 173)

Middle Level Problem Multi Step→ The food bank received 30 cases of 24 cans of soup, and 20 cases of 48 cans of soup. How many packages of 14 cans of soup can be made? (Math Makes Sense Grade 6, Page 71)

High Level Problem Multiplication→ The Fairview Secondary School community of 1854 students and 58 teachers attend a special performance of a play at the local theatre. The theatre has 49 rows of 48 seats each.

a) Were any seats empty? How do you know? b) If your answer to part a) is yes, find the number of empty seats.

(Math Makes Sense Grade 6, Page 67) Division→ Sandra checked and found that the 5 people in her family produced 35kg of garbage in one week. What is the average amount of garbage produced by each member of Sandra’s family in one week? Is this above or below the national average of 5 kg per person during one week? (Source: Intermediate Mathematics 1, Page 1)

43

Junior ­ Grade 6

Curriculum Connection → multiply and divide decimal numbers by 10, 100, 1000,and 10 000 using mental strategies → multiply and divide decimal numbers by whole numbers

Nelson Chapter 9 and 10 ­ Multiplying and Dividing Decimals

Low Level Problem Multiplication→ Gasoline costs $0.58/L. If 50 L fills the tank, how much will it cost to fill the tanks? Division→ A box of cereal has a mass of 0.35kg. Mrs. Graham used it to make 25 cupcakes. How much cereal was in each cake? (Intermediate Mathematics 1 page 78)

Middle Level Problem Multiplication­ Michael needs 4m of fabric to upholster a sofa. He sees two fabrics he likes. One costs $8.59 per metre. The other costs $5.98 per metre. How much will Michael save if he buys the less expensive fabric? (Source: Math Makes Sense, 6 page 150) Division­ Sharma paid $58.50 to board a cat at a kennel for 5 days. Her friend Miles paid $12.50 each day to board his cat at a different kennel. Who got the better deal? Explain how you know? (Source: Math Makes Sense 6, page 154)

High Level Problem Multiplication → Amanda works on a farm out in the hills. It takes her 2.25 hours to drive to town and back. She usually goes to town twice a week to get supplies. How much time does Amanda spend driving if she takes 8 trips to town each month? (Source: Math Connects, Course 1, Page 28) Division → Isabella has found that she stays the most fit by running various distances and terrains throughout the week. On Mondays she runs 2.5 miles, on Tuesdays 4.6 miles, on Thursdays 6.75 miles, and on Saturdays 4.8 miles. What is the average distance Isabella runs on each of the days that she runs? Round to the nearest hundredth of a mile. (Source: Math Connects, Course 1, Page 30)

44

Junior ­ Grade 6

Curriculum Connection → represent, compare, and order fractional amounts with unlike denominators, including proper and improper fractions and mixed numbers, using a variety of tools

Nelson Chapter 12 Fractions, Decimals and Ratios

Low Level Problem Fractions→Toya is looking in her closet. If of her shoes are black and are brown,3

164

does she have more black shoes or more brown shoes? Explain. Ratios→ A reindeer can run 96 miles in 3 hours. At this rate, how far can a reindeer run in 1 hour? Explain. (Source Math Connections Course 1, Pg. 48)

Middle Level Problem Fractions→ After a basketball game, the teams had a pizza party. The Jets ate 3

21

pizzas. The Barracudas ate Which team ate more pizza.1237

(Source: Math Makes Sense 6, page 297) Ratios→ Jenny wants to buy cereal that comes in large and small boxes. The 32­ounce box costs $4.16, and the 14­ounce box costs $2.38. Which box is less expensive per ounce? Explain. (Source Math Connections Course 1, Pg. 48)

High Level Problem Ratio → Katherine has diabetes. At each meal, she must estimate the mass in grams of carbohydrates she plans to eat and inject the appropriate amount of insulin. Katherine needs 1 unit for 15g of carbohydrates. Katherine’s lunch has 60g of carbohydrates. How many units of insulin should Katherine inject? (Source: Math Makes Sense Grade 6, page 326) Fractions → Rob jogged 6 laps of the track. He then walked laps before5

21013

jogging another laps.1011

How many laps did he jog? How many laps did he do in all?

45

Intermediate ­ Grade 7

Curriculum Connection → represent perfect squares and square roots, using a variety of tools → generate multiples and factors, using a variety of tools and strategies → use estimation when solving problems involving operations with whole numbers, decimals, and percents, to help judge the reasonableness of a solution

Nelson Chapter 1 Factors and Exponents

Low Level Problem

Suppose that the Grade 7 students expect to sell between 100 and 150 hotdogs. a) How many pack ages of 12 hot dogs and 8 buns should the students buy if

they don’t want any leftovers? b) If the buns cost $2 a pack and the hot dogs cost $6 a pack how much did they

spend?

Middle Level Problem Kristina had three jobs; she worked at the supermarket and earned $7.50/h; she worked at the recreation centre and earned $8.25/h; she also worked at the movie theatre and earned $9.00/h. One month she worked 20 h at the supermarket, 33 h at the recreation centre, and 15 h at the movie theatre. She also always donated $10 per month to a charity. Create an order of operations expression to calculate her earnings for the month and then solve the expression.

High Level Problem Real world problem involves four numbers. Out of the four numbers, there is at least one composite and one prime number. Solving the problem involves performing two or more operations. A reasonable estimate for the answer is 15. What might the problem be? Think of three or more possibilities. (Source: Open Questions for three­part lesson, by Marian Small, page 179)

46

Intermediate ­ Grade 7

Curriculum Connection → demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units (e.g., speed is a rate that compares distance to time and that can be expressed as kilometres per hour); → solve problems involving the calculation of unit rates (Sample problem:You go shopping and notice that 25 kg of Ryan’s Famous Potatoes cost $12.95, and 10 kg of Gillian’s Potatoes cost $5.78. Which is the better deal? Justify your answer.

Nelson Chapter 2 Ratio, Rate and Percent

Low Level Problem Kylie said that she bought a slice of pizza for $1.50. Liam bought 2 slices for $3.25. Who got the better deal? Are you sure? (Source: Open Questions for three­part lesson, by Marian Small, page 188)

Middle Level Problem Last week, Laura earned $58. She spent $15 on lunch, $21 on a new shirt and $6 on a bus and saved the rest. Michael earned $83 and saved $20. Who saved the greater percent of his earnings.

High Level Problem Jayla and Rhea paid the same amount of money for their sweater at a 40% off sale, and Rhea got hers at a 20% off sale. How were the original prices of the sweaters related? Explain. (Source: Open Questions for three­part lesson, by Marian Small, page 192)

47

Intermediate ­ Grade 7

Curriculum Connection → add and subtract integers, using a variety of tools (e.g., two­colour counters, virtual manipulatives, number lines).

Nelson Chapter 6 ­ Addition and Subtraction of Integers

Low Level Problem The price of a share of stock started the day at $37. During the day it went down $3, up $1, down $7, and up $4.What was the price of a share at the end of the day?

Middle Level Problem Sam checks his investments everyday. One Friday, he missed the news and did not know if there was a gain or a loss. His mother told him that the overall value had dropped $230 during the week. What was his gain or loss on the Friday? Day Monday Tuesday Wednesday Thursday Friday

Gains/Loss ($)

+ 300 ­93 ­125 +51

High Level Problem An elevator can travel several levels below ground. Its first stop is four levels down from the ground floor. It the travels nine levels above the first stop. For the final stop the elevator then travels three floors down from the second stop. Tanya gets on with her son and he presses buttons and they then travel down 5 levels, then down another 3 to finally go back up 7 levels to the final stop. If there are no other changes, where is the elevators location after the final stop?

48

Intermediate ­ Grade 7

Curriculum Connection → add and subtract fractions with simple like and unlike denominators, using a variety of tools (e.g., fraction circles, Cuisenaire rods, drawings, calculators) and algorithms; → add and subtract integers, using a variety of tools (e.g., two­colour counters, virtual manipulatives, number lines).

Nelson Chapter 9­ Fraction Operations

Low Level Problem Addition → Liam spent 2 ⅖ hours on his math homework and 1 ⅗ hours on his science homework. How much time did he spend doing math and science homework? (Source: Math Connects Course 1, Pg.42) Subtraction → Mrs. Smith has 2 ½ dozen eggs on Friday. She used 1 ¼ dozen on the weekend. What fraction does she have left? (Source: Intermediate Mathematics 1, page 141)

Middle Level Problem Addition → To save energy, Lou Ann decided to put extra insulation in the attic of her house. Over the living room she needed 9 packages, over the bathroom 28

743

packages, and above each of the three bedrooms 6 packages were used. How83

many packages of insulation were used in total? (Intermediate Mathematics 1, page 139) Subtraction → Gina wants to make muffins. The recipe for blueberry muffins calls for 2 ¾ cups of flour. The recipe for cornmeal muffins calls for 1 ⅓ cups of flour. How many more cups of flour would Gina need for blueberry muffins than corn muffins? (Source: Math Connects Course 1, Pg.42)

High Level Problem Addition → To save on energy, Alison decided to put extra insulation in the attic of her house. Over the living room she needed 9 ⅞ packages, over the bathroom 2 ¾ packages, and above each of the three bedrooms 6 ⅜ packages were used. How many packages of insulation did she use in total? If insulation sells for $6.00 per package, how much did it cost to insulate the house?

49

Subtraction → A stock which is opened at $21 ⅛ fell $2 ¾ during the trading for one day. What was its closing price? What is the price of the stock if it falls another $1 ⅞? How much has the stock fallen from the original price? (Source: Intermediate Mathematics 2, page 129)

50

Intermediate ­ Grade 8

Curriculum Connection → determine common factors and common multiples using the prime factorization of numbers → solve multi­step problems arising from real­life contexts and involving whole numbers and decimals, using a variety of tools (e.g., graphs, calculators) and strategies (e.g., estimation, algorithms)

Nelson Chapter 1 Number Relationships

Low Level Problem Choose a manipulative that you could use to show a common multiple and a common factor of a two­digit number and one­digit number of your choice. Using the same numbers, repeat with another manipulative. Then, repeat the whole thing with other sets of numbers. (Source: Open Questions for three­part lesson, by Marian Small, page 207)

Middle Level Problem The volume of a rectangular prism with whole­number dimensions has a volume that is a prime number. What are the dimensions? Use an example to support your explanation. (Source: Nelson Mathematics Textbook page 41)

High Level Problem Suppose that you and a partner are playing a game with two dice. You roll the dice and add the numbers. You get 1 point if the sum is a prime. Your partner gets 1 point if the sum is a composite number. Who is more likely to win? Explain your reasoning. (Source: Nelson Mathematics Textbook page 7)

51

Intermediate ­ Grade 8

Curriculum Connection → solve problems involving proportions, using concrete materials, drawings, and variables (Sample problem:The ratio of stone to sand in Hard Fast Concrete is 2 to 3. How much stone is needed if 15 bags of sand are used?); → solve problems involving percent that arise from real­life contexts (e.g., discount, sales tax, simple interest) (Sample problem: In Ontario, people often pay a provincial sales tax [PST] of 8% and a federal sales tax [GST] of 7% when they make a purchase. Does it matter which tax is calculated first? Explain your reasoning.); → solve problems involving rates (Sample problem: A pack of 24 CDs costs $7.99. A pack of 50 CDs costs $10.45. What is the most economical way to purchase 130 CDs?).

Nelson Chapter 2 Proportional Relationships

Low Level Problem Use grocery store flyers to find a product that is available in two different sizes. Decide which size is the better buy. Explain why you think that. (Source: Open Questions for three­part lesson, by Marian Small, page 229)

Middle Level Problem

Mary works at a clothing store. She works 25.5 h/week and makes $7.50/h. Plus, she makes 2% on what she sells. What did she make in one week if she

sold $3,456.60 worth of clothes? Round your answer to the nearest cent. Show all your work.

High Level Problem The world’s biggest cookie was about 34m wide.

a) About how many Oreo, Fudge­OO and Chocolate chip cookies would this be if one Oreo is 3cm, one Fudge­OO is 3.2cm and one Chocolate Chip is 4cm?

b) About how much sugar do you think might have been used to make it if, a batch of 12 Oreo’s uses 1 cup, one Fudge­OO uses ¼ cup and a batch of 48 Chocolate Chip cookies uses 1 ½ cups of sugar.

52

Intermediate ­ Grade 8

Curriculum Connection → represent the multiplication and division of integers, using a variety of tools [e.g., if black counters represent positive amounts and red counters represent negative amounts, you can model 3 x (–2) as three groups of two red counters] → solve problems involving operations with integers, using a variety of tools (e.g., two colour counters, virtual manipulatives, number lines) → solve problems involving operations with integers, using a variety of tools (e.g., two colour counters, virtual manipulatives, number lines)

Nelson Chapter 6: Integer Operations

Low Level Problem You are visiting a friend and their dog gets loose. You chase the dog to try and catch it. You chase it 2 blocks east; it turns and goes 5 blocks west, then 8 blocks east, then another 2 blocks east, 7 blocks west and 1 block east before you finally catch the dog. Use positive numbers to represent east blocks, and negative numbers to represent west blocks. How far are you from your friend’s house when you catch the dog?

Middle Level Problem

One way to keep score in golf is by using integers. Each hole in golf is given a rating called ‘par’. Par for a hole is the number of strokes a good player would take to complete the hole. ‘Par’ is given an integer value of 0. A ‘bogey’ is ‘one over par’ or +1 A ‘double bogey’ is ‘two over par’ or +2 A ‘triple bogey’ is ‘three over par’ or +3 A birdie is ‘one under par’ or ­1 An eagle is ‘two under par’ or ­2 Pete scored as follows on eighteen holes of golf. +2, 0, ­1, +2 , ­2, 0, 0, 0, 0, +3, 0, ­1, ­1 , ­1, ­1, 0, +2, 0 Was Pete’s final score above or below par? If par for the course was 72 strokes, how many strokes did Pete take? (Source: Intermediate Mathematics 2, Page 429)

53

High Level Problem Luke and Seth started out to visit Uncle Arnie. After driving 50 km, they saw a restaurant, and Luke wanted to stop for lunch. Seth wanted to look for something better, so they drove on for 8 km before giving up and going back to the restaurant. After eating they traveled on for 26 more kilometres from the restaurant. Seth saw a sign for a classic car museum, which they decided to visit. The museum was 6 km from their route. After returning to the main road, they drove for another 40 km and arrived at Uncle Arnie’s house. How many kilometres is it from Luke and Seth’s house to Uncle Arnie’s house? How many kilometres did they drive on the way there?

54

Intermediate ­ Grade 8

Curriculum Connection → represent the multiplication and division of fractions, using a variety of tools and strategies → solve problems involving addition, subtraction, multiplication, and division with simple fractions;

Nelson Chapter 9: Fractions Operations

Low Level Problem Multiplication → A department store had a “⅓ off” spring sale. The following list gives the regular price of several items. 10­speed ……. $156.60 Skis……..$213.87 Portable TV….. $192.90 Skates……….. $94.20 Football… $41.55 Determine the sale price of each of the above items. Division → Pete keeps 4 pet sharks in his swimming pool. Each day he throws in 2 ¾ kg of food. What is each shark’s equal share of the food? (Source: Intermediate Mathematics 1, page 147)

Middle Level Problem A screw nail 5 cm long is screwed into 3 boards, each 1 cm thick. How far will the5

2 910

screw nail go into the third board? (Source: Intermediate Mathematics 2, page 149)

High Level Problem In an auditorium, ⅙ of the students are fifth graders, ⅓ are fourth graders, and ¼ of the remaining students are second graders. If there are 96 students in the auditorium, how many second graders are there?

55

Problem Solving in Mathematics Grades 3­6: Powerful Strategies to Deepen Understanding by Alfred S. Posamentier; Stephen Krulik

ISBN­13: 978­1412960670 Organizing Data Problem 2.5 (Grades 3–5) You have a lot of dimes, nickels, and pennies in your pocket.You reach in and pull out three of the coins without looking.What are the different amounts of money you could have taken from your pocket? Solution: Let’s make an organized list. Be sure to include all possible combinations of coins. One solution is to start with the maximum number of dimes, and then reduce the number of dimes by 1 after all possible combinations for that number of dimes have been listed.

Dimes Nickels Pennies Totals

3 0 0 = 30¢

2 1 0 = 25¢

2 0 1 = 21¢

1 2 0 = 20¢

1 1 1 = 16¢

1 0 2 = 12¢

0 3 0 = 15¢

0 2 1 = 11¢

0 1 2 = 7¢

0 0 3 = 3¢ Check to make sure all the amounts are different. In this problem, the list is the actual answer.

56

Answer: There are 10 different sums you could make with three coins. The answers are 30¢, 25¢, 21¢, 20¢, 16¢, 12¢, 15¢, 11¢, 7¢, and 3¢. Problem 2.6 (Grades 3–5) Harlow, Indira, Jessica, and Karl are taking karate lessons.They will work out in pairs. How many different pairs are possible? Solution: Let’s make a list of the possible pairs. We organize the list by considering all possible pairs beginning with Harlow. Harlow—Indira Harlow—Jessica Harlow—Karl Now we consider all the pairs beginning with Indira. Notice, however, that Indira—Harlow is a repeat of Harlow—Indira, already counted. Indira—Jessica Indira—Karl Finally, consider pairs beginning with Jessica that have not already been counted. Jessica—Karl Answer: There are 6 possible pairs of students. Problem 2.11 (Grades 3–4) There is a monorail running around the amusement park. The monorail car has no passengers when it leaves the terminal. At the first stop, it picks up 5 people. At the second stop, 4 people get on and 2 people get off. At the third stop, 5 people get on and no one gets off. At the fourth stop, 1 person gets on and 4 people get off. How many passengers are now on the monorail car? Solution: Let’s make a chart to simulate the events.

Stops People On People Off Number on Monorail

Terminal 0 0 0

57

#1 5 0 5

#2 4 2 7

#3 5 0 12

#4 1 4 9 We can also organize the data by adding all the total number of people getting on the monorail, and then the total of all the people getting off the monorail. This will yield the answer quite quickly: On Monorail = 5 + 4 + 5 + 1 = 15 Off Monorail = 2 + 4 = 6 Remaining on Monorail = 15 – 6 = 9 A Answer: There are 9 people left on the monorail at the end of the trip.

58

Intelligent Guessing and Testing Problem 3.6 (Grades 3–4) The following is the menu in the school cafeteria:

Apple 25¢ Chocolate Milk 30¢ Grilled Cheese Sandwich 75¢

Granola Bar 45¢ Orange Juice 35¢ Veggie Burger $1.10

Ice Cream 50¢ Milk 25¢ Slice of Pizza 85¢

Liu spent $1.30 and bought exactly 2 items.What did he buy for lunch Solution: We use the guess and test strategy: Let’s guess milk and veggie burger. That’s 25¢ + $1.10 = $1.35. Too large. Let’s guess grilled cheese and ice cream. That’s 75¢ + 30¢ = $1.05. Too small. Let’s guess veggie burger and granola bar. That’s $1.10 + 45¢ = $1.55. Too large. Let’s guess pizza and granola. That’s 85¢ + 45¢ = $1.30. That works! Answer: He bought a slice of pizza and a granola bar Problem 3.7 (Grades 3–5) There are 2 baby pandas at the local zoo.They are named Tristan and Isolde.The people are voting to see which one is their favorite. Exactly 105 people voted. Tristan was favored by 2½ times as many people as Isolde. How many votes did each panda receive? Solution: We’ll use the guess and test strategy. A table will help us keep track of our guesses. Let’s start with 50 votes for Isolde and compute 2½ times as many for Tristan.

Isolde Tristan Total Votes

50 125 175 (Too many votes)

40 100 140 (Still too many)

30 75 105 (Yes!)

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We have the answer. The total is 105 and 75 is exactly 2½ times as many as 30. Answer: Isolde received 30 votes; Tristan received 75 votes. Problem 3.11 (Grades 4–6) In my pocket, I have quarters and nickels. I have four more nickels than quarters. Altogether, I have $1.70 in my pocket. How many nickels and how many quarters do I have? Solution: Using the guess and test strategy:

#Quarters Value #Nickels Value Value

1 $0.25 5 25¢ 50¢ (Too small)

2 $0.50 6 30¢ 80¢ (Still too small)

3 $0.75 7 35¢ $1.10 (Getting there)

4 $1.00 8 40¢ $1.40 (Better)

5 $1.25 9 45¢ $1.70 (Yes!) Answer: I have 9 nickels and 5 quarters..

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Solving a Simpler Equivalent Problem Problem 4.6 (Grades 3–5) Maria has been asked to find three consecutive even numbers whose sum is 60. What are the numbers? Solution: We use the simpler problem strategy. We begin with the smallest set of three even numbers, then try the next set, and so on, to see if there is a pattern we can make use of.

2 + 4 + 6 = 12 Set 1 begins with 2 (which is 1 × 2)

4 + 6 + 8 = 18 Set 2 begins with 4 (which is 2 × 2)

6 + 8 + 10 = 24 Set 3 begins with 6 (which is 3 × 2)

8 + 10 + 12 = 30 Set 4 begins with 8 (which is 4 × 2)

Aha! The sum is going up by 6. The sums will be 12, 18, 24, 30, 36, 42, 48, 54, and 60. We want the 9th sum, 60. The three consecutive even numbers that give that sum will begin with 9 × 2, or 18. We will try 18 + 20 + 22 = 60. Answer: The three consecutive even numbers whose sum is 60 are 18, 20, and 22. Teaching Notes: This problem provides practice for those students who need work in addition. They can continue adding the sequences of three consecutive numbers until they actually find all the sums and reach the required 60. We can also show students that the sum of three consecutive even numbers is always three times the middle number. Therefore, 60/3 = 20, which is then the middle number, gives us the three numbers 18 + 20 + 22 = 60. Problem 4.7 (Grades 3–5) Maurice put 7 cubes in a row, tight against one another.Then he spray­painted them with red paint.When they had dried, he took them apart. He noticed that faces of the cube that were touching the table did not get painted, nor did the faces of cubes that

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touched one another. How many faces of the 7 cubes were painted? Solution: We can reduce the number of cubes to a smaller number and see what happens. We can look for a pattern to develop.

Number of Cubes Faces Painted

1 5

2 8

3 11

As we add one cube, the number of painted faces increases by 3. Let’s complete the

table.

Number of Cubes Faces Painted

1 5

2 8

3 11

4 14

5 17

6 20

7 23

Answer: There will be 23 painted faces for 7 cubes. Teaching Notes: Some of your students may notice that the number of faces painted can be found by multiplying the number of cubes by 3 and adding 2. This can be expressed in a number sentence format as F = 3n + 2, where F is the number of painted faces and n is the number of cubes. You may wish to point this out to the class.

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Problem 4.10 (Grades 3–5) Barbara wants to buy some boxes in which to store her toy soldier collection. She buys 5 large boxes. Inside each large box are two medium boxes. Inside each medium box are two small boxes. How many boxes in all did Barbara buy? Solution: We can solve a simpler equivalent problem, by using one large box and then multiplying our answer by 5 (because we actually have to account for 5 large boxes). Let’s find the number of boxes combined if we had only one large box. We can then multiply our answer by 5. One large box + 2 medium boxes + 4 small boxes = 7 boxes. Finally, 7 × 5 = 35. If we combine the above solution method together with the making a drawing strategy, we can easily show one box with two medium boxes inside and two small boxes inside each medium box (see Figure 4.4). We can then count the total number of boxes, 7, and multiply by 5. Insert Here Answer: Barbara bought 35 boxes in all. Teaching Notes: When students attempt this problem, a common error is to forget the original large boxes. You may wish to emphasize counting these boxes.

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Acting it Out or Simulations Problem 5.1 (Grades 3–5) Distribute Cuisenaire rods for this problem.The rods are the cars of a train. Each rod (car) holds the same number of people as its length. 1 = White 2 = Red 3 = Green 4 = Violet 5 = Yellow Thus: You have one rod of each color and length. Show how you can build a train of lengths from 1 through 15. Solution: Many of these trains can be done in more than one way. (Only some are shown.)

1 = W 6 = W + Y or R + V 11 = Y + V + R

2 = R 7 = V + G or Y + R 12 = Y + V + G

3 = G 8 = Y + G or V + G + W 13 = Y + V + G + W

4 = V 9 = Y + V 14 = Y + V + G + R

5 = Y 10 = Y + V + W 15 = Y + V + G + R + W

Students should be encouraged to find alternate ways to represent the various lengths. Those students who find this easy might be asked about how many ways each of these lengths can be represented. Answer: As shown above. Problem 5.3 (Grades 3–5) Mr. Perlman is giving the 19 goldfish from the science lab to three students to care for during the summer. Each student gets an odd number of goldfish. Jack gets the most, Sam gets the smallest number, and Max gets the rest. How many goldfish did each of them take home to care for?

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Solution: An elegant method is to give the children 19 slips of paper or 19 chips to represent the goldfish. Have them act out the problem. The only sets of three different odd numbers that add up to 19 are 3, 7, 9, 3, 5, 11, 1, 5, 13, and 1, 7, 11. Answer: Jack got 9 goldfish to take care of, Max got 7, and Sam got 3; Jack got 11, Max got 5, and Sam got 3; Jack got 13, Max got 5, and Sam got 1; or Jack got 11, Max got 7, and Sam got 1. Problem 5.4 (Grades 3–6) Sharon and Janet counted the number of baseball cards they have together. They have a total of 16 baseball cards. Which of the following statements cannot be true and why? (a) Janet has 13 cards. (b) Sharon has 12 cards. (c) Janet has 1 more card than Sharon. (d) Sharon has 2 more cards than Janet. (e) Sharon has an odd number of cards. (f ) Janet has an even number of cards. Solution: Give the children 16 chips or slips of paper and have them try each of the given statements to see if it can be true. They can try to “act out” each of the possible answers and then come to the conclusion that statement (c) is impossible. Teaching Notes: You might go a little deeper into the mathematics taken up in this activity by noting that if Janet had 1 more than Sharon, then one of them would have an odd number of cards and the other would have an even number of cards. The sum of an odd and an even number is always an odd number. Therefore (c) is impossible, because 16 is not an odd number. Answer: Statement (c) cannot be true. Problem 5.7 (Grades 3–5) There are 18 blocks in three unequal stacks.The first stack has 3 more blocks than the third stack.The third stack has 6 fewer blocks than the second stack. How many blocks are in each stack?

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Solution: Students should take 18 blocks (or any other entities, such as paper squares) and act out the problem. Answer: The first stack has 6 blocks, the second has 9 blocks, and the third has 3 blocks.

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Working Backwards Problem 6.1 (Grades 4–6) Ilana, Jennie, Karl, and Luis are members of the school stamp club. Last week, they traded stamps.When the meeting was over and they had finished trading, Ilana had 28 stamps. She had given 10 to Jennie, and she received 12 from Karl and 7 from Luis. How many stamps did Ilana start with? Solution: Because we know how many stamps Ilana finished with (the end condition) and want to know what she started with (the beginning condition), we can use the working backwards strategy

Ilana finished with 28. She had 28.

She got 7 from Luis. She must have previously had 21.

She got 12 from Karl. She must have previously had 9.

She gave Jennie 10. She must have started with 19.

Answer: She started with 19 stamps. Teaching Notes: To check the answer, begin at the beginning with 19 stamps and follow the action from start to finish. You should end up with Ilana having 28 stamps. Problem 6.3 (Grades 4–5) Ron has twice the number of baseball cards as Stan. Stan has 9 more cards thanTara. Tara has 17 cards. How many baseball cards do the three friends have together? Solution: We know the end of the problem: Tara has 17 cards. We can begin there and work backwards as follows. Tara has 17 cards. Stan has 9 more than she does, so he has 17 + 9 = 26 cards. Ron has two times the number of cards that Stan has, so he has 52 cards. Together, the three of them have 17 + 26 + 52 = 95 cards.

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Answer: They have a total of 95 cards. Problem 6.10 (Grades 3–4) Sam is saving to buy an electronic remote for his Speedo cars. He checked the price on January 1. He checked again in April, and it was now $6.00 higher than in January. He finally bought it in July, and it was twice as high as it had been in April; it was $52.00. How much would he have saved if he had bought it in January? Solution: Let’s work backwards.

July: $52.00 $52.00

April: half as much as July $26.00

January: $6 less than April $20.00 He would have saved $52.00 − $20.00 = $32.00. Answer: He would have saved $32.00 if he had bought it in January Problem 6.13 (Grades 3–5) The four children in the Barnes family decided to have a surprise party for their dad. Lucy paid $12 for paper plates and cups. Mark spent $10 for the decorations. Neville spent $8 on apple juice. Olivia bought the cake for $10.To share the costs equally, how much money should each child give to the others? Solution: We can work backwards by first determining what each person had to pay by finding the total amount required and then dividing by the number of participants. The total spent on the party was $12 + $10 + $10 + $8 = $40. Because there are 4 children, $40 ÷ 4 = $10. Each child should pay $10. Now we can examine what each child spent

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and decide who owes money to whom. Mark and Olivia have each spent exactly $10. Neville should give Lucy $10 – $8 = $2. Answer: Neville should give Lucy $2.00. Teaching Notes: This might be an excellent time to discuss with the students how to make a problem simpler by breaking it up into its parts and solving each part separately

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Finding a Pattern Problem 7.5 (Grades 3–6) What are the next two numbers in the sequence 2, 3, 5, 7, 11, 13, 17, ...? Solution: If we look at these numbers we should be able to recognize a pattern of a different sort; namely, these are consecutive prime numbers— that is, numbers that can be divided only by themselves and 1. Therefore, the next two prime numbers are 19 and 23. Answer: 19 and 23. Problem 7.6 (Grades 3–5) A standard traffic light turns green then yellow then red,then green and so on.What color is the 13th light? Solution: The sequence of lights is green, yellow, red, green, yellow.... Let’s make a table and follow this pattern.

Light # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Color g y r g y r g y r g y r g y r The 13th light would be green. Notice that the sequence of green lights is 1, 4, 7, 10, 13,... Answer: The 13th light is green.

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Logical Reasoning Problem 8.1 (Grades 3–4) A man has a drawer with only black and blue socks in it. He wants to be certain he has a matching pair of socks, but it is dark in the room and he can’t tell the socks apart. How many socks must he pull from the drawer to be sure he has a matching pair? Solution: A logical approach would be to assume the worst­case scenario, namely, that he pulls out one black sock and one blue sock on his first two pulls. The third sock must guarantee that he has a matching pair. Answer: He must draw 3 socks from the drawer. Problem 8.2 (Grades 3–4) Ron, Stan, Tom, and Ursula were entered in a ping­pong tournament. Ron lost to Ursula in the first­round game.Tom won one game and lost one game. Tom played Ursula in the second round.Who won the tournament? Solution: Here we can only use logical reasoning. We begin with Tom’s record. Tom won one game and lost one game. He must have won his first­ round game in order to get into the second round. Ron lost to Ursula in the first round. This enabled Ursula to be in the second round. But Tom must have lost to Ursula in the second round, because he won only one game. Thus, Ursula was the tournament winner. Answer: Ursula won the tournament Problem 8.4 (Grades 4–6) Mrs. Adams planted three kinds of berry plants in her garden. Of these, 1– 2 are blueberries, 1– 4 are strawberries, and the rest are raspberries. She planted 10 raspberry plants. How many berry plants did she plant altogether? Solution: By logically reasoning what we are given and what we are asked to find, we realize that the raspberries must be the remaining quarter of the circle, since . Thus, 1 –

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4 = 10 raspberry plants. Therefore, the number of strawberry plants, which also represents 1 – 4 of the plants, must also be 10. Then 1 – 2 of the plants are blueberry, which must be 20 in number. Answer: She planted 10 + 10 + 20 = 40 plants altogether. Problem 8.8 (Grades 4–6) Three children, Frank, Katie, and Sherrie, are entered in the charity Bike­a­Thon. Their pledges total $2.00 per mile for each of them. Katie, the oldest, rode twice as far as Sherrie. Frank, the youngest child, rode 8 miles. Sherrie rode 5 miles more than Frank. How much money did each child earn for the charity? Solution: We can use each of the clues in the problem to find out how many miles each child rode. Then we can find the amount each one earned for the charity. Notice that the fact that Katie is the oldest, Frank is the youngest, and Sherrie is the middle child in age has nothing to do with the problem. The problem tells us that Frank rode 8 miles. Because Sherrie rode 5 miles more than Frank, she rode 8 + 5 = 13 miles. Katie rode twice as far as Sherrie, so she rode 13 × 2 = 26 miles. We can now find how much money each child earned: Frank: 8 × $2.00 = $16.00 Katie: 26 × $2.00 = $52.00 Sherrie: 13 × $2.00 = $26.00 Answer: Frank earned $16.00, Katie earned $52.00, and Sherrie earned $26.00. Problem 8.12 (Grades 4–6) Mrs. Ross has three children: Louis, Myra, and Howard.The sum of their ages is 34. Louis is the oldest. Myra is not the youngest. Howard is 8 years younger than the oldest who is 16 years old. How old are the three children? Solution: We use logical reasoning. The problem tells us that Louis is the oldest and that he is 16. Myra is not the youngest, which means Howard is the youngest. This makes

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Howard 8 years old (8 years younger than Louis). The middle child is Myra. Because the sum of the ages of the three children is 34, we add 16 + 8 = 24. This leaves 10 years for Myra. Answer: Louis is 16, Myra is 10, and Howard is 8. Problem 8.16 (Grades 3–5) The coach of the high school tennis team has to select two men and two women to join the team. Six people had tried out. But the demands of six students made it difficult for him to choose. (1) Mitch said,“I’ll play only if Sarah plays also.” (2) Sarah said,“I won’t play if Ron is on the team!” (3) Ron said,“I won’t play if Dan or Emily is on the team.” (4) Dan said,“I’ll play only if Amanda plays.” (5) Amanda really didn’t care. Who should he choose? Solution: Let’s see if logical reasoning makes solving this problem easier. The first statement says that Mitch and Sarah must be together. The fourth clue says that Dan and Amanda must be together. If Ron is chosen, clue three tells us that Dan and Emily are eliminated and clue two says Sarah would be eliminated if Ron were picked. So, this means that Ron cannot play, because if he did, three people would not play, leaving only three members of the team. The coach should select Mitch, Sarah, Dan, and Amanda. Answer: The coach should choose Mitch, Sarah, Dan, and Amanda.

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Making a Drawing Problem 9.1 (Grades 3–5) There are 4 rides at the school carnival: the Carousel, the Fish Ride, the Gliding Swans, and the Giant Wheel.The Gliding Swans is on the far left.The Fish Ride is between the Giant Wheel and the Carousel.The Giant Wheel is next to only one other ride. In what order are the rides from left to right? Solution: Make a horizontal drawing in the form of an “arrangement line.” Use the clues given to place the rides along this line. Begin by placing the Gliding Swans on the far left of the line. Because the Giant Wheel is next to only one other ride, it must be at the far right. The Fish Ride is between the Carousel and the Giant Wheel. The order is:

Gliding Swans Carousel Fish Ride Giant Wheel Answer: From left to right, the rides are the Gliding Swans, Carousel, Fish Ride, and Giant Wheel. Problem 9.3 (Grades 3–5) One day, five students rushed to get into the line for lunch. John was ahead of Karen and behind Leila. Leila was ahead of Sharon and behind Neville. Sharon was ahead of John.Who was last in line? Who was first? Solution: Make a diagram of the situation being described, so that you can visualize the students’ arrangement. Put the students’ names into the proper positions, according to the clues given in the problem. The problem will then solve itself as you make the proper placements. Here you can see that the diagram was the only useful strategy. FRONT OF LINE Neville Leila

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Sharon John Karen Answer: Karen was last in line and Neville was first. Problem 9.6 (Grades 3–5) During the summer vacation, 3 out of every 5 plants in our classroom died. We started with 40 plants. How many plants died? Solution: A possible solution is to make a drawing of the situation. The bold slash marks represent the 3 out of 5 plants that died. 11 111 11 111 11 111 11 111 11 111 11 111 11 111 11 111 There are 24 bold slashes out of 40. Teaching Notes: For an older group of students, logical reasoning could provide an effective solution. By asking how many groups of 5 plants there are in 40 plants (8 groups), they then know that each group had three plants that died, or 8 × 3 = 24 plants died. Answer: 24 plants died.

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