extra practice problem 1 - hand2mind€¦ · extra practice problem 1 . paths to problem solving™...

Extra Practice Problem 1 Paths to Problem Solving™ Teacher Resource Pages

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Nancy added 12 coins to her bank. First, she added 2 pennies. Next, she added a coin that was not a penny. Then, she put in the rest of the coins using the same pattern. How many of the 12 coins were pennies?

Money in the BankProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAct It OutLook for a Pattern

Act It Out Two-Color CountersCoins

8 pennies

Find the solution.

Suggested Strategy: Act It Out Facilitate the use of this strategy if students need guidance.

Sample solution:

Let red counters represent pennies and yellow counters represent other types of coins.

Lay out 2 red counters and 1 yellow counter.

Lay out another group of 3 counters. Each group has 2 red and 1 yellow.

Continue laying out groups of 3 counters of this pattern until you have laid out 12 counters.

Count the number of red counters.

There are 8 pennies.

Extra Practice Problem 1


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Seth cut a pizza into 2 halves. Then he cut one of those halves into 2 halves. He cut one of those halves into 2 halves. Finally he cut one of those halves in half. How many pieces of pizza does Seth now have?

Pizza HalvesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a Drawing or DiagramMake a Model

Make a Drawing or Diagram None 5 pieces

Find the solution.

Suggested Strategy: Make a Drawing or Diagram Facilitate the use of this strategy if students need guidance.

Sample solution:

Draw a circle to represent the pizza.Draw a line to cut the circle in half.

Choose one half of the circle.Draw a line to cut that half into 2 halves.

Draw a line to cut one of those halves in half.

Draw a line to cut one of the small sections in half.

Count the number of pieces in the circle.There are 5 pieces.



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Every third lunch tray has an orange wedge on it. 24 students go through the lunch line. How many students get an orange wedge?

Who Gets an Orange?Problem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a TableMake a Model

Make a Table Two-Color Counters 8 students

Find the solution.

Suggested Strategy: Make a Table Facilitate the use of this strategy if students need guidance.

Sample solution:

Make a 2-column table with a column for Student in Line and a column for Orange Wedges.

The first student to get an orange wedge is third in line. Write the number 3 in the first row, under Student in Line. Write the number 1 in the first row under Orange Wedges.

The next student to get an orange wedge is sixth in line. Write the number 6 in the second row under Student in Line. Write the number 2 in the second row under Orange Wedges.

Continue this pattern until you get to 24 in the Student in Line column.

8 students had an orange wedge on their lunch tray.


Student in Line

Orange Wedges

3 1

6 2

9 3

12 4

15 5

18 6

21 7

24 8


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Tanya has 22 guppies in her fish tank. She has green guppies and blue guppies. There are 4 more green guppies than there are blue guppies. How many guppies of each color does Tanya have?

Fish StoryProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionGuess and CheckMake a Model

Guess and Check Color TilesCentimeter Cubes

13 green guppies, 9 blue guppies

Find the solution.

Suggested Strategy: Guess and Check Facilitate the use of this strategy if students need guidance.

Sample solution:

Select two numbers that have a difference of 4. Add the two numbers. Is the sum 22?

1st guess 2nd guess 3rd guess 4th guess

11 and 15Difference: 15 – 11 = 4Add: 11 + 15 = 2626 is too much.Try smaller numbers.

5 and 9Difference: 9 – 5 = 4Add: 5 + 9 = 1414 is too small.Try larger numbers.

10 and 14Difference: 14 – 10 = 4Add: 10 + 14 = 2424 is too much.Try smaller numbers.

9 and 13Difference: 13 – 9 = 4Add: 9 + 13 = 2222 is correct.

Tanya has 13 green guppies and 9 blue guppies.



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Nina bought 3 shirts and 4 shorts. She bought a red shirt, a pink shirt, and an orange shirt. She bought tan shorts, white shorts, brown shorts, and green shorts. How many different outfits can Nina make?

Shirts and ShortsProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake an Organized List Make an Organized List None 12 different outfits

Find the solution.

Suggested Strategy: Make an Organized List Facilitate the use of this strategy if students need guidance.

Sample solution:

Use these abbreviations: shirts: R = red, P = pink, O = orange shorts: T = tan, W = white, B = brown, G = green

Make an organized list showing outfits of one shirt and one pair of shorts.

Outfits with the red shirt Outfits with the pink shirt Outfits with the orange shirt R, T P, T O, T R, W P, W O, W R, B P, B O, B R, G P, G O, G

Nina has 12 possible outfits.



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Mrs. Parsons teaches 4 science classes. She ordered 92 textbooks. She gives 1 textbook to each student. Every science class has the same number of students. How many students are in 1 science class?

Mrs. Parson’s Science ClassProblem


Guess and Check Centimeter Cubes 23 students in each science class

Find the solution.


Sample solution:

Write number sentences that multiply 4 by different numbers until you find a product of 92.

1st guess 2nd guess 3rd guess

20Multiply: 4 x 20 = 8080 is too little.Try a larger number.

25Multiply: 4 x 25 = 100100 is too much.Try a smaller number.

23Multiply: 4 x 23 = 9292 is correct.

There are 23 students in each science class.



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Caroline had a party. She made 4 sub sandwiches for her guests: ham, turkey, chicken, and vegetable. She cut each sub into 8 equal pieces. After the party, she had 1 piece of the ham sub left over. She had 4 pieces of the turkey sub left over. She had 3 pieces of the chicken sub left over. She had no pieces of the vegetable sub left over. How many whole sandwiches were eaten at the party?

Let’s Party!Problem


Make a Drawing or Diagram Snap Cubes® 3 whole sandwiches

Find the solution.


Sample solution:

Draw 4 rectangles to represent the sub sandwiches. Divide each rectangle into 8 pieces.

ham chicken

turkey vegetable

ham chicken

turkey vegetable

Shade the pieces that are left. Place an x in the parts that were eaten.

ham chicken

turkey vegetable

x x x x x x x

x x x x

x x x x x

xx x x x x x x

ham chicken

turkey vegetable

x x x x x x x

x x x x

x x x x x

xx x x x x x x

Now count the number of shaded sections. There are 8 shaded sections.

8 pieces equals one whole sandwich. Subtract from the number of sandwiches to find how many were eaten: 4 – 1 = 3.

3 whole sandwiches were eaten at the party.



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Alan works at a shoe store. He stacks the shoe boxes. He puts 6 boxes on the bottom layer. Then he puts 5 boxes on the next layer. Then he puts 4 boxes on the next layer, and so on. How many boxes can Alan stack in all?

Stacks of BoxesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a Drawing or DiagramLook for a PatternMake a Model

Make a Drawing or Diagram Color Tiles 21 shoe boxes

Find the solution.


Sample solution:

Draw 6 rectangles to show the boxes Alan stacked on the bottom layer.

Draw 5 boxes on the next layer. Then draw 4 boxes, and 3 boxes, and so on until a layer has only 1 box.

Top layer: 1 box 2 boxes 3 boxes 4 boxes 5 boxesBottom layer: 6 boxes

Add the number of boxes in each layer: 1 + 2 + 3 + 4 + 5 + 6 = 21.

Alan stacked 21 shoe boxes in all.



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Maria makes bunches of flowers. Each bunch has 7 tulips and 6 daisies. She has 42 daisies. She has 100 tulips. How many bunches of flowers can Maria make? How many flowers total will she need for that many bunches?

A Bunch of FlowersProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a TableMake a Model

Make a Table Snap Cubes® 7 bunches; 91 total flowers

Find the solution.


Sample solution:

Make a 3-column table with 1 column for bunches, 1 column for tulips, and 1 column for daisies.

In the first row, write the number 1 in the Bunches column. Because 1 bunch has 7 tulips, write the number 7 in the Tulips column. Because 1 bunch has 6 daisies, write the number 6 in the Daisies column.

For 2 bunches, Maria will need 7 more tulips and 6 more daisies: 7 + 7 = 14 and 6 + 6 = 12. Under the heads in the second row, write the numbers 2 (bunches), 14 (tulips), and 12 (daisies).

Add more rows by continuing the pattern of adding 7 to the number of tulips and 6 to the number of daisies. Continue to add rows until you get to 100 in the Tulips column or to 42 in the Daisies column.

Maria can make 7 bunches using 42 daisies. To make 7 bunches will take 49 tulips and 42 daisies. 49 + 42 = 91, so Maria will need 91 total flowers.


Bunches Tulips Daisies

1 7 6

2 14 12

3 21 18

4 28 24

5 35 30

6 42 36

7 49 42


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These are the choices for sundaes at Dipsy’s Dessert Shop.

Ice cream flavorsvanilla strawberry chocolate

Topping choicesbutterscotch fudge marshmallow

How many different sundaes can you order? You can get 1 flavor of ice cream and 1 kind of topping in each sundae.

Dispsy’s Dessert ShopProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake an Organized ListMake a Model

Make an Organized List Snap Cubes®Color Tiles (Rainbow)

9 different sundaes

Find the solution.

Suggested Strategy: Make a Organized List Facilitate the use of this strategy if students need guidance.

Sample solution:

Use these abbreviations for the ice cream flavors and the topping choices: flavors: V = vanilla, S = strawberry, C = chocolate toppings: B = butterscotch, F = fudge, M = marshmallow

Make an organized list showing sundaes made of one ice cream and one topping. vanilla sundae strawberry sundae chocolate sundae V, B S, B C, B V, F S, F C, F V, M S, M C, M

You can get 9 different sundaes.



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Scott went to the gym to exercise. He lifted weights for 45 minutes. He jogged for 30 minutes. When he left the gym, it was 8:45 a.m. What time did Scott get to the gym?

Working OutProblem


Make a Drawing or Diagram None 7:30 a.m.

Find the solution.


Sample solution:

Sketch a clock showing the time 8:45 and work backwards.

111210

39

4857 6

12

12 111210

39

4857 6

12 111210

39

4857 6

Subtract 45 minutes by moving the minute hand back (counterclockwise) 45 minutes.

111210

39

4857 6

12

12 111210

39

4857 6

12 111210

39

4857 6

8:45 – 0:45 = 8:00

Subtract 30 minutes by moving the minute hand back (counterclockwise) 30 minutes.

111210

39

4857 6

12

12 111210

39

4857 6

12 111210

39

4857 6

8:00 – 0:30 = 7:30

Scott got to the gym at 7:30 a.m.



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Erin measured the height of a sunflower. Her results are in this table.

Week Height (cm) 1 5 2 8 3 11 4 14 5 17 6 20 7 23 8 26 9 29 10The plant grows at the same rate each week. How tall will the sunflower be in 10 weeks?

How Tall Will It Get?Problem

Possible Strategies Suggested Strategy Suggested Tools SolutionLook for a PatternMake or Use a Graph

Look for a Pattern Snap Cubes®Graph Paper

32 cm

Find the solution.

Suggested Strategy: Look for a Pattern Facilitate the use of this strategy if students need guidance.

Sample solution:

Find the difference in heights of the plant between the 1st and 2nd weeks: 8 – 5 = 3.

Find the difference in heights of the plant between the 2nd and 3rd weeks: 11 – 8 = 3.

Each difference is 3 cm. Confirm that the differences for weeks 3 through 9 are all 3 cm and continue the pattern to fill in the table for the 10th week.

The plant will be 32 cm tall the 10th week.



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Hank plays a game at the carnival. For every 8 tickets he wins, he gets 2 free tickets. In all, Hank wins 24 tickets. How many free tickets does he get?

Tickets to the CarnivalProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake or Use a GraphAct It Out

Make or Use a Graph Two-Color Counters 6 free tickets

Find the solution.


Suggested Strategy: Make or Use a Graph Facilitate the use of this strategy if students need guidance.

Sample solution:

Make a pictograph with the first row labeled Tickets Won and the second row labeled Free Tickets.Let each ticket picture in the graph equal 2 tickets.

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Decide how many ticket pictures to draw in the graph to represent 8 tickets won: 8 ÷ 2 = 4. Draw the pictures.

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

If Hank wins 8 tickets, he gets 2 free tickets. Decide how many ticket pictures to draw for the 2 free tickets: 2 ÷ 2 = 1. Draw the picture.

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Continue this pattern, 4 in the top row and 1 in the bottom row, until you have 12 ticket pictures drawn in the Tickets Won row. This represents the 24 tickets that Hank won.

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Tickets Won

Free Tickets

Key: 1 = 2 tickets

Count the number of ticket pictures in the Free Tickets row. Multiply that number by 2: 3 x 2 = 6. Hank got 6 free tickets.


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Patti bought two kinds of pencils. One kind came in packs of 4. The other kind came in packs of 6. She had 20 pencils in all. How many packs of each kind did Patti buy?

Packs of PencilsProblem


Guess and Check Snap Cubes® 2 packs of 6 pencils, 2 packs of 4 pencils

Find the solution.


Sample solution:

Write number sentences to find a sum equal to 20 where one addend is a multiple of 4 and the other addend is a multiple of 6.

Select 2 numbers for the first guess. Check. Continue until the solution is discovered.


3 and 1(3 x 4) + (1 x 6) = 12 + 6 = 1818 is too little.Try different numbers, at least 1 larger.

3 and 2(3 x 4) + (2 x 6) = 12 + 12 = 2424 is too much.Try different numbers, at least 1 smaller.

2 and 2(2 x 4) + (2 x 6) = 8 + 12 = 2020 is correct.

Patti bought 2 packs of 6 pencils and 2 packs of 4 pencils.



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Brittany’s 17th birthday is in December. Use these clues to find the date of her birthday. The sum of the digits of the date is 4. The date is an odd number. The date is greater than her age.

Brittany’s BirthdayProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a TableGuess and Check

Make a Table None December 31

Find the solution.


Sample solution:

Use the table given to cross out dates that do not match the clues.

Cross out the dates whose digits do not have a sum of 4. (—)

Of the dates not crossed out, cross out the dates that are even numbers. (X)

Of the dates not crossed out, color over the dates that are less than 17.

There is one number left. Check the answer. Is the date an odd number? Is it greater than 17? Is the sum of its digits equal to 4?

The digits of 31 have a sum of 4: 3 + 1 = 4.31 is an odd number.31 > 17

Brittany’s birthday is December 31.


December

Sun Mon Tue Wed Thu Fri Sat

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31

December

Sun Mon Tue Wed Thu Fri Sat

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31


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You can order 2 items for lunch from this list.

How many different ways can you pick 2 items?

Pick Your LunchProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake an Organized List Make an Organized List Snap Cubes® 10 ways to pick 2 items

Find the solution.

Suggested Strategy: Make an Organized List Facilitate the use of this strategy if students need guidance.

Sample solution:

Step 1: List all 2-item combinations with a turkey club. [4] turkey club and cup of soup turkey club and salad turkey club and fruit cup turkey club and baked potato

Step 2: List all 2-item combinations with a cup of broccoli soup. [3]Do not list cup of soup and turkey club, because that combination was listed in Step 1.

cup of soup and salad cup of soup and fruit cup cup of soup and baked potato

Step 3: List all 2-item combinations with a salad. [2]Do not list salad and turkey club, because that combination was listed in Step 1. Do not list salad a nd cup of soup, because that combination was listed in Step 2.

salad and fruit cup salad and baked potato

Step 4: List all 2-item combinations with a fruit cup. [1]Do not list fruit cup and turkey club, because it was listed in Step 1, or fruit cup and cup of soup from Step 2, or fruit cup and salad from Step 3.

fruit cup and baked potato

Step 5: All the 2-item combinations that include a baked potato have been listed in Steps 1–4. Count or add the number of combinations.

4 + 3 + 2 + 1 = 10There are 10 ways a customer can pick 2 items.


LunchTurkey club sandwichCup of broccoli soupGarden saladFresh fruit cupBaked potato


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Amanda’s school has 4 halls with lockers. Lockers are numbered 1 to 15 in the first hall. Lockers are numbered 16 to 30 in the second hall. Lockers are numbered 31 to 45 in the third hall. Lockers in the fourth hall follow the same pattern. What is the number of the last locker in the fourth hall?

Locker NumbersProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionLook for a PatternMake a Drawing or Diagram

Look for a Pattern None 60

Find the solution.


Sample solution:

Find the difference between the numbers of the last lockers in the first and second halls:30 – 15 = 15.

Find the difference between the numbers of the last lockers in the second and third halls:45 – 30 = 15.

The pattern for determining the number of the last locker in each hall is “add 15.”

Use the pattern to find the number of the last locker in the fourth hall.1st hall: 152nd hall: 15 + 15 = 303rd hall: 30 + 15 = 454th hall: 45 + 15 = 60

The number on the last locker in the fourth hall is 60.



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4 friends each have 12 coins. Jeannie donates 3 _ 4

of her coins. Alicia donates all of her coins. Roy donates 1 _ 2 of his coins,

and Tim donates 1 _ 3

of his coins. Who donated the fewest coins?

Making a DonationProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAct It OutMake a Model

Act It Out Two-Color Counters Tim donated the fewest coins (3)

Find the solution.


Sample solution:

Lay out 12 counters for Jeannie. Because 3 _ 4 means 3 out of 4, arrange the counters into 4 equal groups. Select 3 of the groups. This is 9 of the 12 counters. Jeannie donated 9 coins.

Lay out 12 counters for Alicia. She donated all of her coins. Alicia donated 12 coins.

Lay out 12 counters for Roy. Because Roy donated 1 _ 2 of his coins, arrange the counters into 2 equal groups. Select 1 of the groups. This is 6 of the 12 counters. Roy donated 6 coins.

Lay out 12 counters for Tim. Because 1 _ 3 means 1 out of 3, arrange the counters into 3 equal groups. Select 1 of the groups. This is 3 of the 12 counters. Tim donated 3 coins.

Tim donated the fewest coins, 3.



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Yosef is planting 3 rows of bushes and flowers. He wants to plant 2 flowers for every bush. Each row will have 6 bushes and flowers in all. Each row will look like all the other rows. How many bushes and how many flowers will Yosef plant?

Yosef’s GardenProblem


Make a Drawing or Diagram Two-Color Counters 6 bushes, 12 flowers

Find the solution.


Sample solution:

Use a square to represent a bush and a circle to represent a flower. For the first row of plants, draw a square and draw two circles.

6 items are planted in each row. Add more items to the first row. Draw another square and 2 more circles. Now the first row has six items in it.

There are 3 rows in all. Draw two more rows to match the first row.

Count the number of squares and circles. There are 6 squares and 12 circles.

Yosef will plant 6 bushes and 12 flowers.



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23 students play team sports. 8 play baseball only. 11 play soccer only. How many students play soccer? How many play baseball?

Team SportsProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake or Use a GraphGuess and Check

Make or Use a Graph Two-Color Counters Baseball: 12; soccer: 15

Find the solution.


Sample solution:

Make a Venn diagram.

Draw 2 circles that overlap. Label the left circle Baseball and the right circle Soccer.

8 4 11

8 11

Baseball Soccer

Baseball Soccer

Baseball Soccer

In the left circle, write the number of students who play only baseball. In the right circle, write the number of students who play only soccer.

8 4 11

8 11

Baseball Soccer

Baseball Soccer

Baseball Soccer Figure out how many played both baseball and soccer: 23 – (8 + 11) = 4.

4 students play both baseball and soccer. Write 4 in the overlapping section of the circles.

8 4 11

8 11

Baseball Soccer

Baseball Soccer

Baseball Soccer

To find the number of students who play baseball, add the numbers inside the baseball circle:8 + 4 = 12.

To find the number of students who play soccer, add the numbers inside the soccer circle:4 + 11 = 15.

12 students play baseball and 15 students play soccer.



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A quilt has 16 colored squares in one row. The colored squares are in this order: pink, purple, blue, green, pink, purple, blue, green. This pattern continues. How many pink squares are in 1 row?

Making a QuiltProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionLook for a PatternMake a Model

Look for a Pattern Color Tiles (Rainbow) 4 pink squares

Find the solution.


Sample solution:

The colors of the first 8 squares of the row are given. Continue the pattern until 16 squares are listed.

squares 1–8: pink, purple, blue, green, pink, purple, blue, greensquares 9–16: pink, purple, blue, green, pink, purple, blue, green

Count the number of times “pink” is listed.

There are 4 pink squares in 1 row of the quilt.



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Tyrone bought horns and hats for a school party. The horns cost $0.10 each. The hats cost $0.20 each. He bought 20 items for $2.50. How many of each item did he buy?

Party Hats and HornsProblem


Guess and Check Snap Cubes® 15 horns, 5 hats

Find the solution.


Sample solution:

Find 2 numbers that have a sum of 20 and make a total of $2.50 if one number is multiplied by the cost of a horn and the other is multiplied by the cost of a hat.

Select 2 numbers whose sum is 20. Check. Continue until the solution is discovered.


10 and 1010 horns and 10 hatsThe total cost of these items:10($0.10) + 10($0.20) = $1 + $2 = $3$3 is too high.Try again using a smaller number of hats.

14 and 614 horns and 6 hatsThe total cost of these items:14($0.10) + 6($0.20) = $1.40 + $1.20 = $2.60$2.60 is too high.Try again using a smaller number of hats.

15 and 515 horns and 5 hatsThe total cost of these items:15($0.10) + 5($0.20) = $1.50 + $1.00 = $2.50$2.50 is correct.

Tyrone bought 15 horns and 5 hats.



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The Crazy 25 Relay race starts today. There are 5 members on each relay team. The first runner runs 50 meters. The next runner runs 75 meters. Each team member runs 25 meters farther than the runner who just finished. How far does the last runner have to run?

Crazy 25 Relay RaceProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a TableMake a Drawing or Diagram

Make a Table None 150 meters

Find the solution.


Sample solution:

Make a 2-row table with a row for Runner and a row for Distance.

The first runner runs 50 meters. Write 1 and 50 m in the first column.

Runner 1Distance 50 m

The 2nd runner runs 25 meters farther than the first runner: 50 + 25 = 75. Write 2 and 75 in the second column.

Runner 1 2Distance 50 m 75 m

Continue this pattern until you have completed the table through Runner 5.

Runner 1 2 3 4 5Distance 50 m 75 m 100 m 125 m 150 m

The fifth runner of the relay team ran 150 meters.



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Toma cut a pattern into wood. He cut a square and then a rhombus next to it to make a pattern. He cut 15 figures into the wood. What was the last figure he carved?

Wood CarvingProblem


Make a Drawing or Diagram Two-Color CountersColor Tiles

A square

Find the solution.


Sample solution:

Draw the pattern and then count the figures.

Begin with a square. Then draw a rhombus.

Alternate drawing the 2 figures until you get to the 15th figure.

The 15th figure is a square.



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Madeline has $27. She has only $5 bills and $1 bills. If she has 7 bills, how many of each bill does she have?

How Much Money?Problem


Guess and Check Replica Money Five $5 bills, two $1 bills

Find the solution.


Sample solution:

Find 2 numbers that have a sum of 7 and that make a total of $27 when one number is multiplied by $5 and the other is multiplied by $1.

Select 2 numbers whose sum is 7. Check. Continue until the solution is discovered.

1st guess 2nd guess 3rd guess2 and 52 $5 bills and 5 $1 bills2($5) + 5($1) = $10 + $5 = $15$15 is too low.Try again by using more $5 bills.

6 and 16 $5 bills and 1 $1 bill6($5) + 1($1) = $30 + $1 = $31$31 is too high.Try again by using less $5 bills.

5 and 25 $5 bills and 2 $1 bills5($5) + 2($1) = $25 + $2 = $27$27 is correct.

Madeline has five $5 bills and two $1 bills.



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Lunch costs $2.25. Sigourney has a $10 bill. What is the most number of lunches she can buy? How much money will she have left?

Buying LunchProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a ModelAct It Out

Make a Model Replica Money 4 lunches; $1 left over

Find the solution.

Suggested Strategy: Make a Model Facilitate the use of this strategy if students need guidance.

Sample solution:

Lay out $10 in single dollars. Trade a single dollar for 4 quarters.$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

Take away $2.25. There are 7 single dollars and 3 quarters left.

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $Repeat this action (take away $2.25) until there is less than $2.25 remaining.

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $ $

Count how many times $2.25 was taken away to determine the number of lunches that Sigourney can buy. Count the remaining money.

Sigourney can purchase 4 lunches with $1 left over.



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Tarik wants to display his model planes. He has a 26-inch shelf. Each plane is 4 inches long. He wants the planes to be 2 inches apart. What is the greatest number of planes he can fit on the shelf?

Model PlanesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a Drawing or Diagram Make a Drawing or Diagram None 4 planes

Find the solution.


Sample solution:

Draw a horizontal line to represent the shelf.Indicate the length of a 4-inch plane by writing a 4 on the line.

4

= 28 in.

= 24 in.

= 26 in.2 4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2 4

4 2 Draw a vertical line and write a 2 after it. The 4 is for the plane. The 2 is for the 2-inch space.

4

= 28 in.

= 24 in.

= 26 in.2 4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2 4

4 2

Write another 4, a line, and a 2. Continue this pattern to see how close to 26 you can get.

4

= 28 in.

= 24 in.

= 26 in.2 4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2 4

4 2

Add the numbers on the line. This shelf, for example, is too long because the pattern adds up to 28 inches, and the shelf is only 26 inches long.

Take away the last plane. 28 – 4 = 24 inches. Four planes fit on the shelf.

4

= 28 in.

= 24 in.

= 26 in.2 4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2 4

4 2

Check to see if you can fit more planes by starting with a 2-inch space. Draw a horizontal line and write a 2, a line, and a 4. Continue this pattern to see how close to 26 you can get.

4

= 28 in.

= 24 in.

= 26 in.2 4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2

4 2 4 2 4 2 4 2 4

4 2

This placement also holds 4 planes.

The greatest number of planes that can fit on the 26-inch shelf is 4.



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Sandeep packs math books and spelling books into the same number of boxes. He does not mix the two kinds of books. He can fit 4 times as many spelling books into one box as he can math books. Sandeep can fit 7 math books into 1 box. How many total boxes does he pack to hold 105 books?

Books and BoxesProblem


Make a Drawing or Diagram Snap Cubes® 6 boxes (3 boxes of math books, 3 boxes of spelling books)

Find the solution.


Sample solution:

Draw rectangles to represent the boxes.

Draw 2 rectangles.

7

28

7

28

7

28

7

7

28

Write 7 in one of the rectangles.This is a box of math books.

A box of spelling books has 4 times as many books in it: 7 x 4 = 28.

7

28

7

28

7

28

7

7

28

Write 28 on the other rectangle.The total number of books in these boxes is 35: 7 + 28 = 35.

Continue this pattern until the total number of books is 105.

7

28

7

28

7

28

7

7

28

35 + 35 + 35 = 105

Sandeep can pack 3 boxes of math books and 3 boxes of spelling books, or 6 boxes in all, to hold 105 books.



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Mrs. Thatcher asked her students what their favorite colors are. She wrote their answers on the board. Which color is the most popular?

blue, yellow, purple, blue, red, pink, purple, pink, pink, purple, blue, blue, yellow, red, pink, pink

What’s Your Favorite Color?Problem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake or Use a Graph Make or Use a Graph Color Tiles (Rainbow) Pink

Find the solution.


Sample solution:

Put the data into a line plot. Draw a horizontal line. Write the colors under the horizontal line.

blue yellow purple red pink

Place an X for each color that was chosen.

X

X X

X X X

X X X X X

X X X X X

blue yellow purple red pink

The tallest column of Xs is the most popular color.

Pink is the most popular color.



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Troy, Angel, Singh, and Robert are passing a football. Troy passes to Angel, and Angel passes back to Troy. The boys continue passing the football. Each boy makes 1 pass to and receives 1 pass from every other boy. How many times is the football passed?

Football PassesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAct It OutMake a Drawing or Diagram

Act It Out Object that can be tossed or handed around

12 times

Find the solution.


Sample solution:

Work in a group of 4.

Have the first student pass an object to the second student and the second student pass the object back to the first student.

Have the first student repeat this process with the third and fourth students. Count the number of passes made. [6]

Have the second student pass the object back and forth with the third and fourth students. The first student is not included because that student has already taken his or her turn to start the passing process and has already exchanged passes with all the other students. Count the number of passes. [4]

Have the third student pass the object back and forth with the fourth student. Count the number of passes. [2]

The fourth student will not act out the procedure because he or she has already exchanged passes with the other 3 students.

Count or add the numbers of passes: 6 + 4 + 2 = 12.

There were 12 passes.


extra practice problem 1 - hand2mind€¦ · extra practice problem 1 . paths to problem solving™...

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