extra practice problem 1 - hand2mind.com g5 p1-p3…5 of the cookies. rachel baked 1_ 3 of the...

Extra Practice Problem 1 Paths to Problem Solving™ Teacher Pages

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Jason, Rachel, and Siena baked a total of 15 dozen oatmeal cookies for the bake sale. Jason baked 2 _ 5 of the cookies. Rachel baked 1 _ 3 of the cookies, and Siena baked the rest. Who baked the most cookies?

Cookie Bake OffProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a ModelMake a Drawing or DiagramWrite an Expression or Equation

Make a Model Fraction TilesFraction Tower® Cubes

Jason

Find the solution.

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

Sample solution:

Use fraction tiles or Fraction Tower Cubes to model 2 _ 5 and 1 _ 3 . Place the models next to each other in a way that allows the fractions to be compared. If desired, compare these with a whole piece. Use additional fraction pieces to model the sum of 2 _ 5 and 1 _ 3 . Place these pieces side by side on top of a whole piece to reveal how much “the rest” is and position this model in a way that allows the rest to be compared with 2 _ 5 and 1 _ 3 .

Jason baked the most cookies: 2 _ 5 (Jason) > 1 _ 3 (Rachel) > the rest (Siena).

Extra Practice Problem 1

1 _ 5 1 _ 5

1 _ 5 1 _ 5 1 _ 3

1 _ 3

the rest

1


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Raj is making plaster of paris. He adds 3 _ 4 cup of water to 1 1 _ 2 cups of dry mix to make 2 1 _ 4 cups of plaster. Raj has 9 cups of dry mix. How many cups of plaster can he make?

Plaster Mix-UpProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a TableWrite an Expression or Equation

Make a Table None 13 1 _ 2 cups

Find the solution.

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

Sample solution:

Make a table with 3 columns labeled Cups of Mix, Cups of Water, and Cups of Plaster.In the first row, write 1 1 _ 2 in the Mix column, 3 _ 4 in the Water column, and 2 1 _ 4 in the Plaster column.

Cups of Mix Cups of Water Cups of Plaster

1 1 _ 2 3 _ 4 2 1 _ 4 .

Add a row to the table to show the amounts required for a double batch of plaster. To get the numbers for the second row, add 1 1 _ 2 to the Mix column, add 3 _ 4 to the Water column, and add 2 1 _ 4 to the Plaster column.

Cups of Mix Cups of Water Cups of Plaster

1 1 _ 2 3 _ 4 2 1 _ 4

1 1 _ 2 + 1 1 _ 2 = 3 3 _ 4 + 3 _ 4 = 1 1 _ 2 2 1 _ 4 + 2 1 _ 4 = 4 1 _ 2

To the table, add a row for a triple batch, a row for a quadruple batch, and so on until you have 9 cups in the Mix column (that is, the amount of mix that Raj has).

Cups of Mix Cups of Water Cups of Plaster1 1 _ 2 3 _ 4 2 1 _ 4

3 1 1 _ 2 4 1 _ 2

4 1 _ 2 2 1 _ 4 6 3 _ 4

6 3 9

7 1 _ 2 3 3 _ 4 11 1 _ 4

9 4 1 _ 2 13 1 _ 2

Raj can make 13 1 _ 2 cups of plaster using 9 cups of dry mix.



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For a display in the front hallway, Mrs. Washington hung student pictures in a pyramid. The top row has 1 photo. The second row has 8 photos. The third row has 15 photos. The fourth row has 22 photos, and so on. How many pictures will be in the tenth row of the pyramid?

Picture GalleryProblem

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

Look for a Pattern Color TilesTwo-Color Counters

64

Find the solution.

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

Sample solution:

Look for a pattern in the number of photos by making a table.

Row Number of Photos1 1

2 8

3 15

4 22

Compare the number of pictures in the 1st and 2nd rows of the table: 8 – 1 = 7.

Compare the number of pictures in the 2nd and 3rd rows: 15 – 8 = 7.

The pattern appears to be “add 7” to get the number of pictures in the next row.

Verify the pattern with the fourth row in the table: 15 + 7 = 22.

Extend the table to ten rows. Each row in the table represents a row in the pyramid of photos.

Row Number of Photos5 296 367 438 509 57

10 64

Mrs. Washington will place 64 photos in the tenth row of the picture pyramid.



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Ronaldo has 8 coins worth $1.85. What combinations of coins could he have? He has no dollar coins. He has no half-dollar coins.

Making ChangeProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionGuess and CheckMake an Organized List

Guess and Check Replica Money 7 quarters, 1 dime

Find the solution.

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

Sample solution:

Select a combination of 8 coins and find its value. Try different combinations until you find one that is worth $1.85. Use an organized approach if possible, and use the result of each guess to help you make the next guess after it.

First guess: 8 quarters—8 x $0.25 = $2.00.This is too much. Try fewer quarters.

Second guess: 6 quarters and 2 dimes—6 x $0.25 = $1.50;2 x $0.10 = $0.20.$1.50 + $0.20 = $1.70.This is too little. Try 7 quarters.

Third guess: 7 quarters and 1 dime—7 x $0.25 = $1.75;1 x $0.10 = $0.10.$1.75 + $0.10 = $1.85.This is the correct amount.



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The hallway at the middle school is lined with blue and yellow lockers. The third locker is yellow. Every third locker after that is yellow. How many blue lockers are among the first 50 lockers?

Blue and Yellow LockersProblem

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

Make a Drawing or Diagram Two-Color Counters 34

Find the solution.

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

Sample solution:

Draw an array of rectangles to represent the 50 lockers. Mark the third locker in the first row with a Y for yellow.

Y

Continue to mark every third locker with a Y. The unmarked lockers represent the blue lockers.

Y Y Y

Y Y Y

Y Y Y Y

Y Y Y

Y Y Y

Count the number of rectangles marked with a Y and subtract that number from 50: 50 – 16 = 34.

There are 34 blue lockers in the group of 50 lockers.



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Tanya is making beaded necklaces. She uses 15 glass beads and 12 stone beads for each necklace. Tanya buys a container of 200 glass beads. What is the minimum number of stone beads she should buy to be able to use as many of the glass beads as possible?

How Many Glass Beads?Problem

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

Make a Table Two-Color Counters 156

Find the solution.


Sample solution:

Make a table with 3 columns. Label the first column Necklace, the second column Glass Beads, and the third column Stone Beads. Write the number of beads that Tanya needs to make 1 necklace in the first row. Write the number of beads that Tanya needs to make 2 necklaces in the second row.

Necklace Glass Beads Stone Beads1 15 122 15 + 15 = 30 12 + 12 = 24

Complete the table by adding 15 to the second column and 12 to the third column for each new row. Add rows until the Glass Bead column exceeds 200.

Necklace Glass Beads Stone Beads1 15 122 30 243 45 364 60 485 75 606 90 727 105 848 120 969 135 108

10 150 12011 165 13212 180 14413 195 15614 210 168

Tanya can use 195 glass beads to make 13 necklaces. She needs 156 stone beads.



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Dwayne is buying chlorine for his swimming pool. He can buy a 2.25-lb. package or a 3.5-lb. package. The instructions state to add a quarter pound of chlorine to 1,875 gallons of water. Dwayne’s pool holds 20,000 gallons. Estimate how many pounds of chlorine Dwayne needs. Is your estimate more than or less than the amounts contained in the packages? Which package of chlorine should Dwayne buy?

Swimming Pool ChemistryProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionSolve a Simpler ProblemIdentify a Subgoal

Solve a Simpler Problem None He needs about 2 1 _ 2 lb.; he should buy the 3-lb. package.

Find the solution.

Suggested Strategy: Solve a Simpler Problem Facilitate the use of this strategy if students need guidance.

Sample solution:

Round 1,875 to 2,000 and work the problem in simple steps.

For every 2,000 gallons of water, Dwayne needs 1 _ 4 pound of chlorine. Ask yourself, For how many gallons of water would Dwayne need 1 pound of chlorine?

1 _ 4 pound x 4 = 1 pound2,000 gallons x 4 = 8,000 gallonsFor 8,000 gallons of water, Dwayne would need 1 pound of chlorine.

Find out how many times 8,000 gallons goes into 20,000 gallons to determine how many pounds of chlorine Dwayne needs.

8,000 gallons → 1 pound of chlorine 8,000 gallons → 1 pound of chlorine+ 4,000 gallons → 1 _ 2 pound of chlorine

20,000 gallons → 2 1 _ 2 pounds of chlorine

Dwayne needs about 2 1 _ 2 pounds of chlorine. He should buy the 3-lb. package.



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Ms. Parson hands out award tokens for good study habits. She started doing this on the first school day in November. After that day, she skipped 2 days before she awarded tokens again. Then she skipped 3 days before she awarded tokens again. Then she skipped 4 days, 5 days, and so on. How many times did Ms. Parson hand out tokens in the first 30 school days?

Rewarding Good Study HabitsProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a Drawing or DiagramMake a ModelMake a Table

Make a Drawing or Diagram Two-Color Counters 6

Find the solution.


Sample solution:

Write the numbers from 1 through 30 to represent the 30 days.

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

Ms. Parson handed out tokens on the first day. Circle the number 1. Then Ms. Parson skipped two days (Days 2 and 3) and handed out tokens on Day 4. Cross out the 2 and 3. Circle the 4.

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

Continue crossing out the days that Ms. Parson did not hand out tokens and circling the days that she did.

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

Count the number of circles. Ms. Parson handed out tokens 6 times in the first 30 days.



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Nikki has green, blue, purple, and tan shorts. She has 5 shirts: striped, yellow, red, white, and flowered. How many different outfits can she make?

Nikki’s ChoicesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake an Organized ListWrite an Expression or Equation

Make an Organized List Snap Cubes®Centimeter Cubes

20

Find the solution.

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

Sample solution:

Use these abbreviations to represent the colors for the pieces of clothing:Shorts: G, B, P, TShirts: S, Y, R, W, F

Make an organized list. Begin with the green shorts, or G.Match G with each different shirt.G, S G, Y G, R G, W G, F

Now do the same for each of the other colors of shorts.B, S B, Y B, R B, W B, FP, S P, Y P, R P, W P, FT, S T, Y T, R T, W T, F

Count the number of combinations. Nikki has 20 possible outfits.



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Terrel and Malika walked 4 blocks to school. After school, they walked to the store for a snack. Then they walked 6 more blocks to a friend’s house to do homework. Later, they walked 7 blocks home. If Terrel and Malika walked a total of 18 blocks, how many blocks is the store from their school?

Walk It OffProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionWork BackwardsWrite an Expression or EquationIdentify a Subgoal

Work Backwards None 1 block

Find the solution.

Suggested Strategy: Work Backwards Facilitate the use of this strategy if students need guidance.

Sample solution:

Begin with 18 blocks and work backwards.

First, subtract 7 for the number of blocks Terrel and Malika walked home from their friend’s house: 18 – 7 = 11.

Then, subtract 6 for the number of blocks from the store to their friend’s house: 11 – 6 = 5.

Finally, subtract the distance they walked to school. The difference is the distance the store is from the school: 5 – 4 = 1.

The store is 1 block from the school.



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Dry erase markers come in boxes of 4 and 9. Mr. Meekly wants to purchase exactly 1 marker for each of his 104 students. What combinations of boxes can Mr. Meekly buy?

Markers for Everyone!Problem

Possible Strategies Suggested Strategy Suggested Tools SolutionAccount for All PossibilitiesMake a TableMake a ModelWrite an Expression or Equation

Account for All Possibilities Snap Cubes® 8 boxes of 4, 8 boxes of 9;17 boxes of 4, 4 boxes of 9;26 boxes of 4, 0 boxes of 9

Find the solution.

Suggested Strategy: Account for All Possibilities Facilitate the use of this strategy if students need guidance.

Sample solution:

Number of Boxes of 4 Number of Markers Additional Number of Markers Needed

Can This Be Made With Boxes of 9?

Number of Boxes of 9 Needed

Total Number of Markers

0 0 x 4 = 0 104 – 0 = 104 No1 1 x 4 = 4 104 – 4 = 100 No2 2 x 4 = 8 104 – 8 = 96 No3 3 x 4 = 12 104 – 12 = 92 No4 4 x 4 = 16 104 – 16 = 88 No5 5 x 4 = 20 104 –20 = 84 No6 6 x 4 = 24 104 – 24 = 80 No7 7 x 4 = 28 104 – 28 = 76 No8 8 x 4 = 32 104 – 32 = 72 Yes 8 32 + 72 = 1049 9 x 4 = 36 104 – 36 = 68 No

10 10 x 4 = 40 104 – 40 = 64 No11 11 x 4 = 44 104 – 44 = 60 No12 12 x 4 = 48 104 – 48 = 56 No13 13 x 4 = 52 104 – 52 = 52 No14 14 x 4 = 56 104 – 56 = 48 No15 15 x 4 = 60 104 – 60 = 44 No16 16 x 4 = 64 104 – 64 = 40 No17 17 x 4 = 68 104 – 68 = 36 Yes 4 68 + 36 = 10418 18 x 4 = 72 104 – 72 = 32 No19 19 x 4 = 76 104 – 76 = 28 No20 20 x 4 = 80 104 – 80 = 24 No21 21 x 4 = 84 104 – 84 = 20 No22 22 x 4 = 88 104 – 88 = 16 No23 23 x 4 = 92 104 – 92 = 12 No24 24 x 4 = 96 104 – 96 = 8 No25 25 x 4 = 100 104 – 100 = 4 No26 26 x 4 = 104 104 – 104 = 0 Yes 0 104 + 0 = 104



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During a game of dodge ball, Luis ran 21 feet to his left and 7 feet to his right. Then he ran 5 feet backward and 10 feet forward. After that, he ran back to where he started from. He did this by running back over part of the path that he already ran. Find the shortest path that he might have followed. What is the length of this path?

Find the Fastest RouteProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a Drawing or DiagramAct It Out

Make a Drawing or Diagram None 19 feet

Find the solution.


Sample solution:

Draw and label each of Luis’ movements. Start with the first movement.

21 feet left BEGIN

Now, draw and label an arrow to represent 7 feet to the right.

21 feet left7 feet right BEGIN

Then, draw and label an arrow to represent 5 feet backward.

21 feet left7 feet right

5 feet back

BEGIN

Finally, draw and label an arrow to represent 10 feet forward.

21 feet left7 feet right

5 feet back

BEGIN

END10 ft forward

Examine your diagram. To get back to his beginning position, Luis needs to go back 5 feet and to the right 14 feet: 5 + 14 = 19. The shortest distance to the beginning is 19 feet.



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Tracy has 5 sticks. Their lengths are 6 inches, 10 inches, 4 inches, 5 inches, and 8 inches. Which pairs of sticks cannot be used with the 4-inch stick to make a triangle?

Triangle SticksProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAct It OutMake a ModelAccount for All Possibilities

Act It Out Sticks or strawsRulerScissors

5-inch and 10-inch6-inch and 10-inch

Find the solution.

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

Sample solution:

Use sticks (or straws) to act out the problem.

Measure and cut the lengths needed and label the lengths on the sticks. Systematically combine the 4-inch stick with different pairs of the other sticks. For each combination, check whether or not you can form a triangle.

Lengths (inches) Can form a triangle?4 5 6 Yes4 5 8 Yes4 5 10 No4 6 8 Yes4 6 10 No4 8 10 Yes

For each 3-stick combination, compare the length of the longest stick with the sum of the lengths of the other 2 sticks. Notice that a triangle cannot be formed if the length of the longest stick is not greater than the sum of the lengths of the other 2 sticks. This fact is predicted by the Triangle Inequality Theorem, which states that in any triangle, the sum of the lengths of any 2 sides is greater than the length of the 3rd side.



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Kyle invites 3 friends to go with him to the movies. In how many different ways can the 4 friends sit in a row?

At the MoviesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake an Organized List Make an Organized List Color Tiles 24

Find the solution.


Sample solution:

Use an abbreviation for each person: K for Kyle and 1 through 3 for Kyle’s 3 friends.

Kyle in the 1st seat Kyle in the 2nd seat Kyle in the 3rd seat Kyle in the 4th seat K, 1, 2, 3 1, K, 2, 3 1, 2, K, 3 1, 2, 3, K K, 1, 3, 2 1, K, 3, 2 1, 3, K, 2 1, 3, 2, K K, 2, 1, 3 2, K, 1, 3 2, 1, K, 3 2, 1, 3, K K, 2, 3, 1 2, K, 3, 1 2, 3, K, 1 2, 3, 1, K K, 3, 1, 2 3, K, 1, 2 3, 1, K, 2 3, 1, 2, K K, 3, 2, 1 3, K, 2, 1 3, 2, K, 1 3, 2, 1, K

Count the number of ways the friends can sit. They can sit in 6 x 4 = 24 different ways.



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In every 2 dozen yellow roses that Jo gets from her supplier, 3 roses are tipped in red. Jo decides to sell the tipped roses separately. How many plain yellow roses will she have received when she has collected 5 dozen red-tipped roses?

Special RosesProblem

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

Make a Table Two-Color Counters 420

Find the solution.


Sample solution:

Make a table with 2 rows and a few columns to start with. Label the rows Tipped Roses and Plain Roses. The columns will display the running total of roses 2 dozen at a time. In every 2 dozen, or 24, roses, 3 are tipped in red. So for every 24 roses, 3 are tipped and 21 are plain. Start filling in the table.

Tipped Roses 3 6Plain Roses 21 42

Continue filling in the table according to the pattern. In the Tipped Roses row, each successive cell is 3 more than the previous. In the Plain Roses row, each successive cell is 21 more than the previous. Add columns to the table until 30 is reached in the Tipped Roses row.

Tipped Roses 3 6 9 12 15 18 21 24 27 30Plain Roses 21 42 63 84 105 126 147 168 189 210

The table shows that Jo will receive 210 plain roses for every 30 tipped roses. To calculate how many plain roses she will have received when she has collected 5 dozen (60) red-tipped roses, multiply 210 by 2: 210 x 2 = 420.

To collect 5 dozen red-tipped roses, 420 plain yellow roses will be received.



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There are 40 marbles in a bag. Three out of every 5 marbles are red. How many marbles in the bag are not red?

Marble SortProblem

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

Make a Model Two-Color Counters 16

Find the solution.


Sample solution:

Begin with 5 counters. Because 3 out of 5 marbles are red, make 3 of the counters red and 2 yellow .

Make 8 groups of 5 counters each. In each group, make 3 counters red and 2 counters yellow.

Count the numbers of red counters and yellow counters. There are 24 red counters and 16 yellow counters, so 16 of the 40 marbles are not red.



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Jin picked a number between 1 and 100. She divided the number by 7, added 15, multiplied by 3, and subtracted 21. The result was 45. What number did Jin pick?

Can You Find It?Problem

Possible Strategies Suggested Strategy Suggested Tools SolutionWork BackwardsWrite an Expression or Equation

Work Backwards None 49

Find the solution.

Suggested Strategy: Work Backwards Facilitate the use of this strategy if students need guidance.

Sample solution:

Start with 45, and reverse operations as you work backwards.

Add 21: 45 + 21 = 66;

Divide by 3: 66 ÷ 3 = 22;

Subtract 15: 22 – 15 = 7;

Multiply by 7: 7 x 7 = 49.

Jin’s number was 49.



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Ricco is watching a carnival game. He counts the number of players and the number of winners at one particular game. His data are in the table.

Players 16 30 58 114 226Winners 1 2 3 4 5

How many players will have played when there are a total of 8 winners?

And the Winner Is . . .Problem

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

Look for a Pattern None 1,794

Find the solution.


Sample solution:

Notice that the number of winners increases by 1 from each entry to the next, and look for a pattern in the way the number of players changes.

Compare the 1st and 2nd Players entries: 30 – 16 = 14;Compare the 2nd and 3rd Players entries: 58 – 30 = 28;Compare the 3rd and 4th Players entries: 114 – 58 = 56;Compare the 4th and 5th Players entries: 226 – 114 = 112.The pattern is that each difference is double the previous.

Find the 6th Players entry: 226 + (2 x 112) = 226 + 224 = 450.Find the 7th Players entry: 450 + (2 x 224) = 450 + 448 = 898.Find the 8th Players entry: 898 + (2 x 448) = 898 + 896 = 1,794.

When there are 8 winners, there will have been 1,794 players.



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Ling, Eddie, Blake, and Jade are in line at the cafeteria for lunch. Jade says that she would like to be first and the others agree she can be. How many ways can the 4 students stand in line?

First in LineProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake an Organized ListMake a Drawing or Diagram

Make an Organized List Color Tiles 6

Find the solution.


Sample solution:

Put Jade in front and Ling second.Jade, Ling, Eddie, BlakeJade, Ling, Blake, Eddie

Put Jade in front and Eddie second.Jade, Eddie, Ling, BlakeJade, Eddie, Blake, Ling

Put Jade in front and Blake second.Jade, Blake, Eddie, LingJade, Blake, Ling, Eddie

There are 6 different ways that Ling, Eddie, and Blake can stand in line behind Jade.



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How many squares will be used to make Figure 6 in the pattern?

Six FiguresProblem

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

Look for a Pattern Color Tiles 28

Find the solution.


Sample solution:

Count the number of squares in each figure.Figure 1: 8 squaresFigure 2: 12 squaresFigure 3: 16 squaresFigure 4: 20 squares

Compare the numbers of squares, and look for a pattern. The pattern is that the number of squares increases by 4 from each figure to the next.

Figure 5 will have 20 + 4 = 24 squares.Figure 6 will have 24 + 4 = 28 squares.


Figure 1 Figure 2 Figure 3 Figure 4


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A rectangle has a perimeter of 20 meters and an area of 24 square meters. What are the dimensions of the rectangle?

DimensionsProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionGuess and CheckMake a ModelWrite an Expression or Equation

Guess and Check Color Tiles 4 meters x 6 meters

Find the solution.

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

Sample solution:

Since the area of the rectangle is 24 square meters, the width times the length must be 24. List the possible multiplication facts for the product 24:

1 x 242 x 123 x 84 x 6

Since the perimeter of the rectangle is 20, the width plus the length must be half of 20, or 10. From the list of multiplication facts, find the one for which the sum of the factors is 10.

1 + 24 = 25; the sum is not 10.2 + 12 = 14; the sum is not 10.3 + 8 = 11; the sum is not 10.4 + 6 = 10; the sum is 10.

The rectangle has dimensions 4 meters x 6 meters.



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The scores for the first 3 games of the Rams season are shown in the table. Which game did the Rams win by the most points?

What’s the Score?Problem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake or Use a GraphWrite an Expression or Equation

Make or Use a Graph None Game 2

Find the solution.

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

Sample solution:

Make a double bar graph to display the data in the table.Draw the 2 axes. Label the x-axis Game and the y-axis Number of Points. Put a scale on each axis. Title the graph Rams Season.

Graph Game 1. Make a bar to represent the score of 56 for the Rams. Make another bar of a different color to represent the score of 48 for the opponents. Add a key to the graph.

01 2

Game

Rams Season

Rams

Opponent

Num

ber

of P

oint

s

3

20

40

60

80

Use the rest of the data from the table to graph the other games. Compare the heights of the bars in each pair.

01 2

Game

Rams Season

Rams

Opponent

Num

ber

of P

oint

s

3

20

40

60

80

The greatest difference in the heights of the bars is for Game 2. The Rams won by the most points in Game 2.


Game 1 Game 2 Game 3Team Score Team Score Team ScoreRams 56 Rams 67 Rams 63

Opponent 48 Opponent 52 Opponent 61


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What is the sum of the measures of the angles in a hexagon?

Sum It UpProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionSolve a Simpler ProblemMake a Drawing or Diagram

Solve a Simpler Problem Pattern Blocks 720°

Find the solution.

Suggested Strategy: Solve a Simpler Problem Facilitate the use of this strategy if students need guidance.

Sample solution:

A hexagon has 6 sides. Draw a hexagon.

Divide the hexagon into triangles by drawing a line from one vertex to each other nonadjacent vertex.

Realize that the sum of the measures of the angles in a triangle is 180°. Count the number of triangles within the hexagon.

There are 4 triangles in the hexagon. Add 180° four times:180° + 180° + 180° + 180° = 720°.

The sum of the measures of the angles in a hexagon is 720°.



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Guenther, Ellis, Cooper, and Brittany each play a different musical instrument in the school band. The instruments they play are the trumpet, trombone, drums, and saxophone. Guenther plays the trombone. Ellis does not play the saxophone or drums. Cooper does not play the drums. Which instrument does each person play?

Musical InstrumentsProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAccount for All PossibilitiesMake a TableMake a Drawing or Diagram

Account for All Possibilities Color Tiles Guenther: trombone, Ellis: trumpet, Cooper: saxophone, Brittany: drums

Find the solution.


Sample solution:

Make a table to show what you know and to account for all the possibilities.

Start with Guenther. He plays the trombone and none of the other instruments. Write yes under trombone; write no under trumpet, saxophone, and drums.

Trumpet Trombone Drums SaxophoneGuenther No Yes No NoEllisCooperBrittany

Continue with the other students using the information in the problem.

Ellis, Cooper, and Brittany do not play the trombone. Ellis does not play the saxophone or drums, so Ellis plays the trumpet.

Cooper and Brittany do not play the trumpet. Cooper also does not play the drums, so Brittany plays the drums. Cooper plays the saxophone.

Trumpet Trombone Drums SaxophoneGuenther No Yes No NoEllis Yes No No NoCooper No No No YesBrittany No No Yes No

Guenther plays the trombone. Ellis plays the trumpet. Cooper plays the saxophone. Brittany plays the drums.



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On Tuesday, 350 students ate lunch in the lunch room. It was determined that 42 ate chicken, 105 ate pizza, and 77 ate macaroni and cheese. The rest of the students ate packed lunches from home. Did a larger percent eat packed lunches or pizza?

Lunch on TuesdayProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionWrite an Expression or EquationMake or Use a Graph

Write an Expression or Equation None Packed lunches

Find the solution.

Suggested Strategy: Write an Expression or Equation Facilitate the use of this strategy if students need guidance.

Sample solution:

Find the number of packed lunches: 350 – 42 – 105 – 77 = 126.

Determine the percent for each purchased food.

Chicken: 42 ÷ 350 = 0.12, or 12%Pizza: 105 ÷ 350 = 0.30, or 30%Macaroni and cheese: 77 ÷ 350 = 0.22, or 22%Packed lunch: 126 ÷ 350 = 0.36, or 36%

More students ate packed lunches than ate pizza.



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Huong went to the library to get some DVDs. He looked at comedies, dramas, and adventures. How many different combinations of 3 DVDs could he choose?

So Many ChoicesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAccount for All PossibilitiesMake an Organized ListMake a Drawing or Diagram

Account for All Possibilities Color Tiles 10

Find the solution.


Sample solution:

Use the abbreviations C for comedy, D for drama, and A for adventure.

Begin with the possibilities for 3 of the same type: CCC, DDD, AAA.

Now account for the possibilities for 2 of the same type.2 comedies: CCD, CCA.2 dramas: DDC, DDA.2 adventures: AAC, AAD.

Finally, account for the possibility that all 3 DVDs are different: CDA.

There are 10 different combinations.



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Reggie, Tanika, and Jackie have the same kind of eraser. Reggie has 5 __ 11 of his eraser left. Tanika has 4 __ 10 of hers, and Jackie has 8 __ 15 of hers. Who has used the most of his or her eraser?

Three ErasersProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake a Drawing or DiagramMake a ModelWrite an Expression or Equation

Make a Drawing or Diagram Ruler Tanika

Find the solution.


Sample solution:

Draw 3 rectangles the same size.

Divide one rectangle into 11 equal parts. Shade 5 parts.Divide another rectangle into 10 equal parts. Shade 4 parts.Divide the last rectangle into 15 equal parts. Shade 8 parts.

The shaded regions represent the parts of the erasers that are left. Compare the shaded regions: 4 __ 10 < 5 __ 11 < 8 __ 15

Jackie has the largest piece of eraser left and Tanika has the smallest piece. Tanika has used the most eraser.



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Terrance exercises every day. He runs for 30 minutes. He does yoga for 15 minutes. He swims for 30 minutes. Today, he will do only 2 of the exercises. He will do one or both of them more than once. He wants to exercise a total of 1 1 _ 2 hours. How many different ways can he do this?

Staying FitProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionAccount for All PossibilitiesMake a TableWrite an Expression or Equation

Account for All Possibilities None 28

Find the solution.


Sample solution:

Use these abbreviations for Terrance’s exercise choices:R for running,Y for yoga,S for swimming.

For each exercise, determine how many times the exercise would have to be done to expend 1 1 _ 2 hours.

RRR: 1 _ 2 + 1 _ 2 + 1 _ 2 = 1 1 _ 2 SSS: 1 _ 2 + 1 _ 2 + 1 _ 2 = 1 1 _ 2 YYYYYY: 1 _ 4 + 1 _ 4 + 1 _ 4 + 1 _ 4 + 1 _ 4 + 1 _ 4 = 1 1 _ 2

These are the 3 ways that Terrance could spend 1 1 _ 2 hours doing 1 exercise, but he will do 2 exercises, not 1. Find all the different ways to trade time with a 2nd activity.

RSS RYYYY SYYYYSRS YRYYY YSYYYSSR YYRYY YYSYY YYYRY YYYSY YYYYR YYYYS

RRYY SSYY RYRY SYSY RYYR SYYS

YYRR YYSS YRYR YSYS YRRY YSSY

Count the possibilities. There are 28 different possibilities.


RRSRSRSRR


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Alex had 24 marbles. Todd had 21 marbles. Alex traded 3 _ 8 of his marbles for 3 _ 7 of Todd’s marbles. Which boy has more marbles after the trade than he began with?

Trading MarblesProblem

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

Make a Model Two-Color Counters Neither

Find the solution.


Sample solution:

Lay out 24 counters for Alex. Because 3 _ 8 is 3 out of 8, arrange the counters into 8 equal groups. Select 3 of the groups. This is 9 of the 24 counters.

Lay out 21 counters for Todd. Because 3 _ 7 is 3 out of 7, arrange the counters into 7 equal groups. Select 3 of the groups. This is 9 of the 21 counters.

Each boy traded 9 marbles to the other, so each had the same number of marbles after the trade that he had before the trade.



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The first week Wendy went to the gym, she ran 2 miles. The second week, she ran 2.75 miles. The third week, she ran 3.5 miles. If the pattern continues, about how many miles will she run the tenth week? Estimate the distance to the nearest mile.

Miles and MilesProblem

Possible Strategies Suggested Strategy Suggested Tools SolutionMake or Use a GraphMake a TableWrite an Expression or Equation

Make or Use a Graph None 9 miles

Find the solution.

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

Sample solution:

Make a line graph of the data. Draw the two axes. Label the x-axis Week and the y-axis Miles. Label the scales on the axes. Title the graph Wendy’s Weekly Runs.

Plot the points for Weeks 1–3.

Draw a line through the points and continue the line to the tenth week.

Look at the y-axis to see about how many miles Wendy will run in Week 10.

To the nearest mile, Wendy will run about 9 miles in the tenth week.

010 2 3 4 5 6 7 8 9

Week

Wendy’s Weekly Runs

Miles

10

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1011


extra practice problem 1 - hand2mind.com g5 p1-p3…5 of the cookies. rachel baked 1_ 3 of the...

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