Extra Practice Problem 1
Problem
Money in the Bank
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Act It Out
Look for a Pattern
Act It Out
Two-Color Counters
Coins
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.
© ETA hand2mind™
There are 8 pennies.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 1
Extra Practice Problem 2
Problem
Pizza Halves
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Drawing or Diagram
Make 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.
© ETA hand2mind™
Count the number of pieces in the circle.
There are 5 pieces.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 2
Extra Practice Problem 3
Problem
Who Gets an Orange?
Every third lunch tray has an orange wedge on it. 24 students go through the lunch line.
How many students get an orange wedge?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Table
Make 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.
Student in
Line
Orange
Wedges
3
1
6
2
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.
9
3
12
4
15
5
18
6
Continue this pattern until you get to 24 in the Student in Line column.
21
7
24
8
© ETA hand2mind™
8 students had an orange wedge on their lunch tray.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 3
Extra Practice Problem 4
Problem
Fish Story
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Guess and Check
Make a Model
Guess and Check
Color Tiles
Centimeter 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
11 and 15
Difference: 15 – 11 = 4
Add: 11 + 15 = 26
26 is too much.
Try smaller numbers.
5 and 9
Difference: 9 – 5 = 4
Add: 5 + 9 = 14
14 is too small.
Try larger numbers.
10 and 14
Difference: 14 – 10 = 4
Add: 10 + 14 = 24
24 is too much.
Try smaller numbers.
4th guess
9 and 13
Difference: 13 – 9 = 4
Add: 9 + 13 = 22
22 is correct.
© ETA hand2mind™
Tanya has 13 green guppies and 9 blue guppies.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 4
Extra Practice Problem 5
Problem
Shirts and Shorts
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make 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
R, T
R, W
R, B
R, G
Outfits with the pink shirt
P, T
P, W
P, B
P, G
Outfits with the orange shirt
O, T
O, W
O, B
O, G
© ETA hand2mind™
Nina has 12 possible outfits.
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Extra Practice Problem 5
Extra Practice Problem 6
Problem
Mrs. Parson’s Science Class
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Guess and Check
Make a Model
Guess and Check
Centimeter Cubes
23 students in each
science class
Find the solution.
Suggested Strategy: Guess and Check
Facilitate the use of this strategy if students need guidance.
Sample solution:
Write number sentences that multiply 4 by different numbers until you find a product of 92.
1st guess
20
Multiply: 4 x 20 = 80
80 is too little.
Try a larger number.
2nd guess
25
Multiply: 4 x 25 = 100
100 is too much.
Try a smaller number.
3rd guess
23
Multiply: 4 x 23 = 92
92 is correct.
© ETA hand2mind™
There are 23 students in each science class.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 6
Extra Practice Problem 7
Problem
Let’s Party!
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?
Possible Strategies
Suggested Strategy
Make a Drawing or Diagram
Make a Model
Make a Drawing or Diagram
Suggested Tools
Snap Cubes®
Solution
3 whole sandwiches
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students need guidance.
Sample solution:
Draw 4 rectangles to represent the sub sandwiches. Divide each rectangle into 8 pieces.
ham
ham
chicken
chicken
turkey
turkey
vegetable
vegetable
Shade the pieces that are left. Place an x in the parts that were eaten.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
ham
ham
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
chicken
chicken
x
x
x
x
x
x
x
x
turkey
turkey
x
x
x
x
x
x
x
x
vegetable
vegetable
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.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 7
© ETA hand2mind™
x
x
Extra Practice Problem 8
Problem
Stacks of Boxes
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Drawing or Diagram
Look for a Pattern
Make a Model
Make a Drawing or Diagram
Color Tiles
21 shoe boxes
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students need guidance.
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:
Bottom layer:
1 box
2 boxes
3 boxes
4 boxes
5 boxes
6 boxes
Add the number of boxes in each layer: 1 + 2 + 3 + 4 + 5 + 6 = 21.
© ETA hand2mind™
Alan stacked 21 shoe boxes in all.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 8
Extra Practice Problem 9
Problem
A Bunch of Flowers
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?
Possible Strategies
Suggested Strategy
Make a Table
Make a Model
Make a Table
Suggested Tools
Snap Cubes®
Solution
7 bunches; 91 total flowers
Find the solution.
Suggested Strategy: Make a Table
Facilitate the use of this strategy if students need guidance.
Sample solution:
Make a 3-column table with 1 column for bunches, 1 column for tulips, and
1 column for daisies.
Bunches Tulips
Daisies
1
7
6
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.
2
14
12
3
21
18
4
28
24
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).
5
35
30
6
42
36
7
49
42
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.
© ETA hand2mind™
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.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 9
Extra Practice Problem 10
Problem
Dispsy’s Dessert Shop
These are the choices for sundaes at Dipsy’s Dessert Shop.
Ice cream flavors
vanilla strawberry chocolate
Topping choices
butterscotch 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.
Possible Strategies
Suggested Strategy
Make an Organized List
Make a Model
Make an Organized List
Suggested Tools
Snap Cubes®
Solution
9 different sundaes
Color Tiles (Rainbow)
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
© ETA hand2mind™
You can get 9 different sundaes.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 10
Extra Practice Problem 11
Problem
Working Out
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Drawing or Diagram
Make a Model
Make a Drawing or Diagram
None
7:30 a.m.
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students need guidance.
Sample solution:
Sketch a clock showing the time 8:45 and work backwards.
9
9
9
12
11
12
10 11
10
8 11
8 7
10
7
12
6
6
1
1 2
2
3
1 43
2
5 4
5
3
Subtract
45 minutes
by moving the minute hand back (counterclockwise) 45 minutes.
8
4
9
9
9
12
7
11
6
12
10 11
10
8 11
8 7
10
7
12
6
6
5
1
1 2
2
1 4
4
5 2
5
3
3
3
8 – 0:45 = 8:00
4
8:45
12
11
1
7
5
6
12
1 2
10 11
Subtract
30 minutes
by moving the minute hand back (counterclockwise) 30 minutes.
10
2
9
3
9 11 12 1 3
8
4
8 7
10
2
5 4
6
7
5
9
3
6
4
7
6
© ETA hand2mind™
8
5
8:00 – 0:30 = 7:30
Scott got to the gym at 7:30 a.m.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 11
Extra Practice Problem 12
Problem
How Tall Will It Get?
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
10
The plant grows at the same rate each week. How tall will the sunflower be in 10 weeks?
Possible Strategies
Suggested Strategy
Look for a Pattern
Make or Use a Graph
Look for a Pattern
Suggested Tools
Snap Cubes®
Solution
32 cm
Graph Paper
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.
© ETA hand2mind™
The plant will be 32 cm tall the 10th week.
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Extra Practice Problem 12
Extra Practice Problem 13
Problem
Tickets to the Carnival
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make or Use a Graph
Act 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
Tickets
Won
Free Tickets
Free
Key: 1Tickets = 2 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
Tickets
Won
Free Tickets
Free
Key: 1Tickets = 2 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
Tickets
Won
Free Tickets
Free Tickets
Key: 1
= 2 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
© ETA hand2mind™
Tickets Won
Free Tickets
Free Tickets
Key: 1
= 2 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.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 13
Extra Practice Problem 14
Problem
Packs of Pencils
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?
Possible Strategies
Suggested Strategy
Guess and Check
Make a Model
Guess and Check
Suggested Tools
Snap Cubes®
Solution
2 packs of 6 pencils, 2 packs
of 4 pencils
Find the solution.
Suggested Strategy: Guess and Check
Facilitate the use of this strategy if students need guidance.
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.
1st guess
3 and 1
(3 x 4) + (1 x 6) = 12 + 6 = 18
18 is too little.
Try different numbers,
at least 1 larger.
2nd guess
3 and 2
(3 x 4) + (2 x 6) = 12 + 12 = 24
24 is too much.
Try different numbers,
at least 1 smaller.
3rd guess
2 and 2
(2 x 4) + (2 x 6) = 8 + 12 = 20
20 is correct.
© ETA hand2mind™
Patti bought 2 packs of 6 pencils and 2 packs of 4 pencils.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 14
Extra Practice Problem 15
Problem
Brittany’s Birthday
December
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.
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
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Table
Guess and Check
Make a Table
None
December 31
Find the solution.
Suggested Strategy: Make a Table
Facilitate the use of this strategy if students need guidance.
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. (—)
December
Of the dates not crossed out, cross out the dates that are even numbers. (X)
Sun
Mon Tue
Wed Thu
Fri
Sat
1
2
3
4
5
6
7
Of the dates not crossed out, color over the dates that are less than 17.
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
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
© ETA hand2mind™
Brittany’s birthday is December 31.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 15
Extra Practice Problem 16
Problem
Pick Your Lunch
You can order 2 items for lunch from this list.
How many different ways can you pick 2 items?
Possible Strategies
Suggested Strategy
Make an Organized List
Make an Organized List
Lunch
Turkey club sandwich
Cup of broccoli soup
Garden salad
Fresh fruit cup
Baked potato
Suggested Tools
Snap Cubes®
Solution
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 and cup of soup, because that combination was listed in Step 2.
salad and fruit cup
salad and baked potato
© ETA hand2mind™
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 = 10
There are 10 ways a customer can pick 2 items.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 16
Extra Practice Problem 17
Problem
Locker Numbers
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Look for a Pattern
Make a Drawing or Diagram
Look for a Pattern
None
60
Find the solution.
Suggested Strategy: Look for a Pattern
Facilitate the use of this strategy if students need guidance.
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: 15
2nd hall: 15 + 15 = 30
3rd hall: 30 + 15 = 45
4th hall: 45 + 15 = 60
© ETA hand2mind™
The number on the last locker in the fourth hall is 60.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 17
Extra Practice Problem 18
Problem
Making a Donation
3
4 friends each have 12 coins. Jeannie donates _ of her coins. Alicia donates all of her coins. Roy donates _12 of his coins,
4
1
and Tim donates _ of his coins. Who donated the fewest coins?
3
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Act It Out
Make a Model
Act It Out
Two-Color Counters
Tim donated the fewest
coins (3)
Find the solution.
Suggested Strategy: Act It Out
Facilitate the use of this strategy if students need guidance.
Sample solution:
Lay out 12 counters for Jeannie. Because _34 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 _12 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 _13 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.
© ETA hand2mind™
Tim donated the fewest coins, 3.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 18
Extra Practice Problem 19
Problem
Yosef’s Garden
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Drawing or Diagram
Make a Model
Make a Drawing or Diagram
Two-Color Counters
6 bushes, 12 flowers
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students need guidance.
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.
© ETA hand2mind™
Count the number of squares and circles. There are 6 squares and 12 circles.
Yosef will plant 6 bushes and 12 flowers.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 19
Extra Practice Problem 20
Problem
Team Sports
23 students play team sports. 8 play baseball only. 11 play soccer only. How many students play soccer? How many play baseball?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make or Use a Graph
Guess and Check
Make or Use a Graph
Two-Color Counters
Baseball: 12; soccer: 15
Find the solution.
Suggested Strategy: Make or Use a Graph
Facilitate the use of this strategy if students need guidance.
Sample solution:
Make a Venn diagram.
Draw 2 circles that overlap. Label the left circle Baseball and the right circle Soccer.
Baseball
Soccer
Baseball
Soccer
Baseball
Soccer
In the left circle, write the number of students who play only baseball. In the right circle,
Baseball
Soccer who play only soccer.
write the number
of students
Baseball
Soccer
8
Baseball
8
11
Soccer
11
8
Baseball
11
Soccer
Baseball
Soccer
Figure out how many played both baseball and soccer: 23 – (8 + 11) = 4.
4 students play8both baseball
4
11 and soccer. Write 4 in the overlapping section of the circles.
Baseball Soccer
8
4
11
8
4
11
© ETA hand2mind™
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.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 20
Extra Practice Problem 21
Problem
Making a Quilt
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Look for a Pattern
Make a Model
Look for a Pattern
Color Tiles (Rainbow)
4 pink squares
Find the solution.
Suggested Strategy: Look for a Pattern
Facilitate the use of this strategy if students need guidance.
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:
squares 9–16:
pink, purple, blue, green, pink, purple, blue, green
pink, purple, blue, green, pink, purple, blue, green
Count the number of times “pink” is listed.
© ETA hand2mind™
There are 4 pink squares in 1 row of the quilt.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 21
Extra Practice Problem 22
Problem
Party Hats and Horns
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?
Possible Strategies
Suggested Strategy
Guess and Check
Make a Model
Guess and Check
Suggested Tools
Snap Cubes®
Solution
15 horns, 5 hats
Find the solution.
Suggested Strategy: Guess and Check
Facilitate the use of this strategy if students need guidance.
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.
1st guess
10 and 10
10 horns and 10 hats
The 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.
2nd guess
14 and 6
14 horns and 6 hats
The 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.
3rd guess
15 and 5
15 horns and 5 hats
The total cost of
these items:
15($0.10) + 5($0.20) =
$1.50 + $1.00 = $2.50
$2.50 is correct.
© ETA hand2mind™
Tyrone bought 15 horns and 5 hats.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 22
Extra Practice Problem 23
Problem
Crazy 25 Relay Race
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Table
Make a Drawing or Diagram
Make a Table
None
150 meters
Find the solution.
Suggested Strategy: Make a Table
Facilitate the use of this strategy if students need guidance.
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
Distance
1
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
Distance
1
50 m
2
75 m
Continue this pattern until you have completed the table through Runner 5.
Runner
Distance
1
50 m
2
75 m
3
100 m
4
125 m
5
150 m
© ETA hand2mind™
The fifth runner of the relay team ran 150 meters.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 23
Extra Practice Problem 24
Problem
Wood Carving
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Drawing or Diagram
Make a Model
Make a Drawing or Diagram
Two-Color Counters
Color Tiles
A square
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students
need guidance.
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.
© ETA hand2mind™
The 15th figure is a square.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 24
Extra Practice Problem 25
Problem
How Much Money?
Madeline has $27. She has only $5 bills and $1 bills. If she has 7 bills, how many of each bill does she have?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Guess and Check
Make a Model
Guess and Check
Replica Money
Five $5 bills, two $1 bills
Find the solution.
Suggested Strategy: Guess and Check
Facilitate the use of this strategy if students need guidance.
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 guess
2 and 5
2 $5 bills and 5 $1 bills
2($5) + 5($1) = $10 + $5 = $15
$15 is too low.
Try again by using more $5 bills.
6 and 1
6 $5 bills and 1 $1 bill
6($5) + 1($1) = $30 + $1 = $31
$31 is too high.
Try again by using less $5 bills.
5 and 2
5 $5 bills and 2 $1 bills
5($5) + 2($1) = $25 + $2 = $27
$27 is correct.
© ETA hand2mind™
Madeline has five $5 bills and two $1 bills.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 25
Extra Practice Problem 26
Problem
Buying Lunch
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Model
Act 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.
© ETA hand2mind™
Sigourney can purchase 4 lunches with $1 left over.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 26
Extra Practice Problem 27
Problem
Model Planes
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make a Drawing or Diagram
Make a Drawing or Diagram
None
4 planes
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students need guidance.
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
4
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
4
4
2
2
4
2
4
2
4
2
4
2
4
2
4
= 28 in.
2
4 close 2to 26 you4 can get.
= 28 in.
Write another
4, a2 line, and4 a 2. Continue
this4 pattern2to see how
4
4
4
4
4
2
2
2
4
4
4
2
2
2
4
4
4
2
2
2
4
4
4
2
2
2
4
= 28 in.
= 24 in.
= 24 in.
Add the4numbers2on the line.
for example,
up to 28 inches,
4 This shelf,
2
4
2is too long
4 because
2 the pattern adds
= 24 in.
2
4
2
4
2
4
2
4
2
= 26 in.
4
2
and the shelf
is
only
26
inches
long.
2
4
2
4
2
4
2
4
2
26 in.
= 28
4
2
4
2
4
2
4
2
4
Take away
2 the last4 plane. 28
2 – 4 = 24
4 inches. 2Four planes
4 fit on2the shelf.4
4
4
2
2
4
4
2
2
4
4
2
2
4
4
2
2
4
2
4
2
4
2
4
2
2
4
2
4
2
4
2
4
2
4
= 26 in.
=
= 28
24 in.
in.
= 24 in.
4 can fit 2
4
2
4 a 2-inch
2 space. 4Draw a horizontal
2
= 26line
in. and
Check to2see if you
more planes
by starting
with
write a 2, a line, and a 4. Continue this pattern to see how close to 26 you can get.
= 26 in.
© ETA hand2mind™
2
This placement also holds 4 planes.
The greatest number of planes that can fit on the 26-inch shelf is 4.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 27
Extra Practice Problem 28
Problem
Books and Boxes
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?
Possible Strategies
Suggested Strategy
Make a Drawing or Diagram
Make a Model
Make a Drawing or Diagram
Suggested Tools
Snap Cubes®
Solution
6 boxes (3 boxes of math
books, 3 boxes of spelling
books)
Find the solution.
Suggested Strategy: Make a Drawing or Diagram
Facilitate the use of this strategy if students need guidance.
Sample solution:
Draw rectangles to represent the boxes.
Draw 2 rectangles.
Write 7 in one of the rectangles.
This is a box of math books.
7
7
7
A box of spelling books has 4 times as many books in it: 7 x 47= 28.
Write 28 on the other rectangle.
7
The total number of books in these boxes is 35: 7 + 28 = 35. 28
7
28
28
Continue this pattern
7 until the total
7 number of7 books is 105.
7
7
28
28
7
28
28
7
28
28
7
7
28
28
28
35 + 35 + 35 = 105
© ETA hand2mind™
Sandeep can pack 3 boxes of math books and 3 boxes of spelling books, or 6 boxes in all, to hold 105 books.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 28
Extra Practice Problem 29
Problem
What’s Your Favorite Color?
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
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Make or Use a Graph
Make or Use a Graph
Color Tiles (Rainbow)
Pink
Find the solution.
Suggested Strategy: Make or Use a Graph
Facilitate the use of this strategy if students need guidance.
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
yellow
purple
red
pink
blue
The tallest column of Xs is the most popular color.
© ETA hand2mind™
Pink is the most popular color.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 29
Extra Practice Problem 30
Problem
Football Passes
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?
Possible Strategies
Suggested Strategy
Suggested Tools
Solution
Act It Out
Make a Drawing or Diagram
Act It Out
Object that can be tossed or
handed around
12 times
Find the solution.
Suggested Strategy: Act It Out
Facilitate the use of this strategy if students need guidance.
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.
© ETA hand2mind™
There were 12 passes.
Paths to Problem Solving™ Teacher Resource Pages
Extra Practice Problem 30