1Have students share their strategies for this problem. This will help students solve problem 2.
3There are many answers to this problem. Have students share some of the possibilities. This
reinforces the importance of valuing multiple strategies.
Reaching All Learners
Intervention
Students struggling with problem 1 may need you to suggest a strategy.
Suggest that they list multiples of $0.25 and $ 0.15 and see if they can find a combination that equals $1.10.
Extension
Ask students to find all possible amounts up to 10 pencils and 10 erasers.
English Language Learners
Read the text on this page aloud to English language learners and discuss the context of the school store.
6 Comparing Quantities
Section B: Looking at Combinations 6T
Hints and Comments
Overview
Students determine how many pencils and erasers were purchased at a school store, given the total amounts of money spent on each purchase.
About the Mathematics
Some students may use the letters E and P to stand for the prices of the eraser and pencil. Some students may mostly use guess-and-check. Students are expected to perform the multiplication without a calculator.
Comments About the Solutions
2. Students may not realize that more than one combination is possible. By working in groups, students will discover various solutions. Talking about possible purchases prepares them for problem 3.
3. Some students may realize that there are countless possibilities. Some may notice that three erasers can be exchanged for five pencils (each collection is worth 75 cents).
Some students might build on problem 2 and find
$3.00 and $0.75 (using a doubling and halving strategy) and also $6.00, $4.50, and so on.
Solutions and Samples
1. Four pencils and two erasers. Strategies will vary.
Sample strategies:
• Some students may first multiply and then add different combinations of the products.
Erasers Pencils
1 $0.25 $0.25 1 $0.15 $0.15 2 $0.25 $0.50 2 $0.15 $0.30 3 $0.25 $0.75 3 $0.15 $0.45 4 $0.25 $1.00 4 $0.15 $0.60 5 $0.15 $0.75 6 $0.15 $0.90 7 $0.15 $1.05 After checking various combinations of erasers and pencils, students will see that two erasers and four pencils ($0.50 $0.60) come to a total of $1.10.
• Some students may organize their work in a table and systematically search for the desired total.
2. The amount $1.50 can be obtained three different ways: six erasers, three erasers and five pencils, or ten pencils. Explanations will vary. Students may use one of the strategies shown in problem 1.
3. Answers will vary. Sample student responses:
• $1.90. Seven erasers and one pencil, or four erasers and six pencils, or one eraser and eleven pencils can be bought for $1.90.
• $2.00. Eight erasers, or two erasers and ten pencils, or five erasers and five pencils can be bought for $2.00.
Total Pencils
Erasers
4 erasers, 1 pencil $1.00 $0.15 $1.15 3 erasers, 2 pencils $0.75 $0.30 $1.05 2 erasers, 3 pencils $0.50 $0.45 $0.95 2 erasers, 4 pencils $0.50 $0.60 $1.10
Combination
Looking at Combinations
B
Looking at Combinations BMonica wants to make finding the total price of pencils and erasers easier, so she makes two price lists: one for different numbers of erasers and one for different numbers of pencils.
4. Copy and complete the price lists for the erasers and the pencils.
Erasers Price
0 $0.00
1 $0.25
2 $0.50
3 $0.75
4 $1.00
5 $1.25
6 7
Pencils Price
0 $0.00
1 $0.15
2 $0.30
3 4 5 6 7
One day the box has $1.05 in it.
5. Show how Monica can use her lists to determine how many pencils and erasers have been bought.
Monica and Martin aren’t satisfied.
Although they now have these two lists, they still have to do many calculations. They are trying to think of a way to get all the prices for all the combinations of pencils and erasers in one chart.
6. Reflect What suggestions can you make for combining the two lists?
Discuss your ideas with your class.
Notes
4You may wish to emphasize that each column begins with zero pencils and zero erasers.
5 and 6Have students share their strategies with the class. Ask students which strategies they like best and why?
Reaching All Learners
Accommodation
Copy the tables on this page so students don’t have to spend the time copying the information into their notebook. Some students may need a calculator to complete the tables.
7 Comparing Quantities
Section B: Looking at Combinations 7T
Hints and Comments
Materials
copy of tables from this page;
calculator, optional
Overview
Students continue solving problems in the context of the school store, using information from two lists. They consider possible methods for combining the lists.
About the Mathematics
The lists for the costs of quantities of erasers and pencils are like multiplication tables. The lists start with zero items and zero cost because combinations need not include both pencils and erasers.
Comments About the Solutions
5. Students may pick a number from one list and find a number from the other list so that the sum is $1.05.
6. Students may be overwhelmed if they do not realize that they need to make a chart. Some creative solutions are shown in the Solutions column.
Solutions and Samples
4.
5. Answers will vary. Students may realize that Monica should look for combinations of prices for different quantities of erasers and pencils that total $1.05. Sample student responses:
• Seven pencils cost $1.05.
• Three erasers and two pencils cost $1.05 (3 $0.25 2 $0.15).
6. Answers will vary.
• Students may suggest making a table.
• Or they may suggest a combination chart.
• Some students may create a systematic list.
1E 0.25 1P 0.15 1P 1E 0.40 1P 2E 0.65 1P 3E 0.90 2P 1E 0.55
Erasers Price
0 $0.00
1 $0.25
2 $0.50
3 $0.75
4 $1.00
5 $1.25
6 $1.50
7 $1.75
Pencils Price
0 $0.00
1 $0.15
2 $0.30
3 $0.45
4 $0.60
5 $0.75
6 $0.90
7 $1.05
E P
0 1
1 0
1 1
1 2
2 0
2 1
2 2
Price
$0.25
$0.15
$0.40
$0.65
$0.30
$0.55
$0.80
0 0 1 2 3 4 5 6
$0.00
$0.25
$0.50
$0.75
$1.00
$1.25
$1.50 1
$0.15
$0.40
$0.65
$0.90
$1.15
$1.40
$1.65 2
$0.30
$0.55
$0.80
$1.05
$1.30
$1.55
$1.80 3
$0.45
$0.70
$0.95
$1.20
$1.45
$1.70
$1.95 4
$0.60
$0.85
$1.10
$1.35
$1.60
$1.85
$2.10
Pencils
Erasers
Looking at Combinations
B
B Looking at CombinationsNumber of Erasers Combination Chart
NumberofPencils
0 1 2 3
0 1 2 3
15 40 0 25
Number of Erasers
Costs of Combinations (in cents)
NumberofPencils
0 1 2 3
0 1 2 3
15 40 0 25
Monica and Martin come up with the idea of a combination chart. Here you see part of their chart.
7. a. What does the 40 in the chart represent?
b. How many combinations of erasers and pencils can this chart show?
If you extend this chart, as shown below, you can show more combinations.
Use the combination chart on Student Activity Sheet 1 to solve the following problems.
8. Fill in the white squares with the prices of the combinations.
9. Circle the price of two erasers and three pencils.
Notes
After students have explored the chart on their own, you may want to discuss the patterns in the chart.
Make sure all students can find the patterns.
7bYou may want to point out that each square represents a combination.
Reaching All Learners
Parent Involvement
You may suggest that students explain the combination chart to their families. Students can make up prices of two items and use these prices to partially complete a combination chart. They then can ask family members to fill in prices for certain squares using patterns in the numbers.
Accommodation
Provide graph paper or copies of Student Activity Sheet 4 so students do not have to create their own combination charts.
8 Comparing Quantities
Section B: Looking at Combinations 8T
Hints and Comments
Materials
Student Activity Sheet 1 (one per student);
graph paper, optional;
Student Activity Sheet 4, optional
Overview
Students interpret and use the combination chart to find prices of combinations of pencils and erasers.
About the Mathematics
Using a combination chart is one of the major strategies of the unit for solving systems of equations with two unknowns. Many patterns can be recognized in this chart. Each move from one square to another represents an exchange of items and a change in total value. A similar chart was used in the unit Expressions and Formulas to organize calculations with numbers that did not represent quantities of items.
Comments About the Solutions
7. b. Some students may compute 16 by multiplying 4 4, while other students may need to count.
Students should begin to keep the meaning of the numbers in mind as they use the chart.
8. Students struggling with this problem may need to fill in all the white squares so they can look for patterns. Some students may need to fill in the gray squares to find the value of the square for 8 erasers and 7 pencils. Other students may use the pattern to find this value.
9. In describing this location, students might use the words row and column. Be sure they understand that in a row, the numbers are arranged in a horizontal line. In a column, the numbers are arranged in a vertical line.
Solutions and Samples
7. a. The number 40 represents the cost of
purchasing one eraser and one pencil in cents.
b. 16 combinations, including zero pencils and zero erasers
8.
9. The price of two erasers and three pencils is $0.95, circled in the chart above.
Number of Erasers Costs of Combinations (in cents)
Number of Pencils
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7
30 45 60 75
50 75 100 125 150 175 65 90 115 140 165 190 80 105 130 155 180 95 120 145 170 195 110 135 160 185 210 125 150 175 200 140 165
155
55 70 85 100 115
305
15 40 0 25
Looking at Combinations
B
Looking at Combinations BNumber of Erasers
Number of Pencils
Number of Erasers
Number of Pencils
Number of Erasers
Number of Pencils
a b
Use the numberpatternsin your completedcombination charton Student Activity Sheet 1 to answer problems 10–16.
10. a. Where do you find the answer to problem 1 ($1.10) in the chart?
b. How many erasers and how many pencils can be bought for $1.10?
11. a. Reflect What happens to the numbers in the chart as you move along one of the arrows shown in the diagram?
b. Reflect Does the answer vary according to which arrow you choose? Explain your reasoning.
12. What does moving along an arrow mean in terms of the numbers of pencils and erasers purchased?
13. a. Mark on your chart a move from one square to another that represents the exchange of one pencil for one eraser.
b. How much does the price change from one square to another?
14. a. Mark on your chart a move from one square to another that represents the exchange of one eraser for two pencils.
b. How much does the price change for this move?
15. Describe the move shown in charts a and b below in terms of the exchange of erasers and pencils.
16. There are many other moves and patterns in the chart. Find at least two other patterns. Use different color pencils to mark them on your chart. Describe each pattern you find.
Notes
These problems lead students to reinforce the meaning of the
combination chart. You may want to discuss these problems after students finish working on them.
10Have student use a colored pencil and circle the answer on Student Activity Sheet 1.
13bPoint out that this question is referring to the move they made in 13a.
Reaching All Learners
Intervention
If students are having difficulty with problem 11, you may choose a starting point and demonstrate that moving along any diagonal produces the same result.
English Language Learners
You may want to work with your English language learners in a small group reading each question to them aloud. Be sure they understand each
problem before moving on to the next one.
Vocabulary Building
Have students add the terms combination chart and patterns to the vocabulary section of their notebook.
Assessment Pyramid
16
Discover, investigate, and extend patterns.
9 Comparing Quantities
Section B: Looking at Combinations 9T
Hints and Comments
Materials
Student Activity Sheet 1 (one per student);
colored pencils (same assortment of colors for each student)
Overview
Students continue exploring the combination chart.
They describe patterns in the chart in terms of the pencils-and-erasers context.
About the Mathematics
Moving the same way on the combination chart always results in the same increase or decrease no matter where you start. For example, moving down one square and to the right two squares always increases the number by 35. Noticing patterns can prepare students to understand symbolic expressions.
Comments About the Solutions
10. This problem will reveal any misconceptions students have about using the combination chart.
12.–14.
Students may get so involved with the numbers in the chart that they forget their meanings. These problems help them relate the patterns they find in the chart to the combinations of erasers and pencils.
16. Be sure that students can explain the pattern in the chart in terms of the costs of erasers and pencils.
Solutions and Samples
10. a.
b. Two erasers and four pencils 11. a. The numbers increase by 40.
b. No, each arrow should give the same answer.
All of the arrows show a move of “one square up” and “one square to the right.”
12. One more pencil and one more eraser are purchased, for an increase of $0.40.
13. a.
b. This would result in an increase of $0.10.
14. a.
b. This would result in an increase of $0.05.
15. a. An eraser is exchanged for a pencil. This results in a decrease of 10 cents.
b. A pencil is exchanged for 2 erasers. This results in an increase of 35 cents.
16. Answers will vary. See students’ charts.
Explanations will vary. Sample explanations:
• Decreasing the pencils by one and increasing the erasers by one, increases the price by 10 cents.
• Decreasing the pencils by one and decreasing the erasers by one, decreases the price by 40 cents.
• Increasing the pencils by one and increasing the erasers by two, increases the price by 65 cents.
Number of Erasers
Number of Pencils
0 1 2 3
0 1 2 3
4 110
Number of Erasers
Number of Pencils
Number of Erasers
Number of Pencils
Looking at Combinations B
Anna and Dale are going to remodel a workroom. They want to put new cabinets along one wall of the room. They start by measuring the room and drawing this diagram.
Anna and Dale find out that the cabinets come in two different widths:
45 centimeters (cm) and 60 cm.
17. How many of each cabinet do Anna and Dale need in order for the cabinets to fit exactly along the wall that measures 315 cm?
Try to find more than one possibility.
Looking at Combinations
B