Section B: Looking at Combinations 14T
Hints and Comments
Materials
Student Activity Sheet 4, optional.
Overview
Students read the Summary that emphasizes the benefits and usage of a combination chart. Then students copy a given combination chart that they will use to solve problems 2-4 on the next page.
Students use these Check Your Work problems as self-assessment. The answers to these problems are also provided in the Student Book.
About the Mathematics
In this section, the combination chart is the chief strategy for solving problems that involve systems of equations.
Planning
Have students read the Summary and then start with the Check your Work problems. You may assign these problems as homework.
Comments About the Solutions
1. Note that students copy the table and encourage them to fill in only the numbers they need to solve problems 2–4 on the next page instead of spending a lot of time filling in the complete chart.
Solutions and Samples
Looking at Combinations B
2. How many tickets are needed for two rides on the Loop-D-Loop and three rides on the Whirlybird?
3. Janus has 19 tickets. How can she use these tickets for both rides so that she has no leftover tickets?
4. a. On your combination chart, mark a move from one square to another that represents the exchange of one ride on the Whirlybird for two rides on the Loop-D-Loop.
b. How much does the number of tickets as described in 4a, change as you move from one square to another?
5. Use the combination chart on Student Activity Sheet 3.
a. Write a story problem that uses the combination chart.
b. Label the bottom and left side of your chart. Give the chart a title and include the units.
c. What do the circled numbers represent in your story problem?
Do you think combination charts will always have a horizontal and vertical pattern? Why or why not? What about a pattern on the diagonal?
50 52 54 56 58 60 40 42 44 46 48 50 30 32 34 36 38 40 20 22 24 26 28 30 10 12 14 16 18 20
0 2 4 6 8 10
Notes
For Further Reflection Reflective questions are meant to summarize and discuss important concepts.
Reaching All Learners
Extension
Puzzles involving charts can be used as warm-up activities. Challenge students to create a puzzle that is impossible to solve or that has more than one solution. Students should be able to explain why the puzzle has no solution.
Assessment Pyramid
4b, 5a, FFR 2, 3, 4a, 5bc Assesses Section B Goals
15 Comparing Quantities
Section B: Looking at Combinations 15T c. The circled entry 16 stands for the number of
people traveling on three motorcycles and one minibus. The circled entry 40 stands for the number of people traveling in four minibuses.
For Further Reflection Sample response:
There will always be a horizontal and vertical pattern since the number of quantities increases by the same amount in each direction (one pencil, one eraser, one child, and so on). There should always be some diagonal patterns. For example, an increase of 1 item in each direction should result in the same combined amount.
Solutions and Samples
2. 16 tickets
Different strategies are possible:
• In the chart, you can see that for two Loop-D-Loop rides you need 10 tickets, and for three Whirlybird rides you need six tickets. So altogether you need 16 tickets.
• You can draw arrows that go up one square and to the right one square, like on the chart above.
This move adds seven tickets, and 7 9 16.
3. Janus can go on three Loop-D-Loop rides and two Whirlybird rides or one Loop-D-Loop ride and seven Whirlybird rides.
When you fill out the chart, each entry is either greater or less than 19 except for those two combinations. So all of the other combinations are for either too many or too few tickets.
4. a. Different charts are possible. You should draw an arrow that goes down one square and to the right two squares, like on the chart below.
b. The number of tickets increases by eight.
5. Discuss and check your answers to problem 5 with a classmate.
One example of a story:
a. A motorcycle holds two people, and a minibus holds 10 people.
b.
Numbers of Tickets
0 0
1 2
2
3 5
4 5
7
6 7
9 11 13 15 17 19
8 12
14 16 18 20
9 15 20
22 24 26
25 27 29
30 32 34
35 37 39
40 42
45 47
50 17
19 21 23 25
10 4
6 8 10 12 14 16
10
Number of Loop-D-Loop Rides Number of Whirlybird Rides
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5
0 1 2 3 4 5
Number of Motorcycles Number of People
Number of Minibuses
12 22 32 42 52 54
24 34 44
36 46 56
48 58 60
0 2 4
14 6 16 26
8 18 28 38
10 20 30 40 50
10 20 30 40 50
Hints and Comments
Materials
Student Activity Sheet 3;
Student Activity Sheet 4, optional.
Overview
Students continue working on the Check Your Work problems.
Planning
After students complete Section B, you may assign for homework appropriate activities in the Additional Practice section, located on pages 34–35 of the Student Book.
Comments About the Solutions
5. Given a combination chart, students write a story problem that provides a context for the numbers in the chart. The combination chart does not have labels. This will give students the opportunity to show their understanding of combination charts.
16A Comparing Quantities Teachers Matter
Teachers Matter C
Section Focus
This section focuses on exchange strategies to determine the price of individual items in a purchase. Students also use combination charts to solve equation-like problems.
Pacing and Planning
Day 7: Price Combinations Student pages 16 and 17
INTRODUCTION Problems 1–3 Use exchange strategies to determine
the price of a pair of shorts and a pair of sunglasses given the cost of two separate purchases.
CLASSWORK Problems 4 and 5 Determine the price of an umbrella and
a baseball cap given the costs of two separate purchases.
HOMEWORK Problems 6 and 7 Use exchange strategies to determine the
price of a pencil and a clipboard given the total cost of two separate purchases.
Day 8: Price Combinations (continued) Student pages 18–21
INTRODUCTION Problem 8–10 Use a combination chart to determine
the price of individual items within a single purchase.
CLASSWORK Problems 11–13 Create an equation-like problem and
solve it using a combination chart.
HOMEWORK Check Your Work Student self-assessment: Solve
For Further Reflection various equation-like problems using a combination chart.
Additional Resources: Algebra Tools, pages 1, 2, 4–6; Additional Practice, Section C, Student Book pages 35 and 36
Teachers Matter Section C: Finding Prices 16B
Teachers Matter C
Materials
Student Resources
Quantities listed are per student.
•
Student Activity Sheet 4 Teachers Resources No resources required Student Materials No materials required* See Hints and Comments for optional materials.
Learning Lines
In the previous section, students used the value of one item of each kind to find the value of a com-bination. In this section, students use the value of the combination of items to find the value of one item of each kind. Students create new combina-tions from given ones by adding, subtracting, or extending patterns.
Systems of Equations
In each context problem in this section, students are given two linear equations (in picture or story form) with two unknowns, which are the prices of the two items. For instance, students are given the total price of two umbrellas and a cap and the total price of two caps and an umbrella and must find the price of one cap and one umbrella.
Fair Exchange
To solve this sample problem, students can use exchange strategies: Compared to the first picture, in the second one there is one pair of sunglasses less but two pairs of shorts more. Note that the total price is the same for both combinations.
Thus, one pair of sunglasses can be exchanged for two pairs of shorts. When in the last picture one
pair of sunglasses is exchanged for two pairs of shorts, the result will be: five pairs of shorts for a total price of $50.
Combination Chart
As students relate these pictures to patterns in a combination chart, they discover how patterns in the chart are related to patterns in a succession of equations, like the pattern in the picture equa-tions of the glasses and shorts.
At the End of the Section:
Learning Outcomes
Students have further developed reasoning skills to solve equation-like problems. Students use exchanging, substituting, a combination chart, or other strategies to solve a system of equations, which is presented in pictures or in words.
Students create new combinations from given ones by adding, subtracting, or extending pat-terns. They are able to interpret a mathematical solution in terms of the problem situation.
$50.00
$50.00
$50.00
0 1 2 3 4 5
50
0 1 2 3 4 5
50 20
50 40 30 20 10
Number of Shorts Costs of Combinations
(in dollars)
Number of Glasses
Finding Prices C
So far you have studied two strategies for solving problems that involve combinations of items. The first strategy, exchanging, applied to the problems about trading food at the beginning of the unit. The second strategy was to make a combination chart and use number patterns found in the chart.
In this section, you will apply the strategy of exchanging to solve problems involving the method of fair exchange.