Apples
Order Milk Sandwich Total
$3.40
$
$
1 0
1 1
1 4.20 0
2 1
1 2.80 1
3 4 5 6 7 8 9 10
0
$3.40
$4.20
$2.80
8. In your own notebook, make new combinations until you can determine the price of each item.
Reaching All Learners
Intervention
Some students might not remember what they did to get the next row. If this happens, ask them to explain their steps to the right of the notebook entries. Numbering the lines and recording the operations performed will help students organize their work. You might suggest that students add an extra column and label it “Orders.”
Accommodation
Provide copies of this page or Student Activity Sheet 5.
Assessment Pyramid
8
8
Informally solve problems involving systems of equations using notebook notation.
25 Comparing Quantities
Notes
8Point out that students should begin this problem by putting the information pictured at the bottom of the page into the chart.
If students struggle, point out that this problem can be solved using the same strategy they used in the chicken problem
Have students share their strategies for this problem with the class
Section D: Notebook Notation 25T
Hints and Comments
Materials
Student Activity Sheet 5, optional
Overview
Students use notebook notation to solve a problem involving restaurant orders made up of combinations of three different items.
About the Mathematics
As students manipulate rows in their notebooks, they seek to make a combination with only one kind of item in it. This strategy corresponds to moving to the edges in the combination chart and is similar to strategies used in Section C in which students created combinations of only one item.
Comments About the Solutions
8. This problem assesses students’ ability to use exchanging, substituting, and other strategies to solve problems and to interpret and use
combination charts, notebook notation, and equations.
Solutions and Samples
8. One apple costs $1.00; one milk costs $1.80; one sandwich costs $2.40. Strategies will vary. Sample strategies:
• Some students may use notebook notation as shown below.
• Some students may use the pictures.
Combine pictures 2 and 3 and then subtract picture 1 to find that two milk cost $3.60. So one milk costs $1.80. Substituting that price in picture 2, one sandwich costs $2.40. Also, substituting the $1.80 price for one milk in picture 3, one apple costs $1.00.
1 2 3 4 5 6 7 8
Apples Milk Sandwich Total
1 0 1 $3.40
0 1 1 $4.20
1 1 0 $2.80
2 2 2 $10.40
1 1 1 $5.20
1 0 0 $1.00
0 1 0 $1.80
0 0 1 $2.40
row 1 row 2 row 3 row 4 2
row 5 row 2 row 3 row 6 row 1 row 6
Notebook Notation
D D Notebook Notation
In this section, you explored notebook notation as a good way to get an overview of the information contained in a problem. You can make new combinations in a notebook by:
• adding rows;
• finding the difference between rows; and
• doubling or halving rows; and so on.
The new combinations you create can help you find solutions to new problems.
You can write these combinations of fruits in notebook notation.
1. In your own notebook, make new combinations until you find the price of each item.
$1.10
$1.20
$1.30
Price of Combinations
Apple Banana Pear Price
0 1 1 $1.30
Notes
Students often think they can skip the Summary since there are no problems attached.
Some suggested strategies to over come this:
• Read the Summary aloud to the class or ask a stu-dent to volunteer to read it aloud.
• Ask students to rewrite the Summary in their own words.
• Ask students to find examples of problems in the section that represent the concepts reviewed in the Summary.
Reaching All Learners
Parent Involvement
Have parents review the section with their child to relate the Check Your Work problems to the problems in the section.
Assessment Pyramid
1
Assesses Section D Goals
26 Comparing Quantities
Section D: Notebook Notation 26T
Hints and Comments
Overview
Students read and discuss the Summary and start to solve the first problem of the Check Your Work section. The problem is about using the notebook notation method. Students use these Check Your Work problems as self-assessment. The answers to these problems are also provided in the Student Book.
Solutions and Samples
Answers to Check Your Work
1. Discuss your solution with a classmate.
Different strategies are possible. For example:
Add all combinations (rows 1–3) in the chart.
Then subtract one of the first three rows from row 4.
In this example, the price of one apple is found by subtracting row 1 from row 5.
Answers:
One apple costs $0.50.
One banana costs $0.60.
One pear costs $0.70.
1 2 3 4 5 6
A B P Price
0 1 1 $1.30
1 1 0 $1.10
1 0 1 $1.20
2 2 2 $3.60
1 1 1 $1.80
1 0 0 $0.50
Notebook Notation D
2. Study the following notebook showing lunch orders at Mario’s restaurant.
a. Find the cost of one salad.
Explain how you got your answer.
b. How can you find the cost of one drink? One taco?
3. Can you solve problem 2 by using a combination chart?
Why or why not?
TACO ORDER
1 2 3 4 5 6 7
SALAD DRINK TOTAL 3.00
4 4
4 1
2 8.00
— 2
— 1
11.00
$
$$
Write a description to tell an adult in your family about all of the ways you have learned so far in this unit to solve problems. Show him or her examples of what you have learned.
Notes
For Further Reflection Reflective questions are meant to summarize and discuss important concepts.
Reaching All Learners
Intervention
Students struggling with problem 2a may need the hint to double line 1 to create line 4.
Assessment Pyramid
3, FFR
2
Assesses Section D Goals
27 Comparing Quantities
Section D: Notebook Notation 27T
Hints and Comments
Overview
Students solve more problems about the notebook notation. They compare the notebook notation method to the use of a combination chart, and finish the problems in the Check Your Work section.
Planning
After students complete Section D, you may assign for homework appropriate activities in the Additional Practice section, located on pages 37 of the Student Book.
Solutions and Samples
2. a. One salad costs $2. You may have doubled the first order and then subtracted this from the second order to find the price of a salad, as shown.
b. One drink costs $0.75. One taco costs $1.50.
Compare your work with a classmate’s work.
A sample strategy might be as follows.
From answer a you know that a salad costs
$2.00.
In order 3, there were four salads: 4 $2 $8.00.
The price of the order was $11.00, so four drinks cost $11.00 $8.00 $3.00.
$3.00 4 $0.75 is the price of one drink.
In order 1:
1 taco 2 drinks $3.00 1 taco 2 x $0.75 $3.00 1 taco $1.50 $3.00 So one taco costs $1.50.
3. No, a combination chart cannot be used to solve the problem. A combination chart can be used only for a combination of two items.
For Further Reflection
Although student responses will vary, their descriptions should be illustrated with examples of the various solution strategies such as fair exchange, combination chart, and notebook notation.
Taco
Order Salad Drink Total
$ 3.00
--1
1 2
$ 8.00 1
2 2 4
$ 11.00 4
--3 4
$ 6.00
--2
4 4
$ 2.00 1
--5
--6 7
2
28A Comparing Quantities Teachers Matter
Teachers Matter E
Section Focus
Students write equations to represent combinations of quantities.
They transfer equations into notebook notation and combination charts to solve problems.
Pacing and Planning
Day 12: The School Store Revisited Student pages 28 and 29
INTRODUCTION Problems 1 and 2 Describe the meaning of an equation.
CLASSWORK Problems 3–5 Transfer equations into notebook notation
and combination charts.
HOMEWORK Problems 6–8 Use exchange strategies to rewrite
equations and solve problems.
Day 13: Hats and Glasses (Continued) Student pages 29 and 30
INTRODUCTION Problem 9 Write equations that represent multiples
of a specific combination.
CLASSWORK Problems 10–15 Write equations to represent problems
that involve combinations of quantities.
HOMEWORK Problems 16 and 17 Use notebook notation or exchange
strategies to rewrite equations and find the price of individual items.
Day 15: Tickets Student pages 31–33
INTRODUCTION Problems 18 and 19 Determine the cost of tickets to a movie theater by solving equation-like problems.
CLASSWORK Check Your Work Student self-assessment: Solve systems of
equations for various problem contexts.
HOMEWORK For Further Reflection Reflect on the advantages of representing problems in various forms.
Additional Resources: Additional Practice, Section E, Student Book page 38
Teachers Matter Section E: Equations 28B
Teachers Matter E
Materials
Student Resources
Quantities listed are per student
•
Student Activity Sheet 5 Teachers Resources No resources required Student Materials No materials required* See Hints and Comments for optional materials.