A Final Tip
25 Models You Can Count On
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Hints and Comments
Comments About the Solutions
For Further Reflection
This problem reveals students’ informal understanding of the relative nature of percents. You may refer the students to problem 10 on page 16: this situation is similar: Tank A is fuller but holds less water than Tank B.
Solutions and Samples
c. Answer: 80 minutes
Sample strategy:
Calculate 5% (divided by 3), then calculate 50%
(times ten), and then calculate 100% (double).
d. Answer: 1__14 hours or 75 minutes. Different strategies are possible:
Example 1:
Calculate 40% (halving), and then calculate 20% (halving) and then 100% (times 5).
Example 2:
Using minutes:
Strategy:
Calculate 40% (halving), then calculate 20%
(halving), then 10% (halving), and then 100%
(times 10)
For Further Reflection
Answers and explanations will vary. Sample solutions:
Yes, this is possible but only if the bill for Marisa has a higher amount.
Yes, this is possible but unlikely since breakfast is usually cheaper than dinner.
Yes, this is possible; for example, if the total amount Juan had to pay was $10.00, he gave a 20% tip of
$2.00, and if Marisa had to pay $20.00, her 15% tip was $3.00.
Section B: The Bar Model 25T
0% 15% 100%
80 minutes 12
50%
40
5%
4
0% 20% 100%
1
80%
1 4
40%
1
hour 12hour 1 hour 4hours 4. A percent bar using estimates $20 ($20.10) and
$12 ($11.95) may support your estimations.
10% of $20.10 is about $2.00.
15% of $20.10 is about $2.00 $1.00 $3.00.
20% of $20.10 is about $4.00.
10% of $11.95 is about $1.20.
15% of $11.95 is about $1.20 $0.60 $1.80.
20% of $11.95 is about $2.40.
0% 10% 100%
75 minutes 7.5
80%
60
20%
15
40%
30
10% 100%
$20.00
$2.00
20%
$4.00
10%
5% 100%
$12.00
$1.20
$0.60
20%
$2.40 41.MYCCO.SecB.0629.qxd 06/30/2005 02:42 Page 29
Teachers Matter C
Section Focus
26A Models You Can Count On Teachers Matter
Section Focus
The focus of this section is to introduce the number line as a model to locate places using fractions, in the context of distances on a map.
Students are introduced to the empty number line model to make simple computations with decimals. This section also focuses on decimal place value, comparing and ordering decimals.
Pacing and Planning
Student pages 26 –28, and 37 Day 10: Distances
Use information on road signs to locate the exits on a number line.
INTRODUCTION Problems 1 and 2
Additional practice using information in a road sign to locate exits on a number line.
HOMEWORK Check Your Work,
Problems 1 and 2
Use fraction strips to locate fractions on a number line.
CLASSWORK Problems 3–5
Student pages 28–31 Day 11: Signposts
Use information on a signpost to solve problems about distances.
INTRODUCTION Problems 6 and 7
Use a number line to locate road crossings on a trail.
HOMEWORK Problems 11–14
Locate signposts on a map.
CLASSWORK Problems 8–10
Student pages 31–33 Day 12: The Jump Jump Game
Review homework from Day 11.
Introduction to the Jump Jump Game
INTRODUCTION Review homework.
Problems 15 and 16
Create three new Jump Jump Game scenarios.
HOMEWORK Problem 19
Complete many rounds of the Jump Jump Game with a partner.
CLASSWORK Problems 17 and 18
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Teachers Matter C
Teachers Matter Section C: The Number Line 26B
Student pages 34, 35, 38, and 39 Day 13: Guess the Price
Compare and order decimals (item prices) in the context of a game show.
INTRODUCTION Problems 20–22
Use number sense to balance equations involving decimals.
HOMEWORK For Further Reflection
Student self-assessment: Use the empty number line to compare decimals and solve problems.
CLASSWORK Check Your Work,
Problems 3–6
Materials
Student Resources
Quantities listed are per student.
•
Student Activity Sheets 8–10 Teachers ResourcesNo resources required Student Materials No resources required
* See Hints and Comments for optional materials.
Learning Lines
Models
The number line model helps students develop a conceptual understanding of fractions as numbers.
The empty number line is a tool that is based on the number line: the numbers are placed in the correct order, but not necessarily to scale.
Fractions
Students use their experiences from Section A to order, compare, and reason about the location of (benchmark) fractions on a number line. In Section D, students will solve context problems where they operate with fractions informally.
Decimals
In this section, students’ understanding of place value, their ability to compare and order numbers with one decimal is revisited and further developed.
A fraction bar gives visual support for the decimal notation of tenths.
A method for adding numbers uses an empty number line. By moving from one number to another in the fewest jumps and by jumping only 0.1 space, 1 space, or 10 spaces at the time, students are learning to mentally add and subtract decimals.
This method is known as the compensation method.
For example, jump from 1.6 to 2.5 on the number line: a jump of one to 2.6, and then a jump of 0.1 back. So 2.5 1.6 0.9.
Students’ understanding of decimal numbers with two decimals is supported by the context of money.
Changing dollar amounts into cents is a powerful strategy to help understand decimal place value.
Students’ implicit knowledge of decimals will be expanded in Section D.
At the End of This Section: Learning Outcomes
Students will have further developed a conceptual understanding of fractions, and decimals. They order and compare fractions and decimals on a single number line, and use fractions as numbers and as measures.
Additional Resources: Number Tools; Additional Practice, Section C, Student Book pages 63 and 64
1 mile
0.1 mile
1.6 2.5 2.6
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Notes
1 For some students it may be helpful to use paper strips to find the locations. This will help them understand the fraction strips on the next page.
2a If students have difficulty with this problem, they may have overlooked the information that this second sign is located at the Zoo exit.
2b Students can use the number line and the answer for a to help them solve this problem.
The
Number Line C
Part of Highway 22 is the beltway around Springfield. Signs posted along the road show the distances to the exits. Here is one of these signs.
This line represents the beltway. The mark on the left is the sign. The mark on the right is 1 mile (mi) down the road from the sign.
1. a. Copy the drawing above. Use the information on the sign to position each of the three exits onto the line.
b. Which of these two pairs of exits are farther apart?
i. the exit from Town Centre to the Zoo ii. the exit from the Zoo to Rosewood Forest Show how you found your answer.
The next sign along the beltway is posted at the Zoo exit. Some information is missing in the sign on the right.
2. a. Copy this sign and fill in the missing distances. To fill in the airport distance, you need to know that the Rosewood Forest exit is exactly halfway between the Zoo exit and the Airport exit.
b. How far is the Airport exit from the first sign?
c. Place your Airport exit on the line you drew for problem 1a.