Overview of Chapter 2 In this chapter students use linear equations to represent a sequence of calculations and solve those equations by undoing (working.

Overview of Chapter 2In this chapter students use linear

equations to represent a sequence of calculations and solve those equations by undoing (working backwards)

In chapter 3 students will use the linear equations to model linear growth, graphs, and extend their solution techniques to include balancing.

Lesson 2.1 and 2.2Review previous work with proportions and

introduce students to the idea of undoing to solve a proportion.

Lesson 2.3Develops the idea of deriving linear expressions

from measurement conversions with an emphasis on dimensional analysis

Lesson 2.4Introduces direct variation as an alternative to

solving proportionsStudents create scatter plots of real data, draw a

line through the data points, and find an equation y=kx to fit the data.

Lesson 2.5Students are introduced to the topic of an inverse

functionLesson 2.6

Through an activity students explore a real life problem on gears that relates direct and indirect variations.

Lesson 2.7Students practice the rules for order of operations by

analyzing how the steps in linear expressions describe by a number trick undo each other to end up with the same number.

Lesson 2.8Students write linear equations to represent

sequences of steps and solve those equations by undoing.

Proportions

• Rename fractions as decimal numbers• Write ratios and proportions that express

relationships in data• Solve proportions by multiplying to undo

division• Solve proportions by inverting both ratios• Solve problems using proportions• Review skills in working with percents

ProportionsWhen you say “I got 20 out of 24 questions

correct on the last quiz,” what ratio are you describing?

Picture this ratio using colored cubes.What other names can you give this ratio?How do the cubes help you see those

equivalent ratios?What is the equivalent decimal for this

ratio?

Write 20:24 in a fraction. How do you change this ratio to a decimal?

What are some other ways to write this ratio?

Using the Calculator to convert ratiosCalculator Note 0A shows

How decimal numbers can be converted to fractions

How fractions or ratios can be changed decimal numbers

What is a proportion?A proportion is an equation stating that two

ratios are equal. Check to see if the following ratios are

proportions by finding the equivalent decimal for each.

2 83 12

3 122 8

3 2

12 8

12 83 2

Think about itThe variable M stands for an unknown

number. Replace the variable M to make the

following statement true.

23 6

M

Multiply and Conquer

p. 97

Multiply and Conquer

5619 133M

Step 1: Multiply both sides of the proportion

by 19.

Why can you do this?What does M equal?

Multiply and ConquerStep 2: For each equation, choose a number

to multiply both ratios by to solve the proportion for the unknown number. Then multiply and divide to find the missing value.

21.

35 20Q

a 132

.12 176p

b

30.

30 200L

c 130

.78 15

nd

Step 3: Check that each proportion in Step 2 is true by replacing the variable with your answer.

Step 4: In each equation in Step 2, the variables are in the numerator. Write a brief explanation of one way to solve a proportion when one of the numerators is a variable.

Step 5: The proportions you solved in Step 2 have been changed by switching the numerators and denominators. That is, the ratio on each side has been inverted.

(You may recall that inverted fractions, like and are called reciprocals.)

Do the solutions from Step 2 also make these new proportions true?

12p

12p

35 20.

21a

Q

12 176.

132b

p

30 200.

30c

L

78 15.

130d

n

How can you use what you just discovered to help you solve a proportion that has the variable in the denominator, such as

Why does this work?

Solve the equation.

20 12135 k

Step 7 There are many ways to solve proportions.

Here are three student papers each answering the question “13 is 65% of what number?”

What are the steps each student followed? What other methods can you use to solve

proportions?

Applying what you’ve learnedJennifer estimates that two out of every

three students will attend the class party. She knows there are 750 students in her class. Set up and solve a proportion to help her estimate how many people will attend.

23 750

a

Students who will attend Students who will attend

Students who are invited Students who are invited

Applying what you’ve learnedAfter the party, Jennifer found out that 70%

of the class attended. How many students attended?

70% is one way to write a ratio. What ratio will it equal?

Solve the ratio:

70100 750

s

70% of the class attended.

750 people were in the class.

How can we find out how many people attended the party?

750 70%

750

70750 100X