Download Lesson 2.1

Overview of Chapter 2
 In 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 2.2
 Review previous work with proportions and
introduce students to the idea of undoing to solve
a proportion.
 Lesson 2.3
 Develops the idea of deriving linear expressions
from measurement conversions with an emphasis
on dimensional analysis
 Lesson 2.4
 Introduces direct variation as an alternative to
solving proportions
 Students create scatter plots of real data, draw a
line through the data points, and find an equation
y=kx to fit the data.
 Lesson 2.5
 Students are introduced to the topic of an inverse
function
 Lesson 2.6
 Through an activity students explore a real life
problem on gears that relates direct and indirect
variations.
 Lesson 2.7
 Students 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 2.8
 Students write linear equations to represent
sequences of steps and solve those equations by
undoing.
Proportions
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•
•
•
•
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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
Proportions
 When 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
ratios
 Calculator 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
8

3 12
3 12

2
8
3
2

12 8
12 8

3
2
Think about it
 The variable M stands for an unknown
number.
 Replace the variable M to make the
following statement true.
2 M

3 6
Multiply and Conquer
p. 97
Multiply and Conquer
Step 1: Multiply both sides of the proportion
by 19.
M
56

19 133
Why can you do this?
What does M equal?
Multiply and Conquer
Step 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 Q
a.

35 20
p
132
b.

12 176
L
30
c.

30 200
130
n
d.

78
15
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.
p
(You may recall that inverted fractions, like
12
12
and
are called reciprocals.)
p
Do the solutions from Step 2 also make
these new proportions true?
35 20
a.

21
Q
12 176
b.

p
132
30 200
c.

L
30
78
15
d.

130
n
How can you use what you just discovered to
help you solve a proportion that has the
variable in the denominator, such as
20
12

135
k
Why does this work?
Solve the equation.
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 learned
 Jennifer 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.
Students who will attend
Students who are invited
2
a

3 750
Students who will attend
Students who are invited
Applying what you’ve learned
 After 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:
70
s

100 750
 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
X
70

750 100