T282
Algebra Success
LESSON 14: Discovering y = mx + b
[OBJECTIVE]
The student will determine the slope and y-intercept of a line by examining the
equation for the line written in slope-intercept form.
[MATERIALS]
Student pages S107–S113
Transparencies T287, T289, T291, T293, T295
Wall-size four-quadrant grid
[ESSENTIAL QUESTIONS]
1. What is the y-intercept of a line?
2. Where are the slope and y-intercept of a line located in the equation for the
line?
[GROUPING]
Cooperative Pairs, Whole Group, Individual
[LEVELS
OF
TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Table, Graph, Algebraic Formula, Verbal Description
[WARM-UP] (5 minutes – IP) S107 (Answers on T286.)
• Have students turn to S107 in their books to begin the Warm-Up. Students will
be using several different methods to find slope. They will be finding slope using
the slope formula, the graph of a line, and a table of values. Monitor students
to see if any of them need help during the Warm-Up. Give students 3 minutes
to complete the problems and then spend 2 minutes reviewing the answers as a
class. {Table, Graph, Algebraic Formula}
[HOMEWORK]: (5 minutes)
Take time to go over the homework from the previous night.
[LESSON]: (45–55 minutes – M, GP, IP)
Algebra Success
T283
LESSON 14: Discovering y = mx + b
SOLVE Problem
(2 minutes – GP) T287, S108 (Answers on T288.)
Have students turn to S108 in their books, and place T287 on the overhead.
The first problem is a SOLVE problem. You are only going to complete the S step
with students at this point. Tell students that during the lesson they will learn to
determine the slope and y-intercept of a line by examining the equation of the
line. They will use this knowledge to complete this SOLVE problem at the end of
the lesson. {SOLVE}
Identify Slope and y-intercept
(13 minutes – M, GP) T287, S108
(Answers on T288.)
Use the following modeling activity to model for students how to identify the slope
and y-intercept of lines using graphs of the lines. Optional: Graph the points on
the wall-size four-quadrant grid. {Table, Graph, Algebraic Formula}
MODELING
Slope and y-intercept
Step 1: Have students use the equation y = 2x in Example 1A to find the
y-coordinates for the given x-coordinates in the table. Model how to
plug in each x-coordinate into the equation and evaluate to find the
corresponding y-coordinate. Complete the table as a class.
Step 2: Have students use each ordered pair from the table to plot points on the
coordinate grid. Explain that for the first ordered pair, (-2, -4), students
should start at the origin (0, 0) and then move left 2 units and down 4
units; for the second ordered pair, (-1, -2), students should start at the
origin (0, 0) and then move left 1 unit and down 2 units; and so on.
Step 3: Model for students how to find the slope of the line using two points on
the line. For example, show that to move from the point at (1, 2) to the
point at (2, 4), students must “rise” 2 units and “run” 1 unit. Remind
students that slope is equal to rise , so the slope of the line is 2 , or 2. Then
run
1
work with students to find the slope of the line using the slope formula:
(1, 2) (2, 4)
x1, y1 x2, y2
m = y2 – y1 = 4 – 2 = 2 = 2
x2 – x1
2–1
1
Step 4: Remind students that the y-intercept is the point where the line crosses
the y-axis. Have students identify the y-intercept of the line in Example
1A. The line crosses the y-axis at 0, so the y-intercept of the line is 0.
Repeat the steps above with students to identify the slope and y-intercept
of the line described by the equation y = 2x + 3 in Example 1B.
T284
Algebra Success
LESSON 14: Discovering y = mx + b
More with Slope and y-intercept
(10 minutes – IP) T289, S109
(Answers on T290.)
Have students complete Example 2 on S109 in cooperative pairs or independently.
Make sure students are using the same method as the one modeled in Example
1. Give your students 8 minutes to complete Example 2 and then use 2 minutes
to review the answers. Make sure students have the correct answers on S109,
because they will be referring to their answers later in the lesson. {Table, Graph,
Algebraic Formula}
Slope and y-intercept from Equations
(12 minutes – M, GP, IP)
T288, T290, T291, S108-S110 (Answers on T292.)
3 minutes – M, GP:
Have students use the information and answers on S108
and S109 (T288 and T290) to complete the first table
on S110 as a class. {Table, Algebraic Formula, Verbal
Description}
3 minutes – IP:
After completing the first table, give your students 3
minutes to complete the second table independently. If
they seem to have trouble, question them about the first
table. For example, use questions such as, “Where was the
slope found on each equation in the first table?” (before
(
the x as the coefficient
cient) “Where was the y-intercept found
on each equation in the first table?” (as the constant)
{Table, Algebraic Formula, Verbal Description}
3 minutes – GP:
Complete the paragraph on S110 with your students. Use
T291 to read the paragraph and fill in the blanks. {Table,
Algebraic Formula, Verbal Description}
3 minutes – GP:
Give students 2 minutes to find the slope and y-intercept
of the line described by each equation at the bottom of
the page. Use 1 minute to review answers. {Algebraic
Formula, Verbal Description}
SOLVE Problem
(8 minutes – GP) T293, S111 (Answers on T294.)
Have students turn to S111 in their books, and place T293 on the overhead.
Remind students that the SOLVE problem is the same one from the beginning
of the lesson. Complete the SOLVE problem with your students. Ask them for
possible connections from the SOLVE problem to the lesson. (The function is in
the form y = mx + b. The slope is m and the y-intercept is b.) {SOLVE, Algebraic
Formula, Verbal Description}
Algebra Success
T285
LESSON 14: Discovering y = mx + b
If time permits...
(10 minutes) T295, S112 (Answers on T296.)
8 minutes – IP:
Have students complete Problems 1–3 on S112
independently. Ask students to identify the slope and yintercept first, using the equation y = mx + b and then
complete the table and graph the points to make sure their
answers are correct. {Table, Graph, Algebraic Formula}
2 minutes:
Review the answers to the problems on S112.
[Closure]: (5 minutes)
• To wrap up the lesson, go back to the essential questions and discuss them with
students.
•
What is the y-intercept of a line? (The point where the line crosses the
y-axis.)
•
Where are the slope and y-intercept of a line located in the equation for the
line? (When an equation is in slope-intercept form, y = mx + b, the slope is
the number in front of x, represented by m, and the y-intercept is the number
without a variable, represented by b.)
[Homework]: Assign S113 for homework. (Answers on T297.)
[Quiz Answers] T298–T299
1. B
2. A
3. A
4. D
5. D
6. B
7. D
8. C
9. B
10. C
The quiz can be used at any time as extra homework or to see how students did on
understanding what y = mx + b represents.
T286
Algebra Success
LESSON 14: Discovering y = mx + b
Here is the key to S107.
Warm–Up
Directions: Find the slope of each of the following.
1. the line that connects the points (4, -2) and (3, -5)
(4, -2) and (3, -5)
x1, y1
x2, y2
y –y
-5 – -2
Slope = 3
-3
m = x2 – x1 = 3 – 4 = -1 = 3
2
1
2. the line graphed on the coordinate
plane below
Down 2 units, to the right
3 units
-2
Slope = 3
3. the line whose table is shown below
x
-1
y
-7
0
-2
1
3
2
8
3
13
As the x-coordinates go up
by 1, the y-coordinates go
up by 5, slope = 5
Algebra Success
T287
LESSON 14: Discovering y = mx + b
TRANSPARENCY MASTER
Directions: Complete the following SOLVE problem with your teacher. You will only
complete the S step.
Dan saves money to buy his favorite video game. Dan begins with $5 and saves
$2 per week. The function y = 2x + 5 represents the amount Dan saves, where y
represents the amount of money that Dan has saved and x represents the number
of weeks that have passed. What is the slope and the y-intercept of the function?
S Underline the question.
This problem is asking me to find ___________________________________.
Directions: For each of the following,
• Substitute each x-value into the equation and evaluate. This will give you the
corresponding y-value.
• Use the ordered pairs to plot points.
• Connect the points to form a line.
• Identify the slope and the y-intercept of the line.
Example 1
A. y = 2x
slope = _____
y-intercept = _____
B. y = 2x + 3
x
-2
-1
y
0
1
2
slope = _____
y-intercept = _____
x
-2
-1
0
1
2
y
T288
Algebra Success
LESSON 14: Discovering y = mx + b
Here is the key to S108.
Directions: Complete the following SOLVE problem with your teacher. You will only
complete the S step.
Dan saves money to buy his favorite video game. Dan begins with $5 and saves
$2 per week. The function y = 2x + 5 represents the amount Dan saves, where y
represents the amount of money that Dan has saved and x represents the number
of weeks that have passed. What is the slope and the y-intercept of the function?
function
S Underline the question.
This problem is asking me to find the slope and y
y-intercept of the function.
Directions: For each of the following,
• Substitute each x-value into the equation and evaluate. This will give you the
corresponding y-value.
• Use the ordered pairs to plot points.
• Connect the points to form a line.
• Identify the slope and the y-intercept of the line.
Example 1
A. y = 2x
slope = 2 = 2
1
y-intercept = 0
B. y = 2x + 3
slope = 2 = 2
x
-2
-1
y
-4
-2
0
1
2
0
2
4
x
-2
-1
0
1
2
y
-1
1
3
5
7
1
y-intercept = 3
Algebra Success
T289
LESSON 14: Discovering y = mx + b
TRANSPARENCY MASTER
Example 2
A. y = 1 x – 1
3
slope = _____
y-intercept = _____
B. y = 1 x + 4
3
slope = _____
y-intercept = _____
C. y = 1 x
3
slope = _____
y-intercept = _____
x
-3
0
3
y
x
-3
0
3
y
x
-3
0
3
y
T290
Algebra Success
LESSON 14: Discovering y = mx + b
Here is the key to S109.
Example 2
A. y = 1 x – 1
3
slope =
1
3
y-intercept = -1
B. y = 1 x + 4
3
slope =
1
3
y-intercept = 4
C. y = 1 x
3
slope = 1
3
y-intercept = 0
x
-3
0
3
y
-2
-1
x
-3
0
3
y
3
4
5
x
-3
0
3
0
y
-1
0
1
Algebra Success
T291
LESSON 14: Discovering y = mx + b
TRANSPARENCY MASTER
Directions: Using your answers from S108 and S109, complete the table below.
Problem
Equation
Slope (m
( )
y-intercept (b
( )
Example 1A
Example 1B
Example 2A
Example 2B
Example 2C
Directions: Make some predictions in the table below.
Equation
Slope (m
( )
y-intercept (b
( )
y = 4x – 3
-1
y= 2x+3
y = -x – 2
y = -5x
x+1
y = mx + b
The general equation for a line in slope-intercept form is ____________. The slope
of a line is represented with the letter_______. It is the coefficient of the variable
x when the equation for the line is in slope-intercept form. The y-intercept of a line
is represented with the letter ______ . It is the constant when the equation for the
line is in slope-intercept form.
Directions: Find the slope and y-intercept of the line described by each equation
below.
1. y = 4x – 7
2. y = 1 x + 4
2
3. y = -2x
x–3
T292
Algebra Success
LESSON 14: Discovering y = mx + b
Here is the key to S110.
Directions: Using your answers from S108 and S109, complete the table below.
Problem
Equation
Slope (m
( )
y-intercept (b
( )
Example 1A
y=2
2x
x
2
0
Example 1B
y=2
2x
x+3
2
3
Example 2A
y = 3x – 1
-1
Example 2B
y = 3x + 4
Example 2C
y = 3x
1
3
1
3
1
3
1
1
1
4
0
Directions: Make some predictions in the table below.
Equation
Slope (m
( )
y = 4x – 3
4
y-intercept (b
( )
-3
-1
2
3
-1
y= 2x+3
y = -x – 2
y = –5x
x+1
-1
-5
-2
y = mx + b
m
b
1
The general equation for a line in slope-intercept form is y = mx + b. The slope of
a line is represented with the letter m. It is the coefficient of the variable x when
the equation for the line is in slope-intercept form. The y-intercept of a line is
represented with the letter b. It is the constant when the equation for the line is in
slope-intercept form.
Directions: Find the slope and y-intercept of the line described by each equation
below.
1. y = 4x – 7
slope = 4, y-intercept = -7
2. y = 1 x + 4
slope = 1 , y-intercept = 4
3. y = -2x
x–3
slope = -2, y-intercept = -3
2
2
Algebra Success
T293
LESSON 14: Discovering y = mx + b
TRANSPARENCY MASTER
Directions: Complete the following SOLVE problem with your teacher.
Dan saves money to buy his favorite video game. Dan begins with $5 and saves
$2 per week. The function y = 2x + 5 represents the amount Dan saves, where
y represents the amount of money that Dan has saved and x represents the
number of weeks that have passed. What is the slope and the y-intercept of the
function?
S Underline the question.
This problem is asking me to find ________________________________.
O Identify the facts.
Eliminate the unnecessary facts.
List the necessary facts.
L Choose an operation or operations.
Write in words what your plan of action will be.
V Estimate your answer.
Carry out your plan.
E Does your answer make sense? (Compare your answer to the question.)
Is your answer reasonable? (Compare your answer to the estimate.)
Is your answer accurate? (Check your work.)
Write your answer in a complete sentence.
T294
Algebra Success
LESSON 14: Discovering y = mx + b
Here is the key to S111.
Directions: Complete the following SOLVE problem with your teacher.
Dan saves money to buy his favorite video game.
game Dan begins with $5 |and
saves $2 per week.| The function y = 2x + 5 represents the amount Dan saves,
|where y represents the amount of money that Dan has saved |and x represents
the number of weeks that have passed.| What is the slope and the y-intercept
of the function?
S Underline the question.
This problem is asking me to find the slope and y-intercept of the
function.
O Identify the facts.
Eliminate the unnecessary facts.
List the necessary facts.
Dan begins with $5.
He saves $2 per week.
The function is y = 2
2x
x + 5.
L Choose an operation or operations.
Write in words what your plan of action will be.
Identify the slope and the y-intercept.
V Estimate your answer. the slope and y-intercept will be either 2 or 5
Carry out your plan.
y=2
2x
x+5
y = mx + b
Slope = 2 and y-intercept = 5
E Does your answer make sense? (Compare your answer to the question.)
Yes.
Is your answer reasonable? (Compare your answer to the estimate.) Yes.
Is your answer accurate? (Check your work.) Yes.
Write your answer in a complete sentence.
The slope is 2 and the y-intercept is 5.
Algebra Success
T295
LESSON 14: Discovering y = mx + b
TRANSPARENCY MASTER
Directions: Identify the slope and the y-intercept from the equation. Then use the
table to find points on the line, graph the points, and check your answers for the
slope and y-intercept.
1. y = x
slope (m
( ) = _____
y-intercept (b
( ) = _____
2. y = x + 4
slope (m
( ) = _____
x
-1
0
1
y
x
-1
0
1
y
x
-1
0
1
y
y-intercept (b
( ) = _____
3. y = x – 3
slope (m
( ) = _____
y-intercept (b
( ) = _____
T296
Algebra Success
LESSON 14: Discovering y = mx + b
Here is the key to S112.
Directions: Identify the slope and the y-intercept from the equation. Then use the
table to find points on the line, graph the points, and check your answers for the
slope and y-intercept.
1. y = x
slope (m
( )=1
y-intercept (b
( )=0
x
-1
0
1
y
-1
0
1
x
-1
0
1
y
3
4
5
x
-1
0
1
y
-4
-3
-2
2. y = x + 4
slope (m
( ) =1
y-intercept (b
( )=4
3. y = x – 3
slope (m
( ) =1
y-intercept (b
( ) = -3
Algebra Success
T297
LESSON 14: Discovering y = mx + b
Here is the key to S113.
Homework
For each of the following, write the slope and y-intercept.
1. y = 1 x + 3
4
2.
Slope = -3, y-intercept = -1
Slope = 1 , y-intercept = 3
4
3. y = -x + 3
4.
Slope = -1, y-intercept = 3
5. y = 5x – 4
Slope = -3, y-intercept = 0
4
4
6.
y = 2x + 1
3
Slope = 2 , y-intercept = 1
3
8.
y = 3x – 5
7
Slope = 3 , y-intercept = -5
Slope = 1, y-intercept = 8
9. y = -3x
x
y = 3x – 2
Slope = 3 , y-intercept = -2
Slope = 5, y-intercept = -4
7. y = x + 8
y = -3x
x–1
7
10.
y = 1x – 6
2
Slope = 1 , y-intercept = -6
2
T298
Algebra Success
LESSON 14: Discovering y = mx + b
Name ___________________________________
Quiz
Date ______________
1. What is the slope of the line described by the function y = 4x + 3?
A. 7
B. 4
C. 3
D. 1
__________________________________________________________________
2. What is the y-intercept of the line described by the function y = -2x
x – 3?
A. -3
B. -2
C. 2
D. 3
__________________________________________________________________
3. Which equation describes a line that has a slope of -1?
A. y = -x + 1
B. y = x – 1
C. y = -2x
x–1
D. y = x
__________________________________________________________________
4. Which equation describes a line that has a y-intercept of 3?
A. y = -3x
x–3
B. y = 3x
x
C. y = 1 – 3
3
D. y = x + 3
__________________________________________________________________
5. What is the y-intercept of the line described by the function y = 3 x?
5
A. 5
B. 3
C. 3
5
D. 0
__________________________________________________________________
6. What is the slope of the line described by the function y = 2 x – 5?
A. -5
B.
-2
3
3
C. 2
3
D. 5
Algebra Success
T299
LESSON 14: Discovering y = mx + b
7. Which of the following is true for the function y = -2x
x + 6?
A. The graph is a line crossing the y-axis at -2.
B. The graph is a line with a slope equal to 6.
C. The graph is a line passing through the point (-2, 6).
D. The graph is a line crossing the y-axis at (0, 6).
__________________________________________________________
8. Which of the following is an equation of a line with a slope of -1?
A. y = x + 1
B. y = x – 1
C. y = -x + 2
D. y = 2x + 2
__________________________________________________________
9. What is the slope of the line described by the function y = x + 2?
A. 2
B. 1
C. 0
D. -1
__________________________________________________________
10. What is the y-intercept of the line described by the function y = 2x + 4?
A. -4
B. 2
C. 4
D. 6