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					                            TEKS A.6 CF
                  Minding your m’s and b’s
TAKS Objective 3
The student will demonstrate an understanding of linear functions.

TEKS Math Concepts (A.6)
Linear functions. The student understands the meaning of slope and intercepts of the
graphs of linear functions and zeros of linear functions and interprets and describes the
effects of changes in parameters of linear functions in real-world and mathematical
situations. The student is expected to:
               (A)     develop the concept of slope as a rate of change and determine
                       slopes from graphs, tables, and algebraic representations;
               (B)     interpret the meaning of slope and intercepts in situations using
                       data, symbolic representations, or graphs;
               (C)     investigate, describe, and predict the effects of changes in m
                       and b on the graph of y  mx  b ;
               (D)     graph and write equations of lines given characteristics such as two
                       points, a point and a slope, or a slope and y-intercept;
               (E)     determine the intercepts of the graphs of linear functions and zeros
                       of linear functions from graphs, tables, and algebraic expressions;
               (F)     interpret and predict the effects of changing slope and y-
                       intercept in applied situations; and
               (G)     relate direct variation to linear functions and solve problems
                       involving proportional change.



Overview
Students can graph linear equations written in the form y  mx  b . Students can examine
patterns in the graphs to determine the effects of changing the values of m and b in the
equation y  mx  b.




TAKS Objective 3      page 1                                            TEKS A.6C&F
Instructional Strategies
Discovery & Cooperative Learning, Direct Teaching
Students will work in partners to graph linear functions and discover patterns that
illustrate the effects of increasing/decreasing the slope, m, and the y-intercept, b, have on
the graph of y  mx  b .
Students will extend these patterns they discover to reason what would happen in real-
world situations when a rate of change occurs or the initial starting conditions are
changed.



Lesson Objectives
   1. Students will identify the effects of increasing/decreasing the slope, m, has on the
      graph of a linear equation, y  mx  b.
   2. Students will identify the effects of increasing/decreasing the y-intercept, b, has
      on the graph of a linear equation, y  mx  b.
   3. Given a real-world scenario, students will describe what changing m or b means
      in the context of the problem.



For Teacher’s Eyes Only
This lesson is built on the previous lesson (TEKS A.6 AB How does your slope rate?)
of developing an understanding of slope as the rate of changes that is represented as
a ratio of y to x. This lesson emphasizes the slope intercept form of y=mx+b where m
represents the ratio of y to x.

Changing Slope
Changing the value of the slope of a line has the following effects:
    m  0 line rises from left to right (increasing); as m gets larger the graph of the
       line gets steeper
       m  0 line falls from left to right (decreasing); as m gets smaller the graph of the
       line gets flatter (less steep)
       m  0 line has no rise  horizontal line
       m = undefined ; line has no run  vertical line (recall this is NOT a function)




TAKS Objective 3       page 2                                             TEKS A.6C&F
The y-intercept was introduced in the previous lesson (TEKS A.6 AB How does
your slope rate?) with Job #2. The ‘What Have I Earned? #2’ is meant to build on
the meaning of what is a y-intercept.
Changing y-intercepts
     As b increases, the graph of the line translates up
     As b decreases, the graph of the line translates down




Misconceptions
 Misconception
In the formula, y  mx  b , b represents the x-intercept.


 Mathematics Concept
b represents the y-intercept on the graph of the function and can be found on the y-axis.


Rebuild Concept
Provide students with practice locating and identifying the y-intercept of a linear
function.




 Misconception
A line with a negative slope, for example a slope of -4, is less steep than a line with a
positive slope, for example a slope of 4.


 Mathematics Concept
The steepness of a line is determined by the m . So a line with slope of -4 has the same
steepness as a line with slope of 4 since 4  4  4 . The positive and negative sign only
identify whether the line is rising (increasing) or falling (decreasing).


TAKS Objective 3       page 3                                             TEKS A.6C&F
Rebuild Concept
Provide students with practice identifying the slopes of various lines and the
characteristics of the line that result from the value of the slope.


Student Prior Knowledge
Students have prior experience working with the concept of slope as a rate of change as
well as identifying the slope of a line given the graph of the line (TEKS A.6.A). Students
also have prior experience working with making conjectures and predictions about
functional relationships (TEKS A.1.E).




       Materials
          Activity Sheets
          Map Colors
          Graphing Calculators
          Rulers




TAKS Objective 3       page 4                                           TEKS A.6C&F
                                    5 E’s
ENGAGE
The learner is introduced to a new experience and must draw from prior experiences to
make sense of the engage activity.

   1. Show students Display Sheet #1 Story Lines. Have students work in pairs to read
      and answer the prompt.
   2. Once students have had a chance to answer the prompt, have students share their
      answers with the whole class.




EXPLORE
During the explore activity, the student becomes directly involved with a particular
phenomena by manipulation of materials that are used to discover the phenomena.

   1. Students will need to work in pairs. Distribute a copy of Activity Sheet #1
      Charting Slopes and y-intercepts to each student.
   2. One student in the partner pair will complete Sheet A while the other student
      completes Sheet B.
   3. Once they are finished, both students will compare and discuss their data and then
      answer the questions (#4-#6) that follow.
   4. Once all students have had time to complete the activity, analyze the activity
      using the following questions:
          a. As the slope increases, what happened to the graph of the linear equation?
          b. As the slope decreases, what happened to the graph of the linear equation?
          c. As the y-intercept increases, what happens to the graph of the linear
              equation?
          d. As the y-intercept decreases, what happens to the graph of the linear
              equation?


TAKS Objective 3      page 5                                           TEKS A.6C&F
EXPLAIN
The student communicates in verbal and written form about the information derived from
the learning experience.

In discussing the results of Activity Sheet #1 Charting Slopes and y-intercepts with the
students, ask the students to verbalize what effect changing the slope had on the graph of
 y  mx  b .

As the slope, m, in y  mx  b is changed the “steepness” of the graph changes. This idea
of steepness can be related to how quickly the data is increasing or decreasing. Point out
that both negative and positive slope can characterized the same if we talk about the
change in the m . As m increases, the graph of the line becomes steeper. Absolute value
bars are used to cover both cases when m is a positive and a negative value (when the line
is rising and falling). The sign of the slope simply tells us whether the line is rising or
falling (whether we are increasing or decreasing).

As the y-intercept, b, in y  mx  b is changed the position of the line is changed. As b
increases the graph of the line is translated up on the y-axis. As b decreases the graph of
the line is translated down on the y-axis.

In terms of real-world situations, changes in slope indicate a rate of change in the original
situation has increased or decreased. A change in the y-intercept indicates that an initial
starting condition has changed. Tie these concepts back to Display Sheet #1 Story Lines.
Ask students to elaborate more about their scenarios in terms of changing slopes and y-
intercepts and what they would mean in the context of the scenario.




ELABORATE
During the elaboration phase, students expand their knowledge by making connections
about what they have learned and applying this new knowledge to real world situations.

   1. Distribute Activity Sheet #4 Savings, Slope, and Intercept to each student.
   2. Have students work in pairs following the directions on the activity sheet to
      answer the questions.
   3. After students have had an opportunity to complete the questions, debrief the
      activity.

TAKS Objective 3       page 6                                             TEKS A.6C&F
EVALUATE
Evaluation throughout the learning experience is an ongoing process and has a
diagnostic function.

   1. Distribute Activity Sheet #5 Changing Slopes and Intercepts to each student.
   2. Students will work individually to complete this activity.




TAKS Objective 3     page 7                                         TEKS A.6C&F
TAKS Objective 3   page 8   TEKS A.6C&F
                            A        B




                                      C




Pick any 2 lines from the graph and
write a scenario to model the relations
represented by each linear graph. Be
sure to identify the 2 lines you use in
creating your scenario.




TAKS Objective 3   page 9       TEKS A.6C&F
                        Charting Slopes and y-intercepts

You will work in partners for this activity. Each person will have their own chart
and a graphing calculator.


1. Record the slope and y-intercept of each equation in your chart using a map
   color. You will need a different map color for each equation.

2. Enter each equation into your calculator and graph it. Using the calculator
   (graph or table of values) plot the y-intercept and draw the graph of your
   equation using the same map color that you used to record the slope and the
   y-intercept. Note: all of your lines will be graphed on the same coordinate
   grid.

3. Compare your chart with your partner’s chart. Do you notice any patterns?

You and your partner should answer the following questions together.

4. What happens to the graph of your equation as your slope (m) values in your
   equation increase? decrease?

5. What happens to the graph of your equation as your y-intercept (b) values in
   your equation increase? decrease?

6. What effect does the sign (positive/negative) of the slope have on the graph
   of your equations?




TAKS Objective 3     page 10                                       TEKS A.6C&F
                        Charting Slopes and y-intercepts
                           Record Sheet A – Partner I

Part I
 Equation    Slope   y-intercept
  y x
    1
 y  x
    2
     1
y  x
     2
    1
 y  x
    4
     1
y  x
     4
 y  2x

 y  2x

 y  4x

 y  8x

 y  8x


Part II
 Equation    Slope   y-intercept
  y x

 y  x 2

y  x 5

y  x 10

 y  x 3

y  x 6

y  x  10




TAKS Objective 3     page 11                               TEKS A.6C&F
                       Charting Slopes and y-intercepts
                          Record Sheet B – Partner II

Part I
 Equation   Slope   y-intercept
  y x

 y  x
    2
 y   x
    3
     2
y  x
     3
    1
 y  x
    6
     1
y  x
     6
 y  3x

 y  3x

 y  5x

 y  5x


Part II
 Equation   Slope   y-intercept
  y x

 y  x 2

y  x 5

y  x 8

 y  x 3

y  x 6

 y  x 8




TAKS Objective 3    page 12                               TEKS A.6C&F
                         Savings, Slope, and Intercept

You decide to open a savings account to start saving money for college. After
your first week at your new job, you take $50 of your paycheck and open a
savings account. You continue to deposit $50 at the end of each week into your
savings account.

1. Draw a graph to represent this situation. Label the horizontal axis as “weeks”
   and the vertical axis as “dollars.”




2. What is the slope of the line that you have drawn? What does it represent?


3. What is the y-intercept of your line? What does it represent?


4. If you increased the amount of money that you save each week, how would
   your graph change? What part of the graph does this represent?


5. If you decrease the amount of money that you save each week, how would
   your graph change? What part of the graph does this represent?


6. Suppose you already had a savings account before you got your job. This
   savings account already had $200 in it. How would this change your graph?
   What part of the graph does this represent?




TAKS Objective 3    page 13                                        TEKS A.6C&F
Now suppose instead of putting money into your bank account, you are
withdrawing money. You start with $500 in your account. You withdraw $10 each
day with your debit card for lunch.

7. Draw a graph to represent this situation. Label the horizontal axis as “weeks”
   and the vertical axis as “dollars.”




8. What is the slope of the line that you have drawn? What does it represent?


9. What is the y-intercept of your line? What does it represent?


10. If you increased the amount of money that you withdraw each week, how
    would your graph change? What part of the graph does this represent?


11. If you decrease the amount of money that you withdraw each week, how
    would your graph change? What part of the graph does this represent?


12. Suppose you started with $200 in the savings account instead of $500. How
    would this change your graph? What part of the graph does this represent?




TAKS Objective 3    page 14                                        TEKS A.6C&F
                            Changing Slopes and Intercepts

        1. How does changing the value of m affect the graph of y  mx  b ?




        2. How does changing the value of b affect the graph of y  mx  b ?




     On the left you are given the graph of a linear equation. Some changes are made
     to the slope and the y-intercept. Redraw the graph using the information that you
     are given.

3.




                                   Slope is increased by 3
                                    y-intercept stays the
                                             same




                                                             Equation___________________
Equation___________________




     TAKS Objective 3    page 15                                     TEKS A.6C&F
4.




                                  Multiply the slope by -1
                                   y-intercept stays the
                                            same




Equation___________________                                  Equation___________________
5.




                                   Slope stays the same
                                   y-intercept increases
                                           by 4




Equation___________________                                  Equation___________________
6.




                                      Slope is doubled
                                   y-intercept increases
                                            by 2




Equation___________________                                  Equation___________________


     TAKS Objective 3   page 16                                     TEKS A.6C&F

				
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