lesson 15: deriving the equation of a line objectiventnmath.kemsmath.com/level h lesson notes/grade...

Mathematics Success – Grade 8 T379

LESSON 15: Deriving the Equation of a Line

[OBJECTIVE]The student will derive the equation of a line and use this form to identify the slope and y-intercept of an equation.

[PREREQUISITE SKILLS]Slope

[MATERIALS]Student pages S185–S197Ruler or straight-edgeColored paper (1 sheet per student for foldable)

[ESSENTIAL QUESTIONS]1. Where is the slope represented in the equation of a line? 2. Where is the y-intercept represented in the equation of a line? 3. What is the slope-intercept form of the equation of a line? Explain the meaning

of the equation.

[WORDS FOR WORD WALL]y-intercept, equation of a line: y = mx, y = mx + b, undefined slope, zero slope, horizontal, vertical

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)*For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson.

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer, Algebraic Formula, Graph

[WARM-UP] (IP, WG) S185 (Answers on T391.)Have students turn to S185 in their books to begin the Warm-Up. Students will identify the unit rate or slope represented by equations, tables or graphs. Monitor students to see if any of them need help during the Warm-Up. After students have completed the Warm-Up, review the solutions as a whole group. {Graphic Organizer, Graph, Table, Algebraic Formula}

[HOMEWORK]Take time to go over the homework from the previous night.

[LESSON][1 – 2 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 8T380

SOLVE Problem (WG, GP) S186 (Answers on T392.)

Have students turn to S186 in their books. 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 how to derive the equation of a line and use this form to identify the slope and y-intercept of an equation. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer}

Slope as Rise over Run (M, GP, CP, WG) S186 (Answers on T392.)

M, GP, CP, WG: After students complete the S Step of the SOLVE problem, direct them to the chart at the bottom of the page. Students will use this chart to review the concept of rise over run as a way to determine slope. Be sure to identify Partners A and B for activities. {Verbal Description, Graphic Organizer, Pictorial Representation, Graph}


MODELING

Slope as Rise over Run

Step 1: Direct students’ attention to the first graph on S186. • Partner A, what are we looking for as we examine the line on the graph?

(the slope of the graph or the vertical change over the horizontal change)

• Partner B, what is another word for the vertical change? Justify your answer. (The rise is the vertical change from one point on the graph to another point on the graph.)

• Let’s identify two points on the line that are on the graph: (1, 1) and (2, 3).

• Partner A, what is the vertical change from (1, 1) to (2, 3)? Explain your answer. [2 units – The vertical change from (1, 1) to (2, 3) is 2 units] Record as the rise.

• Partner A, what is another word for the horizontal change? (The run is the horizontal change from one point on the graph to another point on the graph.)

• Partner B, what is the horizontal change from (1, 1) to (2, 3)? Explain your answer. [1 unit – The horizontal change from (1,1) to (2, 3) is 1 unit] Record as the run.

• Partner A, can we simplify 2 over 1? (Yes) What is the slope? (2) Record.

Step 2: Direct students’ attention to the second graph on S186. • Partner B, what are we looking for in this graph? (rise over run) • Let’s identify two points on the line that are on the graph: (-3, 5) and

(0, 3). Plot the points on the line.



MODELING

Derive the Equation of a Line

Step 1: Have students direct their attention to Step 1 on S187. • Partner A, explain how we can complete the table. (Substitute the

x-value into the function to arrive at an output value, y.) • Partner B, if we substitute -2 in for x, what is our result? Explain your

thinking. [3(-2) = -6] Record. • Have students complete the table by substituting in the x-values in

the equation y = 3x. • Review the answers in the table as a class.

Step 2: Direct students to Step 2. • Have student pairs discuss how to plot the line in Step 2. (Graph the

line by plotting the points we created.) • Partner A, how do we identify the points from the table? (Use the

x-coordinate and the y-coordinate to create five ordered pairs.) • Have students plot the points from Step 1 and then review what the

graph should look like as a class.

Step 3: Direct students’ to Step 3. • Partner A, what is Step 3 asking us to find? (the rise over the run, or

the slope)

Derive the Equation of a Line (M, GP, IP, CP, WG) S187, S188, S189 (Answers on T393, T394, T395.)

M, GP, CP, WG: Have students turn to S187 in their books. Use the following activity to help students derive the equation of the line. Be sure students know their role as Partner A or Partner B. {Verbal Description, Graphic Organizer, Pictorial Representation, Graph, Algebraic Formula}

• Partner A, what is the vertical change from (-3, 5) to (0, 3)? (-2 units) Record as the rise.

• Partner B, what is the horizontal change from (-3, 5) to (0, 3)? (3 units) Record as the run.

• Partner B, what is the rise over the run? (-2 3) Record.

• The rise over the run is one way to identify and write the (slope) of the line.

• In algebra we use the letter (m) to represent slope. • For Questions 1 and 2, go back and write the slope by using the

notation of “m =.”


• Partner B, what is one way that we can find the rise over the run? (We can count the vertical and horizontal distance between two points on the graph and write the values as a ratio of rise over run.)

• Partner A, what is another way we can find the rise over the run? (We can use two points by subtracting their y-coordinates and subtracting their x-coordinates.)

• Have students use one of the methods to complete Step 3.

*Teacher Note: The solution provided uses two points to find the rise over the run, but the rise and run are marked on the graph.

Step 4: Direct students to Step 4. • Partner B, what is Step 4 asking us to find? (where the graph crosses

the y-axis) • Give students a moment to identify where their graphs cross the

y-axis. [(0, 0)] Record. • Partner A, what is the second part of Step 4 asking us to find? (the

y-intercept) • Have student pairs discuss their ideas about the meaning of the

y-intercept. • Partner B, what does “intercept” sound like? (intersect) • Partner A, what do you predict a y-intercept is? (a place where the

graph intersects the y-axis) • Partner B, what is the y-intercept (b)? Explain your thinking. (Zero,

because this is the point where the line crosses the y-axis.) Record. • According to the last question, what other notation can we use to

describe a y-intercept? (b)

Step 5: Direct students to Part B on the bottom of S187. • Complete the table, graph, slope and y-intercept questions for Part B

using the questioning from Steps 1 – 4. • Have students work in pairs to complete Part C and Part D on S188.

After students have completed the table, graph, slope and y-intercept questions for both problems, review the answers as a whole group.

Step 6: Direct students to look at the chart on the top of S189. • Ask students to work together to complete the table at the top.

Students will copy their solutions from Parts A – D into the table.

Step 7: Direct students to the “Slope” organizer. • Partner A, which step did we use to find the slope? (Step 3) • Partner B, which step did we use to find the y-intercept? (Step 4) • Partner A, what do you notice about the slopes for Parts A and B?

(They are the same; they are both 3.) Record.




Identifying Slope and y-intercepts to Write Equations (M, GP, CP, IP, WG) S190 (Answers on T396.)

M, GP, CP, WG: Students will derive the general equation for a line. After developing that concept, students will be able to identify the slope and y-intercept from an equation. Additionally, they will be given slopes and y-intercepts and will be asked to write the related equation. Be sure students know their roles as Partner A and Partner B. {Algebraic Formula, Verbal Description, Graphic Organizer}

MODELING

Identifying Slope and y-intercepts to Write Equations

Step 1: Have students discuss Questions 1 and 2. • Partner B, based on the conclusions we made on S189, what is

a general equation that can be used for the equation of a line? (y = mx + b or y = mx) Record.

• Partner A, what does the x represent? (the independent variable) Record. • Partner B, what does the y represent? (the dependent variable) Record.

• Partner B, what do you notice about the slopes for Parts C and D? (They are the same; they are both -2.) Record.

• Partner A, what do you notice about the equations for Parts A and B? (They both have a 3 as a coefficient for x, the independent variable.) Record.

• Partner B, what do you notice about the equations for Parts C and D? (They both have a -2 as a coefficient for x, the independent variable.) Record.

• What can you conclude about the slope within an equation of a line? (The slope will be the coefficient of the x, or the independent variable.) Record.

Step 8: Direct students to the “y-intercept” organizer. • Partner A, what do you notice about the y-intercepts for Parts A and

C? (They are the same; they are both 0.) Record. • Partner B, what do you notice about the y-intercepts for Parts B and

D? (Part B’s is 1 and Part D’s is -3.) Record. • Partner A, what do you notice about the equations for Parts A and C?

(They do not have anything added to or subtracted from x.) Record. • Partner B, what do you notice about the equations for Parts B and D?

(Part B has 1 added to x and Part D has 3 subtracted from x.) Record. • What can you conclude about the y-intercept within an equation of a

line? (The y-intercept will be added to or subtracted from x.) Record.


• Partner A, what does the m represent? (the slope of the equation) Record.

• Partner B, what does the b represent? (the y-intercept of the equation) Record.

*Teacher Note: If the y-intercept is 0, the equation of the line will be y=mx.

Step 2: Direct students’ attention to the first table on S190. • Partner A, what are we asked to find in this activity? (the slope and

y-intercept for each equation) • Have partners look at Question 3. Partner B, where is the slope located?

(It is the coefficient of x. It is 2.) Record. • Partner A, where is the y-intercept located? (It is added to or subtracted

from the x. It is -5.) Record.

Step 3: Direct students to Question 6. Partner A, what is different about this equation? (There is not an x in the equation.)

• Partner B, what is the coefficient of x: (0) Explain. (Zero times the x is equal to zero.)

• Partner A, what is the slope of this line? (0) Record. • Partner B, what is the y-intercept? (9) Record.

Step 4: Direct students to Question 7. Partner A, what is different about this equation? (The equation is not in y = mx + b form.)

• Partner B, where is the slope always located? (It is always the coefficient of the independent variable.) What is the slope? (-3) Record.

• Partner A, what is the y-intercept? (6) Record.

Step 5: Direct students’ attention to the second table on S190. • Partner B, what are we asked to find in this activity? (the equation of

the line) • Have students look at Question 9. Partner B, what does the m

represent? (slope) • Partner A, where is the slope located? (It is the coefficient of x.) • Partner B, what does the b represent? (y-intercept) • Partner B, where is the y-intercept located? (It is added to or subtracted

from the x.) • Partner A, how can we write the equation for this line? (y = 5x + 4)

Record.


IP, CP, WG: Students will complete both of the tables. Remind students that they derived the general form of an equation of a line in Question 1 and they may refer to that as they complete the tables. Take a few moments to review student solutions as a whole group. {Algebraic Formula, Verbal Description, Graphic Organizer}



MODELING

Equations for Horizontal and Vertical Lines

Step 1: Have students name the four points on the line for the first graph. [(0, 4), (1, 4), (-2, 4), (3, 4)] Record.

• Partner A, how are the points similar? (The y-coordinates are all 4.) Record.

• Choose two coordinates and find the slope. [Example uses (0, 4) and (1, 4), The slope is 0.] Record.

• Partner B, what is the y-intercept of the line? (4) Record. • What is the equation of the line? (y = 0x + 4, or y = 4) Record. • This is a (horizontal) line. • Have student pairs draw another horizontal line anywhere on the

graph and select two points on that line to determine the slope. Have student pairs share the y-intercept of their line and the slope.

• What is the slope of any of the horizontal lines? (The slope of all (horizontal) lines is zero slope.) Record.

• Have students discuss what it means that the slope is zero. (When the slope is positive the line goes up from left to right and when the slope it negative it goes down from left to right. When a line is horizontal is does not go up or down, so the value of the slope is zero.)

• The equation of all (horizontal) lines is in the form (y = a number). Record.

Step 2: Have students name the four points on the line for the second graph. [(1, 3), (1, 2), (1, 1), (1, 0)] Record.

• Partner B, how are the points similar? (The x-coordinates are all 1.) Record.

• Choose two coordinates and find the slope. [Example uses (1, 1) and (1, 0), The slope is undefined because we are dividing by 0.] Record.

• Partner A, what is the y-intercept of the line? (There is no y-intercept; the line does not cross the y-axis.) Record.

• What is the equation of the line? (x = 1) Record. • This is a (vertical) line. • Have student pairs draw another vertical line anywhere on the graph

and select two points on that line to determine the slope. Have student pairs share the equation of their lines. (The slope is undefined because we are dividing by 0.)

Equations for Horizontal and Vertical Lines (M, GP, CP, WG) S191 (Answers on T397.)

M, GP, CP, WG: Students will explore more with equations by looking at equations for horizontal and vertical lines. Be sure students know their role as Partner A and Partner B. {Verbal Description, Pictorial Representation, Graph, Algebraic Formula}


• The slope of all (vertical) lines is (undefined). Record. • Have students discuss what it means that the slope is undefined.

(When the slope is positive the line goes up from left to right and when the slope is negative it goes down from left to right. A vertical line has an undefined slope because all points on the line have the same x-coordinate. As a result the formula used for slope has a denominator of 0, which makes the slope undefined.)

• The equation of all (vertical) lines is in the form (x = a number). Record.

MODELING

Using a Graph to Write the Equation of a Line

Step 1: Direct students’ attention to S192. Have students take a look at the top left box.

• Partner A, what two pieces of information do we need to write the equation of a line? (the slope and the y-intercept)

• Partner B, how can we quickly identify the y-intercept? Explain your answer. (We can find where the graph intersects the y-axis.)

• Partner A, what is the y-intercept? Justify your thinking. (2; This is where the line crosses the y-axis.) Record.

• Partner A, how can we find the slope of the line? (Identify another point on the line and use the rise over run method.)

• Partner B, what is the additional point that we can identify on the line? [(1, 4)]

• Partner A, what is the rise from the y-intercept to (1, 4)? (2) Record. • Partner B, what is the run from the y-intercept to (1, 4)? (1) Record. • What is the slope of the line? (2 over 1 which simplifies to 2) Record. • Have partners discuss how they can write the equation of the line

using the slope and y-intercept. (y = mx +b) • Substitute in the values and write the equation. (y = 2x + 2) Record.

Step 2: Direct students’ attention to the top right box on S192. • Partner B, what two pieces of information do we need to write the

equation of a line? (the slope and the y-intercept)


Using a Graph to Write the Equation of a Line (M, GP, CP, IP, WG) S192 (Answers on T398.)

M, GP, CP, WG: Students will expand their knowledge of equations and lines by using a graph to identify the slope and y-intercept, then use this information to write the equation of a line. Be sure students know their roles as Partner A and Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer, Graph, Algebraic Formula}



MODELING

Using an Equation to Graph a Line

Step 1: Direct students’ attention to S193. • Partner B, what is the slope of the equation? (5) Explain your answer.

(It is the coefficient of x.) • Partner A, what is the y-intercept of the equation? (1) Defend your

answer. (It is added to x.) • Partner B, what does the y-intercept tell us? (It tells us the point

where the graph crosses the y-axis.) • Partner A, what is the ordered pair for the y-intercept? [(0, 1)] Plot

the point. • Partner B, what does the slope tell us? (It tells the rise over run to use

to get to the next point on the line.)

• Partner A, how can we quickly identify the y-intercept? (We can find where the graph intersects the y-axis.)

• Partner B, what is the y-intercept? (-3) Record. • Partner B, how can we find the slope of the line? (Identify another

point on the line and use the rise over the run method.) • Partner A, what is the additional point that we can identify on the line?

[(4, -2)] • Partner B, what is the rise from the y-intercept to (4, -2)? (1) Record. • Partner A, what is the run from the y-intercept to (4, -2)? (4) Record.

• What is the slope of the line? (14) Record.

• When we substitute in your values for slope and the y-intercept, what is our equation? (y = 1

4x – 3) Record.

IP, CP, WG: Students will complete the bottom two sections on S192. They will continue to write equations for the lines that are graphed. Take a moment to review the solutions with students and be sure they understand how to translate the graph of a line to an equation. {Graphic Organizer, Pictorial Representation, Algebraic Formula, Graph}

Using an Equation to Graph a Line (M, GP, CP, IP, WG) S193, S194 (Answers on T399, T400.)

M, GP, CP, WG: Students will complete this section to learn how to translate from an equation to a graph. Encourage students to work backward from the previous section as they think about this activity. Be sure students know their roles as Partner A and Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer, Algebraic Formula, Graph}


• Partner A, is there a way we can write 5 in the form of rise over run? (Yes, 5 over 1.)

• Partner B, what does the 5 tell us to do? (Rise 5 units.) Have students place their pencils on the y-intercept and from there, count up 5 units.

• Partner A, what does the 1 tell us to do? (Move 1 unit to the right.) Have students drag their pencils one unit to the right and plot a point there.

• If we want to continue the line to the left, what movement should we use starting from the y-intercept? (Move 5 units down and one unit to the left.) Have students plot a point here.

• Connect the points and graph the line.

Step 2: Direct students’ attention to the bottom of S193. • Take a look at the second equation. Let’s analyze this equation before

we begin starting to graph. Have students make predictions of what they think the line will look like.

• Partner A, what is the slope of the equation? (0) How do you know? (There is no x in the equation.)

• Partner B, what is the y-intercept of the equation? (-3) How do you know? (It is the constant number that would be added to x if we write the slope as 0x.)

• Partner A, what does the y-intercept tell us? (It tells us the point where the graph crosses the y-axis.)

• Partner B, what is the ordered pair for the y-intercept? [(0, -3)] Plot the point.

• Partner A, what does the slope tell us? (It tells the rate, or the rise over the run, to use to get to the next point on the line.)

• Partner B, is there a way we can write 0 in the form of rise over run? (Yes, 0 over 1.)

• Partner A, what does the 0 tell us to do? (We do not need to move up or down to get to the next point.)

• Partner B, what does the 1 tell us to do? (Move 1 unit to the right.) Have students drag their pencils one unit to the right and plot a point there.

• If we want to continue the line to the left, what movement should we use starting from the y-intercept? (Move one unit to the left.) Have students plot a point here.

• Connect the points and graph the line. Students should identify this line as a horizontal line.




MODELING

Foldable

Step 1: Model for students as they create the foldable. • Have students put the paper flat on the desk, horizontally. Fold the

top down so that the top flap is a little shorter than the bottom flap.

Step 2: Fold the paper from left to right into thirds. Cut along the dotted lines on the top flap only to make three sections.

• On the first section on the left write, “Slope.” On the second flap write, “Deriving the Equation of a Line.”

Slope Deriving the Equation of a Line

Step 3: Lift the first flap and examine the graph. Write the steps for determining the slope of a line using the rise over the run.

• Lift the second flap and write the directions for determining the y-intercept and the slope of the line. Use that information to write the equation of the line.

run

rise

run

rise

IP, CP, WG: Students will complete S194 with a partner. They will continue to graph lines from the given equation. Take a moment to review the solutions with students and be sure they understand how to translate the equation of a line to a graph of the line. {Verbal Description, Graphic Organizer, Pictorial Representationm, Algebraic Formula, Graph}

Foldable (WG, GP, M)

Give each student one piece of colored paper. Use the following instructions to create a foldable for Slope and Writing the Equation of a Line. {Verbal Description, Graphic Organizer, Graph, Algebraic Formula}

lesson 15: deriving the equation of a line objectiventnmath.kemsmath.com/level h lesson notes/grade...

Documents