PRIMARY CONTENT MODULE Algebra - Linear Equations ...

1 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-37/H-37 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONWhat does the number m in y = mx + bmeasure?To find out, suppose (x1, y1) and (x2, y2) are twopoints on the graph of y = mx + y1 = mx1 + b and y2 = mx2 + Algebra to simplify y2 y1x2 x1 And give a geometric this! PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-38 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONA nswer:y2 y1x2 x1=mx2+b() mx1+b()x2 x1=mx2 mx1+b bx2 x1=mx2 mx1x2 x1=m(x2 x1)x2 x1 distributive property=mNo matter which points (x1,y1) and (x2, y2) arechosen, m = y2 y1x2 what does this mean?

2 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-39 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONM eaning of m = y2 y1x2 x1 in y = mx + bm = y2 y1x2 x1 is the rise ( y2 y1) over the run ( x2 x1) andm is called the slope. (x2, y2)y2 y1x2 x1(x1, y1) PRIMARY CONTENT MODULEA lgebra I - Linear Equations & InequalitiesT-40/H-40 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeFind the slope, m, of the line whose graphcontains the points (1,2) and (2, 7).

3 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-41 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING Equations olutionm = y2 y1x2 x1 = 7 22 1m = 51m = 5 The rise over the run, or slope, of the line whosegraph includes the points (1,2) and (2,7) is CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-42 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONWhat does it mean if the slope, m, is negative iny = mx + b?The negative slope means that y decreases asx some examples.

4 X2 x1(x2,y2)y2 y1(x1,y1)m = y2 y1x2 x1 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-43 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONxy = -2xy = -2x + 2y = -2x 20-2 0 = 0-2 0 + 2 = 2-2 0 2 = -21-2 1 = -2-2 1 + 2 = 0-2 1 2 = -4y = 2xy = 2x 2y = 2x + 2 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-44/H-44 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONDEFINITIONSD efinition 1In the equation y = mx + b for a straight line, thenumber m is called the slope of the 2In the equation y = mx + b for a straight line, thenumber b is called the y-intercept of the CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-45 1999, CISC.

5 Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONM eaning of the y-intercept, b, iny = mx + bLet x = 0, then y = m 0 + b,so y = number b is the coordinate onthe y-axis where the graph crosses PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-46 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xample:y = 2x + 3 What is the coordinate on the y-axis where thegraph of y = 2x + 3 crosses y-axis?Answer: 33 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-47 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONThe Framework.

6 The following idea must be clearlyunderstood before the student can progressfurther:A point lies on a line given by, forexample, the equation y = 7x + 3, ifand only if the coordinates of thatpoint (a, b) satisfy the equation whenx is replaced with a and y is replacedby b. (page 159)Review this statement with the people at yourtable and discuss how you would present this tostudents in your CONTENT MODULEA lgebra - Linear Equations & InequalitiesH-48 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONV erify whether the point (1,10) lies on the liney = 7x + CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-48 1999, CISC.

7 Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONV erify whether the point (1,10) lies on the liney = 7x + : If a point lies on the line, its x and ycoordinates must satisfy the x = 1 and y = 10 in the equationy = 7x + 3, we have 10 = 7 1 + 310 = 10 which is true, therefore the point (1,10)lies on the line y = 7x + CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-49/H-49 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeTell which of the lines this point (2,5) lies on:1.

8 Y = 2x + 12. y = 12x + 43. y = 3x + 14. y = 3x + 15. y = 4x + 13 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-50 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xampleSuppose we know that the graph of y = 2x + bcontains the point (1, 2).What must the y-intercept be?Answer: Substitute x = 1 and y = 2 iny = 2x + b, and then solve for = 2 1 + b2 = 2 + b4 = b b = 4 PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-51/H-51 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeFind b for the given lines and points on y = 3x + b;(2,7)2.

9 Y = 5x + b;( 1, 3)3. y = 12x + b;(4,5) PRIMARY CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-52/H-52 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONG raph y = 3x + 1 by plotting two points andconnecting with a straight CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-53/H-53 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xample: y = 2x 5. Use the properties of they-intercept and slope to draw a CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-54 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING Equations olution:Use b.

10 In the equation y = 2x 5, the y-intercept, b, is 5. This means the line crossesthe y-axis at 5. What is the x coordinate forthis point?The coordinates of one point on the line are(0, 5), but we need two points to graph a ll use the slope to locate a second the equation, we see that m = 2 = 21. Thistells us the rise over the run . We will moveover 1 and up 2 from our first point. The newpoint is (1, 3). rise of 2 run of 1 Verify that (1, 3)satisfies the CONTENT MODULEA lgebra - Linear Equations & InequalitiesT-55 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING Equations tandard 7 Algebra I, Grade 8 StandardsStudents verify that a point lies on a line givenan equation of a line.