Lesson 3: Solving Problems Using Coordinate Methods

Lesson 3: Solving ProblemsUsing Coordinate MethodsSelected Content StandardsBenchmark Addressed:G-6-M Demonstrating an understanding of the coordinate system (e.g., locatepoints, identify coordinates, and graph points in a coordinate plane torepresent real-world situations)G-3-H Solving problems using coordinate methods, as well as synthetic andtransformational methods (e.g., transform on a coordinate plane adesign found in real-life situations)GLEs Addressed:Grade 729. Plot points on a coordinate grid in all 4 quadrants and locate thecoordinates of a missing vertex in a parallelogram (G-6-M) (A-5-M)Grade 924. Graph a line when the slope and a point or when two points areknown (G-3-H)Lesson FocusIn this lesson, students will develop skills in solving problems using coordinatemethods. It includes the following aspects:• Locating points in the coordinate plane• Developing a simple method to determine slope• Using slope in data interpretationGEE 21 ConnectionThe skills that will be addressed in this lesson include the following:• Identify points on the coordinate plane• Demonstrate an understanding of slope• Understand the relationship between coordinate geometry and algebraicgeometryTranslating Content Standards into InstructionA. The first thing students must understand is how to locate points on thecoordinate plane.1. Show students the plane with an x- and y-axis. Emphasize which is which.Be sure to label the axes each time.2. Explain that every point on the plane begins at the origin and ends at thepoint. Some students have a tendency to count the x-coordinates andthen return to the origin to count the y-coordinates.Focused Learning Lessons for Mathematics16Geometry

3. Be sure students understand they are counting spaces between the pointson the plane and not the points themselves. Relate this to the numberline where zero is the beginning number for all numbers.4. Remind students that positive moves are to the right or up (when thingsin your life are up and going right you are a very positive person) and thatnegative moves are to the left or down (when your best friend has left youoff the party list you are feeling pretty down and negative about theworld).5. Students must understand that the ordered pair is in order alphabetically.This must be emphasized to reduce misunderstanding as we reverse theorder for slope.6. Have students demonstrate understanding by completing StudentWorksheet #1.B. Our next goal is to help students develop a simple method to determineslope. Use the Teacher Blackline #1 to demonstrate these concepts.1. It is important that students remember that it takes 2 points to determinea line. For that reason, we will be using two ordered pairs to find slope.(8,5) and (3,2)2. Many students will remember the formula from algebray2 − y1= slope. This is sometimes called the rise over the run.x2 − x1Students don’t need to memorize this formula. They just need tounderstand it. Another way of saying rise over run is the change in y overthe change in x. The change means the difference in the values of y andthe difference in the values of x so we use subtraction. Encouragestudents who are proficient with this formula to continue using it.Slope =RiseRun=Difference in yDifference in x3. Another way to remember how to find the slope is to compare the verticalchange over the horizontal change. The y values only move vertically, upand down, while the x values only move horizontally, left and right.4. Using our coordinates, (8,5) and (3,2) find the slope of the line formed bythese ordered pairs.Slope = 5 – 28 – 3Slope = 35Focused Learning Lessons for Mathematics17Geometry

In this example, we let the first y in the numerator be 5 and the first x inthe denominator be 8. These numbers came from the first ordered pairlisted. What if we decided we wanted to start our formula with the otherordered pair? Show the students that you still get the slope of 3 , so it5doesn’t matter which ordered pair you use first in the slope formula.5. Demonstrate the accuracy of this slope by plotting the original points,and then examining the aspects of the line.(a) Line goes up left to right (positive slope)(b) Demonstrate by using the slope ratio that you could find manymore points in the line by starting at a point on the line andgoing up 3 units, then to the right 5 units.6. Find the slope of the line through the ordered pairs (-3,6) and (-5,9). The3slope of this line is − , which is negative. Lines with a negative slope fall or godown from left to right.27. An example of coordinates on a horizontal line would be (14,3) and(-11,3). If a line is horizontal, then the y-values on the line must be thesame. Find the slope of this line, which is zero. You will always get azero in the numerator of the slope formula when subtracting the y-valuesof horizontal lines, the slope of all horizontal lines is zero.8. An example of coordinates on a vertical line would be (5,7) and (5,10). Ifa line is vertical, all x-values on the line must be the same. Find the slopeof this line which is undefined. Division by zero is impossible so dividingby zero here makes the slope undefined. You will always get a zero inthe denominator of the slope formula when subtracting the x-values ofvertical lines, the slope of all vertical lines is undefined.9. Summary(a) line with a positive slope – rises from left to right(b) line with a negative slope – falls from left to right(c) slope = 0 - horizontal line(d) slope = undefined - vertical lineProvide students with Student Worksheet #2 Let them work in groups of 2or 3 and compare their results as they go along. Encourage students to workindependently. Then compare results, so that corrections can be madebefore there are too many mistakes.C. Once students have an understanding of slope, they can begin interpretingwith the slope-intercept form of the equation of a line. The graphingcalculator is nice to use here, but the examples are given as if a student didnot have access to the graphing calculator.Focused Learning Lessons for Mathematics18Geometry

1. Introduce the slope-intercept form y = mx+ b, explaining that x and yrepresent each ordered pair on the line (x,y), m is the slope, and b is thepoint where the line crosses the y-axis(the y-intercept).2. Show students examples of graphs, when given the slope and y-intercept. Use Teacher Blackline #2 to demonstrate these concepts.m=3 b=4y= 3x+4“b” is the y-intercept, which is the point at which the line crosses the y-axis. When the line crosses the y-axis, the value of x, at that point, willalways be zero. The “b” is a shortcut for writing the ordered pair whichis (0,b). In the equation y = 3x + 4, the y-intercept is the ordered pair(0,4). This point should be plotted first, when graphing the line.You need a second point to connect the two points and have your line.Use the slope to find the second point. The slope of this line is 3. Sincethe slope represents the rise/run, we need a number for both thenumerator and denominator which would be the fraction 3 . 3 is the1change in y and 1 is the change in x. From the point (0,4), go 3 spacesup, then 1 space to the right. You now have your new point (7,1).Encourage the students to follow the same steps each time for lessconfusion. Since the numerator shows up first, calculate the change in yfirst, then the change in x. If the numerator is negative, you go down. Ifthe denominator is negative, you go to the left.3. Encourage them to always graph a 3 rd point to check for accuracy.Use Teacher Blackline #3 to demonstrate the following concepts.4. Show students how to find the slope-intercept form of the equation of aline, when given the slope and the y–intercept. The slope is –3 and they-intercept is 7. The equation is y = -3x +7. This one is just a matter ofsubstitution, because you were given the two numbers you needed.5. Show students how to find the slope-intercept given a point and theslope. Solve for b given the point (6,2) and a slope of 1 2 .y = mx + b2= 1 2 (6) + b2 = 3 + b-1 = bUse substitution to rewrite the equation in proper form:y = 1 2 x – 1Focused Learning Lessons for Mathematics19Geometry

6. Find an equation of a line through the ordered pairs (3,-3) and (-5,-1).Write in slope-intercept form. Find the m and the b to complete theequation. Use the ordered pairs to first find the slope, m.m =−3−−13−−5− 3+ 1 −2 −1= = =3+5 8 4Now use the slope and one ordered pair to find b. Either ordered pairmay be used since both points lie on this line. Encourage the studentsto pick the ordered pair which will allow the easiest computation. Theone with the smaller numbers and the fewer negatives would be the bestchoice. I will use (3,-3)y = mx + b1-3 = − (3) + b43-3 = − + b41− 2 = b 4y= x−2Use substitution to rewrite in proper form:−1 14 4Have students do Student Worksheet #3, where they will graph and writeequations in slope-intercept form.Sources of Evidence about Student LearningA. Have students complete worksheets provided with the lesson.B. Have students determine the slope of stairs at home or school by measuringthe rise and run of various sets.C. Have students determine the slope of handicap ramps in your communityusing a line level, string, and tape measure. This takes a little time, butreally brings slope to life.GEE 21 ConnectionSee attachment at the end of this unit for sample questions related to the GEE21.Focused Learning Lessons for Mathematics20Geometry

Attributes of Student Work at the “Got-It” levelA. When students are able to take any combination of two parts of the slopeinterceptform and complete the graph of the line.B. When students are able to look at the graph of a line and determine theequation for the line.Focused Learning Lessons for Mathematics21Geometry

Lesson 3: Solving Problems Using Coordinate MethodsStudent Worksheet # 1Use the coordinate plane below to name the ordered pair for each point.A = P = Q = B = M = O =I = F = W = D = S = C =N = K =Fill in the blanks.1) Point C lies on the __________ axis.2) The ______ axis is horizontal.3) The line passing through points Q and ______ have the same x-coordinate.4) The line passing through points B and N is a _________________ line.Focused Learning Lessons for Mathematics22Geometry

Lesson 3: Solving Problems Teacher Blackline #1Using Coordinate Methods1) Find the slope of the line formed by the ordered pairs (8,5) and (3,2).2) Find the slope of the line formed by the ordered pairs in reverse order(3,2) and (8,5). Is the slope the same for problems 1 and 2?3) Find the slope of the line formed by the ordered pairs (-3,6) and (-5,9).4) Find the slope of the horizontal line through the ordered pairs (14,3) and(-11,3).5) Find the slope of the vertical line through the ordered pairs (5,7) and(5,10).Focused Learning Lessons for Mathematics23Geometry

Lesson 3: Solving Problems Student Worksheet #2Using Coordinate MethodsI. Describe the slope of the following lines:1) Lines that rise to the right2) Lines that fall to the right.3) Lines parallel to the x-axis.4) Lines perpendicular to the x-axis.5) Vertical lines6) Horizontal lines7) Non-vertical, parallel lines8) Non-vertical, perpendicular linesII.Use the coordinate plane below for these questions:suuur.1) Describe the slope of EFsuuur.2) Find the slope of EF3) Find the slope of suuuur CD .suuur4) Describe the slope of AB .5) State the slope of a line parallel to suuur EF .6) State the slope of a line perpendicular to suuur EF .Focused Learning Lessons for Mathematics24Geometry

Lesson 3: Solving Problems Teacher Blackline #2Using Coordinate Methods1) Graph y = 3x +4 using the slope and the y-intercept.2) Graph1y=− x− 3 using the slope and the y-intercept.2Focused Learning Lessons for Mathematics25Geometry

Lesson 3: Solving Problems Teacher Blackline #3Using Coordinate Methods1) Find an equation of a line in slope-intercept form that has a slope of –3 anda y-intercept of 7.2) Find an equation of a line in slope-intercept form that has a slope of 1 2 andgoes through the ordered pair (6,2).3) Find the equation of a line that goes through the ordered pairs (3,-3) and(-5,-1). Write in slope-intercept form.Focused Learning Lessons for Mathematics26Geometry

Lesson 3: Solving Problems Student Worksheet #3Using Coordinate Methods1) Graph using the slope and y-intercept:2y= x+23Write equations of lines in slope-intercept form using the giveninformation.2) slope = -7 and y-intercept = 53) m = 4, one point on the line is (-3,4)4) (5,2) (7,6)Focused Learning Lessons for Mathematics27Geometry

Lesson 3 : Solving ProblemsUsing Coordinate MethodsAnswer KeysStudent Worksheet #1A (-6,4) P (1,5) Q (3,3) B (-4,-3) M (4,0) O (0,0) I (-1,-4) F (-6,-5) W (8,4)D (8,-5) S (0,-5) C (-5,0) N (3,-3) K (0,2)1) x2) x3) N4) horizontalTeacher Blackline #11) 3/5 2) yes; 3/5 3) -3/2 4) zero 5) undefinedStudent Worksheet #2Part I1) positive 2) negative 3) zero 4) undefined 5) undefined 6) zero7) have the same slope 8) opposite reciprocalsPart II1) positive 2) 4/4 or 1 3) 0 4) undefined 5) 1 6) –1Teacher Blackline #2Graphs are at the end of this answer keyTeacher Blackline #31) y = -3x + 7 2) y = ½ x – 1 3) y = - ¼ x – 2 ¼Student Worksheet #31) graph at the end of answer key2) y = -7x + 5 3) y = 4x + 16 4) y = 2x –8Focused Learning Lessons for Mathematics28Geometry

Lesson 3: Solving ProblemsUsing Coordinate MethodsAnswer keysTeacher Blackline #21)2)Focused Learning Lessons for Mathematics29Geometry

Lesson 3: Solving ProblemsUsing Coordinate MethodsAnswer KeyStudent Worksheet #31)Focused Learning Lessons for Mathematics30Geometry