HCPSS Worthwhile Math Task

Let’s Race!

Common Core Standard

Analyze and solve pairs of simultaneous linear equations.

8.EE.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x  2y  5 and 3x  2y  6 have no solution because 3x  2y cannot simultaneously be 5 and 6.

8.EE.8c Solve real-world and mathematical problems leading to two linear equations in two  variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

The Task

Olivia and her brother William had a bicycle race. Olivia rode at a speed of 20 feet per second, while William rode at a speed of 15 feet per second. To be fair, Olivia decided to give William a 150-foot head start. The race ended in a tie. How far away was the finish line from where Olivia started?

(Adapted from Glencoe Algebra 1 Student Word Problem Workbook 5-1 http://glencoe.mcgraw-hill.com/sites/0078738229/student_view0/student_workbooks.html)

Facilitator Notes

1. Introduce the task to the students. Allow students a few minutes to read the task and begin to develop a strategy for solving. Give students graph paper, blank paper, rulers, graphing utilities, etc. 2. Have students work in pairs or small groups to solve the problem, and have groups chart their strategies. 3. As groups work, circulate to monitor what strategies are being used to solve the problem. Have students record solutions on chart paper or display solutions appropriately, based on strategies used to solve, so that they are presentation-ready. 4. Once groups have had an opportunity to solve the task, have groups compare solutions and strategies (either through a gallery walk, a jigsaw, or through group presentations).

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5. Make sure to highlight the various ways students solved the problem. Encourage students to use appropriate vocabulary (elimination, graphic solution, substitution, etc.).

Follow-Up Questions

1. Have students explain what x and y represent in the context of this problem. 2. How long does it take them both to finish the race? 3. If Olivia and William both started from the start line, how much sooner would Olivia cross the finish line than William? How do you have to change your work to answer this question? 4. Imaging that their cousin Marty joins the race, too. Write an equation to represent his position at any time during the race if after 5 seconds, he had gone 150 feet and after 20 seconds, he had finished the race.

Solutions

The finish line was 600 feet from where Olivia started. Accept multiple strategies for solving.

One possible strategy: Substitution Method

Olivia: y  20x William: y 15x 150

Since we are trying to find out where the finish line is and we know they crossed the finish line at the same time (x values are equal), we solved each equation for x and then set them equal and solved for the y value where this occurs (the position).

y y 150 Olivia: x  William: x  20 15

y y 150  20 15   15y  20(y 150) 15y  20y  3000 3000  5y 600  y



Another possible strategy: Graphing

(Image generated with GeoGebra)

Follow Up Questions: 1. Explain what x and y represent in the context of this problem.

x represents the time elapsed in seconds and y represents the position in feet at any time during the race.

2. How long does it take them both to finish the race?

They cross the finish line at 30 seconds.

3. If Olivia and William both started from the start line, how much sooner would Olivia cross the finish line than William? How do you have to change your work to answer this question?

To solve this problem, William’s position would now be represented by the equation y 15x because his starting position is 0. Olivia would cross the finish line 10 seconds before William.

 Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. HCPSS Worthwhile Math Task

4. Imagine their cousin Marty joins the race, too. Write an equation to represent his position at any time during the race if after 5 seconds, he had gone 150 feet and after 20 seconds, he had finished the race.

600 150 m   30, b=0 so y  30x . 20  5

 

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.