Human Graphing with Proportions

This resource was created by Big Ideas in Beta, a Big Ideas Fest project, with acknowledgement to Pat C. Browne

LEARNING OUTCOMES:

• Students will relate and compare graphic, symbolic, numerical representations of proportional relationships. 
• Students will calculate the constant rate of change/slope of a line graphically. 
• Students will understand that all proportional relationships start at the origin. 
• Students will recognize and apply direct variation.

COMMON CORE STANDARDS ADDRESSED:

Expressions and Equations (8.EE)

8.EE.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

TIME REQUIRED FOR LESSON:

45 minutes - 50 minutes

TIME REQUIRED FOR TEACHER PREPARATION:

15 minutes - 20 minutes

MATERIALS FOR LESSON:

• Whiteboard/ chalkboard 
•  Markers Two 20 ft. pieces of colored electrical tape t(red and black( taped on the classroom (or hallway) floor to create a large graph with x and y axes 
• Two sets of large index cards numbered 1 – 20 (one written in red, the other in black) 
• A ball of heavy yarn (approximately 25 feet) 
• Graph paper and pencils

OVERVIEW OF LESSON:

1. Review that a  proportion is an equation that two ratios are equal.  Review the different ways of writing proportions - arithmetically (4 to 1, 4:1, and  4/11) and algebraically (a to b, a:b, and  a/b).  Explain that the colon notation is read the same way one reads analogies in English. Write a:b = c:d  Say: “ais to b, as c is to d.”

2.  Explain to students that today they are going to explore how equivalent ratios (fractions) and can be solved and graphed.   Divide the class into three groups.  Explain that students in the first group will represent numbers on the x axis of  graph; the second the numbers on the y axis; and students  the third group will plot points on the graph.

3.  Direct the students with the red numbered cards to each take a card and stand in order on the x axis. Direct the students with the  black numbered cards to each take a card and stand in order on the y axis.

4.  As students to look at the following set of equivalent ratios (write on the board):

2:3             4:6       6:12

Direct three students from the plotting group to stand where their ratio would fall on the graph. 

Direct the remaining students in this group to plot the slop of the graph by stretching the yarn from student to student.

Ask: “What is unique about the slope of the line?”  (Students should recognize that proportional relationships start at and follow the line of origin.)

5.  Repeat the process asking students to rotate to a different group.  This time direct the students to plot these ratios (write on the board:

3:4             6:8       14:16

Ask: “Do these plotted ratios start at and follow the line of origin? “  Explain that these do not because the ratios are not proportional (equivalent).

6.  Allow the students to demonstrate and test three more sets of ratios:

1.      5:6             10:12               15:18  (yes; follows the origin)

2.      3/7             6/14                 9/18  (no; doesn’t follow origin)

3.      4/5             8/10                 12/15 (yes; follows origin)

7.  Ask students to return to their seats.  Assess students’ understanding by asking them to graph the following proportions (written on the board) on graph paper:

Which of the following ratio proportions are equivalent?  How do you know?

(1)        6:9       12:18   18:27 (yes; follows the origin)

(2)        5:8       10:14   15:24  (no; doesn’t follow origin)

(3)        2:3       8/12     14:21 (yes; follows the origin)

(For enrichment, write the ratios in alternative forms, such as fractions or decimals)

ADDITIONAL INFORMATION:

For additional information on teaching proportional reasoning, refer to:

Sousa, D. A. How the Brain Learns Mathematics, Corwin Press, 2008.

Van de Walle, j. A., K. S. Karp, and J. M. Bay-Williams, Elementary and Middle School Mathematics – Teaching Developmentally, Allyn & Bacon, 2010.