Graphing Rela-onships Session NCTM Interac-ve Ins-tute, 2016 Name - PowerPoint PPT Presentation

Graphing Rela-onships Session NCTM Interac-ve Ins-tute, 2016 Name Title Affilia-on Email address

Common Core Standards This session will address the following: 8.EE.5 Graph propor9onal rela9onships, interpre9ng the unit rate as the slope of the graph. Compare two different propor9onal rela9onships represented in different ways. 8.F.2 Compare proper9es of two func9ons each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descrip9ons). 8.F.4 Construct a func9on to model a linear rela9onship between two quan99es. Determine the rate of change and ini9al value of the func9on from a descrip9on of a rela9onship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and ini9al value of a linear func9on in terms of the situa9on it models, and in terms of its graph or a table of values. 8.NS Know that there are numbers that are not ra9onal, and approximate them by ra9onal numbers. 2

Warm Up What does slope mean? Describe a context that you use in your classroom to explore the meaning of slope. 3

Consider the Following: The equa9on for the speed (not the height) of a ball that is thrown straight • up in the air is given by v = 128 – 32t, where v is the velocity (in feet per second) and t is the number of seconds aTer the ball is thrown. With what ini9al velocity was the ball thrown? What is the meaning of the slope? The average lifespan of American women has been tracked, and the model • for the data is y = 0.2t + 73, where t = 0 corresponds to 1960. Explain the meaning of the slope and y-intercept. Fisherman in the Finger Lakes Region have been recording the dead fish they • encounter while fishing in the region. The Department of Environmental Conserva9on monitors the pollu9on index for the Finger Lakes Region. The model for the number of fish deaths "y" for a given pollu9on index "x" is y = 9.607x + 111.958. What is the meaning of the slope? What is the meaning of the y-intercept? From hap://www.purplemath.com/modules/slopyint.htm 4

Body Measurements Lab • Get in Groups of 4-5, when you have found your group check in with me by having each of your group members raise his or her hand. CHECK IN BEFORE MOVING ON 5

Body Measurements Lab • Working with your group, collect the body measurements data and record on your lab sheet. • ATer data collec9on is complete, check in with me. • You may select the units you use within your group. CHECK IN BEFORE MOVING ON 6

Body Measurements Lab • Record a sample of your measurements on the chart paper in the front of the room. • Let’s look at the data we’ve collected. CHECK IN BEFORE MOVING ON 7

Body Measurements Lab • Let’s consider a line that would “fit” this data. • Think about what the slope is and interpret what it means. CHECK IN BEFORE MOVING ON 8

Discussion • What does the slope of this “best fit” line mean? • Why was the paaern with knuckles less discernable than the forearm/hand paaern? 9

haps://www.youtube.com/watch? v=0tAZe6pP-FM 10

Golden Ra-o 11

Precision. . . • How close are our slopes to the Golden Ra9o? • What are some possible sources of “error”? 12

Think-Pair-Share Using the data we’ve collected, graph a func9on that fits the data. What factors might account for our error? 13

Discuss What is an irra9onal number? Why are func9ons with irra9onal slopes tricky? (to graph, to discuss, to describe, etc.) What are some other situa9ons that are represented by irra9onal numbers? 14

A circular track? 15

A circular track 16

What if we add a walk to the track? 17

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Tony’s Walk Adapted from Navigating through Algebra in Grades 19 6-8 (2001). NCTM: Reston, VA.

Tony’s Walk – Student 1 Adapted from Navigating through Algebra in Grades 20 6-8 (2001). NCTM: Reston, VA.

Tony’s Walk – Student 2 21

Tony’s Walk – Student 3 22

Graphs to Stories • In your group, consider the graph on your card. • On your poster, write a short story and create a table of data that corresponds to your graph. • As you finish, please hang your poster. 23

Graphs to Stories - Discussion What are some things you no9ced about the graphs of other groups? Which graphs were the most difficult to interpret and create data for? The easiest? 24

Mathema-cal Prac-ces Through today’s tasks, which mathema9cal prac9ces have you been engaged in? 25

Reflec-on Individual wri-ng – Pair discussion • What was the role of representa9ons in the Tony’s Walk task? • How do the representa9ons support student understanding? • How do you feel about exploring irra9onal numbers through interpre9ng func9ons?

Reflec-on ( Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 29)

Summary – Big ideas from this session • Graphing rela9onships • Interpre9ng slope • Irra9onal slope / meaning of irra9onal numbers • Linking descrip9ons, tables, and graphs 28

Exit Ticket • Think about the big ideas from this session. 1. Which ideas resonated with your thinking? 2. Which ideas are you s9ll pondering? 3. Which ideas/tasks challenged your thinking? 29

Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM ’ s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council. 30

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