Multiple Representations of Functions

Anne Bartel & Barb Everhart

GoalsGoals1. Distinguish among different types of

functions using tables, graphs, equations, and real life situations.

2. Make connections among the different representations of functions.

3. See slope in tables, graphs, equations, and real life situations.

4. Investigate slope as a rate of change.5. Help children make connections

between y = mx and y = mx + b.

Review of EquationsReview of Equations

What are the different ways that children solve linear equations?

Basic factsGenerate and evaluateUndoingCover-upTransposingEquivalent equations

Summative AssessmentSummative AssessmentLook at each item

Predict: Which strategies would your students use?

Share in Grade-level GroupsWhat strategies did

you teach?

What were your reasons for teaching those strategies?

What did your students learn?

Baby GirlsBaby GirlsAccording to growth tables used by pediatricians the median birth weight of baby girls increases at a constant rate each month for the first six months of life. The median birth weight of baby girls is 7.0 pounds and they typically gain 1.5 pounds per month for the first six months after birth.

Make a table for this data; then make a scatter plot.

Scatter-Plot and TableScatter-Plot and Table

Age (months

)

Weight (pounds

)

0 7

1 8.5

2 10

3 11.5

4 13

5 14.5

6 16

Rule of 3 Plus 1Rule of 3 Plus 1

Rule of 3 Plus 1- Rule of 3 Plus 1- Babies ActivityBabies Activity

Rule of 3 Plus 1Rule of 3 Plus 1

The growth of understanding about mathematical ideas runs along a continuum. The more a student understands something, the more connections s/he can make among representations.

Turtle Race Sorting ActivityTurtle Race Sorting Activity

The six teams at Anthony Middle School were having a turtle race. The turtles raced along a track. Some of the turtles ran faster than other turtles in previous races so they started at different places along the track to be fair. The six teams were supposed to keep track of their turtle’s progress using a graph, table, and equation. They were asked to also write a description of their turtle’s progress.

Turtle Race Sorting ActivityTurtle Race Sorting Activity

It was windy on the day of the race so several of the groups lost some of their data. Please cut out the different graphs, tables, equations, and descriptions below and match them to the appropriate turtle. Fill in any missing tables, graphs, equations, and/or descriptions.

Turtle Race Sorting ActivityTurtle Race Sorting ActivityDescribe the graphs:

◦Gary’s?◦Puff’s?◦Bob’s?

Read your stories for Bob

How are Sandi’s and Puff’s graphs different?

Think - Pair - Share:What follow-up questions would

you ask your students in this activity to pull out the mathematics?

What is slope?What is slope?

Write your definition of slope of a line:

Student Difficulty with Student Difficulty with SlopeSlopeA) Which line is steeper?

A B

12 += xy 121 += xy

Student Difficulty with Student Difficulty with SlopeSlopeB) Situation - Steepness

Which ramp is steeper?

A B

Student Difficulty with Student Difficulty with SlopeSlopeB) Rate of Change -- speed

Who is walking faster?

A B

Student Difficulty with Student Difficulty with SlopeSlopeC) Rate of Change -- Table

What is the slope in each of these tables? (Children tend to focus on difference rather than ratio)

Student Difficulty with Student Difficulty with SlopeSlopeD)Equation -- What it goes up by,

start, etc.What is the slope of the line described by the equation y = 3x + 2?

Students need to make connections between these values and the situation.

GearsGears

The Connected Gears ProblemYou have two gears on your table. Gear A (orange) has 8 teeth, and gear B (blue) has 10 teeth. Answer the following questions.

1. If you turn gear A a certain number of times, does gear B turn more revolutions, fewer revolutions, or the same number of revolutions? How can you tell?

GearsGears

The Connected Gears ProblemYou have two gears on your table. Gear A (orange) has 8 teeth, and gear B (blue) has 10 teeth. Answer the following questions.

2. Devise a way to keep track of how many revolutions gear A makes. Devise a way to keep track of gear B’s revolutions. How can you keep track of both at the same time?

The Connected Gears The Connected Gears ProblemProblem

How many rotations will gear B go through if gear A goes through one rotation?

How will this activity help your students develop a more sophisticated understanding of slope?

Revolutions of

gear A (8 teeth)

Revolutions of

gear B(10

teeth)

10 8

5 4

20 161 ?

Leaky FaucetLeaky FaucetCassandra decided to see how fast her bathtub faucet was leaking. She got a large container and put it under her faucet when she got up in the morning, and then checked periodically during the day to see how much water was in the container. She wrote the times and the amounts down. They are in the table on the right.

Time Amount of Water

7:00 a.m. 2 ounces

8:15 a.m. 12 ounces

9:45 a.m. 24 ounces

2:30 p.m. 62 ounces

5:15 p.m. 84 ounces

6:00 p.m. 90 ounces

9:30 p.m. 118 ounces

What do we want to What do we want to emphasize? emphasize?

TablesTablesUse a mix of well-ordered tables

and not well-ordered tables. Avoid using “it goes up by”

language. Focus on relationships across

measures. Multiplicative thinking is key.

What do we want to What do we want to emphasize?emphasize? EquationsEquations Focus on meaning of each term.

What do we want to What do we want to emphasize?emphasize?

GraphsGraphs Use slope triangles.

What do we want to What do we want to emphasize?emphasize? RepresentationsRepresentationsMake connections among

representations.

What do we want to What do we want to emphasize?emphasize?

Graphing CalculatorGraphing CalculatorUsing the Graphing Calculator:

Problem with “It goes up by.”

Slope – Rate of Change?Slope – Rate of Change?Paper Bridge Problem

from CMP Thinking with Mathematical Models

Fold 15 sheets of paper individually; fold 1 inch up on each side.

Stack the sheets to build paper bridges one at a time:1 layer thick2 layers thick3 layers thick, etc.

Slope – Rate of Change?Slope – Rate of Change? Place each bridge between 2 books or 2

tables of equal height. Place pennies, one at a time, into a cup in

the center of the bridge. Record the number of pennies that crumples the bridge. (Be sure to throw out used bridges.)

Make a mathematical model to describe the relationship between thickness and breaking weight. Use the model to estimate the breaking weight for 6- and 7-layer bridges.

Slope – Rate of ChangeSlope – Rate of Change

Thickness

(# sheets)

Breaking Weight

(# pennies)

1 5

2 14

3 24

4 39

5 41

Slope – Rate of ChangeSlope – Rate of Change

49 −= xy

The 9 means it takes 9 pennies to break each layer of the bridge.

The 9 means it takes 9 pennies to break each additional layer of the bridge.

Apples ProblemApples Problem

GoalsGoals1. Distinguish among different types of

functions using tables, graphs, equations, and real life situations.

2. Make connections among the different representations of functions.

3. See slope in tables, graphs, equations, and real life situations.

4. Investigate slope as a rate of change.

5. Help children make connections between y = mx and y = mx + b.

EvaluationOn an index card, please record:

P: one positive from today’s work

M: one ‘minus’ or concern from today’s work

I: something that you found interesting or intriguing from today’s work.

Thanks for your feedback.