C A R E

Curriculum Assessment Remediation Enrichment Algebra I Honors

Mathematics CARE Package #5 SPS Online

Domain Functions: Interpreting Functions

Standards MAFS.912.F-IF.2.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table)

over a specified interval. Estimate the rate of change from a graph.

Item Spec Clarifications:

Students will interpret the average rate of change of a continuous function that is

represented algebraically, in a table of values, on a graph, or as a set of data (with a real-

world context.)

Students will interpret the y-intercept of a linear model that represents a set of data with a

real-world context.

Domain Functions: Building Functions

Standards MAFS.912.F-BF.1.1 Write a function that describes a relationship between two quantities.

Item Spec Clarifications: Students will write a linear function, an arithmetic sequence, an exponential function, or a geometric

sequence when given a graph, a verbal description, or a set of ordered pairs that models a real-world

context.

Students will write an explicit function, define a recursive process, or complete a table of

calculations that can be used to mathematically define a real-world context.

Students will write a function that combines functions using arithmetic operations and relate the

result to the context of the problem.

Students will write a function to model a real-world context by composing functions and the

information within the context.

Students will write a recursive definition for a sequence that is presented as a sequence, a graph, or a

table.

Domain Algebra: Reasoning with Equations and Inequalities

Standards MAFS.912.A-REI.4.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the

coordinate plane, often forming a curve (which could be a line).

Item Spec Clarifications: Students will verify if a set of ordered pairs is a solution of a function.

CURRICULUM

Performance Task –

The High School Gym Alignment: MAFS.912.F-IF.2.6

The school assembly is being held over the lunch hour in the school gym. All the teachers and students are there by noon and the assembly begins. About 45 minutes after the assembly begins, the temperature within the gym remains a steady 77 degrees Fahrenheit for a few minutes. As the students leave after the assembly ends at the end of the hour, the gym begins to slowly cool down.

Let T denote the temperature of the gym in degrees Fahrenheit and M denote the time, in minutes, since noon.

a. Is M and function of T? Explain why or why not.

b. Explain why T is a function of M, and consider the function T = g(M). Interpret the meaning of g(0) in the context of the problem.

c. Becky says: “The temperature increased 5 degrees in the first half hour after the assembly began.” Which of the following equations best represents this statement? Explain your choice.

i. g(30) = 5

ii. 𝒈(𝟑𝟎)−𝒈(𝟎)

𝟑𝟎 = 5

iii. g(30) – g(0) = 5

iv. T = g(30) - 5

d. Which of these choices below represents the most reasonable value for the

quantity 𝒈(𝟕𝟓)−𝒈(𝟔𝟎)

𝟏𝟓 ?

i. 4

ii. 0.3

iii. 0

iv. -0.2

v. -5

PERFORMANCE TASK Considerations & Solution Guide

ASSESSMENT

The Mini-MAF includes standards in MAFS.912.F-IF.2.6, MAFS.912.F-BF.1.1, and

MAFS.912.A-REI.4.10. Use the following table to assist in remediation efforts.

Questions Standards Lesson from CORE

1, 4, 6 MAFS.912.F-IF.2.6 4-3, 4-4

3, 7, 8 MAFS.912.F-BF.1.1 4-5

2, 5, 9 MAFS.912.A-REI.4.10 4-2

REMEDIATION / RETEACH

Key Vocabulary- Functions / Rate of Change

Remediation/Reteaching Resources–Functions / Rate of Change

(Resource Links Description Linear Functions and Slope This is a complex, five-part lesson including videos and

activities. Topics include slope, independent and dependent

variables, rates, and graphs.

Equation Grapher (simulation)

Interactive simulation allows students the experience of

graphing different equations and seeing the effect on an

equation when you change the coefficients and constants.

Functions and Everyday Situations

Students are asked to verbally articulate a relationship between

variables.

Stacking Cups: writing equations from data

Teacher guide

Student activity

Hands-on lab where students find a linear relationship between

the number of cups stacked and the height. This gives insight

into slope as a rate of change.

Function Flyer

Sliders on graphing software allows the students to adjust the

function and see the affect each change has on the parent

function.

Function Explicit Function

Rate of Change x-intercept

Average Rate of Change y-intercept

Recursive Function Parent Function

Slope Solution

ENRICHMENT

Name _______________________________ Date___________

The graph represents the velocity of souvenirs shot from an air cannon into the crowd during a football game.

When shot, the souvenir has an initial velocity of 260 feet per second (fps).

1. During which one-second interval is the rate of

change the greatest? Justify your answer.

2. What is the average rate of change in velocity

over the interval from 1-3 seconds? Show

your work below.

3. What is the average rate of change in velocity over the interval from 3-5 seconds if at 5 seconds the velocity

is 17 fps? Show your work below.

Mathematics Formative Assessment System

Time (in sec)

Air Cannon

Vel

oci

ty (

in f

ps)

Florida Center for Research in Science, Technology, Engineering, and Mathematics

Copyright ©2014 - All Rights Reserved

TASK RUBRIC

Getting Started

Misconception/Error

The student does not understand the concept of average rate of change and how it is calculated.

Examples of Student Work at this Level

The student is unable to determine the one-second interval during which the rate of change is the greatest. In addition, the student is

unable to calculate the average rates of change over the specified intervals. Instead, the student:

Subtracts the y-coordinates of the endpoints of the given intervals and does not take into account the corresponding changes

in x.

Calculates a difference over another difference but uses the wrong values.

Averages the y-coordinates of points with whole number coordinates within the given interval.

Questions Eliciting Thinking

What is average rate of change? How does the average rate of change differ from the constant rate of change of a linear function?

What do you look for in a nonlinear graph to determine rate of change over an interval?

How do you calculate the rate of change of a linear function?

How do you calculate average rate of change of a non-linear function?

Can you identify the endpoints on the graph for the given intervals? Can you use these points to find the average rate of change for

each interval?

Instructional Implications

Provide additional instruction on the concept of rate of change. Initially, consider linear relationships and relate the rate of change to

the slope of the line that models a linear relationship between two variables. Ask the student to calculate the rate of change of a

linear function using several different ordered pairs and guide the student to observe that the rate of change (like the slope) of a

linear relationship is the same regardless of the ordered pairs used to calculate it. Remind the student that a defining attribute of

linear relationships is that the rate of change is constant. Emphasize that the average rate of change over any interval will always be

the same as the rate of change (or slope of the line) that represents the graph of the relationship.

Next, introduce the student to the concept of average rate of change in the context of nonlinear relationships. Begin with a relatively

simple relationship such as . Ask the student to determine the change in y for several consecutive one unit intervals of x and

to compare them. Relate the different rates of change in y to the steepness of the graph. Provide instruction on calculating the

average rate of change over larger intervals.

Provide the student with a nonlinear graph that models the relationship between two variables. Ask the student to calculate the

average rate of change over several different intervals. Ensure the student can identify the points on the graph represented by the

endpoints of each interval. Have the student draw secant lines that contain the endpoints of the intervals. Relate the average rate of

change calculation to the calculation of the slope of the secant lines.

Provide additional opportunities to calculate and interpret average rate of change over specified intervals for both linear and non-

linear functions.

Moving Forward

Misconception/Error

The student demonstrates some understanding of the concept of average rate of change but makes major errors in calculating it.

Examples of Student Work at this Level

The student:

Calculates the reciprocals of the average rates of change.

Disregards the scales of the axes and calculates the rates of change as if each axis were scaled by one.

Questions Eliciting Thinking

Can you explain how you found the rate of change for these intervals? What is the unit of measure for the x-coordinates? What is

the unit of measure for the y-coordinates?

What is the meaning of the rate of change you calculated, and how does it relate to the context of this problem?

Instructional Implications

Ask the student to describe the scale used on each axis. Discuss why different scales may have been used for the x- and y-axes.

Ask the student to reconsider his or her calculations.

Provide the student with the graph of another nonlinear function. Describe average rate of change as a change in y over the

corresponding change in x. Relate the average rate of change calculation to the calculation of the slope of a secant line that contains

the endpoints of an interval. Ask the student to calculate the average rates of change for several intervals by calculating the slopes

of the secant lines. Guide the student to observe any differences in the slopes of the secant lines.

Provide additional opportunities to calculate and interpret average rates of change over specified intervals for both linear and non-

linear functions.

Almost There

Misconception/Error

The student makes a minor mathematical error when calculating the average rates of change.

Examples of Student Work at this Level

The student identifies the interval from 0 – 1 second as the one-second interval with the greatest rate of change, but the student:

Does not indicate that the rates of change are negative.

Makes a minor error in one of the calculations.

Does not realize the coordinate (5, 17) was given in question #3 and uses an estimated y-coordinate to calculate the average rate

of change.

Questions Eliciting Thinking

In what order did you subtract the y-coordinates? Did you subtract the corresponding x-coordinates in the same order?

I think you made a small mistake in your work. Can you try to find and correct it?

Can you reread the third question? What did you overlook when you did your calculations?

Instructional Implications

Provide the student with a few examples of common errors made when finding average rates of change, including those described

at this level, and have him or her identify and correct those errors.

Provide additional opportunities to calculate and interpret average rate of change over specified intervals for both linear and non-

linear functions.

Got It

Misconception/Error

The student provides complete and correct responses to all components of the task.

Examples of Student Work at this Level

The student:

Indicates that the greatest one-second rate of change occurs during the interval from 0 – 1 seconds because this is the interval in

which the graph is decreasing the most.

Calculates a rate of change of -50 fps for question #2.

Calculates a rate of change of -16.5 fps for question #3.

Questions Eliciting Thinking

Why are the average rates of change you calculated negative? What does that mean in the context of this problem?

How do the rates of change that you calculated for this function differ from the rates of change for two different intervals of a linear

function?

Instructional Implications

Present the student with a similar problem that involves an air cannon with a velocity that may be represented by the

equation . Have the student find the average rate of change from x = 0 to x= 1. Compare the average rate of change

for this interval for the first souvenir and the second souvenir.

Ask the student to draw secant lines through the endpoints of the two intervals on the graph. Relate finding average rate of change

over these intervals to finding the slope of the secant lines that contain the endpoints of the intervals. Guide the student to observe

how the slopes of the secant lines relate to the steepness of the graph at each of the intervals.

Have the student calculate the average rate of change for intervals with the same left endpoint but of decreasing length

(e.g. , then , then ) for the function . Have the student draw secant lines through the endpoints of

each interval. Encourage the student to observe how the secant lines get progressively closer to the line tangent to the graph at (1,

1). If available, use a graphing utility to illustrate this concept.