Compare, Represent, and Evaluate Functions ABLE Teacher Academy

Compare, Represent, and Evaluate Functions

ABLE Teacher Academy

Goals• Participants gain content understanding and

strategies in areas of mathematics.• Participants can access resources for areas in

mathematics.• Participants can confidently share with other

teachers the mathematics content covered.• Participants can begin to structure the sequence

in which to teach new mathematics topics.

Rationale for Topics• 2014 GED® Instructor Survey

– Survey you completed last spring

• Shifts in the 2014 GED® Test1

• Other contributing factors

2014 GED® Assessment Guidelines2

A.7. Compare, represent, and evaluate functions. (Day 1)– A. Compare two different proportional relationships represented

in different ways. Examples include, but are not limited to: compare a distance-time graph to a distance-time equation to determine which of two moving objects has a greater speed.

– D. Compare properties of two linear or quadratic functions each represented in a different way (algebraically, numerically in tables, graphically, or by verbal descriptions). Examples include, but are not limited to: given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Q.7. Calculate and use mean, median, mode, and weighted average. (Day 1, afternoon)

A.3. Write, manipulate, solve, and graph linear inequalities. (Day 2)– A. Solve linear inequalities in one variable with rational number

coefficients.– B. Identify or graph the solution to a one variable linear

inequality on a number line.– C. Solve real-world problems involving inequalities.– D. Write linear inequalities in one variable to represent context.

2014 GED® Assessment Guidelines2

Building Staircase Activity

Building Staircase ActivityYour ABLE program is in the middle of building a new center. Unfortunately, funding has been cut and you are now responsible for helping finish the construction. As a math instructor, you have been charged with building the back staircase.

Currently, you don’t have any information on how many steps must be in the staircase. Explore the following questions to help guide your newly assigned responsibility.

Building Staircase Activity1. How many squares are needed, if only one

step is required?

2. How many squares does it take to build only the second step?

3. How many squares are required to form only the third step?

Building Staircase ActivityDebrief . . .

1.How many squares are needed, if only one step is required?

One square is needed to make a single step.


2.How many squares does it take to build only the second step?

It only requires two squares to build second step. If you think of the steps as a column, you should be able to see the two squares stacked on top of each other.


3.How many squares are required to form only the third step?

Just as in the prior question, if we look at the third column, we can see that there are three squares in the third step.

Building Staircase Activity4. How many total squares are necessary to

construct this 3-step staircase?

5. How many total squares are needed to build a 5-step staircase?

6. How many squares does it take to make only the eleventh step?


4.How many total squares are necessary to construct this 3-step staircase?

To determine the total number of squares needed to make the staircase, we simply add the number of squares together. (Step 1 + Step 2 + Step 3 = 1 + 2 + 3 = 6.) We observe that there are six squares needed to construct a 3-step staircase.


5.How many total squares are needed to build a 5-step staircase?

Using either square tiles, graph paper, pictures, visualization, or another technique, you should be able to deduce that there are 15 squares needed to build a 5-step staircase.


6.How many squares does it take to make only the eleventh step?

If we are interested in the eleventh step, then we are only concerned with the number of tiles in that eleventh column (eleven).

Building Staircase Activity7. How many squares are required to build

only the 98th step?

8. Determine a rule to identify the number of squares needed to make only the nth step.

9. Establish a rule to find the total number of squares required to make a staircase with n number of steps. Explain how you determined this rule.


7.How many squares are required to build only the 98th step?

If we examine the pattern, the number of squares required to build a specific step is the same amount as the value of the step. Thus, the 98th step would require 98 squares.

What is a function3?





8.Determine a rule to identify the number of squares need to make only the nth step.

If we examine the pattern, the number of squares required to build a specific step is the same amount as the step number. Therefore, the number of squares needed to build the nth step is n. If we write this in function notation we would see that f(n)=n.


9.Establish a rule to find the total number of squares required to make a staircase with n number of steps. Explain how you determined this rule.

Given the staircase size, n, the rule to find the total number of squares is f(n)=[n(n+1)]/2. You may have been able to arrive at this by guess and check, or some other heuristic.


9. . . . In case you didn’t, you may consider the following method. If we were to make a copy of a staircase (let’s use a 3-step staircase) and fit in with the original staircase, a rectangle is constructed.


•What content was addressed in this first portion of the activity?

•Where can you find resources that support your own content knowledge and math instruction?

Building Staircase ActivityPart II

Building Staircase Activity10.Here is a different set of staircases. How many

total cubes are needed to make a staircase with four steps?


10.Here is a different set of staircases. How many total cubes are needed to make a staircase with four steps?

Using cubes, pictures, visualization, or any other technique, we would find that we need 40 cubes to build a staircase with four steps.

Building Staircase Activity11.Following this pattern, how many cubes make

up only the top step of a staircase with five steps?

Building Staircase ActivityDebrief . . . 11.Following this pattern, how many cubes make up only the top step of a staircase with five steps?

If we are interested in the top step, then we are only focused on the number of cubes in that single step (five).

Building Staircase Activity12.How many cubes are required to build only the

top step of a staircase with n steps?


12.How many cubes are required to build only the top step of a staircase with n steps?

If we examine the pattern, the number of cubes required to build the top step is the same amount as the value of the number of steps in the staircase. Note that this is similar to examining the number of squares needed to build a step in the previous example. Thus, the top step of a staircase with n steps would require n cubes, or f(n)=n.

Building Staircase Activity13.How many cubes are used to build the base of a

staircase with n steps?


13.How many cubes are used to build the base of a staircase with n steps?

If we examine the staircase with two steps, the base is two cubes wide and two cubes deep. For three steps, we see that the base is three cubes wide and three cubes deep. If we continue the pattern we see that the base of a staircase with n steps has a width of n and a depth of n. That is the base is n by n, or n2.

Building Staircase Activity14.Determine a function that identifies the number

of cubes required to build a staircase with n steps. Justify your function.


14.Determine a function that identifies the number of cubes required to build a staircase with n steps. Justify your function.

Given the staircase size, n, the rule to find the total number of cubes is f(n)=[n*n*(n+1)]/2=[n2(n+1)]/2. You may have been able to arrive at this by guess and check, or some other heuristic. In the case you didn’t, you may consider using the method that we previously used in #9.

Building Staircase Activity15.Compare and contrast the difference between

the different staircases and components.


15.Compare and contrast the difference between the different staircases and components.

We can use the worksheet to compare the different situations. To begin with, it may be intuitive that the second (3-dimensional staircase) increases more rapidly than the 2-dimensional staircase or a column of steps.


Compare

tables


Compare graphs


Compare the functions

Mathematical Reasoning• What strategies were used to discover the

mathematics?

• What techniques were used to deliver the mathematics?

• What content can be incorporated into this activity?

If you’re still speaking to us . . .

Jaime Kautz

[email protected]

Brenda McMahon

[email protected]

Session Resources

1GED Testing Service®. 2014 GED® Test. Accessed August 15, 2013. http://www.GEDtestingservice.com.

2GED Testing Service®. Assessment Guide for Educators. Accessed August 15, 2013. http://www.GEDtestingservice.com.

3mathisfun.com. What is a function? Accessed on August 21, 2013. http://www.mathsisfun.com/sets/function.html.

Documents

Compare, Represent, and Evaluate Functions ABLE Teacher Academy