Lesson 1 Scott’s March Motivation Develop Understanding

Jump Start

Revisiting the Past

Last year, Scott started going to a gym and getting in shape. He began by doing push-ups. On the first day, he could do $3$ push-ups, and then he found that he could do $2$ more push-ups each day. He kept track of the number of push-ups he did each day using this diagram:

Make a table and write explicit and recursive equations for the function $f (n)$ that shows the number of push-ups Scott did each day during the month of March last year. Explain how your equations and table connect to the diagram.

Learning Focus

Model patterns of growth with tables, equations, graphs, and diagrams.

Make conjectures about function rates of change.

What patterns do you notice, and how do these patterns connect to our understanding of functions we have studied?

Open Up the Math: Launch, Explore, Discuss

A big promotion at the gym last year required Scott to keep track of the total number of push-ups he had done in March. The March Motivation promotion raised money for charity by finding sponsors to donate based on the number of push-ups completed. Scott used the diagram and table below to show the number of push-ups he did each day and the total number of push-ups he did in the month. He completed three push-ups on day one and five push-ups (for a combined total of eight push-ups) on day two. Scott continued this pattern throughout the month.

$n$ Days	$f (n)$ Push-ups each day	$g (n)$ Total number of push-ups in the month
$1$	$3$	$3$
$2$	$5$	$8$
$3$	$7$	$15$
$4$	$9$	$24$
$5$	$11$	$35$
$\dots$	$\dots$
$n$

1.

Write the recursive and explicit equation for $g (n)$ , the accumulated total number of push-ups Scott completed by any given day during the March Motivation promotion last year.

Total March Motivation

Scott was proud of the money he raised last year.

This year, Scott decided to take March Motivation to a whole new level! He plans to look at the total number of push-ups he completed for the month last year $(g (n))$ and do that many push-ups each day $(m (n))$ .

2.

How many push-ups will Scott complete on day four? How did you come up with this number?

Last Year			This Year
$n$ Days	$f (n)$ Push-ups each day last year	$g (n)$ Total number of pushups in the month	$m (n)$ Push-ups each day this year	$T (n)$ Total push-ups completed for the month
$1$	$3$	$3$	$3$
$2$	$5$	$8$	$8$
$3$	$7$	$15$	$15$
$4$	$9$	$24$
$5$
$\dots$	$\dots$
$n$

3.

How many total push-ups will Scott complete for the month on day four?

4.

Write the recursive equation for $T (n)$ to represent the total number of push-ups Scott will complete for the month on any given day.

5.

Without finding the explicit equation, make a conjecture as to the type of function that would represent the explicit equation for the total number of push-ups Scott would complete on any given day for this year’s promotion.

6.

How does the rate of change for this function compare to the rates of change for the function in the Jump Start and problem 1?

7.

Test your conjecture from problem 5 and justify that it will always be true (see if you can move to a generalization for all polynomial functions).

Ready for More?

Find the explicit equation for $T (n)$ , the function that models the total number of push-ups Scott did in March this year.

Takeaways

	Linear Function	Quadratic Function	Cubic Function
Rate of Change
Recursive Equation
Explicit Equation

Vocabulary

cubic function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we modeled situations with quadratic and cubic functions with recursive and explicit equations. We learned that the rate of change of a cubic function is quadratic, the rate of change of a quadratic function is linear, and the rate of change of a linear function is constant.

Retrieval

1.

Place the appropriate inequality symbol between the two expressions to make the statement true.

If $a > b$ , then $\frac{1}{a}$ $\frac{1}{b}$ .

2.

Use long division to find the quotient without using a calculator. If you have a remainder, write the remainder as a whole number.

$23 71,496$

3.

Is $23$ a factor of $71,496$ ? Justify your answer.