## Successive Discounts

- Lesson

In this lesson, students will examine numeric, algebraic, and graphical representations of compositions of function in the context of successive discounts at a retail store.

Throughout this lesson, students will think about some questions
individually, then share solutions and discuss their mathematical
thinking with a partner. Afterwards, the pairs will share their work
with the class. For obvious reasons, this structure is known as *think-pair-share*, and it provides students with an opportunity to become actively engaged in learning mathematics.

As a large group, introduce the lesson by discussing the context of successive discounts at a retail store. Solicit input from students who work at retail stores that offer successive discounts and from students who have received successive discounts at a retail store.

Distribute the Successive Discounts Activity Sheet, and instruct students to read the context. Tell students that the purpose of the context will be to illustrate the mathematical concept of composition of functions.

Successive Discounts Activity Sheet

Allow students enough time to complete the first two problems on the activity sheet. Tell students to work individually first, then have them share their solutions with a partner. Instruct students to share with their partners how they determined the solutions.

After students have shared with their partners, as a large group,
ask students how they determined the sale price for the jeans. It is
likely that students will calculate a 25% discount by multiplying the
price by 0.25 and subtracting that amount from the price. For the
purpose of creating a function, though, have students think about how
to determine the price after a 25% discount with just one operation. If
students become stuck while trying to find such a function, ask them to
think about how much they pay when saving 25%. Alternatively, you can
have students write the final price in terms of the original price and
the discount, which should lead to a one‑operation function: *x* – 0.25*x* = 0.75*x*.

After students have discussed solutions and strategies for the first two problems, instruct them to complete Questions 3‑6. Again, tell students to work individually first, then have them share their solutions with their partners. Instruct them to share with their partners how they determined the solutions. If students ask if their answers are correct during the partner time, ask students to discuss with their partner whether or not their answers make sense. Allow partners time to discuss the process. After students have shared with their partners, ask the class to share the functions that they wrote. Allow plenty of time for discussion. Ask students if they have different algebraic representations for any of the functions. Focus on the process of applying one function first, and then applying another function second.

*Solutions to Questions 1-6*

- $23.25
- $22.00
*f*(*x*) =*x*– 5*g*(*x*) = 0.75*x**r*(*x*) = 0.75*x*– 5*s*(*x*) = 0.75(*x*– 5)

Instruct students that they will now use a graphing calculator to examine a graphical representation of the situation. Individually, students should complete Questions 7‑8. After students have completed these questions, instruct them to share their viewing windows with a partner. Have students share their reasons for choosing a particular domain. After students have shared with a partner, as a large group have students share their domains. Tell students that it is acceptable to have different domains, as long as they are reasonable based on the context. [A reasonable domain will not include negative values, and it will be large enough to include the $36 price of jeans as well as prices beyond.]

*Possible Solutions for Questions 7-8*

- Domain: [0, 75]
- Sample viewing window, with XMIN=0, XMAX=75, YMIN=0, YMAX=75:

Individually, students should complete Question 9 on the activity sheet. Students can use the TRACE function to identify the point of intersection. Ask students about the significance of the point of intersection. Students should determine that the point of intersection represents the point at which the sales price of jeans with just the 25% discount is equal to the sales price with just the $5 off coupon.

*Sample Graph for Question 9*

- The graph below left shows the two functions in the window from
Question 8, and it is obvious that the graphs intersect. The detail in
the graph, below right, shows that the point of intersection occurs
near
*x*= 20,*y*= 15:

Questions 10‑13 on the activity sheet connect the lesson to
composition of functions. Instruct students to work with a partner on
Questions 10‑13. Allow students plenty of time to discuss how to use
only *y*_{1} and *y*_{2} to write equivalent representations for *r*(*x*) and *s*(*x*). If students are having difficulty, have them think about the order in which the discounts are applied and which discounts *y*_{1} and *y*_{2}
represent. To review Questions 10‑13, have two groups of partners
combine to make groups of four. Have the groups share their functions
and graphs. After students have worked in groups, review answers as a
large group. Have students explain why the functions *y* = *y*_{1}(*y*_{2}) and *y* = *y*_{2}(*y*_{1}) are equivalent to *y* = *r*(*x*) and *y* = *s*(*x*), respectively.

*Solutions for Questions 10‑13*

- The graph of
*y*=*r*(*x*) will always fall below the graphs of*y*_{1}=*f*(*x*) and*y*_{2}=*g*(*x*) when*y*≥ 0, because two discounts will result in a lower price than only one discount. - The function
*y*=*r*(*x*) is equivalent to the function*y*=*y*_{1}(*y*_{2}). In this case,*y*_{2}, which represents the 25% discount, is applied first, and*y*_{1}, which represents the $5 off, is applied second. The significance of the graph is that since the functions*y*=*r*(*x*) and*y*=*y*_{1}(*y*_{2}) are equivalent, they will produce the same graph. When two functions produce the same graphs, the graphs overlap. Students can use the TRACE function to move along the graphs of*y*=*r*(*x*) and*y*=*y*_{1}(*y*_{2}) to convince themselves that the lines actually overlap. The graph below left shows the TRACE along the graph of*y*=*r*(*x*), and the graph below right shows the TRACE along the graph of*y*=*y*_{1}(*y*_{2}). - The graph of
*y*=*s*(*x*) will always fall below the graphs of*y*_{1}=*f*(*x*) and*y*_{2}=*g*(*x*) when*y*≥ 0, because two discounts will result in a lower price than only one discount. - The function
*y*=*s*(*x*) is equivalent to the function*y*=*y*_{2}(*y*_{1}). In this case,*y*_{1}, which represents the $5 off, is applied first, and*y*_{2}, which represents the 25% discount, is applied second. The significance of the graph is that since the functions*y*=*s*(*x*) and*y*=*y*_{2}(*y*_{1}) are equivalent, they will produce the same graph. When two functions produce the same graphs, the graphs overlap. Students can use the TRACE function to move along the graphs of*y*=*r*(*x*) and*y*=*y*_{2}(*y*_{1}) to convince themselves that the lines actually overlap. The graph below left shows the TRACE along the graph of*y*=*s*(*x*), and the graph below right shows the TRACE along the graph of*y*=*y*_{2}(*y*_{1}).

To connect the graphical representation to the algebraic representation, tell students that by applying the functions *f*(*x*) and *g*(*x*) in succession, they have created compositions of the two functions. On Questions 5 and 6, have students use *f*(*x*) and *g*(*x*) instead of *y*_{1} to write equivalent expressions for *r*(*x*) and *s*(*x*). Students should arrive at the following:
*r*(*x*) = *f*(*g*(*x*)) and
*s*(*x*) = *g*(*f*(*x*)).

Question 14 can be used as a concluding activity for the lesson. Students should realize that applying the percent discount first will result in the lowest price, because the percent discount is taken from a larger amount.

*Solution to Question 14*

If *y* = *r*(*x*) and *y* = *s*(*x*) are graphed in the same viewing window, the lines are parallel and *s*(*x*) > *r*(*x*), which means that the sales price represented by the function *y* = *s*(*x*) will always be greater than the sale prices represented by the function *y* = *r*(*x*),
although the difference is small. (For the graph below, XMIN=35,
XMAX=45, YMIN=20, and YMAX=30, to highlight the difference between the
lines. Using the TRACE function, students should notice that the sales
price of the jeans will always be $1.25 less if the percent discount is
taken first. This result can be verified algebraically by subtracting: *r*(*x*) – *s*(*x*) = 0.75*x* – 5 – [0.75(*x* – 5)] = 0.75*x* – 5 – 0.75*x* + 3.75 = 1.25.)

- Successive Discounts Activity Sheet
- Graphing Calculators

**Assessment Options**

*f*(

*x*) and

*g*(

*x*), have students determine if the compositions

*f*(

*g*(

*x*)) and

*g*(

*f*(

*x*)) result in equivalent expressions. Students should use algebraic and graphical representations to justify their answers.

2. Have students respond to a journal prompt such as the one below:

A department store is holding its annual end-of-year sale. Featured items are marked 40% off. In addition, a flyer sent to the newspapers includes a coupon for $10 off the purchase of any featured item.

- Let
f(x) be the sale price of a featured item after the 40% discount. Write an expression forf(x) in terms ofx. Explain why this finds the sale price of a featured item.- Let
g(x) be the sale price of a featured item after the $10 off coupon. Write an expression forg(x) in terms ofx. Explain why this finds the sale price.- Find
f(g(x)). Explain how this relates to the order of the discounts for the sale price of a featured item.- Find
g(f(x)). Explain how this relates to the order of the discounts for the sale price of a featured item.- Refer to parts
candd. Explain why one composition will always result in a lower price, regardless of the original price of the featured item.

**Extensions**

- Incorporate a third discount, such as save an additional 10% by using the store credit card. Have students determine compositions of three functions.
- Provide students with composite functions such as
*h*(*x*) = 0.6(x – 2). Have students determine functions,*f*(*x*) and*g*(*x*), such that*h*(*x*) =*f*(*g*(*x*)).

**Questions for Students**

- Compare and contrast the algebraic representations for r(x) and s(x). How does the order of the discounts affect the algebraic representations?
- How did you determine an appropriate domain for the problem situation? (This is an interesting question to explore, because if the jeans are cheap enough, it is possible to have a negative price after both discounts. One thing to discuss is that the store would not pay the customer, so if the discounts would result in a negative price, the item would just be free.)
- What is the significance of the point of intersection of the graphs of y = f(x) and y = g(x) as related to the problem situation?
- How did you determine functions equivalent to y = r(x) and y = s(x)?
- How do the graphs of y = r(x) and y = s(x) illustrate that the sale prices represented by the function y = s(x) will always be greater than the sale prices represented by the function y = r(x)?

**Teacher Reflection**

- How did the different representations for composition of functions enhance student understanding?
- What were some of the ways that the students illustrated that they were actively engaged in the learning process?
- How did technology promote understanding of composition of functions?

### Learning Objectives

Students will:

- Determine numeric and symbolic representations for composition of functions.
- Determine equivalent expressions for composition of functions.
- Use technology to develop graphical representations for composition of functions.

### NCTM Standards and Expectations

- Understand relations and functions and select, convert flexibly among, and use various representations for them.

- Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.

- Use symbolic algebra to represent and explain mathematical relationships.

- Draw reasonable conclusions about a situation being modeled.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.