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Unit 5, Lesson 4

Using Function Notation to Describe Rules (Part 1)

Algebra 1

4.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

This Warm-up familiarizes students with a new way of using function notation and gives them a preview of the work in this lesson.

The prompt also gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is how the values in the and the columns in each table correspond to the expression describing each function.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first use less formal or imprecise language and then restate their observation with more precise language in order to communicate more clearly.

Launch

Arrange students in groups of 2. Display the two tables for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

What do you notice? What do you wonder?


1	8
1.5	7
5	0
-2	14


-2	-8
0	0
1	1
3	27

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the tables. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

4.2

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

In this activity, students are introduced to the idea that some functions can be defined by a rule, and the rule can be described in words or with expressions and equations. Students examine some simple rules and make connections between their verbal and algebraic representations. Doing so prompts them to look for and make use of structure (MP7).

The algebraic statements are written in function notation, so the work also reinforces students’ understanding of the notation and expands their capacity to use it to describe functions.

Launch

Display an image of a “function machine” with “cube the input” as the rule.

Tell students that a function takes any input and cubes it to generate the output. Ask students to

Find the output when the inputs are 0, 1, 3, and .
Write the input-output relationship using function notation and name the function .

If not mentioned by students, point out that these equations describe the same function as that shown by the second table in the Warm-up.

Explain to students that some functions have a specific rule for getting its output. The rule can be described in words (such as “cube the input”) or with expressions (such as ). Tell students that they’ll now look at some rules expressed in both ways.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “The equation _____ matches _____ because . . . .” and “I noticed _____, so . . . .” Encourage students to challenge each other when they disagree.
Advances: Speaking, Representing

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the expressions and descriptions to start with, and hold a brief small-group or whole-class discussion once students have completed the first match. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Social-Emotional Functioning

4.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students analyze functions that relate two quantities in a situation and work to define the relationship between the quantities with a rule. They do so by creating a table of values and generalizing the process of finding one quantity given the other. Students also plot the values in each table to see the graphical representation of the functions.

The mathematical reasoning here is not new. Students have done similar work earlier in the course, when investigating expressions and equations. What is new is seeing these relationships as functions and using function notation to describe them.

Students are likely to graph the functions by plotting the values in the tables and then connecting the points with a curve. As students work on the second set of questions about a perimeter function, which is linear, look for those who relate to a linear equation, namely, , and then graph a line with a vertical intercept of and a slope of 2. Invite them to share their thinking during the whole-class discussion.

Lesson Synthesis

Display for all to see the equations and . Ask students,

“How would you describe to a classmate who is absent today what each equation means? What would you say to help them make sense of these?” (Each equation gives the rule of a function. The rule for says that, to get the output, we multiply the input by 5 and add 3. The rule for says that the output is 10 times the input, minus 4.)
“How do the rules help us find the value of or ?” (If we substitute 10 for in each equation and evaluate the expression, we would have the value of or at , which are 53 and 96, respectively.)
“Is it possible to graph a function described this way? How?” (We could create a table of values and find the coordinate pairs at different -values. Or, if a rule is expressed as a linear equation, we could use it to identify the slope and vertical intercept of the graph.)

to access Cool-down.

Student Lesson Summary

Some functions are defined by rules that specify how to compute the output from the input. These rules can be verbal descriptions or expressions and equations. For example:

Rules in words:

Rules in function notation:

To get the output of function , add 2 to the input, then multiply the result by 5.

To get the output of function , multiply the input by and subtract the result from 3.

Some functions are defined by rules that relate two quantities in a situation. These functions can also be expressed algebraically with function notation.

Suppose function gives the cost of buying pounds of apples at \$1.49 per pound. We can write the rule to define function .

To see how the cost changes when changes, we can create a table of values.

pounds of apples,	cost in dollars,
0	0
1	1.49
2	2.98
3	4.47

Plotting the pairs of values in the table gives us a graphical representation of .

<p>Points plotted. Horizontal axis, 0 to 8, n, pounds of apples. Vertical axis, 0 to 10, cost in dollars. Points at 0 comma 0, approximately 1 comma 1 point 5, 2 comma 3, 3 comma 4 point 5, 4 comma 6.</p>

Launch

Arrange students in groups of 2. Give students a few minutes of quiet time to work on the first set of questions and then a moment to discuss their responses with their partner. Then, pause for a brief discussion before students proceed to the second set of questions.

Invite students to share their rule for the area function. Some students may have written , while others . Ask students who wrote each way to explain their reasoning. Highlight explanations that point out that is the name of the function and that function notation requires specifying the input, which is .

Clarify that in the past, we may have used a variable like to represent the area, but in this case, is used to name a function to help us talk about its input and output. If we wish to also use a variable to represent the output of this function (instead of using function notation), it would be helpful to use a different letter.

Activity

None

A square that has a side length of 9 cm has an area of 81 cm². The relationship between the side length and the area of the square is a function.
1. Complete the table with the area for each given side length.
  
  Then, write a rule for a function, , that gives the area of the square in cm² when the side length is cm. Use function notation.
  
  side length (cm) area (cm²)
  
  1
  
  2
  
  4
  
  6
2. What does represent in this situation? What is its value?
3. On the coordinate plane, sketch a graph of this function.
A roll of paper that is 3 feet wide can be cut to any length.
1. If we cut a length of 2.5 feet, what is the perimeter of the paper?
2. Complete the table with the perimeter for each given side length.
  
  Then, write a rule for a function, , that gives the perimeter of the paper in feet when the side length in feet is . Use function notation.
  
  side length (feet) perimeter (feet)
  
  1
  
  2
  
  6.3
  
  11
3. What does represent in this situation? What is its value?
4. On the coordinate plane, sketch a graph of this function.

Student Response

to access Student Response.

Building on Student Thinking

If students struggle to graph the functions, suggest that they use the coordinate pairs in the tables to help them.

side length (cm)	area (cm²)
1
2
4
6

side length (feet)	perimeter (feet)
1
2
6.3
11

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Lesson 4

Using Function Notation to Describe Rules (Part 1)

Warm-up

Standards Alignment

Building On

Addressing

HSF-IF.A.2

Building Toward

HSF-BF.A.1

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

8.F.A.1

Addressing

HSF-IF.A.1

Building Toward

Standards Alignment

Building On

Addressing

HSF-BF.A.1.a

HSF-IF.A.2

HSF-IF.B.4

HSF-IF.C

Building Toward