4.1: Notice and Wonder: Two Functions (5 minutes)
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 \(f(x)\) and the \(g(x)\) 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.
Display the tables for all to see. Tell students that their job is 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 the things they notice with their partner, followed by a whole-class discussion.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the tables. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
4.2: Four Functions (10 minutes)
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.
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 \(x\).
- Write the input-output relationship using function notation and name the function \(g\).
\(g(0)=0\\ g(1)=1\\ g(3)=27\\ g(x)=x^3\)
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 (like “cube the input”) or with expressions (such as \(x^3\)). Tell students that they’ll now look at some rules expressed in both ways.
Supports accessibility for: Conceptual processing; Organization
Here are descriptions and equations that represent four functions.
A. To get the output, subtract 7 from the input, then divide the result by 3.
B. To get the output, subtract 7 from the input, then multiply the result by 3.
C. To get the output, multiply the input by 3, then subtract 7 from the result.
D. To get the output, divide the input by 3, and then subtract 7 from the result.
- Match each equation with a verbal description that represents the same function. Record your results.
- For one of the functions, when the input is 6, the output is -3. Which is that function: \(f, g\), \(h\), or \(k\)? Explain how you know.
- Which function value—\(f(x), g(x), h(x)\), or \(k(x)\)—is the greatest when the input is 0? What about when the input is 10?
Are you ready for more?
Mai says \(f(x)\) is always greater than \(g(x)\) for the same value of \(x\). Is this true? Explain how you know.
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Invite students to briefly share how they matched the equations and verbal descriptions in the first question. Discuss questions such as:
- “The expressions for functions \(f\) and \(g\) both involve multiplying by 3 and subtracting 7. How are they different?” (The order in which the operations happen is different. Function \(f\) first multiplies the input by 3, and then 7 is subtracted from the result. Function \(g\) first subtracts 7, then multiplies the result by 3.)
- “The expressions for \(h\) and \(k\) both involve subtracting 7 and dividing by 3. How did you decide which one corresponds to description A and which one corresponds to D?” (By looking at what is done to \(x\) first. In \(h\), \(x\) is divided by 3 before 7 is subtracted, so it must correspond to D.)
Next, ask students how they determined which function has \((6, \text-3)\) as an input-output pair and which function has the greatest output when \(x\) is 0 and when \(x\) is 10. Highlight explanations that mention evaluating each function at those input values and seeing which one generates -3 for the output or gives the greatest output.
The functions in this activity are given without a context. Tell students that they will now look at rules that describe relationships between quantities in situations.
Design Principle(s): Support sense-making
4.3: Rules for Area and Perimeter (20 minutes)
Previously, students interpreted rules of functions only in terms of the operations performed on the input to lead to the output. 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 \(P(\ell)=2\ell + 6\) to a linear equation, namely \(y=2x+6\), and then graph a line with a vertical intercept of \((0,6)\) and a slope of 2. Invite them to share their thinking during the whole-class discussion.
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 \(A = s^2\), while others \(A(s)=s^2\). Ask students who wrote each way to explain their reasoning. Highlight explanations that point out that \(A\) is the name of the function and that function notation requires specifying the input, which is \(s\).
Clarify that in the past, we may have used a variable like \(A\) to represent the area, but in this case, \(A\) 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.
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.
Complete the table with the area for each given side length.
Then, write a rule for a function, \(A\), that gives the area of the square in cm2 when the side length is \(s\) cm. Use function notation.
side length (cm) area (cm2) 1 2 4 6 \(s\)
- What does \(A(2)\) represent in this situation? What is its value?
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.
If we cut a length of 2.5 feet, what is the perimeter of the paper?
Complete the table with the perimeter for each given side length.
Then, write a rule for a function, \(P\), that gives the perimeter of the paper in feet when the side length in feet is \(\ell\). Use function notation.
side length (feet) perimeter (feet) 1 2 6.3 11 \(\ell\)
- What does \(P(11)\) represent in this situation? What is its value?
On the coordinate plane, sketch a graph of this function.
If students struggle to graph the functions, suggest that they use the coordinate pairs in the tables to help them.
Select students to share the rule they wrote for the perimeter function (from the second set of questions) and how they determined the rule. Students may have written expressions of different forms for \(P(\ell)\):
\(\ell + \ell + 3+3\)
\(2 \ell + 2 (3)\)
\(2 (\ell + 3)\)
\(6 + 2\ell\)
Record and display the variations for all to see and discuss whether they all give the value of \(P(\ell)\). Ask students to explain how they know these expressions are equivalent and define the same function.
Next, discuss how students sketched the graph of the function. If no students made a connection between the slope and vertical intercept of the graph of \(P\) to the parameters in their equation, ask them about it. For example, display the graph of \(P\) and ask students to use it to write an equation for the line.
Display for all to see the equations \(f(x) = 5x + 3\) and \(g(x)=10x-4\). 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 \(f\) says that, to get the output, we multiply the input by 5 and add 3. The rule for \(g\) says that the output is 10 times the input, minus 4.)
- “How do the rules help us find the value of \(f(10)\) or \(g(10)\)?” (If we substitute 10 for \(x\) in each equation and evaluate the expression, we would have the value of \(f\) or \(g\) at \(x=10\), 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 \(x\)-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.)
4.4: Cool-down - Perimeter of a Square (5 minutes)
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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:
- To get the output of function \(f\), add 2 to the input, then multiply the result by 5.
- To get the output of function \(m\), multiply the input by \(\frac12\) and subtract the result from 3.
Rules in function notation:
- \(f(x) = (x + 2) \boldcdot 5\) or \(5(x+2)\)
- \(m(x) = 3 - \frac12x\)
Some functions that relate two quantities in a situation can also be defined by rules and can therefore be expressed algebraically, using function notation.
Suppose function \(c\) gives the cost of buying \(n\) pounds of apples at \$1.49 per pound. We can write the rule \(c(n) = 1.49n\) to define function \(c\).
To see how the cost changes when \(n\) changes, we can create a table of values.
|pounds of apples, \(n\)||cost in dollars, \(c(n)\)|
Plotting the pairs of values in the table gives us a graphical representation of \(c\).