# Lesson 10

Interpreting Inputs and Outputs

These materials, when encountered before Algebra 1, Unit 4, Lesson 10 support success in that lesson.

## 10.1: A Function Riddle (10 minutes)

### Warm-up

The purpose of this warm-up is to get students thinking about inputs and outputs for a function. The rules for functions do not need to be simple mathematical expressions. Students examine a table of inputs and outputs to try to discover the function linking them. Students may request additional outputs for suggested inputs.

If students continue to struggle to discover the function, encourage them to suggest inputs other than numbers.

The purpose of the warm-up is to get students thinking about inputs and outputs for a function. It is not essential that they solve the riddle, so you may tell them the solution at the end of the allotted time or allow them to think about it and return to it the next day or later.

### Launch

Arrange students in groups of 2.

Display the table for all to see. Tell students to think of a function that has these inputs and outputs, and after they have had a minute to think, you may supply additional outputs upon request. Ask them to give a quiet signal when they think they have discovered the function.

The function’s output is the number of letters in the English word for the input. Repeat the student suggestion for an input in correct mathematical language, and use that to compute the output. For example, if a student says “a hundred,” you should say “for one hundred, the output is 10.” If a student says “four over five,” you should say “four fifths has an output of 10.” A request for “minus five” should have the response “negative five has an output of 12.”

### Student Facing

The table shows inputs and outputs for a function. What function could it be?

input output
1 3
2 3
3 5
4 4
5 4
10 3
11 6

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to get students thinking more about inputs and outputs as they relate to functions. Ask students,

• “Is the rule connecting the input and output a function? Explain your reasoning.” (It might depend on what inputs you allow. For example, with an input of 4.2 do you count the letters of “four point two” or “four and two tenths” or “forty two tenths” or some other way of spelling the number?)
• “Are there any inputs that cannot be used with this function? Are there any outputs you cannot get with this function?” (If I can describe the thing in English, there is probably a way it can be input into the function. The outputs must be positive numbers, so a number like -3 or something that is not a number like “a house” cannot output from this function.)

## 10.2: What’s the Input? (20 minutes)

### Activity

In this activity students identify a likely input variable from a pair of variables then have the option to write an equation or draw a graph that represents the relationship between 2 variables. In the associated Algebra 1 lesson, students examine familiar functions to begin work on discussing domain and rage. A familiarity with how functions are connected between situations, equations, and graphs as well as differentiating input and output supports students in thinking about domain and range.

### Student Facing

1. For each pair of variables, which one makes the most sense as the input? When possible, include a reasonable unit.
1. The number of popcorn kernels left unpopped as a function of time cooked.
2. The cost of crab legs as a function of the weight of the crab legs.
5. $$f(t) = 5t + 8$$ where $$t$$ represents the time that a bike is rented, in hours, and $$f(t)$$ gives the cost of renting the bike.
6. $$g(n) = 7n+4$$ where $$n$$ represents the number of pencils in a box and $$g(n)$$ represents the weight of the box of pencils in grams.
2. Write the equation or draw the graph of a function relating the 2 variables.
1. Input: side length of a square, output: perimeter of the square
2. Input: time spent walking (minutes), output: distance walked (meters)
3. Input: time spent working out (minutes), output: heart rate (beats per minute)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to understand conventions about input and output as well as connecting representations of functions. Select students to share their solutions. After each solution shared, ask if there are different possible answers. Ask students,

• “What units make the most sense for the input of the popcorn problem? Explain your reasoning. Do any other units make sense?” (The time for the popcorn problem makes the most sense to measure in seconds. It only takes about 3 minutes to pop a batch of popcorn, so seconds would be accurate without the numbers getting too large. Minutes could also work, but the measurements would need to include fractional or decimal parts.)
• “For a situation involving two variables in which variable A is dependent on the value of variable B, which one makes the most sense to use as the input variable?” (It is convention to use the independent variable, B in this case, as the input.)
• “Which of the equations you wrote or graphs you drew could be different depending on the person, and which must be a certain thing?” (The perimeter of the square must be of the form $$f(s) = 4s$$ where $$s$$ is the side length, but the other 2 situations could depend on what is happening. For example, it doesn’t say how fast the person is walking or how intense the workout is.)

## 10.3: Matching Possible Inputs (15 minutes)

### Activity

In this partner activity, students take turns matching functions with possible inputs from a list. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

### Launch

Arrange students in groups of 2. Tell students that for each function in column A, one partner goes through the list of inputs in column B explaining whether the inputs are possible to use in the function or not. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next function in column A, the students swap roles. If necessary, demonstrate this protocol before students start working.

### Student Facing

For each function in column A, find which inputs in column B could be used in the function. Be prepared to explain your reasoning for whether you include each input or not.

1. Take turns with your partner to match a function with its possible inputs.
1. For each function, explain to your partner whether each input is possible to use in the function or not.
2. For each input, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
1. $$f(\text{person}) = \text{the person’s birthday}$$
2. $$g(x) = 2x + 1$$
3. $$h(\text{item}) = \text{the number of chromosomes in the item}$$
4. $$P(\text{equilateral triangle side length}) = 3 \boldcdot (\text{side length})$$
5. $$C(\text{number of students}) = 9.99 (\text{number of students}) + 15$$
• Martha Washington (the first First Lady of the United States)
• an apple
• 6
• 9.2
• 0
• -1

For each function, write 2 additional inputs that make sense to use. Write 1 additional input that does not make sense to use. Be prepared to share your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Much discussion takes place between partners. Invite students to share how they determined whether an input was possible or not. Select students to share their additional inputs that make sense and the input that does not as well as their reasoning for their values. Ask students,

• “Describe any difficulties you experienced and how you resolved them.” (I wasn’t sure whether 0 could be a number of students in the group or not. It might depend on the context, like non-students could be in the group, so 0 could work. Or maybe this is for a field trip that will not happen if 0 students are in the group going, so it does not make sense to use 0 as an input in that context.)
• “Can you think of a function or rule that would accept all of the listed inputs?” (A function that takes in any thing that can be typed and outputs the horizontal length (in millimeters) of the typed item using a certain font.)