Unit 5, Lesson 10
Interpreting Inputs and Outputs
Extra Support Materials for Algebra 1
10.1
Warm-up
Standards Alignment
None
Building On
Addressing
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
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 that links 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.
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. Students make sense of problems and persevere in solving them while grappling with the relationship between the inputs and outputs of this function (MP1).
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 or term that was used 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,” the response should be “for one hundred, the output is 10.” If a student says “four over five,” the response should be “four-fifths has an output of 10.” A request for “minus five” should have the response “negative five has an output of 12.”
Activity
NoneThe 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
Activity Synthesis
The purpose of the discussion is to get students thinking more about inputs and outputs as they relate to functions. Here are some questions for discussion:
- “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 be an output from this function.)
10.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students identify a likely input variable from a pair of variables and 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 range. A familiarity with how functions are connected among situations, equations, and graphs as well as differentiating input and output supports students in thinking about domain and range. Students reason abstractly and quantitatively as they reason about the relationship of inputs and outputs (MP2).
Launch
NoneActivity
None10.3
Activity
Materials
NoneActivity Narrative
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).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Arrange students in groups of 2. Display the task for all to see. Tell students that for each function in Column A, one partner goes through the list of inputs in Column B explaining whether or not the inputs are possible to use in the function. 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.
Before students begin, use Critique, Correct, Clarify to give students an opportunity to improve a sample written response evaluating if the inputs 4, -1, and a tree could be used in the function by correcting errors, clarifying meaning, and adding details.
- Display this first draft:
All of the inputs could be used in the function because inputs go in the parentheses next to the .
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement. - Give students 2–4 minutes to work with a partner to revise the first draft.
- Display and review these criteria:
- Check each input to make sure it makes sense in the function.
- Explain why each input does or does not make sense.
- Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.