Lesson 4

Tables, Equations, and Graphs of Functions

4.1: Notice and Wonder: Doubling Back

CCSS Standards

• 8.F.A.1

Warm-up: 5 minutes

The purpose of this warm-up is to familiarize students with one of the central graphical representations they will be working with in the lesson. As students notice and wonder, they have the opportunity to reason abstractly and quantitatively if they consider the situation the graph represents (MP2).

Launch

Tell students they will look at a graph, and their job is to think of at least one thing they notice and at least one thing they wonder about the picture. Display the graph for all to see and give 1 minute of quiet think time. Ask students to give a signal when they have noticed or wondered about something.

Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.

Student Facing

What do you notice? What do you wonder?

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Activity Synthesis

Ask students to share things they noticed and wondered about the graph. Record and display these ideas for all to see. For each of the things students notice and wonder, ask them to reference the graph in their explanation. If no one notices that at every distance from the starting line, there are two associated times (except at 200 meters), bring that to their attention.

If there is time, ask students to make some estimations or guesses for the wonders that refer to the information in the graph. For example, if someone wonders what the title of the graph may be, ask them to create a title that would make sense for this context.

4.2: Equations and Graphs of Functions

CCSS Standards

• 8.F.A.1
• 8.F.A.3

Activity: 15 minutes

The purpose of this activity is for students to connect different function representations and learn the conventions used to label a graph of a function. Students first match function contexts and equations to graphs. They next label the axes and calculate input-output pairs for each function. The focus of the discussion should be on what quantities students used to label the axes and recognizing the placement of the independent or dependent variables on the axes.

Monitor for students who recognize that there is one graph that is not linear and match that graph with the equation that is not linear.

Launch

Arrange students in groups of 2. Display the graph for all to see. Ask students to consider what the graph might represent.

After brief quiet think time, select 1–2 students to share their ideas. (For example, something starts at 12 inches and grows 15 inches for every 5 months that pass.)

Remind students that axes labels help us determine what quantities are represented and should always be included. Let them know that in this activity the graphs of three functions have been started, but the labels are missing and part of their work is to figure out what those labels are meant to be.

Give students 3–5 minutes of quiet work time and then time to share responses with their partner. Encourage students to compare their explanations for the last three problems and resolve any differences. Follow with a whole-class discussion.

Heavier Support: MLR 5 (Co-Craft Questions and Problems). Use this routine to help students understand the contexts of the equations before relating them to the graphs and determining input-output pairs. First, present the equations one at a time (or all three together, if appropriate) with their respective situations. Ask students to write down possible mathematical questions that they think are answerable by looking at the equation and situation (1–2 minutes). In pairs, students then compare their questions (1–2 minutes). Then invite students to share their questions with the whole class, with some brief discussion (2–3 minutes). Once students have a chance to understand each context, give them the graphs and the prompts to complete.
Conceptual Processing: Processing Time. Check in with individual students, as needed, to assess for comprehension during each step of the activity.

Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

Student Facing

The graphs of three functions are shown.

1. Match one of these equations to each of the graphs.
1. $d=60t$, where $d$ is the distance in miles that you would travel in $t$ hours if you drove at 60 miles per hour.

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Student Lesson Summary

Student Facing

Here is the graph showing Noah's run.

The time in seconds since he started running is a function of the distance he has run. The point (18,6) on the graph tells you that the time it takes him to run 18 meters is 6 seconds. The input is 18 and the output is 6.

The graph of a function is all the coordinate pairs, (input, output), plotted in the coordinate plane. By convention, we always put the input first, which means that the inputs are represented on the horizontal axis and the outputs, on the vertical axis.