Lesson 19

Tables, Equations, and Graphs, Oh My!

19.1: Matching Equations and Tables (10 minutes)

Activity

The purpose of this activity is to familiarize students with the 9 relationships they will continue to work with for the rest of this lesson. In this activity, students match equations and tables that represent the same relationship, without knowing what real-world situations the relationships represent. Students can make use of structure (MP7) as they narrow down which tables could possibly match each equation, such as recognizing whether the values for the dependent variable should be greater or less than the corresponding values for the independent variable, based on the operation in the equation.

Launch

Arrange students in groups of 2. Ask students to take turns: one partner identifies a match and explains why they think it is a match. The other 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. The students swap roles for the next equation. If necessary, demonstrate this protocol before students start working. Also, consider demonstrating productive ways to agree or disagree, for example, by explaining your mathematical thinking or asking clarifying questions.

Consider allowing students to use calculators to ensure inclusive participation in the activity.

Student Facing

Match each equation with a table that represents the same relationship. Be prepared to explain your reasoning.

\(S - 2 = T\)

\(G = J + 13\)

\(P = I - 47.50\)

\(C + 273.15 = K\)

\(e = 6s\)

\(m = 8.96V\)

\(y = \frac{1}{12}x\)

\(t = \frac{d}{2.5}\)

\(g = 28.35z\)


Table 1:

independent
variable
dependent
variable
20 8
58.85 23.54
804 321.6


Table 2:

independent
variable
dependent
variable
5 18
36 49
75 88


Table 3:

independent
variable
dependent
variable
2.5 22.4
20 179.2
75 672


Table 4:

independent
variable
dependent
variable
20 \(1\frac23\)
36 3
804 67


Table 5:

independent
variable
dependent
variable
58.85 11.35
175.5 128
804 756.5


Table 6:

independent
variable
dependent
variable
2.5 275.65
20 293.15
58.85 332

Table 7:

independent
variable
dependent
variable
5 3
20 18
36 34

Table 8:

independent
variable
dependent
variable
2.6 73.71
20 567
36 1,020.6

Table 9:

independent
variable
dependent
variable
2.6 15.6
36 216
58.85 353.1

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle to relate the variables in the equation to the columns of the table. Remind them that when we have one variable expressed in terms of the other variable, we call the former the dependent variable and the latter the independent variable. For example, in the equation \(a = b + 6\) we say that \(b\) is the independent variable and \(a\) is the dependent variable, because \(a\) is expressed in terms of \(b\).

Activity Synthesis

Much of the discussion will take place between partners. Invite students to share how they identified tables and equations that match.

  • What characteristics of the equation or table helped you narrow down the potential matches?
  • Were there any matches you and your partner disagreed about? How did you work to reach an agreement?

19.2: Getting to Know an Equation (15 minutes)

Activity

In the previous activity students looked at 9 different equations. In this activity, each student focuses on just one of those equations to learn about the real-world situation it represents. They interpret the variables and values in terms of the situation (MP7). Adding to the table they saw in the previous activity gives students an opportunity to practice both evaluating expressions and solving equations.

This activity also asks students to create a graph of their assigned relationship on graph paper. In previous activities, students were given a grid with the axes already labeled and numbered whenever they were asked to create a graph of a relationship. Here they must decide for themselves how to scale each axis.

Launch

Decide whether you want to assign the equations or allow students to select which equation they will work with. Make sure there is at least one student working with each equation.

Give students 5–10 minutes of quiet work time followed by partner and whole-class discussion. If there is more than one student working with the same equation, consider letting them share their answers and reasoning with each other, after they have had some time to work on their own.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. Check in with students within the first 2-3 minutes of work time. Look for students who need additional support deciding how to scale each axis for table. Invite students to share strategies they have used so far, as well as any questions they have before continuing. 
Supports accessibility for: Memory; Organization

Student Facing

The equations in the previous activity represent situations.

  • \(S - 2 = T\) where \(S\) is the number of sides on a polygon and \(T\) is the number of triangles you can draw inside it (from one vertex to the others, without overlapping)
  • \(G = J + 13\) where \(G\) is a day in the Gregorian calendar and \(J\) is the same day in the Julian calendar
  • \(P = I - 47.50\) where \(I\) is the amount of income and \(P\) is the profit after \$47.50 in expenses
  • \(C + 273.15 = K\) where \(C\) is a temperature in degrees Celsius and \(K\) is the same temperature in Kelvin
  • \(e = 6s\) where \(e\) is the total edge length of a regular tetrahedron and \(s\) is the length of one side
  • \(m = 8.96V\) where \(V\) is the volume of a piece of copper and \(m\) is its mass
  • \(y = \frac{1}{12}x\) where \(x\) is the number of eggs and \(y\) is how many dozens that makes
  • \(t = \frac{d}{2.5}\) where \(t\) is the amount of time it takes in seconds to jog a distance of \(d\) meters at a constant speed of 2.5 meters per second
  • \(g = 28.35z\) where \(g\) is the mass in grams and \(z\) is the same amount in ounces

Your teacher will assign you one of these equations to examine more closely.

  1. Rewrite your equation using words. Use words like product, sum, difference, quotient, and term.
  2. In the previous activity, you matched equations and tables. Copy the values from the table that matched your assigned equation into the first 3 rows of this table. Make sure to label what each column represents.

    independent variable:
    ________________________
    dependent variable:
    ________________________
    60
    300
  3. Select one of the first 3 rows of the table and explain what those values mean in this situation.
  4. Use your equation to find the values that complete the last 2 rows of the table. Explain your reasoning.
  5. On graph paper, create a graph that represents this relationship. Make sure to label your axes.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle to understand what each step is asking them to do. Consider preparing an example to share with them, such as this for the equation \(2.54i = c\), and point out which part of the example corresponds to the question they are on.

  1. The number of centimeters is the product of the number of inches and 2.54.
  2. independent variable:
    length (inches)
    dependent variable:
    length (centimeters)
    5 12.7
    36 91.44
    75 190.5
    60  
      300
  3. If something is 36 inches long, this is equivalent to 91.44 when measured in centimeters.
  4. In the table above, fill in 152.4 in the right column, next to the 60, because \(2.54 \boldcdot 60 = 152.4\). Fill in 118.11 in the left column, next to the 300, because \(2.54 \boldcdot 118.11 \approx 300\)
  5. A graph with “length (inches)” on the horizontal axis, “length (centimeters)” on the vertical axis, and points at \((5, 12.7), (36, 91.44), (75, 190.5), (60, 152.4)\), and \((118.11, 300)\)

Some students may struggle with setting up their graph from scratch on graph paper. Prompt them to think about the maximum value they want to represent on each axis and what number they could count by to get to that maximum value in the amount of space they have.

Activity Synthesis

Continue to the next activity. The discussion of these relationships occurs after students have created their visual displays.

19.3: Sharing Your Equation with Others (15 minutes)

Activity

The purpose of this activity is for students to use multiple representations to share with each other what they have learned about their assigned relationship.

Launch

Distribute tools for making a visual display. Give students 5–10 minutes of quiet work time, followed by a whole-class discussion or gallery walk. If time permits, consider having students research more information about their situation to add to their displays.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students consider audience when preparing to display their work. Display the list of items that should be included each display. Ask students, “what kinds of details could you include on your display to help a reader understand what you have learned about your assigned relationship?” Record ideas and display for all to see. Examples of these types of details or annotations include: the order in which representations are organized on the display, attaching written notes or details to certain representations, using specific vocabulary or phrases, or using color or arrows to show connections between representations. If time allows, after the gallery walk, ask students to describe specific examples of additional details that other groups used that helped them to interpret and understand their displays.
Design Principle(s): Maximize meta-awareness; Optimize output

Student Facing

Create a visual display of your assigned relationships that includes:

  • your equation along with an explanation of each variable
  • a verbal description of the relationship
  • your table
  • your graph

If you have time, research more about your relationship and add more details or illustrations to help explain the situation.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Conduct a gallery walk so students have a chance to read and discuss each other’s displays. Ask students to reflect on the following prompts:

  • What is the same about their relationship and your relationship? What is different?
  • What is the independent variable in their relationship? What is the dependent variable?
  • What could they add to the display to make their explanation of the relationship even clearer?

Have students use sticky notes to leave questions or comments for the person who created the display. Give students a moment at the end to review any questions or comments left on their display.