Students use their knowledge about linear models, equations, and inequalities to reason about related sets of linear data.

Content
ACE
Materials
Resources

Problem 2.5

(1 day)

26-34, 44-55

Focus Question

When the graphs of two linear functions intersect, what do the coordinates of that intersection piont tell you?

Prior Knowledge

  • Linear Functions, Equations, and Inequalities
  • Mathematical Modeling
  • Variability in Data
  • The overarching goal of this Problem is to introduce the concept of systems of linear equations in the context of intersecting graphs for linear functions. This Problem gives students a chance to put together what they learned and reviewed in the previous Problems.

Video Launch

  • No Video Launch

Launch

  • What variables affect attendance at each attraction?
    Once students have brainstormed some ideas, focus specifically on the rain forecast with questions like these.
  • How might the probability of rain affect attendance at an amusement park?
    Attendance is likely to be higher when the chance of rain is low.
  • How might the probability of rain affect attendance at a movie theater?
    Attendance is likely to be higher when the chance of rain is high.
  • Do your ideas match the trends in the data?
    Problem 2.5 asks them to analyze the patterns in the data and to make predictions.

Explore

  • What would a plot of these data look like?
    two intersecting lines
  • What do you need to know to find the equation of a line that fits data?
    two points from each line
  • If students write the probability in decimal form, point out that the probabilities in the table are given as percents. The equation will produce a probability as a percent, and there is no need to use decimals.

 

  • What evidence will you use in the summary to clarify and deepen understanding of the Focus Question?
  • What will you do if you do not have evidence?

Summarizing

  • When are the two functions equal in value and when is one function greater than the other? How can you answer such a question using tables, graphs, and exact reasoning?
    Two functions are equal in value when their graphs intersect. For these two functions, that point is when p is approximately 74. You can answer such a question by estimating the coordinate of the intersection point. Or, you can find the intersection point algbraically by asking the question, what value of p makes the values of F and R equal, and then solve by setting  1000−7.5p=300+2p .
    Have students display their graphs and share the strategies they used to find equations for the two attendance functions.
  • How did you use those functions to answer Question B?
    In answering Question B, students may use numerical or graphical approximation methods (perhaps by using a graphing calculator to generate and trace tables and graphs), or they may use symbolic reasoning alone. Be sure both estimation and exact methods are discussed by asking a question such as this one:
  • What strategies can you use to find solutions to linear equations?

    Trace a graph or table to get estimates of the answers or use symbolic reasoning to find exact answers.

    If only one method is offered, ask students for ideas about how the other approaches would be useful, and carry them out to see whether the results agree. In general, numerical or graphical approaches will yield only approximations.

    The discussion of Question B will allow you to assess how confident your students are at solving linear equations. For part (4) of Question B, do not expect students to solve the system of equations symbolically; they likely will not. (Solving systems of equations will be taught formally in It’s in the System.) It is a good idea to present the possibility of doing so, however. Students may be able to understand how to represent the situation symbolically, even if they cannot yet solve it that way.

    Because students are looking for the point at which attendance at the two attractions is the same, they could set the equations equal to one another and solve the resulting linear equation.

    Big Fun=Get Reel 1,0007.5r=300+2r

    To prompt their thinking about the meaning and solution of this equation, you might ask:
    What question does this equation represent?
    For what probability of rain will the predicted attendance be the same at both the amusement park and the theater?

  • How could you estimate the solution using a table?
    For a calculator table, enter  y 1 =1,000−7.5 x  and  y 2 =300+2 x  and find a row for which the values of  y 1  and  y 2  are the same. Thex‑value for this row is the solution. For a hand-drawn table, find y‑values for x‑values in regular intervals. You might have to check between table values to get an accurate solution.
  • How could you estimate the solution using a graph?
    For a calculator graph, enter the function rules and display the graphs in the same window. Trace to find the point where the lines intersect. The x‑‑coordinate of the intersection point is the answer. For a hand-drawn graph, graph both equations on the same axes and estimate the x‑‑coordinate of the intersection point.
  • How could you find the solution symbolically?
    If  1,000−7.5 r=300+2 r , then  700=9.5 r , and r is approximately 74.