Lesson 5

The type of situation in this warm-up is similar to the first activity, except instead of generating representations, students are presented with graphs and asked to interpret them.

Possible questions for discussion:

• "How did you decide what \(x\) and \(y\) represented in each graph?" (Sample response: I noticed that the vertical intercept of the first graph was \((0,0)\), and each time \(x\) increases by 4, \(y\) increases by 6. This corresponds to adding $1.50 per week.)
• "Choose a point on each graph. What do its coordinates mean in this situation?" (Sample response: On the second graph, \((0,22)\) represents that at week 0, Kiran still needs $22.)
• "Where, on each graph, can you see how long it takes Kiran to reach his savings goal?" (On the first graph, you can see that week 15 is the first time he has more than $22. On the second graph, the \(x\)-intercept means that between 14 and 15 weeks, he needs no additional money to purchase the game.)

In this activity, students complete tables to help them write two different linear equations. They also sketch graphs that represent the situation. Note that these are intended to be rough sketches (possibly with the assistance of graphing technology), where the important features to be labeled are the intercepts. It’s not the intention that students carefully plot points to create the graphs.

This work will support similar activities in the Algebra 1 lesson that focus on quadratic relationships. When students determine the output for several numerical inputs and then write an expression for a variable input, they are expressing regularity in repeated reasoning (MP8).

Arrange students in groups of 2 and monitor their progress as they complete the first question. If needed, call the entire class together and display the tables after a few minutes so students can correct errors before writing their equations. Emphasize that the task is asking for a rough sketch of each graph. Just get the general shape right (a line with a positive or negative slope), and label the coordinates of each intercept.

Consider making graphing technology available. If students choose to use it to sketch their graphs, they are using appropriate tools strategically (MP5).

The goal is to make sure students understand that equations, tables, and graphs represent the same situation seen from a different starting point. In the Algebra 1 lesson, students will examine a free-falling object in a similar way using tables, graphs, and equations. Display a copy of the completed tables, equations, and graphs for all to see. Point out connections between these representations and discuss the following questions:

• “Describe how the tables are related to each other.” (In one, the distances increase by 280 feet, and in the other, the distances decrease by 280 feet.)
• “Describe how the graphs are related to each other.” (Both are lines. The lines have opposite slopes and different \(y\)-intercepts.)
• “Suppose the park were 1,200 feet away.
  • How would the tables look different?” (The table that represents the distance from the park would start with 1,200 instead of 2,437, and the other table would stay the same.)
  • How would the graphs look different?” (The graph that represents the distance from the park would intersect the vertical axis at 1,200 instead of 2,437, and the other graph would stay the same.)
• “Suppose the person was walking at a speed of 200 feet per minute.
  • How would the tables look different?” (The distances would increase or decrease by 200 feet for each additional minute instead of by 200 feet.)
  • How would the graphs look different?” (The graphs would not be as steep (assuming the same graphing window) because the slope of the line representing each equation would be 200 (or -200) instead of 280 (or -280).)

The first part is an opportunity to practice evaluating a function at given input values, and interpret a situation similar to the one in the previous activity. It is parallel to an activity in the associated Algebra 1 lesson, except the situation calls for a linear function instead of a quadratic function.

The second part is an opportunity to revisit perfect squares, so that these numbers are more recognizable in the associated Algebra 1 lesson. Monitor for students who struggle with fraction and decimal operations. Consider providing a calculator or reviewing multiplication strategies for those students.

Give students a few minutes of quiet work time on the first question, and then invite them to discuss their work with a partner. Consider arranging students in groups of 2 to work on the second question to encourage them to check each other’s work.

Display the two tables that give distance for all to see, and discuss how they are alike and how they are different. Point out that in both cases, the distances are a linear function of time.

Next, remind students that \(n^2\) is an example of a quadratic expression because it contains a squared term. If time permits, discuss different strategies students used when squaring a number or finding the square root of a number.