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.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they determined how long it would take to reach the park. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

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 -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 280 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 goal of this discussion is to see how a situation can be represented multiple ways and explore squares of numbers that are not only integers. Display the two tables that give distance for all to see. Here are some questions for discussion:

• “How are the two tables alike?” (Both tables are changing by 250 and measure time in minutes and distance in feet.)
• “How are the two tables different?” (One table begins at a time and distance of 0 and increases by 250 feet each minute, while the other begins at a time of 0 and a distance of 4,000 and decreases by 250 feet each minute.)
• “When would someone want to use the first table? The second table?” (The first table shows how far someone has traveled from home, so they might use it to see how far they are at a specific time. The second table shows how far the person is from school, so they might use it to see how far they still need to travel or how long it takes them to get to school.)

Point out that in both cases, the distances are a linear function of time.

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