Lesson 5

Distance To and Distance From

These materials, when encountered before Algebra 1, Unit 6, Lesson 5 support success in that lesson.

5.1: Saving Up (5 minutes)

Warm-up

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.

Student Facing

Kiran is saving up to buy a game for \$22. He starts with no money saved and adds \$1.50 to his savings each week. Both of these graphs represent the situation.

Coordinate plane, x, 0 to 18 by 2, y, 0 to 28 by 2. Straight line graphed from the origin to 20 comma 30.
Graph of a line, origin O. X axis 0 to 18, by 2s. Y axis 0 to 28, by 2s. Line begins at 0 comma 22, passes through 4 comma 16, 12 comma 4, and ends on the x axis around 14 point 5.

Describe what \(x\) and \(y\) represent on each graph.

Student Response

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

Activity Synthesis

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.)

5.2: A Walk to the Park, or a Walk Away from Home? (20 minutes)

Activity

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).

Launch

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).

Student Facing

  1. A person is walking from home to a park that is 2,473 feet away. They are walking 280 feet per minute.
    1. How far away from home are they after 0, 1, 2, 3, 5, \(t\) minutes? 
      minutes 0 1 2 3 5 \(t\)
      distance from home            
    2. How far away from the park are they after 0, 1, 2, 3, 5, \(t\) minutes? 
      minutes 0 1 2 3 5 \(t\)
      distance from park            
  2. Create an equation that relates \(t\) to:
    1. the distance from home
    2. the distance from the park
  3. Create a rough sketch of a graph of each equation. Label the coordinates of any horizontal or vertical intercepts. 
    horizontal axis, t. unlabeled quadrant 1. 
    horizontal axis, t. unlabeled quadrant 1. 
  4. Which is the closest to the number of minutes it takes the person to reach the park: 6, 8, 9, or 12? Explain how you know.

Student Response

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

Activity Synthesis

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).)

5.3: Walking to School (20 minutes)

Activity

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.

Launch

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.

Student Facing

  1. A person walks from home to school. The function \(d(t) = 250t\) gives the distance from home as a function of time, \(t\), in minutes. The school is 4,000 feet from home.

    1. How far does the person walk in 30 seconds?
    2. Here are two tables representing the person’s walk. How are the tables alike? How are they different? 
    3. Complete the tables.
      time(minutes) 0 1 2 3 4 \(t\)
      distance from home (feet) 0 250 500      
      time (minutes) 0 1 2 3 4 \(t\)
      distance from school (feet) 4,000 3,750 3,500      
  2. The square of a number refers to the product of the number and itself. For example, the square of 3 is 9, because \(3^2=9\). Complete the table showing squares and positive square roots of different numbers. 

    \(n\) 4 8 0.8   \(\frac{1}{10}\) 12        
    \(n^2\) 16     81     1.96 256 \(\frac{1}{289}\) 400

Student Response

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

Activity Synthesis

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.