Lesson 5

This warm-up encourages students to look for regularity in how the number of tiles in the diagram are growing. This relates naturally to the work that they are doing with understanding the linear relationships as two of the three patterns students are likely to observe are linear and, in fact, proportional. 

Arrange students in groups of 2. Display the image for all to see and ask students to look for a pattern in the way the collection of red, blue, and yellow tiles are growing. Ask how many tiles of each color will be in the 4th, 5th, and 10th diagrams if the diagrams keep growing in the same way. Tell students to give a signal when they have an answer and strategy. Give students 1 minute of quiet think time, and then time to discuss their responses and reasoning with their partner.

Invite students to share their responses and reasoning. Record and display the different ways of thinking for all to see. If possible, record the relevant reasoning on or near the images themselves. After each explanation, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to what is happening in the images each time.

Time permitting, ask students which patterns represent a linear relationship? Which ones represent a proportional relationship? The patterns for the blue and red blocks are both proportional (hence linear) while the pattern for the yellow blocks is neither proportional nor linear. 

In the previous lesson, students analyzed the graph of a linear, non-proportional relationship (number of cups in a stack versus the height of the stack). This task focuses on interpreting the slope of a graph and where it crosses the \(y\)-axis in context. Students are given cards describing situations with a given rate of change and cards with graphs. Students match each graph with a situation it could represent, and then use the context to interpret the meaning of the slope. They find where the line crosses the vertical axis, i.e., the vertical intercept, and interpret its meaning in each situation. They also decide if the two quantities in each situation are in a proportional relationship.

Make sure that students draw the triangle they use to compute the slope. There are strategic choices that can be made to make the computation easier and more precise. Watch for students who use different triangles for the same slope computation, and ask them to share their reasoning during the whole-group discussion. 

In the whole-group discussion at the end of the task, discuss and emphasize the meaning of the terms slope and vertical intercept or \(y\)-intercept (in situations where the name of the variable graphed on the vertical axis is \(y\)).

You will need the Slopes and Graphs blackline master for this lesson.

Tell students that they will match a set of cards describing different relationships with a set of cards showing graphs of lines. The axes on the graphs are not labeled (since this could be used as an aid in the matching). Instruct students to add labels to the axes as they make their matches.

Arrange students in groups of 2. Distribute a set of 12 cards to each group. 10 minutes of group work and then whole-class discussion. 

The slopes of the 6 lines given on the situation card are all different, so the matching part of the task can be accomplished by examining the slopes of the different lines. Invite students who have made strategic choices of slope triangles for calculating the slopes (for example, Graph 6 contains the points \((0,0)\) and \((5,20)\), which give a value of \(\frac{20}{5}\) for the slope), and ask them to share.

Next, focus the discussion on the interpretation of the point where the line crosses (or touches) the \(y\)-axis. For some situations, the contextual meaning of this point is abstract. For example, in Situation B, a square with side length 0 is just a point that has no perimeter and so it “makes sense” that \((0,0)\) is on the graph. Some students may argue that a point is not a square at all but if we consider it to be a square then it definitely has 0 perimeter. In other situations, the point where the line touches the \(y\)-axis has a very natural meaning. For example, in Situation A, it is the amount Lin’s dad spent on the tablet and 0 months of service: so this is the cost of the tablet.

Define the vertical intercept or \(y\)-intercept as the point where a line crosses the \(y\)-axis. Note that sometimes “\(y\)-intercept” refers to the numerical value of the \(y\)-coordinate where the line crosses the \(y\)-axis. Go over each of the situations and ask students for the meaning of the vertical intercept in the situation (A: cost of the device, B: perimeter of a square with 0 side length, C: amount of money in Diego’s piggy bank before he started adding \$5 each week, D: money Noah saved helping his neighbor, E: amount of money in Elena's piggy bank before she started adding money, F: amount Lin’s mom has paid for internet service before her service begins.

Students have just learned the meaning of the \(y\)-intercept for a line and have been interpreting slope in context. In this activity, they investigate the \(y\)-intercept and slope together and investigate what happens when their values are switched. 

In contexts like this one, the \(y\)-intercept and the slope come with natural units and understanding this can help graph accurately. Specifically, the \(y\)-intercept is the number of pages Lin read before she starts gathering data for the graph. The slope, on the other hand, is a rate: it’s the number of pages Lin reads per day. 

Watch for students who understand the source of Lin’s error (confusing the \(y\)-intercept with the slope) and invite them to share this observation during the discussion.

Work time followed by whole-class discussion.

Ask students:

• “How did your graph compare to Lin’s?” (It is steeper but starts off at 30 instead of 40 pages.)
• “Which point does your graph have in common with Lin’s?” (The point \((1,70)\), 70 pages read after one day.)

Invite selected students to share what’s the likely source of Lin’s error (confusing the \(y\)-intercept with the slope). In this context, the \(y\)-intercept is the number of pages Lin read before she starts counting the days (30), and the slope is the number of pages Lin reads per day (40). 

Emphasize how the \(y\)-intercept and slope influence the graph of a line.

• The \(y\)-intercept indicates where the line touches or crosses the \(y\)-axis.
• The slope indicates how steep the line is.