Lesson 1

A Different Kind of Change

1.1: Notice and Wonder: Three Tables (5 minutes)

Warm-up

This warm-up encourages students to notice a new pattern of change (quadratic) by contrasting it to two familiar patterns (linear and exponential). In the table showing a quadratic relationship, students are not expected to recognize how the input and output values are related. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the output values don’t change by equal amounts or equal factors over equal intervals, and that the output values increase and then decrease.

Student Facing

Look at the patterns in the 3 tables. What do you notice? What do you wonder?

\(x\) \(y\)
1 0
2 5
3 10
4 15
5 20
\(x\) \(y\)
1 3
2 6
3 12
4 24
5 48
\(x\) \(y\)
1 8
2 11
3 10
4 5
5 -4

Student Response

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Activity Synthesis

Invite students to share what they noticed and wondered. Record and display their responses for all to see. After all responses have been recorded without commentary or editing, ask students, “Is there anything on these lists that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, point out contradicting information, etc.

If no one wondered about the rule for the relationship in the third table or how the outputs are changing, ask students to consider these questions. Tell students that in this unit they will investigate relationships such as shown in the third table.

1.2: Measuring a Garden (20 minutes)

Activity

This activity gives students a concrete experience with a quadratic relationship in a familiar geometric context. Given a rectangle with a fixed perimeter, students experiment with how changes to one side of the rectangle affect its area. Along the way, they notice that as one length increases, the area does not continue to increase. Instead, at some point it begins to decrease.

Students are not expected to write an equation such as \(\ell \boldcdot (25 - \ell)\) for the area of the rectangle. At this point, informal observations on how the values are changing are sufficient. Students will have ample opportunities throughout this unit to examine the formal structure of quadratic relationships in depth.

As students work, encourage them to consider side lengths that are not whole numbers. Look for students who organize their work in different systematic ways and select them to share their work later.

Considering the relationship between the dimensions and area of a rectangle given a fixed perimeter requires students to reason abstractly and quantitatively (MP2). Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Display a rectangle for all to see and label the sides with some lengths. Ask students to find the perimeter of the rectangle and the area of the region enclosed by this rectangle. Then, ask for the definitions of perimeter and area. Before beginning the activity, make sure students are clear about the distinction of the two measures—that the perimeter is the distance all the way around a region and the area is the number of unit squares that cover a region without gaps or overlaps.

Give students access to graph paper and tell students that they can use graph paper for the first question if they wish. Also provide access to calculators. Some students may benefit from using them to get to the interesting part of the task.

For the second question, some students may choose to create a spreadsheet to keep track of (and perhaps sort) the lengths, widths, and areas of the rectangle. Make spreadsheet tool available, in case requested.

Representation: Internalize Comprehension. Differentiate the degree of difficulty or complexity by beginning with an example that illustrates how to use graph paper to determine the length and width of each side. Also, demonstrate how the spreadsheet tool can be used to illustrate the length and width of each rectangle. 
Supports accessibility for: Conceptual processing

Student Facing

Noah has 50 meters of fencing to completely enclose a rectangular garden in the backyard.

  1. Draw some possible diagrams of Noah’s garden. Label the length and width of each rectangle.

    vegetable garden
  2. Find the length and width of such a rectangle that would produce the largest possible area. Explain or show why you think that pair of length and width gives the largest possible area.

Student Response

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Anticipated Misconceptions

Some students may exclude a rectangle with side lengths 12.5 and 12.5 from their diagrams of Noah's garden, possibly because they think a square is not a rectangle, or possibly because they only generate whole numbers. Emphasize that a square is a type of rectangle that happens to have four sides of equal length. Prompt them to think of the definition of a rectangle and explain why a square meets all the criteria for a rectangle.

Activity Synthesis

Display the work of a student who organized lengths, widths, and areas in a table. If no students did so, generate a table as a class. An example is shown here. (Pairs of length and area values will be needed in the next activity.) 

length (meters) width (meters) area (square meters)
5 20 100
10 15 150
12 13 156
12.5 12.5 156.25
18 7 126
20 5 100
24 1 24

 Discuss with students:

  • “What do you notice about the relationship between the length and the width?” (They add up to 25. Each time the length increases by 1, the width decreases by 1. The relationship is linear.)
  • “What do you notice about the relationship between the length and the area?” (As the length increases from 1 to 24, the area increases and then decreases. The relationship is not linear.)

Ask students to describe the rectangle they found to have the largest area and how they went about finding it. It is likely that many students will say that it has side lengths of 12 and 13, since these are the whole-number values that produce the greatest area. If no students tried 12.5 and 12.5, ask them to compute this area.

Solicit some ideas from students on how the area is related to the length. Ask questions such as:

  • “How do we know if 12.5 and 12.5 would indeed produce the greatest area?”
  • “Do you think going from 5 to 8 meters in length would produce a rectangle with a greater area? What about going from 15 to 18? Why or why not?”

Tell students that we’ll now try to get a better idea of what’s happening between the side lengths and the area of the rectangle by plotting some points.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

1.3: Plotting the Measurements of the Garden (10 minutes)

Activity

In this activity, students plot the points that represent the relationship between a side length and the area of a rectangle with a perimeter of 50 meters. They encounter a graph where as one quantity increases, a second quantity increases and then decreases. Making sense of the graph in context helps them see why it is reasonable to expect the second quantity to decrease after a certain point.

As students work, look for those whose graphs include enough points to hint at a quadratic shape. Also monitor for those who can articulate why \((1,25)\) cannot represent the length and area of the garden. Invite them to share their work later.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Keep a table from the previous activity displayed. If students created their own table, encourage them to use the values in their table.

If students used a spreadsheet tool to organize the lengths and area for the earlier activity, they may choose to use graphing technology to plot the data.

Student Facing

  1. Plot some values for the length and area of the garden on the coordinate plane.
    Blank coordinate plane. Horizontal axis, length of garden in meters, 0 to 30 by 5’s. Vertical axis, area of garden in square meters, 0 to 200 by 25’s.
  2. What do you notice about the plotted points?
  3. The points \((3,66)\) and \((22,66)\) each represent the length and area of the garden. Plot these 2 points on coordinate plane, if you haven’t already done so. What do these points mean in this situation?
  4. Could the point \((1,25)\) represent the length and area of the garden? Explain how you know.

Student Response

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Student Facing

Are you ready for more?

  1. What happens to the area when you interchange the length and width? For example, compare the areas of a rectangle of length 11 meters and width 14 meters with a rectangle of length 14 meters and width 11 meters.
  2. What patterns would you notice if you were to plot more length and area pairs on the graph?

Student Response

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Activity Synthesis

Select a student to present their graph, or display a graph with some points already plotted and amend it with additional points students provide. Discuss with students:

  • “Is the graph linear? Is it exponential?” (It is neither.)
  • “If we plot a bunch more points—say for every whole-number length between 0 and 25—what do you think the graph would look like?” (Possible predictions: A stretched out upside-down U, or an arch. A line with a positive slope that then turns into one with a negative slope. A curve. Some students might even recall the term parabola.)
  • “How can we tell whether \((1,25)\) does or does not represent the length and area of the garden?” (By multiplying 1 and 24 (if the length is 1, the width is 24) to see if it gives 25. Because 1 times 24 is 24, \((1,25)\) does not represent the length and area.)
  • “Can we tell if a point represents the length and area of the garden simply by plotting it on the graph? For example, does \((1.5, 200)\) represent the length and area? What about \((23,60)\)?” (It depends. If a point is far away from other points (for example, (\(1.5, 200)\), we can probably tell it does not. If it seems to belong with the general trend of the other points, we may need to check if the input times the width equals the output value.)

Tell students that this unit will focus on functions that are like that relating the length and area of the garden. The output of the function may both increase and decrease, so we know they are neither linear nor exponential, but they also don’t change in a random way.

Engagement: Develop Effort and Persistence. Provide a graphic organizer that focuses on increasing the length of on-task orientation in the face of distractions. For example, during the whole-class discussion create a graphic organizer that includes the questions that will be asked in one column and a blank space to record the answers in the other column. 
Supports accessibility for: Attention; Social-emotional skills

Lesson Synthesis

Lesson Synthesis

Invite students to reflect on how the relationship between the side lengths and the area of a rectangle differs from other relationships they’ve seen. Consider asking students to comment on:

  • the values in the table relating the length and the area of the rectangle
  • the graph representing the length-area relationship
  • the rule that relates the input and output of the function

It is not essential that students frame their observations in precise ways at this point. Their capacity to do so will be developed in the coming lessons.

1.4: Cool-down - 100 Meters of Fencing (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

In this lesson, we looked at the relationship between the side length and the area of a rectangle when the perimeter is unchanged.

If a rectangle has a perimeter of 40 inches, we can represent the possible lengths and widths as shown in the table.

We know that twice the length and twice the width must equal 40, which means that the length plus width must equal 20, or \(\ell + w = 20\).

length (inches) width (inches)
2 18
5 15
10 10
12 8
15 5

To find the width given a length \(\ell\), we can write: \(w= 20- \ell\).

The relationship between the length and the width is linear. If we plot the points from the table representing the length and the width, they form a line.

Points plotted. Horizontal axis, length in inches, 0 to 24 by 4’s. Vertical axis, width in inches, 0 to 40 by 10’s. Points at 2 comma 18, 5 comma 15, 10 comma 10, 12 comma 8, 15 comma 5.

What about the relationship between the side lengths and the area of rectangles with perimeter of 40 inches?

Here are some possible areas of different rectangles whose perimeter are all 40 inches.

length (inches) width (inches) area (square inches)
2 18 36
5 15 75
10 10 100
12 8 96
15 5 75

Here is a graph of the lengths and areas from the table:

Points plotted. Horizontal axis, length in inches, 0 to 24 by 4’s. Vertical axis, area in square inches, 0 to 125 by 25’s. Points at 2 comma 36, 5 comma 75, 10 comma 100, 12 comma 96, 15 comma 75.

Notice that, initially, as the length of the rectangle increases (for example, from 5 to 10 inches), the area also increases (from 75 to 100 square inches). Later, however, as the length increases (for example, from 12 to 15), the area decreases (from 96 to 75).

We have not studied relationships like this yet and will investigate them further in this unit.