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Unit 7, Lesson 2

How Does It Change?

Algebra 1

2.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

This Warm-up encourages students to quantify the number of squares in a geometric pattern by looking for and making use of structure (MP7). The thinking involved here prepares students to reason about other patterns that illustrate quadratic relationships.

Launch

Arrange students in groups of 2. Give students a minute of quiet think time, and then ask them to share their thinking with their partner.

Activity

None

<p>An image of a block of squares. There are eight squares on rows 1, 2, 3 and 4 and five squares on rows 5 and 6. There are six squares missing from the upper right corner of the block.</p>

Expression A:
Expression B:
Expression C:
Expression D:

Which expression best matches how you would describe the number of small squares in the figure?
Select at least one other expression and explain how it represents the number of small squares in the figure.

Student Response

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Building on Student Thinking

Activity Synthesis

Make sure that students see how each expression tells us about a way the squares can be counted or how the area of the figure can be determined. If time permits, ask students if they could generate a different expression that also represents the number of squares in the figure.

2.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students contrast two patterns of dots—one pattern shows a linear change and the other shows a quadratic change. They analyze the visual patterns and represent them using tables of values, expressions, and graphs.

The chosen quadratic pattern enables students to easily see the “squaring” of one quantity—visually, in the way the dots are arranged, and numerically, in the values 1, 4, 9, etc. in the table. They also notice that when the points representing the relationship are plotted, a curve is formed. The term “quadratic” will be introduced in the next activity.

Students reason repeatedly about the number of dots at different step numbers and then generalize their observations with variable expressions (MP8). They look for and make use of structure to extend the visual patterns (MP7). If students opt to use spreadsheet or graphing technology, they practice choosing appropriate tools strategically (MP5).

Launch

Some students may choose to use a spreadsheet tool to study the pattern, and subsequently to use graphing technology to plot the data. Make these tools accessible, in case they are requested.

Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use graphing technology, some students may benefit from a checklist or a list of steps that help them use the calculator or software.
Supports accessibility for: Organization, Memory, Attention

2.3

Activity

Instructional Routines

MLR8: Discussion Supports Notice and Wonder

Materials

None

Activity Narrative

In this activity, students encounter another visual pattern that changes quadratically. The number of squares in the pattern is related to the step number, , by the rule . Because of the addition of 2 to , the squaring of may not be as apparent as it was in the earlier activity (where the output values were perfect squares). This prompt gives students opportunities to see and make use of structure (MP7). A specific structure they might notice from the visual pattern is noticing the 2-by-2 square in Step 2 and the 3-by-3 square in Step 3.

As in the previous activity, students reason repeatedly, extend the pattern, and then write a general expression describing it (MP8). They also consider whether the pattern is linear or exponential and explain why it is or isn’t (MP3).

As students work, look for students who draw a picture (especially for the 5th step), those who describe the picture (especially for the 10th or 12th step), and those who rely on numerical (rather than visual) reasoning. Invite them to share their strategies during the synthesis of this activity.

Lesson Synthesis

Invite students to summarize their understanding so far of what a quadratic expression (or relationship) is and is not. Discuss questions such as:

“In what ways is the relationship between the step number and the number of small squares different from linear functions we’ve seen so far?” (When one quantity increases by a certain amount, the other quantity does not change by the same amount. The graph relating the two quantities is not a straight line. The rule that relates the two quantities is different from a linear expression.)
“In what ways is the relationship between the two quantities different from the exponential functions we’ve encountered?” (When the input increases by an amount, the output does not always change by the same factor. The graph representing this relationship is a curve, but it is different from the graph of an exponential function. Even though there may be an exponent in the expression (for example, ), the rule that relates the two quantities is not like exponential expressions we’ve seen, where the input was the exponent itself.)
“How would you describe ‘quadratic relationship’ to someone who is unfamiliar with it?”

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Student Lesson Summary

In this lesson, we saw some quantities that change in a particular way, but the change is neither linear nor exponential. Here is a pattern of shapes, followed by a table showing the relationship between the step number and the number of small squares.

<p>Three steps of a growing pattern.</p>

step	total number of small squares
1	2
2	5
3	10

The number of small squares increases by 3, and then by 5, so we know that the growth is not linear. It is also not exponential because it is not changing by the same factor each time. From Step 1 to Step 2, the number of small squares grows by a factor of , while from Step 2 to Step 3, it grows by a factor of 2.

From the diagram, we can see that in Step 2, there is a 2-by-2 square plus 1 small square added on top. Likewise, in Step 3, there is a 3-by-3 square with 1 small square added. We can reason that the th step is an -by- arrangement of small squares with an additional small square on top, giving the expression for the number of small squares.

The relationship between the step number and the number of small squares is a quadratic relationship, because it is given by the expression , which is an example of a quadratic expression. We will investigate quadratic expressions in depth in future lessons.

<p>Pattern. Step 0,1 dot. Step 1, 3 dots, 2 by 2 with upper left dot removed. Step 2, 5 dots, 3 by 2 with upper left dot removed. Step 3, 7 dots, 4 by 2 with upper left dot removed.</p> — Pattern 1

<p>Pattern. Step 0, 0 dots. Step 1, 1 dot. Step 2, 4 dots, arranged 2 by 2. Step 3, 9 dots, arranged 3 by 3.</p> — Pattern 2

Study the two patterns of dots.
1. How are the number of dots in each pattern changing?
2. How would you find the number of dots in the 5th step in each pattern?

Complete the table with the number of dots in each pattern.

step	number of dots in Pattern 1	number of dots in Pattern 2
0
1
2
3
4
5
10

Plot the number of dots at each step number.

Pattern 1

Pattern 2
Explain why the graphs of the two patterns look the way they do.

Launch

Display the pattern of squares for all to see. Ask students to be prepared to share one thing they notice and one thing they wonder about. Give them a moment to share their observations and questions with a partner.

The last question asks whether the pattern grows exponentially. If needed, remind students that a quantity that changes exponentially grows or decays by equal factors over equal intervals.

Activity

None

Here is a pattern of squares.

Is the number of small squares growing linearly? Explain how you know.

Complete the table.

step	number of small squares
1
2
3
4
5
10
12

Is the number of small squares growing exponentially? Explain how you know.

Student Response

to access Student Response.

Building on Student Thinking

Some students may struggle to draw the next step in the visual pattern. Prompt them to compare Steps 2 and 3 and describe what stays the same and what is changing. For example, the two small squares on either side stay the same and the square in the middle increases. If needed, show them a drawing for Step 4 (one 1-by-1 square on each side, with a 4-by-4 square in the middle) and ask them to draw Step 5.

Some students may not notice that the table goes from Step 5 to Step 10, and they may record 38 (the number of small squares for Step 6) instead of 102. Prompt them to draw a picture of Step 10 and count the number of small squares in their drawing. Emphasize that when the step number increases by more than 1 (for example, from Step 5 to Step 10), that means we are skipping ahead several steps and our pattern needs to reflect that as well. Show students a table with the missing steps included so they can see the growth pattern.

Activity Synthesis

Select students to share their strategies for determining how the pattern was growing. Start from concrete approaches, relying on images, and work toward more abstract representations, using equations. Help students make connections between their conclusions by asking where the can be seen in the picture and where they see the term. After the expression comes up, introduce the term quadratic.

Explain that the relationship between the step number and the number of squares is a quadratic relationship, which includes one quantity (the step number, in this case) being multiplied by itself to obtain a second quantity (the number of squares).

In middle school, we expressed area of squares with side lengths of as . The relationship between the side length () and the area of the square () is a quadratic relationship. So, is an example of a quadratic expression. The expression we wrote in this activity, is also an example of a quadratic expression. A quadratic expression can be written using a squared term, but it can be written in other ways as well.

If not already mentioned in students’ explanations, point out that:

A quadratic relationship is not like a linear relationship because as one quantity increases by a certain amount, the second quantity doesn’t increase by the same amount.
A quadratic relationship is not like an exponential relationship, because as one quantity increases by a certain amount, the second quantity doesn’t change by the same factor. (In this case, if the number of squares grew exponentially, this would mean that, from each step to the next, the number of squares would be multiplied by the same factor.)

MLR8 Discussion Supports. For each observation that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Advances: Listening, Speaking

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of quadratic relationships. Terms may include “quadratic,” “quadratic relationship,” and “quadratic expressions.”
Supports accessibility for: Conceptual Processing, Language

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Lesson 2

How Does It Change?

Warm-up

Standards Alignment

Building On

Addressing

HSA-SSE.A.1

HSA-SSE.B.3

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-SSE.A.1

HSF-BF.A.1.a

HSF-IF.A.3

Building Toward

Standards Alignment

Building On

Addressing

HSA-SSE.A.1

HSF-BF.A.1.a

Building Toward