Lesson 2

How Does it Change?

2.1: Squares in a Figure (5 minutes)

Warm-up

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.

Student Facing

How does each expression represent the number of small squares in the figure?

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.
  • Expression A: \(6 \boldcdot 8 - 2 \boldcdot 3\)
  • Expression B: \(4 \boldcdot 8 + 2 \boldcdot 5\)
  • Expression C: \(8+8+8+8+5+5\)
  • Expression D: \(5 \boldcdot 6 + 3 \boldcdot 4\)

Student Response

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

Activity Synthesis

Make sure 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: Patterns of Dots (15 minutes)

Activity

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 requested.

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.  
Supports accessibility for: Organization; Conceptual processing; Attention

Student Facing

Pattern 1

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.

Pattern 2

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.
  1. Study the 2 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?
  2. 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    
    \(n\)    
  3. Plot the number of dots at each step number.

    Pattern 1

    Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “step number”. Vertical axis from 0 to 30 by 5’s, labeled “number of dots”.

    Pattern 2

    Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “step number”. Vertical axis from 0 to 30 by 5’s, labeled “number of dots”.
  4. Explain why the graphs of the 2 patterns look the way they do.

Student Response

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

Activity Synthesis

Invite students to describe how they see the number of dots change in each pattern. Focus the discussion on the connections between students’ descriptions, the changes in the table, and the graphs. Make sure these points are uncovered in the discussion:

  • For the first pattern, each successive step adds two dots. The growth is constant. The points on the graph form a straight line.
  • For the second pattern, the number of dots that are added changes (and grows) at each step. As a result, the number of dots for the second pattern curves upward when graphed.

Highlight that, in the second pattern, the expression for the number of dots at the Step \(n\) can be expressed as \(n^2\), which can be read as “\(n\) squared”. Point out that the “squaring” is also evident in the visual pattern.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. Give students quiet think time to consider what is the same and what is different about each of the pairs of representations (patterns, tables and graphs). Invite students to discuss what they notice with a partner. Listen for and amplify mathematical language students use to describe connections they notice across and between the different representations. 
Design Principle(s): Cultivate conversation

2.3: Expressing a Growth Pattern (15 minutes)

Activity

In this activity, students encounter another visual pattern that changes quadratically. The number of squares in the pattern is related to the step number \(n\) by the rule \(n^2 +2\). Because of the addition of 2 to \(n^2\), the squaring of \(n\) may not be as apparent as 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 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.

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.

Student Facing

Three steps of a growing pattern.

Here is a pattern of squares.

  1. Is the number of small squares growing linearly? Explain how you know.
  2. Complete the table.
    step number of small squares
    1  
    2  
    3  
    4  
    5  
    10  
    12  
    \(n\)  
  3. Is the number of small squares growing exponentially? Explain how you know.

Student Response

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

Student Facing

Are you ready for more?

Han wrote \(n(n+2) - 2(n-1)\) for the number of small squares in the design at Step \(n\).

  1. Explain why Han is correct.
  2. Label the picture in a way that shows how Han saw the pattern when writing his expression.

Student Response

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

Anticipated Misconceptions

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 (two 1-by-1 squares on each side with a 4-by-4 square in the middle) and ask them to draw Step 5.

Some students may not notice the table goes from Step 5 to Step 10 and 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 size increases by more than 1, 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 \(+2\) can be seen in the picture and where they see the \(n^2\) term. After the expression \(n^2+2\) 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 \(s\) as \(s^2\). The relationship between the side length and the area of the square is a quadratic relationship. \(s^2\) is an example of a quadratic expression. The expression we wrote in this activity, \(n^2+2,\) 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 is multiplied by the same factor.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each strategy that is shared, invite students to restate and/or revoice what they heard using precise mathematical language. Consider providing students with time to restate what they hear to a partner, before selecting one or two students to share with the class. 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 will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making
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, quadratic relationship, and quadratic expressions. Provide examples of both visual patterns and quadratic equations in the display.
Supports accessibility for: Conceptual processing; Language

Lesson Synthesis

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 than 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 than a linear expression.)
  • “In what ways is the relationship between the two quantities different than 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 than the graph of an exponential function. Even though there may be an exponent in the expression (for example, \(n^2\)), the rule that relates the two quantities is not like exponential expressions we’ve seen, where the input was the exponent.)
  • “How would you describe ‘quadratic relationship’ to someone who is unfamiliar with it?”

2.4: Cool-down - Comparing Types of Growth (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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.

Three steps of a growing pattern.
step total number of small squares
1 2
2 5
3 10
\(n\) \(n^2+1\)

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 \(\frac{5}{2}\), 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 \(n\)th step is an \(n\)-by-\(n\) arrangement of small squares with an additional small square on top, giving the expression \(n^2 + 1\) 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 \(n^2 + 1\), which is an example of a quadratic expression. We will investigate quadratic expressions in depth in future lessons.