Lesson 2

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.)