Lesson 3
Building Quadratic Functions from Geometric Patterns
3.1: Quadratic Expressions and Area (10 minutes)
Warm-up
This warm-up introduces some quadratic functions that arise naturally in finding the areas of different shapes. While students are not explicitly asked to find the area of geometric patterns in this lesson, the expressions that they write for the number of small squares in the patterns are effectively formulas for area (in terms of the number of small squares).
Launch
Arrange students in groups of 2. Ask both partners in each group to write the area expressions for Figure A and complete the first column of the table and discuss their responses. Then, ask one partner to write the expressions for Figure B and the other partner to write the expressions for Figure C.
Student Facing
Figure A is a large square. Figure B is a large square with a smaller square removed. Figure C is composed of two large squares with one smaller square added.
Figure A
Figure B
Figure C
Write an expression to represent the area of each shaded figure when the side length of the large square is as shown in the first column.
| side length of large square |
area of A | area of B | area of C |
|---|---|---|---|
| 4 | |||
| \(x\) | |||
| \(4x\) | |||
| \((x+3)\) |
Student Response
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Activity Synthesis
Invite students to share their expressions and record and display them for all to see. Include all equivalent expressions. Students may notice that all the expressions have a variable or a term that is squared. Explain that all of the expressions are quadratic expressions. Some expressions may seem familiar (for example, the expressions representing the area of Figure A) and others may seem quite foreign, but we know that each of them represents the area of a figure given a particular side length of the square.
3.2: Expanding Squares (10 minutes)
Activity
In earlier activities, students wrote expressions to describe visual patterns by first creating tables and reasoning repeatedly about how the pattern changes at incrementally greater step numbers. Here they write an equation to describe a pattern without creating a table of values. They look more directly at the structure of the visual pattern and its connections to the parts of the equations, and then reason in the other direction: given an equation, they generate a visual pattern. Students also begin to frame the relationship between the quantities as a function with inputs and outputs leading to a definition of quadratic function.
Some students are likely to view the \(n\)th step of the given pattern as an \(n\)-by-\(n\) square with four small squares added at the corners, leading to the expression \(n^2 + 4\) for the total number of small squares. Others may view this as an \(n+2\) by \(n+2\) square with 4 rectangles removed, each consisting of \(n\) squares. The latter leads to the (equivalent) expression \((n+2)^2 - 4n\). In the next activity, students will look closely at how equivalent expressions arise when analyzing geometric patterns, but if equivalent expressions come up in this activity, consider inviting students to share them.
This prompt gives students opportunities to see and make use of structure (MP7) when writing an equation from a visual pattern. The specific structure they might notice is an inner square with a side length equal to the step number along with a small square on each corner. Later, generating a pattern given an equation requires them to reason abstractly and concretely (MP2).
Launch
Give students a moment to observe the pattern from the activity and ask them what they notice and what they wonder. Then, ask students to sketch the next step in the pattern and share their sketch with a partner.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
- If the pattern continues, what will we see in Step 5 and Step 18?
- Sketch or describe the figure in each of these steps.
- How many small squares are in each of these steps? Explain how you know.
- Write an equation to represent the relationship between the step number \(n\) and the number of squares \(y\). Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)
- Sketch the first 3 steps of a pattern that can be represented by the equation \(y = n^2 - 1\).
Student Response
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Student Facing
Are you ready for more?
- For the original step pattern in the statement, write an equation to represent the relationship between the step number \(n\) and the perimeter, \(P\).
- For the step pattern you created in part 3 of the activity, write an equation to represent the relationship between the step number \(n\) and the perimeter, \(P\).
- Are these linear functions?
Student Response
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Anticipated Misconceptions
Some students may wonder how to draw a pattern given the equation \(y=n^2-1\). Show them the warm-up problems where they subtracted 1 to remove the small square in Figure B and added 1 when there was an extra small square in Figure C. Prompt them to describe subtracting 1 as removing one small square from each step. Some students may find it easier to start drawing step 2 or step 3. They can work backwards to draw step 1 which would have 0 squares since \(1^2-1=0\). Emphasize that making a table can help them figure out exactly how many small squares are needed in each step.
Activity Synthesis
Make sure students see the connection between the equation \(y=n^2 +4\) and composition of the squares in the pattern: that regardless of what \(n\) is, the figure at Step \(n\) is composed of a square that is \(n\) by \(n\), plus 4 small squares at each corner.
Next, help students relate the work so far to the idea of functions. Discuss with students:
- “In each pattern we’ve seen, is the relationship between the step number and the number of squares a function? How do you know?” (Yes. The step number is the input, and the number of squares is the output. For every step number, there is a particular number of squares.)
- “How can the relationship be expressed using function notation?” (If the function \(f\) gives the number of squares at step number \(n\), we can define it as \(f(n) = n^2+4\).)
Introduce quadratic function as a function that is defined by a quadratic expression. Like other functions, it can be represented with an equation, a table of values, a graph, and a description.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
3.3: Growing Steps (15 minutes)
Activity
This activity further develops students’ ability to distinguish a quadratic function and to articulate that distinction. Given a pattern of squares, students critique a statement about how the pattern is changing and make a case for whether two given equations both represent the same pattern. They notice that an expression without a squared term, for example, \(n(n+2)\), can also be a quadratic expression. Along the way, students practice constructing logical arguments and supporting them (MP3).
As students discuss the first question, look for those who can explain that the given pattern is not linear because the number of squares does not increase by equal numbers over equal intervals.
As they discuss the second question, identify students who:
- understand that applying the distributive property to \(n(n+2)\) gives \(n^2 +2n\)
- can show this equivalence visually: by marking the side length of each rectangle with \(n\) and \(n+2\), and by showing the \(n\)-by-\(n\) square plus 2-by-\(n\) rectangle in each rectangle
Ask these students to share their explanations during whole-class discussion.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Supports accessibility for: Language; Organization
Student Facing
- Sketch the next step in the pattern.
- Kiran says that the pattern is growing linearly because as the step number goes up by 1, the number of rows and the number of columns also increase by 1. Do you agree? Explain your reasoning.
- To represent the number of squares after \(n\) steps, Diego and Jada wrote different equations. Diego wrote the equation \(f(n)=n (n+2)\). Jada wrote the equation \(f(n) = n^2 + 2n\). Are either Diego or Jada correct? Explain your reasoning.
Student Response
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Activity Synthesis
Briefly discuss students’ sketches for Step 4. Then, select students to present their explanations of the second question. Make sure students see that when both the length and width of the rectangle grow at each step, the increase in the number of squares (or in area) from one step to the next is no longer constant, so the growth is not linear.
For the last question, if no students reason about the equivalence of the two expressions visually, demonstrate it. Take the diagram for any step number and show where the \(n^2\) and \(2n\) are, and then where the \(n\) and \((n+2)\). Emphasize that because they both represent the same quantity the expressions \(n^2 + 2n\) and \(n (n+2)\) must be equivalent.
If not already mentioned by students, point out that we can also see that these expressions are equivalent without using the picture, by applying the distributive property, which gives \(n(n+2) = n^2 + 2n\). In other words, both expressions define the same function. This means that \(n (n+2)\) is also a quadratic expression even though it does not have a squared term.
Lesson Synthesis
Lesson Synthesis
To help students consolidate the ideas in the lesson, discuss questions such as:
- “How do you know if the relationship between two quantities represents a quadratic function? What clues would you look for?” (There is only one output for every input. In the relationship, one quantity is in some way squared or multiplied by itself to obtain the second quantity. The relationship can be expressed with a squared term, among other ways. A pattern that changes linearly in two directions, such as a rectangle that grows linearly in length and width.)
- “What does it mean when we say that two expressions define the same function?” (The two expressions describe the same relationship between two quantities, and there’s a way to show that this is the case.)
- “Can we say that \(x(x+5)\) and \(x^2 + 5x\) define the same function? How can you show if they do or don’t?” (Yes. Using the distributive property shows that \(x(x+5)=x^2+5x\). A diagram can also show that the two are equivalent. A rectangle with side lengths \(x\) and \(x+5\) has area \(x(x+5)\). If we decompose the rectangle into two sub-rectangles that are \(x\) by \(x\) and \(x\) by 5 and calculate their areas and combined them, we get \(x^2 + 5x\).)
3.4: Cool-down - A Quadratic Function? (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Sometimes a quadratic relationship can be expressed without using a squared term. Let’s take this pattern of squares, for example.
From the first 3 steps, we can see that both the length and the width of the rectangle increase by 1 at each step. Step 1 is a 1-by-2 rectangle, Step 2 is a 2-by-3 rectangle, and Step 3 is a 3-by-4 rectangle. This suggests that Step \(n\) is a rectangle with side lengths \(n\) and \(n+1\), so the number of squares at Step \(n\) is \(n(n+1)\).
This expression may not look like quadratic expressions with a squared term, which we saw in earlier lessons, but if we apply the distributive property, we can see that \(n(n+1)\) is equivalent to \(n^2 + n\).
We can also visually show that these expressions are the equivalent by breaking each rectangle into an \(n\)-by-\(n\) square (the \(n^2\) in the expression) and an \(n\)-by-\(1\) rectangle (the \(n\) in the expression).
The relationship between the step number and the number of squares can be described by a quadratic function \(f\) whose input is \(n\) and whose output is the number of squares at Step \(n\). We can define \(f\) with \(f(n) = n(n+1)\) or \(f(n) = n^2 + n\).