Unit 4, Lesson 3
Building Quadratic Functions from Geometric Patterns
Integrated Math 2
3.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
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, 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.
Activity
NoneFigure 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 | |||
Student Response
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Building on Student Thinking
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.
Student Lesson Summary
Sometimes a quadratic relationship can be expressed without writing a squared term that appears as a variable raised to the second power (like or ). Let’s take this pattern of squares, for example.
Three steps of a growing pattern. Step 1: Total of two squares, stacked one atop the other. Step 2: Total of six squares, two squares on row 1, row 2 and row 3. Step 3: Total of twelve squares, three squares on row 1, row 2, row 3 and row 4.
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 is a rectangle with side lengths of and , so the number of squares at Step is .
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 is equivalent to .
We can also visually show that these expressions are the equivalent by breaking each rectangle into an -by- square (the in the expression) and an -by- rectangle (the in the expression).
Three steps of a growing pattern. Step 1: Total of two squares, stacked one above the other with a space between the rows. Step 2: Total of six squares, two squares on row 1, row 2 and row 3. Row 3 is above the others with a space between rows 2 and 3. Step 3: Total of twelve squares, three squares on row 1, row 2, row 3 and row 4. Row 4 is above the others with a space between rows 3 and 4.
The relationship between the step number and the number of squares can be described by a quadratic function, , whose input is and whose output is the number of squares at Step . We can define with or with .