Building Quadratic Functions from Geometric Patterns
10 min
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.
Student Task
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) |
Sample Response
Expressions that are equivalent to the following are also acceptable.
| side length of large square |
area of A | area of B | area of C |
|---|---|---|---|
| 4 | 42 | 42−1 | 2(42)+1 |
| x | x2 | x2−1 | 2x2+1 |
| 4x | 16x2 or (4x)2 | 16x2−1 or (4x)2−1 | 32x2+1 or 2(16x2)+1 |
| (x+3) | (x+3)2 or (x+3)(x+3) | (x+3)2−1 | 2(x+3)2+1 |
Activity Synthesis (Teacher Notes)
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.
Standards
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- HSF-BF.A.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
10 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Building Quadratic Functions from Geometric Patterns
10 min
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.
Student Task
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) |
Sample Response
Expressions that are equivalent to the following are also acceptable.
| side length of large square |
area of A | area of B | area of C |
|---|---|---|---|
| 4 | 42 | 42−1 | 2(42)+1 |
| x | x2 | x2−1 | 2x2+1 |
| 4x | 16x2 or (4x)2 | 16x2−1 or (4x)2−1 | 32x2+1 or 2(16x2)+1 |
| (x+3) | (x+3)2 or (x+3)(x+3) | (x+3)2−1 | 2(x+3)2+1 |
Activity Synthesis (Teacher Notes)
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.
Standards
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.
- HSF-BF.A.1.a·Determine an explicit expression, a recursive process, or steps for calculation from a context.