Lesson and Assessments - Quadratic Growth

Progression of Skills (outline)

Before the lesson:

• Students should be familiar with graphing an equation, making a table of values, finding points on a graph, substituting values into variables and some knowledge of perimeter and area of a rectangle.

During the lesson:

• Students will find patterns of how the area changes for a rectangle when provided a fixed perimeter.
• Students will identify patterns in triangular numbers and then create tables and graphs

After the lesson:

• Students will be able to understand graphs of a nonlinear function with a predictable pattern of change in the y-values (areas)

Lesson 1

Directions for Lesson 1

Fencing: In this activity students will be creating an animal pen for rabbits using the same amount of fencing and determining the pen which has the maximum (most) area. Note: This section can also be used as a pre-test if desired.

1. Instructors: begin with asking students about linear growth situations. What would a table of values look like? Describe a situation in words, like the number of gallons of gas with total cost of gas or time driving at a constant speed with the total distance traveled.
2. Make some rectangles with a perimeter of 16 feet.
3. Make a table of all the possible whole-number length of sides that have a perimeter of 16 feet (make a separate entry for each length of each pair). Then, find the area of each one. Make a graph using the entries in the table. (x=length of each side, y=area of rectangle)
4. Make some noticings and wonderings in the table and in your graph.
5. Encourage students to calculate the differences in the areas and compare those differences. How are they different from the differences in the linear situations?

Anticipate Misconceptions:

• Students might find side lengths that multiply to 16.
• Students might find just 2 sides that add up to 16, not 4

Lesson 2

Directions for Intervention Activity 2

Making a Garden: In this activity students will be creating a rectangular garden from a square garden according to building regulations.

Your original plan is to make a square garden. However, the site regulations dictate that the garden must be a rectangle with a width that is two feet less than the original square’s side length and a length that is two feet more than the original square’s side length.

1. Make a drawing of your original square using a side length of 4 feet. Determine its area.
2. Construct the corresponding rectangle from the regulations above. Determine its area.
3. What is the symbolic description (equation) of the areas?
4. Now we will repeat this process starting with different sized squares. Make a table of values for the results starting with a square with side length of 2 feet, then 3 feet, and continue until ending with a square with side length of 6 feet. The table should have a column for the area of the square and a column each for the width, length, and area of the rectangles.
5. Calculate the difference of the area of the square and the area of its corresponding rectangle. Add a final column to your table to input each of these calculations
6. What do you notice about the final column? Is this surprising? Why do you think this happened?

Lesson 3

Directions for Intervention Activity 3

Triangular Number Pattern: In this activity students will be building a pattern noted by the ancient Greeks. Interestingly, they thought of numbers geometrically

1. Triangular Number Pattern. Directions in worksheet
2. The student(s) can use counters to model the situation if it’s helpful
3. Make a table to show from 1 to 10. Graph the relationship.

Progress Monitoring:

• Triangular Numbers: 1,3,6,10,15,21,28,36,45,55.
• What is the pattern of change as the triangular number increases by one? (The difference goes up by 1)
• Follow up question: How many counters would there be for the 15th triangular number? How do you know? What about the 100th? What about the nth (unknown) number? What is the relationship between the nth triangular number and the number of counters?

• Optimal Landing Zone for a Drone: A drone needs to land in a rectangular zone with limited fencing available to outline the landing area. Given a maximum perimeter, determine the dimensions that create the largest possible landing zone for safety.
• Farm Irrigation Design: A farmer is designing a rectangular plot that will be irrigated from a central pipe. They have a limited length of hose to surround the plot and want to maximize the planting area. What are the ideal dimensions?
• Optimizing Window Size for Maximum Light: A builder wants to create a rectangular window with a perimeter of 16 feet to let in the maximum amount of light. What dimensions would maximize the window’s area?
• Volume of an Open-Top Box: Given a 12x12-inch piece of cardboard, cut equal squares from each corner to fold and form an open-top box. What size squares should be cut to maximize the volume of the box?

• National Council of Teachers of Mathematics. (April 25, 2018). Catalyzing Change in High School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
  • Pg. 28 - Developing a positive mathematical identity and agency
  • Pg. 57 - Quantitative Literacy
    • Create and interpret visual representations
• Cuoco, A.A. (Ed.) (2001). The Roles of Representation in School Mathematics. (2001 Yearbook). Reston, VA: National Council of Teachers of Mathematics.
  • “Focus 2” - Visualizing and summarizing data
  • Preface (ix) importance of the perspective of students (given in context of how they learn but relevant to the experiences of the student)
  • Chapter 19 (pg. 238) Realistic Mathematics Principles
    • 1. Learning math involves construction
    • 3. Math learning is promoted by reflection
• Universal Design for Learning (UDL) Guidelines:
  • Optimize individual choice and autonomy (checkpoint 7.1)
  • Optimize relevance, value, and authenticity (checkpoint 7.2)
  • Foster collaboration and community (checkpoint 8.3)
  • Develop self-assessment and reflection (checkpoint 9.3)