Project AMP Dr. Antonio R. Quesada – Director, Project AMP

Inquiry Unit: Modeling Maximums and Minimums

Abstract:

Students will engage in an investigative approach using three modes of representation to derive the maximum area for a rectangular patio while minimizing the cost factor of construction. There are a series of three lessons that are co-dependent upon each other, beginning with the investigation of parabolic maximization through the investigation of plausible rectangular dimensions. Students will then analyze cost functions form separate companies through linear representations to minimizing construction costs. Finally, students will collaborate by combining the knowledge gained in the first two lessons to generalize optimal patio dimensions that fit the constraints.

This unit is designed for Middle School grades 7-8. Extensions available for beginning high school.

Students should be able to/already have: (assumed prior knowledge)

  • Work with order of operation problems involving computations of rational numbers.
  • Move between three modes representations (graphical, numerical, algebraic)
  • Plot points and lines on a coordinate plane with and without technology.
  • Be able to identify and write a slope and y-intercepts of linear functions.
  • Be proficient in basic area formulas.
  • Basic knowledge of intervals through inequalities.
  • Sound knowledge of working with graphing calculator’s list editor, equation editor, and stat plots.
  • Experience capturing screenshots using graphing calculator technologies available.

Lesson Objectives:

  • Connect physical, verbal, and symbolicrepresentations of integers, rational numbers, and irrational numbers (NS.8-10.D)
  • Use problem solving techniques and technology as needed to solve problems involving length, weight, perimeter, area, volume, time and temperature (M.5-7.E)
  • Analyze and explain what happens to area and perimeter or surface areas and volume when the dimension of an object is changed. (M.5-7.F)
  • Use coordinate geometry to represent and examine the properties of geometric figures (G.8-10.D)
  • Use algebraic representation such as tables, graphs, expressions, functions, and inequalities to model and solve problem solving situations (A.8-10.D)
  • Solving systems of linear equations involving two variables graphically and symbolically. (A.8-10.H)

NCTM Focal Points:

  • Problem Solving
  • Reasoning & Proof
  • Communications
  • Connections
  • Representations

Assessment:
Rubric to grade following inquiry based activity

Activity 1: Maximizing the Area of a Rectangular Space

You have just built a new home and plan on building an enclosed rectangular space on the back of the house using 52 sections of fence you have purchased. Each section of fence is onefoot in length. The enclosure will utilize the backside of the house as one of the sides of the enclosure (see figure 1). You decide to build a rectangular enclosure that is no longer than the back side of the house, using all 52 sections of fence.

Figure 1

1. List all possible widths your enclosure can be.

2. What possible restrictions did you discover about the widths of your enclosure?

3. Are there any above widths that would not make sensible enclosures? Explain your reasoning.

4. Give at least two unique ways of describing the “size” of your enclosure. Which of these two do you believe is better for comparing? Explain your reason.

5. If your width uses six sections of fencing, what would the corresponding length need to be? How “large”
would your enclosure be?

6. Can you make the enclosure “larger”? Find another rectangular dimension that is “larger” than question 5.
Explain how do you know it is “larger”:

7. Fill in the table below with all possible enclosure dimensions.

Width
(ft) / Length
(ft) / Area
(sq. feet)
1
2
3
4
5
6

8. What observations can be made about the area?

9. Generalize a model (equation) for calculating the length (l) in terms of the width (w).

Hint: work from perimeter: P=2l +2w

10. Complete the remaining table for all possible dimensions.

Width
(ft) / Length
(ft) / Area
(sq. feet)

11. Using your graphing calculator and the fill down function, create a spreadsheet with the following fields and their appropriate formulas. Remember that both length l and area A were represented in terms of width w.

(A): Widths

(B): Lengths

(C): Area

12. Sketch a scatter plot of possible enclosure sizes. You will again need to remind yourself of what the
independent and dependent variable are for the sketch.

13. What observations do you make from this sketch?

14. Looking at the graph is there a “largest” or maximum patio size? (Hint count in from the ends). If so what are the dimensions and size? Does this agree with your table values?

Before you purchase the materials you decide to read your cities zoning restrictions on the city hall website. Much to your dismay you read that “any enclosed space in front or behind a resident’s house must be no smaller than 300 square feet and nolarger than 325square feet”.

15. Will this zoning restriction alter your plans for construction? Explain.

16. Create an inequality that represents these zoning restrictions and superimpose this onto your scatter plot.
Re-sketch this graph in a window of [6 , 20] x [250, 350].

17. Based on all this information, what size enclosures are available and which will you build? Clearly explain your answer.

Activity 2: Cost Function Analysis Comparisons on an Interval

(available as a .tns file)

Now that you have found a patio which fits the zoning requirements and also maximizes the area restricted by the length and width chosen, you must cover the entire ground surface with paving bricks. You have done extensive research, and have narrowed your choices to two different companies.

Company A sells their bricks for $6.13 per square foot plus a $99.70 delivery charge. As we have discovered in the past the cost, f(x), of x number of square feet can be found by the linear equation f(x)=6.13x + 99.70 for company A.

Company B sells their bricks for $5.80 per square foot plus a $202.00 delivery charge. As we have discovered in the past the cost, f(x), of x number of square feet can be found by the linear equation g(x)=5.80x + 202 for company B

  1. Graph both cost functions on a standard window to show the cost of each company.

Why can’t you see company B?

  1. What window dimension allows you to see both cost functions? Sketch the functions on this window. Do you think these cost functions will intersect?
  1. See your teacher at this time to check progress
  1. What do the three regions represent (above line, on line, below line) of each cost function? What does the slope indicate? What does the y-intercept tell you?
  1. Explain when you would buy from company A; f(x). Show this on your graph.
  1. Explain when you would buy from company B; g(x). Show this on your graph.
  1. Is there a patio size where company A and company B would cost you the same amount of money? If so, describe this patio and its cost.
  1. Is this patio size an option for you to build based on what you calculated in activity 1? Explain.
  1. What size patios are options under company A’s interval? How do you know this?
  1. What size patios are options under company B’s interval? How do you know this?
  1. Write several sentences to generalize what you have learned about the fixed cost and the per square foot cost when you compared the two companies:

Activity 3: Maximization with Restrictions

1. You have budgeted $2,100 for this project. Considering the restrictions applied to your patio by the zoning committee, what size patio appears to be the most cost affective? Why? Use any method to prove your stance.

2. Would a 6% sales tax (tax only applied to pavers….not delivery fee) change you selection above? Prove this
using any method known.

Possible Extensions:

1. Have students’ discover what happens if you build an enclosure using two sides of the house

2. Have students’ alter the shape of the enclosure to a triangle or any other geometric shape and see if they can create a patio larger than the rectangular one discovered.

3. Create different size patio tiles or fencing lengths and have students recalculate. Example: 16 inch square tile, 8 inch square tile, or 4in x 8in rectangular tiles.

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