Algebra I Items to Support Formative Assessment
Unit 3: Quadratic Functions and Modeling
Part I: Graphical Analysis and Modeling with Quadratic Functions
F.BF.A.1 Write a quadratic function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context
b. Combine standard function types using arithmetic operations.
Task F.BF.A.1 a & b
We have a small company that makes and sells figures like the ones below
PART A
1. Find an expression that will help us determine the number of square tiles in any figure (n).
Write a separate expression that will help us determine the number of triangular tiles in any
figure (n). Justify your expressions.
2. We have 150 square tiles available to make a figure. What is the largest figure we can make?
How many triangular tiles would this figure have? Justify your answer.
PART B
The manager at a local mall wants to purchase one of our figures to decorate a wall at the mall.
1. She wants to spend $28.00 on triangular tiles for one figure. How much will the square tiles
for the same figure cost? Justify your answer.
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2. After looking at the space she has decided she wants a larger figure to fit the wall. It appears
the 30th figure would be the correct size.
a. How much money will we need to have to buy all the tiles to make this figure.
Explain how you determined your answer?
b. To make this easier in the future, write an expression that would determine the cost of
the tiles needed for any figure (n). Explain why your expression would work.
Solution:
PART A
1. The number of square tiles is n2. The center of the pattern is a square whose side is formed
by the number of tiles equal to the figure number. The number of triangular tiles is 4n. The
first figure has 4 triangular tiles and each figure after that adds four more tiles.
2. The largest figure we can make with 150 square tiles is figure 12. 122=144 and 132=169.
We don’t have enough square tiles to make figure 13. We would need 48 triangular tiles for
figure 12 because 4(12)=48.
PART B
1. If we spend $28.00 on triangular tiles for a figure, we would have to spend $180.00 for the
square tiles. 28 ÷ 0.35 = 80, so we have 80 triangular tiles. 80 ÷ 4 = 20 so we are making
figure 20. For figure 20 we need 202 = 400 square tiles and 400(0.45) = 180.
2.
a. $447.00 Figure 30 needs 30(4) = 120 triangular tiles and 302 = 900 square tiles. 120(0.35) =
$42 and 900(0.45)= $405. 42 + 405 = $447.
b. 0.45n2 + .35(4n) or .45n2 + 1.4n
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product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
F.BF.A.1 a & b
A farmer has 850 feet of fence to build a rectangular area for his cows adjacent to a river as
shown below. Write an expression to represent the area of the enclosed field as a function of the
width of the field. Using your expression, provide dimensions that would make sense for the
cows to graze comfortably and still have access to the river.
Solution: x(850 - 2x) or -2x 2 + 850x
Possible Explanation:
I think that the width of the fence could be 212.5 feet and the length could be 425 feet. This will provide
plenty of space for the cows to graze.
F.BF.A.1 a & b
An open box is made from a tin sheet 7 in. square by cutting out identical squares from each
corner and bending up the resulting flaps, determine the surface area of the box as a function of
the size of the squares cut out.
Solution: 49 - 4x 2
F.BF.A.1 a & b
The owner of a theater is considering raising ticket prices. He has found that he will sell 700
tickets if the ticket price is $25. He has also discovered that for every $1.00 the ticket price
increases there will be 6 less people in attendance.
1. Write an expression in simplest form that represents the income as a function of the
amount the price is increased.
2. The owner asks your opinion on how to maximize his revenue given what he has learned
about ticket sales.
Solution:
1. Price increase is x
Price is 25 + x
Number of tickets sold is 700 - 6x
Income is (25 + x)(700 - 6x) or -6x2 +550x +17500
2. To maximize revenue I would recommend increasing ticket prices to $70.83 (based on a
price increase of $45.83). This would result in a total revenue of $30,104.17.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.