UbD Stages 1-3

HS.F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Essential Questions:

Where will we find real-world answers on a graph?

What does a function represent to real-world situations?

How can we use a function to help make us the best choice in a decision?

Enduring Understandings:

Consider the given company's profit function. At what price should they sell their product? What is the profit made at that price? What price gives the company the least profit?

Given the following function, what does f(0) tell us? Is there any other way we can represent this value?

Find the x-intercept, y-intercept, intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums. *lame

*not sure if these are applicable to this standard...but they fit closely

What does it mean for a function to be proportional?

Given a function is proportional, is it always linear?

Given a function is linear, is it always proportional?

Facet 1: What is implied when f(0) = 50?

Facet 2: What does point A represent? *graph with A at max

Facet 3: What types of things can we model using a function?

Facet 4: An open-topped box is cut from a rectangular piece of paper. Each corner is cut out using the same size cut as modeled in the below picture. The sides are then folded up, making the box. The volume of the box as a function of the size of the cutout, x, is shown here. Based on the graph what is a reasonable size cutout to get the largest volume for the box? *place points on real and non-real answers of the function

Facet 5: What is the function helping us see in the data?

Facet 6: How do I know my answer is correct, or close to it, without knowing the answer?

Activities:

This will be a small group activity that I expect will take a few days/week. Probably make a project out of it. The students will get colored paper(s) to work with so they can see the box as they do the problems. After writing this I realize it might work better when doing volumes – or leading into volumes.

1.

a) Create an open box(without a top) by: I) cutting four equal-sized square corners from an 8.5 by 11-inch sheet of paper and ii) folding up the sides and taping them together at the edges. Is it necessary that the cutout be square? Explain. *picture to go along with it

b) List the relevant quantities for determining the volume of a box made by cutting four equal-sized square corners from a sheet of paper and folding up the sides.

100. Make a cutout of 1 inch. Use a ruler to measure the length of the sides of your square cutout (measured in inches).

cutout length:

box's height:

length of the box's base:

width of the box's base:

500. Calculate the volume of the box. Describe your method for computing the box's volume when all you know is the length of the side of the square cutout, and you are unable to use a ruler.

Volume:

2.

a) Define a formula to relate the width formula of the box and the length of the side of the square cutout. Be sure to define your variables.

b) Define a formula to relate the length formula of the box and the length of the side of the square cutout. Be sure to define your variables.

100. Define a formula to relate the area of the box's base and the length of the side of the square cutout. Be sure to define your variables.

500. Define a formula V, to relate the volume of the box and the length of the side of the square cutout. Be sure to define your variables.

e) Let x represent the length of the side of the square cutout; let w represent the width of the box's base; let l represent the length of the box's base. Complete the following table.

Quantity	Smallest Possible Value	Largest Possible Value
x
w
l

f) Using a graphing calculator, create a graph that represents the volume of the box. When determining the window setting on your calculator, consider the possible values of x and the possible values of V. Construct the graph below. *maybe just have it graphed already?

g) Use your volume formula to find the volume of the box at when x=0, x=1, x=2, x=3, x=4. Do these points correspond to points on your graph?

h) Using your graphing calculator, find the maximum volume of the box. Be sure to consider the domain of x. What is the volume at that maximum value? What size cutout gives us the largest volume?

i) Using interval notation, when is the volume increasing? When is the volume decreasing?

j) Why does the volume of the box increase and then decrease as the size of the cutout increases? Shouldn't the volume always increase when the cutout gets larger?

k) Why don't we want to use values of x greater than 4? Such as a cutout of 8 inches?

Congratulations! Your now an engineer!