Lesson 6 You Nailed It! Practice Understanding

Jump Start

1.

Which One Doesn’t Belong?

Here is a list of different units. Share with a partner your responses to the question, “Which one doesn’t belong?”

Miles

Gallons

Hours

$\frac{miles}{gallon}$ (miles per gallon)

$\frac{miles}{hour}$ (miles per hour)

$\frac{gallons}{hour}$ (gallons per hour)

2.

Newer automobiles have an odometer that measures speed in miles per hour and an instrument that measures fuel efficiency in miles per gallon. How could you use the readout from these two devices to calculate your fuel consumption in gallons per hour?

Learning Focus

Apply geometric modeling to solve a real-world problem.

How can I use quantities and units to guide my computational thinking in a modeling context?

How can I model geometric contexts using properties of shapes and measurement?

Open Up the Math: Launch, Explore, Discuss

Tatiana is helping her father purchase supplies for a deck he is building in their back yard. He wants the deck to be a rectangle with an area of $96 square feet$ , but he hasn’t decided on the actual dimensions. The deck will be made of planks that are $8 ft$ long and $\frac{1}{2} ft$ wide. These planks will be attached to the framing joists with 16d nails. She would like to design the deck to minimize the cost of the planks and nails.

Nails are sold by the pound at the local hardware store, so Tatiana needs to figure out how many pounds of 16d nails to tell her father to buy. She has gathered the following information.

Each decking plank costs $$ 12$ , and they are only sold in full $8 ft$ lengths. (Tatiana’s dad doesn’t own a table saw, so he doesn’t want to cut any planks.)
Each plank requires $9$ nails to attach it to the framing joists; one set of three nails at each end, and a set of $3$ nails in the middle of the plank.
16d nails are made of steel that has a density of $4.65 \frac{oz}{{in}^{3}}$ .
There are $16 ounces$ in a pound.
The hardware store sells nails by the pound, and 16d nails cost $$ 8.00$ per pound. It is possible to buy a fraction of a pound.

Tatiana has also found the following drawing of a cross-section of a 16d nail. She knows she can use this drawing to help her find the volume of the nail, treating it as a solid of revolution. (Note: The scale on the $x$ - and $y$ -axis is in inches.)

1.

Help Tatiana design a deck that will minimize the cost of the materials. Show the work that supports your final design.

2.

Calculate the volume of a nail. Show the computations that support your work.

3.

Calculate the cost of the deck. Show the computations that support your work.

Ready for More?

How much of the total weight of the nails is due to the seemingly insignificant weight of the pointed tip of the nails?

Takeaways

An example of geometric modeling:

We can calculate the weight of an object that doesn’t physically exist if we know

Adding Notation, Vocabulary, and Conventions

What is the difference between a direct measurement and a derived measurement? Use examples in your description.

What is the difference between a quantity and a unit? Use examples in your description.

What do I think about when using units to reason about computation?

Vocabulary

density
quantity
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we calculated the weight of a solid of revolution by knowing the cross-sectional region that defined the solid and the density of the material from which the solid would be made. This is an example of geometric modeling.

Retrieval

1.

Find the trigonometric ratios for $△ A B C$ .

Write your answer as a fraction. Then write each ratio as a decimal rounded to thousandths place.

$\sin A =$

$\cos A =$

$\tan A =$

2.

Quadrilateral $A B C D$ is a parallelogram.

Given: $m ∠ A = (2 x + 41) °, m ∠ B = (3 x - 46) °$

Find $m ∠ C$ and $m ∠ D$ .