R Values: What are you getting for your money?

Math Goal:  To look at pricing formulas by different manufacturers of insulation.  Students will write the equation of a line using a line of best fit. Students will learn how to interpret the real life meaning of slope, y intercept, and equations.

CTE Goal:  Students will become familiar with R Values and learn about types of insulation available.

Procedure: Students are to select a manufacturer of insulation (foam or batt).  The manufacturer should have 3 or more thicknesses of the product.  The product should be the same material except for thickness.  Examples include fiberglass batts, blown in, rigid foam board, blown in foam, blue jean insulation batts, rockwool, etc.  Students need to record the R value and the cost for each thickness of the product.  

Graph the data with R value along the x axis and the cost along the y axis.  Draw a line of best fit.  Write an equation (algebraically) of the line.  Identify the y intercept and slope.  Explain the real world meaning of each.  Is it a direct variation?   Predict the cost of a new thickness of the product.  What other factors, besides cost, would a builder consider when choosing an insulation product?  Please site your sources of cost.

Write a 1 page summary of the data and your findings.

  

Are You Torqued?

Math Goal:  Students will write the equation of a line using a line of best fit.  Students will learn how to interpret the real life meaning of slope, y intercept, and equations.

 CTE Goal:  Students will become familiar with a torque wrench and learn the importance of torque.

Teacher notes and material needs are at the bottom.

Procedure:

 1. Start the nut on the bolt and hand tighten.

 2. Place the bolt and nut assembly in a vise.  The vise should be clamped down on only the nut.  Be sure the bolt can turn.

 3. Place the protractor on the bolt.  Using a marker, mark a line across the bolt head marking the 0 degree mark.

 4. Set the digital torque wrench to its minimum torque setting in foot-pounds.   Tighten the bolt.   Record the amount of turn in degrees from 0 degrees and the digital torque wrench reading in foot pounds.   This data point is marks the end of the ”snugging zone”.    Compare how far you turned (degree) the bolt to someone else’s bolt.  What can you determine about your grip strength?   Be sure to label this point on the graph.  Hint:  Use a ruler to extend the line drawn on the bolt head to help read the protractor.

 5. Increase the digital torque wrench setting by about 10 foot-pounds.  Turn the bolt until the torque is reached.  Record the amount of turn in degrees from 0 degrees and the digital torque reading.  Note:  the actual digital torque reading will be different than what you set on the wrench.  Record the actual reading.

 6. Repeat step 5 increases the digital torque wrench by 10 foot pounds each time.  Continue increasing until you have material failure.  This can look like a broken bolt, stripping metal, bolt turning in the vise, etc.  Record this data point and label it as “material failure”.  Create a data table with Degree Turn (X) and Foot-pounds (Y)


 7. Graph your data.

 8. Are there data points that do not fit the majority of the points?

 a. If yes, give some reasons why these points do not fit the pattern.

 b. What do you do with any points that do not fit the pattern?

 9. Draw a line of best fit.

 10.   Calculate the slope of the line.

 11.   Calculate the y intercept

 12. Write the equation of the line.

 13. What is the real life meaning of:
a. Slope

 b. Y intercept

 14.   What is the domain and range of this problem?

 Materials Needed:   For each group of students you will need:   A 9/16” bolt (1 ½ “ long) and nut;  digital torque wrench is needed and can be shared between 2 groups.; vise attached to table top;  marker; ruler; protractor with a hexagon (size of bolt head) cut out of the center(see pattern at the end of document).

 Teacher Notes:
This can be done with or without graphing calculators.  Be sure to talk about the CTE situation so students can answer the question “Why do I care about this?”.

Packaging and Shipping

Goal: Collect data regarding weight versus volume for shipping of products. You will create/select an ideal box to ship an egg with foam packing peanuts. A contest will be conducted.

Procedure:
1. Label each of your boxes with A, B, C, etc.

2. Find the volume of each of your boxes. Measure to the nearest ¼ inch. Record the volume in the table below.

3. Fill each box with your shipping item, and foam packing peanuts. Be sure the box can close easily but the item does not shift. Weigh each box (ounces) and record below.

Box Label ( Weight , Volume ) Ratio of weight to volume
A
B
C
D
E






4. Graph the data (weight, volume) on a piece of graph paper.


5. Draw a line of best fit.


6. Calculate the slope.




7. Using y = mx + b , calculate the y-intercept.





8. Write the equation of the line of best fit.





9. Explain the real life meaning of slope.





10. Explain the real life meaning of the y-intercept.





11. Using your equation from #8 above, predict the weight of a package that is 1 feet tall, 2 feet wide, and 3 feet tall.




12. Calculate the ratio weight to volume for each of your boxes and record the decimal value in the last column of the table in #3 above.





13. As a manufacturer responsible for shipping, which of your ratios would be the best assuming your product will arrive safely.


Teacher Notes:

In the manufacturing process, many of the items once completed will need to be shipped to other companies (for additional assembly), shipped to warehouses for distribution, or directly to the customer. During the shipping process, it is critical that the item arrive in perfect condition. If damaged during the shipping process, companies must repair or replace the item. In addition, if products are damaged often, customers will search out other companies to fulfill the order request.

One method of providing cushion for shipped items is with packing peanuts. These foam-packing fragments are made of a variety of materials and in different shapes.

Shipping costs have historically been calculated on the basis of gross weight in kilograms or pounds. By charging only by weight, lightweight, low-density packages become unprofitable for freight carriers due to the amount of space they take up in the truck/aircraft/ship in proportion to their actual weight. The concept of Dimensional Weight has been adopted by the transportation industry worldwide as a uniform means of establishing a minimum charge for the cubic space a package occupies.

Dimensional weight favors shippers of dense objects and penalizes those who ship lightweight boxes. A box of unpopped corn kernels will likely be charged by gross weight; a box of popcorn will probably be charged by its dimensional weight. This is because the large box of popcorn takes up a lot of space but does not fill up a vehicle's capacity in terms of weight, making it an inefficient use of space.
Shippers avoid dimensional weight charges by using smaller boxes, by compressing their goods, and by reducing the use of packing materials.

CTE Goal: Students will understand the packaging needs of manufacturing companies.

Math Goal: Students will collect data and use the line of best fit to create an equation to predict weight of packing materials.

Materials Needed: Each group needs 4-5 small boxes, packing peanuts (enough to fill the largest box), access to a digital scale, small item for shipping (does not have to be the same for all groups), and a raw egg per group if you do the contest. Ideally, the small item for shipping can be a plastic Easter egg with a little weight inside. Packing peanuts can be recycled at your school by reclaiming them in the packages received by your school if you let staff know that you need them



CTE Situation: Teachers need to summarize/demonstrate the information found under Teacher Notes. Then continue with this. Students are to find the “best” foam packing peanuts package for an egg to be shipped. The box is to be dropped (to simulate shipping) from 12 feet. To win the contest between students (groups), the egg must survive unbroken and the ratio of weight in ounces to volume in inches should be the largest decimal value. This is designed for students to find the optimum volume for shipping without damaging the product. The egg contest can be skipped but is a fun extension.

Similar CTE Situation: Discuss other packaging methods such as bubble wrap, shrink wrap, cardboard, etc. Why do companies choose what they do?


Egg Drop Contest: A fun extension is to have each group create/find the ideal box, fill with foam packing peanuts and an egg. The winner will be the group that drops their box without breaking the egg, and has the largest ratio (#12 & 13). To keep the packing peanuts clean, place the egg in a zip lock bag.

Carbon Monoxide Safety

Goal: If the same engine was operated in this room, how fast will the safe ceiling of 200 ppm be reached. Make an estimation before continuing.



Procedure:

1. Looking at the graph above, how many minutes into the activity did the room become unsafe?


When did the room return to safe levels of carbon monoxide (after the engine stopped)?


2. What is the size of the room in the graph above?



3. What are the dimensions of this room (the one you are in now)? Use your tape measure and calculators.



4. What is the volume of this room?



5. Write a proportion and then solve to find the answer to this problem.
If the same engine was operated in this room, how many minutes will the safe ceiling of 200 ppm be reached?






Teacher Notes: CARBON MONOXIDE DANGERS

CTE Goal: Understanding the effects of carbon monoxide poisoning.


Math Goal: Students will be reviewing volume and proportions.


CTE Situation: Show the included power point through slide #9 for today’s goal. Do not tell the students what they are studying today....let them guess with the power point slides. Discuss the need for proper ventilation with combustible engines in an enclosed space.




Similar CTE Situation: Many home today require a carbon monoxide detector. Most detectors will sense 70 ppm in an hour or 400 ppm in 4 minutes. A faulty furnace can produce up to 1600 ppm which can cause headaches and nausea in 20 minutes and death in 1 hour. How many deaths are there from carbon monoxide poisoning in U.S. homes in 1year?


Materials Needed: For each group, you will need a tape measure and calculators (if allowed)




Teacher Notes: This lesson can be applied in multiple mathematic areas including quadratics, exponentials, and piecewise functions.. This lesson is given early in the year when safety is often taught. Therefore, a simple review of volume is the first objective. If taught with Algebra 1 or Algebra 2, it should be extended into interpretation of graphs, domain and range, and identifying pieces of the piecewise function. All of these topics would be considered introductory in the Algebra 2 classroom (or at the very least review).

Water Conservation

Many homes and commercial buildings have rain gutters. The gutters were originally designed to divert water away from the foundation of the building and thus prevent water damage.

Most gutters are fabricated on the construction site. This allows the gutters to be as long as needed (no seams) to fit the building. Assume you have 12 inch wide flat sheet metal to bend upwards to form right angles to create a commercial rain gutter as shown in the drawing above.

In recent years there has been a surge in installing rain harvesting systems. In these systems, rain is collected via rain gutters and stored in large tanks. This water is then used to water gardens and sometimes for household use.

What is the largest (cross sectional area) gutter you can create?

If our house has a roof area of 1200 sq ft, how many gallons of water can we collect in a 1 inch rainfall?

Design a tank to hold the water.