Geometry in Construction Math Problem of the Month

Are You Torqued?

2012-04-25T18:48:00.001-07:00

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

2012-03-17T19:13:00.009-07:00

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

2012-02-19T16:11:00.000-08:00

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

2012-01-16T16:31:00.000-08:00

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.

Driving and Texting

2011-11-27T17:04:00.001-08:00

Goal: Is Car and Driver correct? Learn the difference in the stopping distance of a car when driving while texting vs. not texting.

Drivers today have many distractions that can keep them from applying braking as soon as possible. One of the biggest distractions while driving is texting. Many times drivers believe that a quick, short text is doable while driving. In this lab, you will collect data, and then determine how the stopping distance of a typical car is affected.

Procedure:

Assume you just received a text from your best friend while you are driving.

1. Determine a short 3-5 word response.

2. Text the response one handed (this is how you do it while driving). Record the time it takes to text the message. Be sure to time yourself on the first attempt at texting. Note: your speed in texting will increase and flaw your results.

3. Find 9 additional people to text the same response. Record their time to text.

_________, _________, _________, _________, _________,

_________, _________, _________, _________, _________,

4. Find all 3 averages of the time (mean, median, mode). Decide which average best represents your data.

Mean __________ Median __________ Mode __________

5. Using the internet, (record your source here:____________________________) find the stopping distance of a car traveling 35 mph and 70 mph on dry pavement with normal reaction times.

35 mph stopping distance__________
70 mph stopping distance__________

6. How many additional feet will a driver travel while sending the text. Use the mean, median, or mode from part 4 above to determine the travel distance.

7. How many football fields will the driver travel in the total stopping distance of the car?

8. Is Car and Driver correct?

Cross Gable Framing Angles

2011-10-17T18:32:00.000-07:00

Many homes have small gables along the front of the home. The photo at right shows an example of the type of gable above the front door that we are constructing for this year's house. The width of the gable immediately above the front door is 10 feet.

The main roof has a 3/12 pitch. The small gable will have a 3/12 pitch. It may be helpful to build a model of this portion of the roof out of balsa wood.

1. Find the measure of the angle that needs to be cut on the end of the board found in oval #1.

2. Find the measure of the angle that needs to be cut on the end of the rafter found in oval #2.

3. In oval #3, there are 2 angles that must be cut on the end of the rafter. This is called a compound angle. What are the measures of each of the two angles?

Note: Students will need to know the pythagorean theorem, slope, and right triangle trig.

The Law of Diminishing Marginal Returns

2011-09-10T13:37:00.000-07:00

A Fun Look at Economics and the Parabola

Goal: Students will simulate a factory that has the ability to add workers but can not add capital resources. Students will create a product called Nutterflutters in the class. The production process of the product will be analyzed mathematically.

Background information for the teacher: All CTE areas include an element of business and how to be successful in the world of work. In business, seldom are all of your resources unlimited and usually small changes are put into place to increase production (profits). If the marginal benefits (profits) are greater than the marginal (additional) costs, then the change was successful. Some resources such as additional labor can be added (temporary or long term) easily (relatively inexpensive investment) while other resources take planning and substantial ongoing expense. For example, expensive modifications/additions to floor space in a factory or additional production machinery may be cost prohibitive or may require extensive long range planning. We will be looking at The Law of Diminishing Marginal Returns which predicts that labor productivity will eventually fall as you add additional (marginal) workers to the production process, while keeping other resources fixed.

Materials needed: 1 large jar of peanut butter, 1 large jar of marshmallow cream, 1 box of graham crackers, 2 plastic knives, paper plates, and a roll of paper towels.

Time to complete activity: 45 minutes

Process:
1. Let students know that you are going to examine the Law of Diminishing Marginal Returns from economics and how it affects business decisions. Model at a desk/table how to make the Nutterflutter snack (I have heard them called peanut smores, flutternutters, etc). Let them know they will be eating the products at the end of the activity.

2. How to make a snack: Spread out 2 – 4 paper towels on the table. The paper towels represent the work area (factory space) and the work can not be on the bare table (health codes). Break a large graham cracker into fourths (small rectangles). Spread peanut butter on a graham cracker using a knife (must use a knife). Spread marshmallow cream on the other graham cracker using the other knife (must use a knife for “health codes”). Stack the 2 rectangle graham crackers to make a sandwich and then place it on the paper plate (represents your warehouse).

3. Choose a student (or two) to be quality control inspectors. Discuss what is a “good” snack and what would be rejected. Choose another student to be the timekeeper.

4. Select a student to be the worker at the factory. He/she will be making the snacks. He/she is to make as many complete snacks as possible in 1 minute. When the time is up, the inspector(s) will inspect the products to decide if they pass quality control. Count the number of completed snacks that pass inspection and record in the production table. Unacceptable ones are not to be counted. Marginal output is the number of additional products produced because another worker has been added. See the table below for an example of how to calculate marginal output.

Production Table

# workers.....# snacks produced.........................Marginal Output

.......0............................0............................................... N/A
.......1..................3 as an example..............................3 as an example
.......2..................7 as an example............................. 4 as an example
.......3
.......5
.......8
......etc

Your second worker added 4 more Nutterflutters, called the marginal output.

5. Select an additional student to join the already employed student (there are now 2 people on the production line). Remove the previous snacks made and replace with an empty paper plate. Students can not expand the work area (can not add paper towels), or use additional knives….remember the resources are fixed…the only variable is the number of workers. They are to repeat step 4 above by adding another worker.

6. Continue adding workers, being sure not to increase work area (paper towels represent the floor of the factory) and do not increase the number of knives or supplies. Continue adding students until the marginal output begins to drop or perhaps even go negative.

7. Graph the number of workers and marginal output. This should approximate a parabola. If your data is poor, be ready to adjust as needed. Extensions include writing the equation of a parabola, predicting using your equation, and general terminology.

8. Questions: How did specialization/division of labor/assembly line affect the production? Were the additional workers lazy? In this example, when would the factory hire and when would they quit hiring?

This activity has been around for a long time….done differently with different products produced. So, if you have done something similar, I do not want you to think I am taking credit for the creation of this activity. It is a lot of fun, and gives a real life example of the use of quadratics.

Fascia Board Angles

2011-03-20T13:48:00.000-07:00

The fascia board is a vertical board along the edge of the roof. It is designed as a way of enclosing the end of the rafters of a home. The white board in the photo is a fascia board. Because it is used as a trim/decorative board, the cuts and joints must be precise. The angle (located in the red circle) can be a tough angle to calculate. Many carpenters do not calculate this angle, but instead have it memorized for different roof pitches or use a t-bevel. A t-bevel is a tool that copies angles.

Please calculate the cut angle of the fascia for a 4/12, 5/12, 6/12 pitch roof. What degree measure does the cross cut miter saw need to be set at for your calculated angle? What is the cut at the peak of the roof for the same fascia board? Please show all drawings and calculations.

Is Your Can Efficient?

2011-02-12T20:12:00.000-08:00

In recent years the size and shape of drink cans have been varied. The traditional looking pop can is no longer the only option in purchasing your 12 oz (355 ml) soda. Half size pops, Monster drinks, coffee drinks, etc have filled the grocery shelves.
Your task is to design a cylindrical can (355 ml) that minimizes can production costs. The production costs can be minimized by using the smallest amount of material possible. When cans are manufactured, any material that does not get used in the can will be shredded and recycled. What should the dimensions of the new can be? Hint: 1 mL = 1 cm3

This problem lends itself to using a spreadsheet and graph from the spreadsheet. Please show work/formulas used in the spreadsheet as well as a graph.

High School Mini Zoo Design

2011-01-09T13:09:00.000-08:00

The high school plans to allow the Biology class to build a mini-zoo for small animals such as mice, hamsters, snakes, tarantulas, guinea pigs, etc. Due to budget limitations, costs will be a big concern and design economies must be considered. The cages will be built using 4 ft by 8 ft plywood sheets and 4 ft by 8 ft Plexiglas sheets as the materials. Eight cages will be built. Each cage needs to have a minimum of 360 square inches of floor space. Each cage will be 15 inches tall and one face (side) will have a window for viewing. The window must be at least 10 inches wide.
Your job is to design the “best” mini-zoo. Your solution should include scale drawings of the cages and a written description of the advantages of your zoo design. A material list of will need to be included telling how many sheets of Plexiglas and how many sheets of plywood. Show a cut diagram for how you plan to cut each sheet so that you have all of the pieces needed to construct the zoo cages. Show all math sub-problems used to verify that your solution meets the requirements of the mini zoo.

Stair Design

2010-11-16T07:50:00.000-08:00

Stairs are a challenge to design. The outdoor deck shown is 11 feet 4 inches above the concrete base ( 4 feet by 4 feet by 4 inches deep). The treads are 11 inches wide. The building code states the stairs are to be at most 30 degrees with the ground and the riser height should be between 6 inches and 8 inches. The rise must be the same for all steps. Note that building codes vary by region.
Please find the following.
1. The length of the 2 by 12 used for the stringer. Remember lumber typically comes in increments of 2 feet. Show the design (rise and run) of the stair designed. Include mathematics to show that the stairs are meeting the code of 30 degrees.
2. Where you would place the concrete base so that the stairs rest on the base and at least 3 feet is showing in front of the stairs.
3. Calculate the number of cubic yards of concrete needed for the concrete base.
4. Present at least 2 possible solutions. Explain what the advantages are to each of your 2 solutions.

Concrete Slump Test March 2010

2010-03-07T14:22:00.000-08:00

Concrete Slump Test
Testing Concrete for Strength

Water Amount_________ (Given by the teacher)
The concrete slump test is, in essence, a method of quality control. For a particular mix being used on a construction site, the slump should be consistent. A change in slump height would demonstrate an undesired change in the ratio of the concrete ingredients. If this were to happen, the proportions of the ingredients would be adjusted to keep a concrete batch consistent. This homogeneity improves the quality and structural integrity of the cured concrete. Too much or too little water will weaken the concrete.

Goal: You will be measuring how much “slump” or sag there is in your concrete mix.

Materials Needed per pair: 1 ruler, 1 cup (18 oz)with the bottom cut out, 1 mixing bucket, 1 baggy of 2.5 lbs concrete, 1 scrap piece of OSB approximately 1 foot square, 1 graduated cylinder, and 1 stir stick.

Procedure Per Pair of Students:
1. Take your premeasured ready mix concrete that you received in the food storage baggy and pour it into the mixing bucket provided.

2. Add water to the concrete mix that is in the mixing bucket. Be sure to measure the water amount carefully. This measure is in milliliters. If your water measurement is off slightly, your results will suffer, and thus your grade.

3. Use stir stick and stir the concrete for 3 minutes or until dry spots of concrete are gone.

4. Measure, in inches to the nearest eighth inch, the vertical height of the cup and record. This height is the perpendicular height of the cup. This is the initial height of the concrete needed for #10 below.

5. Pour half the mix into the upside down cup (with the bottom cut out). Be sure cup is on the scrap piece of OSB. Tamp down the concrete by using stir stick to tamp it 25 times. Add the rest of the mix and tamp. Remember to keep the cup full to the top.

6. Scrap off excess concrete that is above the top of the cup.

7. Remove the “form” AKA cup from around the concrete carefully by lifting the cup straight up.

8. After the concrete stabilizes (stops slumping) measure the height of the concrete. The concrete will have various heights so you will need to measure the average height.

9. Find the amount of slump by taking the initial height – the ending height.

10. Record: (amount of water________ , amount of slump_______)

11. Find the average slump of all pairs of students with the same water amount in the class.
Record Average (amount of water________ , amount of slump_________)

12. Collect data from all the groups in class and record in data table below.
Water 110 120 130 140 150
Slump

13. Label the axis appropriately and graph the data on your own graph paper.

14. Write an equation for the line of best fit.

15.Explain the real life meaning of slope.

16. What type of predictions could you make from your equation

17. If you had a 60 lb bag of concrete, how much water would you need to add to give the same ratio of water to concrete mix as you initially had? Hint: You will not need your graph for this.

18. Take your cement to the designated disposal area/person. Wash all of the equipment and return equipment to teacher.

Teacher Instructions:
It is best if you pre bag the ready mix concrete. Bag 2.5 lbs in each bag. Water must be added using milliliters in the amounts of 110 ml, 120 ml, 130 ml, 140 ml, and 150 ml. Each pair of students will do one water amount or one slump test. To clarify, each pair of students will do one water amount (either 110, 120, 130, 140, 150 ml) Repeat the water amounts as needed depending on the number of groups you have.

We do one slump test per pair of students with at least 2 pairs doing each water amount. A 60 lb bag of ready mix will provide about 24 bags of concrete.

The line of best fit will predict the amount of slump per ml of water. Realize that this equation will only be useful for a specific domain. That is, a concrete mix with too little or too much water will not be a mix. The equation gives you an opportunity to discuss the concept of domain of a function. The slump test is one measure of a concrete’s strength. Once again too much or too little water will significantly weaken the concrete.

Make stepping stones with the left over experiments.

Shopping Carts February 2010

2010-02-08T12:50:00.000-08:00

Mathematics appears in unusual places. When grocery stores are designed, an architect must calculate the space needed for storage of the shopping carts. The cart above is 39” long, 22” wide and has a nesting distance of 13.5”. Nesting distance is how much extra length a cart takes when the carts are stacked together. One cart takes up a 39” length and 2 carts take up 52.5” length. Write a length formula that would compute how much length to allow for storing the carts. Assume that there are 5 rows (nests) of carts.

LED Lighting January 2010

2010-01-14T18:20:00.000-08:00

You probably noticed the increase use of LEDs for Christmas lighting. LEDs are beginning to be used in traditional applications for the home. EarthLED thinks that they can produce a LED “bulb” that will be equivalent to a 100 watt incandescent bulb for a suggested retail of $80. The LED will use 12 watts. An incandescent bulb sells for $1. The LED is expected to last 50,000 hours while an incandescent bulb will last 1000 hours. A kilowatt hour cost 8 cents. Prepare a cost analysis for the most used light bulb in a typical household which has an average use of 5 hours a day.

Archways November 2009

2010-01-14T18:19:00.000-08:00

Archways are a popular feature in many construction projects. Many older archways were created using stones called keystones. In the photo above, an archway is shown with the setting of a keystone at the top of the arch.
Archeologists have unearthed a stone from an archway built many years ago. The stone is 25 inches thick. The front surface is in the approximate shape of an isosceles trapezoid. The longer base is 16 inches long and the shorter base is 12 inches long. The legs measure 24 inches. Archeologists know that typically all the other stones in the archway were congruent to the stone that was found. How many stones similar to the one described would it have taken to produce the original archway? What was the interior diameter of the archway?

Water Cistern October 2009

2010-01-14T18:17:00.000-08:00

In Shaker communities of the past, cisterns were built to store water as shown in the photo above. They were built as a cylinder with a hemisphere on each end partially buried underground. The diameter was 10 feet. The water was collected from large barn roofs. Even today, cisterns are still used extensively. Using the Shaker design described above, calculate the size of a cistern so that it could hold all the collected water from a 90 ft by 150 ft barn roof given a 20 inch rainfall. How many gallons will your cistern hold?

Drippping Faucet September 2009

2010-01-14T18:16:00.000-08:00

Water is one of our most valuable resources. Many times it is wasted without us even thinking about it. Suppose a faucet leaks one drop per second. How much water will be wasted by the leak in one year? How many feet of half inch water pipe would that equal? Design an experiment to find the answer.