A Different Kind of Mixture Problem

Math Standards: A.CED.2; A.CED.3; A.REI.11

CTE Standards:  FPP.03.01; FPP.04.01; FPP.04.03

CTE Situation (opener):

When making maple confections the amount of invert sugar will affect the quality of the confection. Invert sugar syrup is a mixture of glucose and fructose; it is obtained by splitting sucrose into these two components. Inverted sugar is sweeter and its products tend to retain moisture and are less prone to crystallization making it valuable to bakers.  By measuring the invert sugar and blending different batches of syrup you will get the optimal invert sugar outcome for the confection.

The ideal invert sugar measurement for maple candy is 1%. If we have light syrup that has .5% invert sugars and dark syrup with 2.2% invert sugars. What mixture of light and dark syrup do you need to reach the desired invert sugar level of 1% to make maple candy?

Solution using Pearson Square Method:

 

The value in the middle of the square (the goal of the mixture) must be intermediate between the two values that are used on the left side of the square.  The numbers on the right side of the square are obtained by subtracting diagonally smallest from largest.  The denominator is the sum of the numbers on the right side of the square.

Solution using systems of equations:

Solve the following system of equations:

         .5L + 2.2D= 1

         L + D= 1

Solution:                                             Answer:

         .5L + 2.2D=1                                    29.4% Dark

              -.5L – .5D= -.5                                    70.6% Light

                   1.7D=.5

                  D=.294        

                 

                  L + .294=1

                  L=.706

Thanks to Erin McCaffrey and Jeannie McLean of Stockbridge Valley Schools of Munnsville NY

Shortcuts, Do They Work?

Mathematical Practice Standards:  Reason abstractly and quantitatively.  Construct viable arguments and critique the reasoning of others.

Aaron needs to rip (cut length wise) a 2 x 10 into 3 equal strips.  He does not like working with the ugly number that is the width of the 2 x 10, which is 9 ¼”.  Fractions are not his friends and he doesn’t want to divide a fraction by 3 (or any number).  He claims he can pick a number that is easy to divide by three (15, for example) and measure that distance diagonally across the board.  Then mark the board at 5 inch increments (5, 10, and 15”) and his marks will divide the board into equal widths.

Does this method work or is this a shortcut that Aaron has fabricated that is a result of his wishful thinking?

Provide an argument supporting why this works or why it doesn’t’ work.  Use diagrams and words combined with your detailed mathematics to support your argument.

Hint for answer:  Use what you know about congruent triangles.  In this problem, you created 3 of them.

What do cantilevers and pulling nails have in common?

Math Standard:

A.CED.1 Create equations in one variable and use them to solve problems.

G.CO.2 Represent transformations. 

CTE Standard:

            Students will understand how levers and fulcrums are used in construction.

Teacher Notes/Materials Needed:

Ruler, cups, fulcrum, fun size candy bars, 2 x 10 x 10, 

CTE Situation (opener):

What principle is involved in deciding how far you can cantilever a deck or 2^nd floor, or how big a hammer or crow bar is needed to pull a nail?  Why is holding a hammer close to the head less efficient than holding it close to the end of the handle? 

Lesson Sequence:  Can 1 candy bar lift 6 candy bars?

1.     Show a visual demonstration of the lever/fulcrum principle by using a ruler, pen (fulcrum), 2 plastic cups, fun size candy bars.  Model this with the fulcrum at the center of the ruler with a cup containing 1 candy bar placed on one end of the ruler and a cup containing 6 candy bars on the other end (note it does not balance).

a.     Discuss and experiment with ruler lever/fulcrum so that 1 candy lifts 6 candies.

b.     An extension, using a 2 X 10 x 10, have a lightweight student lift a larger (football player type) student.

c.      Process with students using the photos above, where is the fulcrum and lever located?  This is a great place to talk about the “center of rotation” and “translation” of the fulcrum.

2.     Transition into developing a math expression/equation to represent the lever scenario

longer distance X lighter person = shorter distance X heavier person

or express as a proportion

Thanks to John Gregory & Steven Davis of Norwich, NY for letting this problem be reprinted.

We Demand (and Supply) Candy

Math Standard:

Label ordered pairs, finding domain and range, writing inequalities

Solve polynomial systems by graphing

CTE Standard:

Define business related terms:  demand (elastic), supply, equilibrium point; Create an Excel chart from table data; Graph a set of points on Excel – labeling via a text box; Change the axis formats in Excel as necessary

Teacher Notes/Materials Needed:

Bags of fun size candy (5 different types)

Excel on a projector with a table file for each type of desired candy

White boards, markers, & mini erasers

CTE Situation

The day prior to this lesson ask students for their favorite fun size candy (given them choices such as Skittles, Starburst, Twix, Milky Way, Reeses, etc) – take a straw poll vote at the end of the class to find the most popular.   Students may ask why and respond with “you’re such a GREAT class I MAY have a treat for you tomorrow AND you may have the chance to purchase candy from me on some days”.  Write the choices on the board and leave them overnight.

Upon entry the next day, students will ask “do we get candy”.  If possible, set up the various choices in piles on tables and have students choose their favorite one piece as a free sample.  State the objectives.  Now that you’ve had a nibble of the delicious candy, would you like more?  However, you will have to buy them TOMORROW.

Day 1 Process:

Students gather around the SmartBoard and teacher enters data as follows:

1.  State that the teacher will offer some Candy #1 (most popular) for $.01

2.  Each student writes on his/her whiteboard the number of pieces of Candy #1 desired to purchase for $.01

3.  Total all of the student’s request to purchase Candy #1

4.  Enter the amount into the Excel table

5.  Increase the amount to $.05 (make up a story about why…. to offer at $.05”)

6.  Repeat steps #2-5 until you have no demand for the particular candy type, increasing $.05 for the purchase price.

7.  Students return to computers and together we create an Excel data series chart for the demand.  Price will be the Y axis and quantity desired will be the X axis. (CTE)

8.  Label the ordered pairs using textboxes, using proper mathematics language.

9.  Below the chart, use set notation for the domain and for the range 

10.  For an extra free piece of candy:  State at the bottom of the chart, “what is the meaning of the domain and range?”

11.  Complete 4 candy charts (for the most popular 4 candies) using steps #1-10.   Create graphs using Excel or any other method.

DAY 2 Process

Prior to class, teacher must determine the supply amounts for the prices determined for the demand.  “While I’m very generous, I will be offering this candy at a specific price for each type.”

1.  Open Candy #1 file and create a Column C labeled Supply  

2.  Give students the figures (teacher generates how much they will supply at each sale value), ask the students for the reasons why at each increase of price, the teacher is willing to offer more pieces of candy; lead the students to conclude that this is the inverse of demand.

3.  Give a piece of graph paper to each student.   Students will duplicate the demand curve from previous day.  Review concepts as necessary, restating domain and range.

4.  Using the teacher generated numbers for supply, have students graph the supply curve on the same piece of graph paper

5.  Determine where the 2 curves meet called the equilibrium point; “this is how you solved systems by graphing”   

6.  Use Excel to create a supply curve (this is a repeat of #4 except now using technology.

7.  Label the equilibrium point

8.  Complete the remaining candy charts in Excel; Sell the candy for the equilibrium price.

Using the table below write the ordered pairs.

How many pieces wanted                    Price

	0.01
	0.05
	0.10
	0.15

State the domain and range of the following relation.

Solve the following system by graphing:

Extension:  Write the equation of the supply and demand curves.  Solve the system by algebraic substitution.

Thanks to Jean Keenan & Joanne Costa for sharing this lesson.

Writing About Rafter Tables

The following table is called a rafter table in the construction industry.  The table is used by carpenters to calculate the length of rafters if you know the slope of a roof.   

Roof Slope	Rafter
2 in 12	1.014
3 in 12	1.031
4 in 12	1.054
5 in 12	1.083
6 in 12	1.118
7 in 12	1.158
8 in 12	1.202
9 in 12	1.25
10 in 12	1.302
11 in 12	1.357
12 in 12	1.413

Mathematics:

Show the calculations used to generate the numbers in the second column.

Technical Writing:

Write out the process for using the rafter table.

Illustrate your process with one example.

How Much Is Enough?

Goal:  You will be able to calculate the linear feet of window trim needed for any dimension window.

Interior window trim (casing) is a molding that is nailed to the finished window frame to give the window a “finished” look.  The trim is located on the inside of the home around the perimeter of the window frame and is often cut at a 45º angle.  See picture.   You can think of it as a picture frame around the window.  The same idea is also used around doors.

Problem 1:

Your window frame (inside edge) is 38” wide by 54”.  There is to be a 3/16” reveal on all edges.  Your corners are mitered at 45º.  The trim (casing) is 2 1/4” wide.  How many inches of trim do you need?  You can ignore saw blade width but do need to consider casing profile. 

Problem 2:

Problem 2 is a generalization of problem 1.  What is a formula (shortcut) for finding the length of trim needed if the window size is “w” inches wide and “h” high with a trim width of “t”?  Again, assume you have a casing profile to consider.

Teacher Notes:

Students may need to research terms such as reveal, casing, and casing profile unless you provide a demo.  You will need to decide if you need to give hints regarding 45-45-90 special right triangles.  You can extend this to building a octagonal frame using 30-60-90 special triangles.  Trim carpenters know the rule by heart but do not call it a “formula”.

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.

Driving and Texting

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

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

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.