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.