Thursday, March 20, 2014

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

Sunday, January 26, 2014

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.