AMATYC November 11 th, 2011

Transcription

1 AMATYC November 11 th, 2011 Dynamic Solution Exercises Dr. Elliott Ostler UNO Mathematics Education Professor Michael Flesch Metropolitan Community College Math Instructor Slide 4-1

2 Welcome Introductions History of the Project Contents of your Packet Slide 4-2

3 Website for Materials All contents of your packet can be found at: You will need to locate the class from the pull down menu on the left side. Math 0910 for Basic Math materials Math 0960 for Beginning Alg. materials Math 1310 for Intermediate Alg. materials Math 1420 for College Alg. materials Slide 4-3

4 Addresses Michael Flesch - mflesch@mccneb.edu Dr. Elliott Ostler eliottostler@gmail.com Slide 4-4

5 Suggestions for Making Vertical Alignment Between Courses Tangible throughout the Mathematics Curriculum Using: Dynamic Solution Exercises (DSE s) Slide 4-5

6 Overview of Session Our presentation justifies the need for, and offers some suggestions on the selection and implementation of mathematical problems known as Dynamic Solution Exercises. Our intent is to help provide insight into how mathematics teachers can go about making vertical articulation a cooperative and tangible part of the mathematics curriculum. Slide 4-6

7 Overview Continued Some sample Dynamic Solution Exercises are provided based on research at Metropolitan Community College in Omaha NE. Some strategies for selecting and building a DSE instructional environment will be discussed. Slide 4-7

8 Dr. Elliott Ostler Filmed at the University of NE at Omaha Slide 4-8

9 DSE Example - Mixture Problem You have been asked to mix two different strength weed killers together to obtain 10 liters of weed killer that has a 61% strength. On hand you have one barrel of weed killer that is 40% strength and another and another barrel that is 70%. How many liters of each should be mixed to obtain 10 liters that is 61% weed killer? Slide 4-9

10 Basic Mathematics This problem could be done by having the students investigate the different possibilities that will occur. The students will perform the calculations fill in a table, and then make conclusions based on the results. Slide 4-10

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12 Questions At what combination of Amount 1 and Amount 2 was the percent of weed killer 61%? At what combination of Amount 1 and Amount 2 was the percent of weed killer the least? The most? Based on amounts used and the percent of weed killer in each, what generalizations might you make? Slide 4-12

13 Beginning Algebra We now use the variable (x) to represent Amount 1 and (10 x) to represent Amount 2. This allows us to write an equation in terms of x. We can see how we might write an algebraic equation to solve with any desired percent in the final solution. Slide 4-13

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15 2 nd Part of Question 70%*x + 40%*(10-x) = 6.1 liters 61%*10 Liters = 6.1 Liters You now want 10 liters of a mixture of the two weed killers with a final strength of 49%. Without having to construct another table, write one equation in terms of x using the model from above that will allow you to find the correct combinations of the two liquids? Slide 4-15

16 Solution Is An Equation 70%*x + 40%*(10-x) = 4.9 liters 49%*10 Liters = 4.9 Liters Solve this equation to get x = 3 and y = 7 We can now found the correct combination for any percent mix we are looking for algebraically. Slide 4-16

17 Intermediate Algebra Introduces a second variable allowing us to set up a system of two equations and unknowns and find a solution by both an algebraic or geometric process. It allows us to write the two constraints as linear equations and find the solutions by graphing the two constraints and finding the intersection. Slide 4-17

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19 Algebraic Solution We have two equations and two unknowns to describe the system: x + y = *x *y = 0.61*10 = 6.1 Solve this system using the substitution or elimination method. Solution will again be x = 7 and y = 3 Slide 4-19

20 Geometric Solution Now solve the system by graphing. 0.7x + 0.4y = 6.1 and x + y = 10 y = -1.75x and y = -x + 10 can be graphed to find the solution. Plot these equations and find the intersection. Slide 4-20

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22 Part 2 You have been directed to put together 10 liters of a mixture of the two different weed killers that will have a final strength of 52%. Can you construct a system of two equations and two unknowns using the model from above that will allow you to find the correct combinations of the two liquids? Solve the system algebraically and geometrically by graphing. Slide 4-22

23 Algebraic Solution The two equations are: x+y = 10 & 0.70x y = 0.52*10 = 5.2 The algebraic solution can be accomplished with the substitution of y = -x+10 into the other equation to get: 0.70x (-x + 10) = 5.2 The solution is x = 4 and y = 10-x = 6 Slide 4-23

24 Geometric Solution The geometric solution can be found by solving each equation for y and graphing. y = -x + 10 and 0.70x y = 5.2 so y = -7/4*x + 13 Slide 4-24

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26 College Algebra Algebraic and geometric interpretation of of an inconsistent system. This example will have a solution, but it will fall outside the allowable physical constraints so thus there will be no solution. The algebraic and geometric process will yield a solution, but not one that is within the defined possibilities for x and y. The student needs to recognize and interpret their results to the algebraic and geometric solutions. Slide 4-26

27 Statement of the Problem You have been asked to mix two different strength weed killers together to obtain 10 liters of weed killer that has a 36% strength. On hand you have one barrel of weed killer that is 80% strength and another and another barrel that is 40%. How many liters of each should be mixed to obtain 10 liters that is 36% weed killer? Slide 4-27

28 Algebraic Solution Through substitution, we have: 0.80*x (10-x) = 3.6 x= -1 and since x + y = 10, y = 11 so our ordered pair is (-1, 11) Our values for x and y are outside the physical constraints of allowable values for x and y which must be between 0 and 10. Slide 4-28

29 Geometric Solution Solve for y and graph each line. y = -x + 10 and 0.80x y = 3.6 y = -x + 10 and y = -2x + 9 We see how the intersection is outside the allowable physical constraints of x and y. Slide 4-29

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31 Some DSE s for Basic Math Concept #1 Division by Zero Concept #2 Comparing addition verses multiplication Concept #3 - Location of Negative Sign in Fractions Concept #4 The Distributive Property Slide 4-31

32 Some DSE s for Basic Math Concept #5 - Distributive Property Factoring Concept #6 - Reducing Fractional expressions to lowest terms by factoring Concept #7 - Multiplication Rewriting to use Distributive Prop. Concept #8 - Quantities Equal to -1 Slide 4-32

33 Some DSE s for Elementary Algebra Concept #1 Comparing Addition verses Multiplication Concept #2 - Location of Negative Sign in a Fraction Concept #3 Division by Zero Concept #4 The Distributive Property Concept #5 - Distributive Property in Reverse Factoring Slide 4-33

34 Some DSE s for Elementary Algebra Concept #6 - Distributing -1 to a Quantity Concept #7 Interpreting fractional Coefficient - Variable Term Concept #8 - Clearing Fractions in an Equation Concept #9 - Multiplication Rewriting using Distributive Prop. Slide 4-34

35 Some DSE s for Elementary Algebra Concept #10 - Simplifying Rational Expressions Find Restricted Values Concept #11 Simplifying Rational Expressions with Numerical Values Concept #12 - Simplifying Rational Expressions Equal to 1 Slide 4-35

36 Some DSE s for Elementary Algebra Concept #13 Simplifying Rational Expressions with Variable Terms Slide 4-36

37 Some DSE s for Intermediate Algebra Concept #1 The Distributive Property Concept #2 Comparing Addition verses Multiplication Concept #3 - Distributing -1 to Quantity Concept #4 - Multiplication Rewriting to use Distributive Prop, Concept #5 - Using Distributive Prop. In Reverse Factoring Slide 4-37

38 Some DSE s for Intermediate Algebra Concept #6 Division by Zero Concept #7 - Simplifying Rational Expressions & Functions, Identify Restricted Values Concept #8 Simplifying Rational Expressions with Numerical Values Concept #9 - Simplifying Rational Expressions Equal to One Slide 4-38

39 Some DSE s for Intermediate Algebra Concept #10 Simplifying Rational Expressions with Variable Terms Concept #11 - Location of Negative Sign in a Fraction Concept #12 - Clearing Fractions in an Equation Slide 4-39

40 Some DSE s for Intermediate Algebra Concept #13 Adding Rational Expressions with Variable Terms Concept #14 Solving Rational Equations Slide 4-40

41 DSE s in Detail for Basic Mathematics We will take a look at some examples of what was developed in your packet of materials concerning the DSE concepts for a Basic Mathematics Course. Slide 4-41

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54 DSE s in Detail for Intermediate Algebra We will take a look at some examples of what was developed in your packet of materials concerning the DSE concepts for an Intermediate Algebra Course. Slide 4-54

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77 Key Ideas for a DSE: Use what best supports Primary Need Standards Large in Scope and often evaluated by checklist Repeated through successive courses/years Focus on Relationships not Process Forces Synthesis of mathematical ideas

78 Discussion & Final Comments Slide 4-78

79 Good DSE s: Sundial: On a rectangular, horizontal sundial, mark the Dial Plate. Evaporating Solution: We want to keep the amount of an antibacterial solution in a swimming pool at a constant level. Show this using multiple iterations of a linear function. Optimize the volume: Removing a wedge from a circle would create a cone. What is the best size piece to remove that would allow for the maximum cone value? Find the Shortest Path: An insect travels at 8ft./sec over dry ground and 5ft./sec over water. If he sees food 28 feet down the shore and 18 feet from the shore. How fast can it get the food?

80 Adding to Forms of One How about: or: = or: Does the sum of and equal? or:. 3x +.7x = 1x 2 2 or: sin x + cos x = 1 2 x 3 3 or: lim lim = x + x x x

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82 Simple to complex: Linear Equations Key ideas: y-axis intercept: b Slope: m Points on a line: (x,y) Arithmetic Sequences Key ideas: First term of the sequence: a1 Common Difference: d General term: an at the nth term Formula: y = mx + b Formula: an = a1 + (n-1)d

83 Statement of the problem You are working for a lawn service and have been asked to mix two different strength weed killers together to obtain 10 liters of weed killer that has a 36% strength. On hand you have one barrel of weed killer that is 80% strength and another and another barrel that is 40%. How many liters of each should be mixed to obtain 10 liters that is 36% weed killer? Slide 4-83

84 A Key for ranking Mathematical Tasks: Simple Complex Task Algorithmic Interpretive Constructive Concrete Abstract Concrete Abstract Single Multiple Single Multiple Single Multiple Single Multiple