Making Sense of Solving Linear and Quadratic Equations with Mapping Diagrams
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1 Making Sense of Solving Linear and Quadratic Equations with Mapping Diagrams Martin Flashman Professor of Mathematics Humboldt State University CMC 3 Conference December 12, 2015 flashman@humboldt.edu
2 Mapping diagrams provide a valuable tool for visualizing functions and connects function concepts to solving equations in many contexts. In this presentation both linear and quadratic equations will be solved using mapping diagrams to make sense visually of the functions and steps used in common algebraic approaches to these problems. GeoGebra will be used as a dynamic tool to connect the concepts with technology.
3 Equations, Functions, and Mapping Diagrams in Common Core Links: ntations/cmc/cmc3.md.links.html
4 Mapping Diagram Sheets Work/Spreadsh eets Section from MD from A B to C and DE (Drafts) GeoGebra YouTube Videos Mapping Diagram blanks (2 axis diagrams) Worksheet.pdf Visualizing Functions: An Overview Sketch to Visualize Solving a Linear Equation using Mapping Diagrams Using Mapping Diagrams to Visualize Linear Functions (10 Minutes) Mapping Diagram blanks (2 and 3 axes) Spreadsheet Template (Linear Functions) Linear Functions (LF) Quadratic Functions(QF) Mapping Diagrams for Solving a Quadratic Equation Solving Linear Equations Visualized with Mapping Diagrams. (10 Minutes)
5 Background Questions Are you familiar with Mapping Diagrams to visualize functions? Have you used Mapping Diagrams to teach functions? Have you used Mapping Diagrams to teach content besides function definitions?
6 Main Resource Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s. A Reference and Resource Book on Function Visualizations Using Mapping Diagrams (Preliminary Sections- NOT YET FOR publication)
7 Mapping Diagram Prelim Examples of mapping diagrams Worksheet 1.a Make tables for m(x) = 2x and s(x)= x+1 x m(x) =2x x s(x) =x+1
8 Mapping Diagram Prelim Examples of mapping diagrams Worksheet 1.b On separate diagrams sketch mapping diagrams for m(x) = 2x and s(x)= x+1 x m(x) =2x x s(x) =x
9 Worksheet 1.b Mapping Diagram: m(x) = 2x x m(x) =2x
10 Worksheet 1.b Mapping Diagram: s(x) = x+1 x s(x)=x
11 Mapping Diagram Prelim Examples of mapping diagrams Worksheet 2 a. First make table for q(x) = x 2. x q(x) =x
12 Mapping Diagram Prelim Examples of mapping diagrams Worksheet 2 a. First make table for q. x q(x) =x b. Sketch a mapping diagram for q(x) = x 2.
13 Mapping Diagram Prelim Worksheet 2.b. Mapping Diagram for q(x) = x 2 x q(x) =x
14 Worksheet 3.a.Complete the following table for the composite function f(x) = s(m(x)) = 2x +1 x m(x) f(x)=s(m(x))
15 Worksheet 3.a.Complete the following table for the composite function f(x) = s(m(x)) = 2x +1 x m(x) f(x)=s(m(x))
16 Mapping Diagram Prelim Worksheet 3.b Use the table 3.a and the previous sketches of 1.b to draw a composite sketch of the mapping diagram with 3 axes for the composite function f(x) = h(g(x)) = 2x + 1
17 Worksheet 3.b Draw a sketch for the mapping diagram with 3 axes of f(x) = 2 x + 1. x m(x) f(x)=s(m(x))
18 Worksheet 3.b Draw a sketch for the mapping diagram with 3 axes of f(x) = 2 x + 1. x m(x) f(x)=s(m(x))
19 Worksheet 3.c Draw a sketch for the mapping diagram with 2 axes of f(x) = 2 x + 1. x m(x) f(x)=s(m(x))
20 Worksheet 4 Mapping Diagram: q(x) = x 2 x q(x) =x
21 Worksheet 4.a Complete the following tables for q(x) = x 2 and R(x) = s(q(x)) = x x q(x) R(x)=s(q(x))
22 Worksheet 4.a Complete the following tables for q(x) = x 2 and R(x) = s(q(x)) = x x q(x) R(x)=s(q(x))
23 Worksheet 4.b 4.b Using the data from part a), sketch mapping diagrams for the composition R(x) = s(q(x)) = x with three axes. x q(x) R(x)=s(q(x))
24 Worksheet 4.b 4.b Using the data from part a), sketch mapping diagrams for the composition R(x) = s(q(x)) = x with three axes. x q(x) R(x)=s(q(x))
25 Worksheet 4.b 4.b Using the data from part a), sketch mapping diagrams for the composition R(x) = s(q(x)) = x with two axes. x q(x) R(x)=s(q(x))
26 An Old Friend: Solving A Linear Equation Worksheet 5.a Solve a linear equation: 2x + 1 = 5 Find x.
27 An Old Friend: Solving A Linear Equation Worksheet 5.a Solve a linear equation: 2x + 1 = 5-1 = -1 2x = 4
28 An Old Friend: Solving A Linear Equation orksheet 5.a Solve a linear equation: 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2
29 An Old Friend: Solving A Linear Equation Worksheet 5.a Solve a linear equation: heck! 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2! 2x+1 = 2*2 + 1 = 5
30 Linear Equations Use Linear Functions! Linear Equations 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2 Check: 2x + 1 = 2*2 + 1 = 5 Linear Functions f(x) = 2x + 1
31 Linear Equations Use Linear Functions! Linear Equations 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2 Check: 2x + 1 = 2*2 + 1 = 5 Linear Functions f(x) = 2x + 1 m(x) = 2x; s(x) = x +1 f(x) = s(m(x))
32 Worksheet 5.b Solving 2x + 1 = 5 visualized with a mapping diagram Algebra: 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2 How does the MD for the function VISUALIZE the algebra?
33 Worksheet 5.b Solving 2x + 1 = 5 visualized with a mapping diagram Algebra: 2x + 1 = 5-1 = -1 2x = 4 Function: f(x)=s(m(x)) = 5 Undo s m(x) = 4
34 Worksheet 5.b Solving 2x + 1 = 5 visualized with a mapping diagram Algebra: 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2 Function: f(x)=s(m(x)) = 5 Undo s m(x) = 4 Undo m x = 2
35 Worksheet 5.b Solving 2x + 1 = 5 visualized with a mapping diagram Algebra: 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2 Function: f(x)=s(m(x)) = 5 Undo s m(x) = 4 Undo m x = 2 CHECK! f(2)=5!
36 Worksheet 5.b Solving 2x + 1 = 5 visualized on GeoGebra Algebra: 2x + 1 = 5-1 = -1 2x = 4 1/2(2x) =1/2(4) x = 2 Function: f(x)=s(m(x)) = 5 Undo s m(x) = 4 Undo m x = 2 CHECK! f(2)=5!
37 Challenge: Solve 2(x-3) = 9 with a mapping diagram
38 Worksheet 6.a Solve 2(x-3) = 9 with a mapping diagram Understand the problem 2(x-3) is a function of x. P(x) = 2(x-3) Find any and all x where P(x) = 9. 2(x-3) is a composition of functions P(x) = s(m(q(z(x)))) where z(x) = q(x) = m(x) = s(x) =
39 Worksheet 6.a Solve 2(x-3) = 9 with a mapping diagram Understand the problem 2(x-3) is a function of x. P(x) = 2(x-3) Find any and all x where P(x) = 9. 2(x-3) is a composition of functions P(x) = s(m(q(z(x)))) where z(x) = x-3; q(x) = x 2 ; m(x) = 2x; s(x) = x+1.
40 Worksheet 6.a Solve 2(x-3) = 9 with a mapping diagram. Make a plan Find any and all x where P(x) = 9. Construct mapping diagram for P as a composition of function : P(x) = s(m(q(z(x)))) Undo P(x) = 9 by undoing each step of P Undo s(x) = x+1 Undo m(x) = 2x Undo q(x) = x 2 Undo z(x) = x-3 Check results to see that P(x) = 9
41 Worksheet 6.b Solve 2(x-3) = 9 with a mapping diagram. Execute the plan Construct mapping diagram for P as a composition of function : P(x) = s(m(q(z(x))))
42 Worksheet 6.b Solve 2(x-3) = 9 with a mapping diagram. Execute the plan Construct mapping diagram for P as a composition of function : P(x) = s(m(q(z(x))))
43 Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Execute the plan Find any and all x where P(x) = 9.
44 Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Execute the plan Undo s(x) = x+1 5 1
45 Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Execute the plan Undo m(x) = 2x 5 1
46 Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Execute the plan Undo q(x) = x 2 5 1
47 Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Execute the plan Undo z(x) = x-3 5 1
48 Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Reflect on the problem?! 5 1
49 Challenge: Worksheet 6.c Solve 2(x-3) = 9 with a mapping diagram Execute the plan with GeoGebra 5 1
50 Technology Examples Excel examples Geogebra examples
51 Overtime? Simple Examples are important! f(x) = x + C Added value: C f(x) = mx Scalar Multiple: m Interpretations of m: slope rate Magnification factor m > 0 : Increasing function m < 0 : Decreasing function m = 0 : Constant function
52 Simple Examples are important! f(x) = mx + b with a mapping diagram -- Five examples: Back to Worksheet Problem #7 Example 1: m =-2; b = 1: f(x) = -2x + 1 Example 2: m = 2; b = 1: f(x) = 2x + 1 Example 3: m = ½; b = 1: f(x) = ½ x + 1 Example 4: m = 0; b = 1: f(x) = 0 x + 1 Example 5: m = 1; b = 1: f(x) = x + 1
53 Visualizing f (x) = mx + b with a mapping diagram -- Five examples: Example 1: m = -2; b = 1 f (x) = -2x + 1 Each arrow passes through a single point, which is labeled F = [- 2,1]. The point F completely determines the function f. given a point / number, x, on the source line, there is a unique arrow passing through F meeting the target line at a unique point / number, -2x + 1, which corresponds to the linear function s value for the point/number, x.
54 Visualizing f (x) = mx + b with a mapping diagram -- Five examples: Example 2: m = 2; b = 1 f(x) = 2x + 1 Each arrow passes through a single point, which is labeled F = [2,1]. The point F completely determines the function f. given a point / number, x, on the source line, there is a unique arrow passing through F meeting the target line at a unique point / number, 2x + 1, which corresponds to the linear function s value for the point/number, x.
55 Visualizing f (x) = mx + b with a mapping diagram -- Five examples: Example 3: m = 1/2; b = 1 f(x) = ½ x + 1 Each arrow passes through a single point, which is labeled F = [1/2,1]. The point F completely determines the function f. given a point / number, x, on the source line, there is a unique arrow passing through F meeting the target line at a unique point / number, ½ x + 1, which corresponds to the linear function s value for the point/number, x.
56 Visualizing f (x) = mx + b with a mapping diagram -- Five examples: Example 4: m = 0; b = 1 f(x) = 0 x + 1 Each arrow passes through a single point, which is labeled F = [0,1]. The point F completely determines the function f. given a point / number, x, on the source line, there is a unique arrow passing through F meeting the target line at a unique point / number, f(x)=1, which corresponds to the linear function s value for the point/number, x.
57 Visualizing f (x) = mx + b with a mapping diagram -- Five examples: Example 5: m = 1; b = 1 f (x) = x + 1 Unlike the previous examples, in this case it is not a single point that determines the mapping diagram, but the single arrow from 0 to 1, which we designate as F[1,1] It can also be shown that this single arrow completely determines the function.thus, given a point / number, x, on the source line, there is a unique arrow passing through x parallel to F[1,1] meeting the target line a unique point / number, x + 1, which corresponds to the linear function s value for the point/number, x. The single arrow completely determines the function f. given a point / number, x, on the source line, there is a unique arrow through x parallel to F[1,1] meeting the target line at a unique point / number, x + 1, which corresponds to the linear function s value for the point/number, x.
58 Simple Examples are important! f(x) = x + C Added value: C f(x) = mx Scalar Multiple: m Interpretations of m: slope rate Magnification factor m > 0 : Increasing function m < 0 : Decreasing function m = 0 : Constant function
59 Function-Equation Questions with linear focus points (Problem 8.a) Use a focus point in the mapping diagram to solve a linear equation: 2x+1 = 5
60 Function-Equation Questions with linear focus points (Problem 8.a) Use a focus point in the mapping diagram to solve a linear equation: 2x+1 =
61 Function-Equation Questions with linear focus points (Problem 8.a) Use a focus point in the mapping diagram to solve a linear equation: 2x+1 =
62 Function-Equation Questions with linear focus points (Problem 8) Suppose f is a linear function with f (1) = 3 and f (3) = -1. Without algebra 8.b Use a focus point to find f (0). 8.c Use a focus point to find x where f (x) = 0.
63 Function-Equation Questions with linear focus points (Problem 8) Suppose f is a linear function with f (1) = 3 and f (3) = -1. Without algebra 8.b Use a focus point to find f (0). 8.c Use a focus point to find x 2 2 where f (x) =
64 Function-Equation Questions with linear focus points (Problem 8) Suppose f is a linear function with f (1) = 3 and f (3) = -1. Without algebra 8.b Use a focus point to find f (0)
65 Function-Equation Questions with linear focus points (Problem 8) Suppose f is a linear function with f (1) = 3 and f (3) = -1. Without algebra 8.c Use a focus point to find x where f (x) =
66 Thanks The End! Questions?
67 References
68 Solving Linear Equations Visualized with Mapping Diagrams (YouTube) by M. Flashman Function Diagrams. by Henri Picciotto Excellent Resources! Henri Picciotto's Math Education Page Some rights reserved Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s. A Reference and Resource Book on Function Visualizations Using Mapping Diagrams (Preliminary Sections- NOT YET FOR publication) Mapping Diagrams and Graphs Visualizing linear functions using mapping diagrams and graphs. tube.geogebra.org Martin Flashman
69 Thanks The End! REALLY!
ATM 2016 Conference Using Mapping Diagrams to Make Sense of Functions and Equations. March 31, Martin Flashman
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