Working backwards

Judah L Schwartz, Apr 19, 2016

The applets in this collection offer opportunities to explore [b][i]synthetic[/i][/b] rather than [b][i]analytic[/i][/b] approaches to problems in arithmetic, algebra and geometry. In the first chapter we look at addition and multiplication computation problems through the lens of the inverse operations. For example, what two numbers, when added, will yield a sum between two given numbers? We also look at prime factorization by asking the user to compose numbers by multiplying primes rather than to decompose them by finding prime factors. In the second chapter we explore building polygons by linking segments to one another. For example, give three line segments that intersect at a point that subdivides each line segment in the ratio of 2:1 can one build a triangle for which the given line segments are the medians? Several of the applets allows the user to explore under what circumstances breaking a stick into n pieces the resulting pieces can be assembled into an n-sided polygon. In the third chapter we explore the building of functions – from concatenating operations and transformations on the constant and the identity functions, or by using attributes such as locations of sign changes of a function and its rate of change. Working backwards is not always possible - as the applet on non-invertable operations makes clear. In the next chapter users can explore three simple differential equations - f(x)=1/f'(x), f(x)/f'(x)=1 and f(x)+f'(x)=1. In the final two chapters we explore the UNsolving of equations and inequalities– i.e., the generating of the equivalence class of equations from the given solution set - for example, find the set of all quadratic inequalities whose solution set is -2 < x < 3. [b]This is a collection of applets drawn from the archive website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/b]

Table of Contents

  1. ...with numbers
    1. Build a million numbers
    2. 'between-ness' in addition/subtraction
    3. 'between-ness' in multiplication/division
    4. DIY estimation - multiplication
    5. DIY estimation - addition
    6. Common Denominator Finder
    7. Common Factor Finder
    8. Inverse Prime Factoring
    9. 'Between-ness' with rational numbers
  2. ... with shapes
    1. Build a triangle from medians?
    2. Build a Triangle from Medians - II
    3. Build a triangle from Angle Bisectors?
    4. Build a triangle from Altitudes?
    5. Point in an Equilateral Triangle - revisited
    6. Point in a Square
    7. Quadrilateral factory ?
    8. Polygon Synthesizer
    9. Parametric polygon synthesizer
    10. Circle between quarter circles
  3. ... with functions
    1. Function Family Builder
    2. Function Attribute Explorer
    3. power drill - factoring & FOIL
    4. Even and Odd Functions - I
    5. Even & Odd Functions - II
    6. Algebraic Tessellations
    7. 'Round trips' in maths? non-invertable operations
  4. ...with functions & their derivatives
    1. f(x) f'(x) puzzle
    2. f(x) / f'(x) puzzle
    3. f(x) + f'(x) puzzle
  5. ... with 1 variable equations & inequalities
    1. UNsolving Linear Equations & Inequalities
    2. UNsolving Quadratic Equations & Inequalities
    3. UNsolving Cubic Equations
    4. UNsolving linear equations - two views
  6. ...with 2 variable equations & inequalities
    1. UNsolve 2 variable linear equations & inequalities
    2. UNsolve 2 variable quadratic equations & inequalities

1.

...with numbers

In this chapter we look at addition and multiplication computation problems through the lens of the inverse operations. For example, what two numbers, when added, will yield a sum between two given numbers? The next two applets in the chapter explore finding common denominators and finding common factors. We also look at prime factorization by asking the user to compose numbers by multiplying primes rather than to decompose them by finding prime factors.

  • 1. Build a million numbers
  • 2. 'between-ness' in addition/subtraction
  • 3. 'between-ness' in multiplication/division
  • 4. DIY estimation - multiplication
  • 5. DIY estimation - addition
  • 6. Common Denominator Finder
  • 7. Common Factor Finder
  • 8. Inverse Prime Factoring
  • 9. 'Between-ness' with rational numbers

1.1.

Build a million numbers

[color=#ff0000][color=#000000]An interesting question might be how many palindromic numbers can you make with this applet ?[/color][br][i][b][br][br]What questions could/would you put to your students based on this applet ?[/b][/i][/color]
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– Judah L Schwartz

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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1.9.

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2.

... with shapes

In this chapter we explore building polygons by linking segments to one another. For example, give three line segments that intersect at a point that subdivides each line segment in the ratio of 2:1 can one build a triangle for which the given line segments are the medians? Several of the applets allows the user to explore under what circumstances breaking a stick into n pieces the resulting pieces can be assembled into an n-sided polygon.

  • 1. Build a triangle from medians?
  • 2. Build a Triangle from Medians - II
  • 3. Build a triangle from Angle Bisectors?
  • 4. Build a triangle from Altitudes?
  • 5. Point in an Equilateral Triangle - revisited
  • 6. Point in a Square
  • 7. Quadrilateral factory ?
  • 8. Polygon Synthesizer
  • 9. Parametric polygon synthesizer
  • 10. Circle between quarter circles

2.1.

Build a triangle from medians?

Three line segments ([b][i]dotted lines[/i][/b]) intersect at a point that divides the lengths of each of the segments in the ratio of 2 to 1.[br][br]These three line segments could be the medians of a triangle.[br][br]You can use the sliders to set the size of these three line segments. The changed lengths will maintain the 2:1 ratio. Try to set them so that points A and B, B and C, and C and A are joined by straight line segments.[br][br]Alternatively, you can try to drag the points A,B and C to where you think the vertices of the target triangle are. When A, B and C lie on the vertices of the target triangle the angle between two segments of the same color will be 180 degrees.[br][br]Can a triangle always be made this way? Can you prove it?
– Judah L Schwartz

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2.3.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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3.

... with functions

In this chapter we explore the building of functions – from concatenating operations and transformations on the constant and the identity functions, or by using attributes such as locations of sign changes of a function and its rate of change. Working backwards is not always possible - as the applet on non-invertable operations makes clear.

  • 1. Function Family Builder
  • 2. Function Attribute Explorer
  • 3. power drill - factoring & FOIL
  • 4. Even and Odd Functions - I
  • 5. Even & Odd Functions - II
  • 6. Algebraic Tessellations
  • 7. 'Round trips' in maths? non-invertable operations

3.1.

Function Family Builder

This environment allows you to explore families of linear, quadratic and absolute value functions. [br][br]Can you make any linear function with this environment?[br]Can you make any quadratic function with this environment?[br]Can you make any absolute value functions with this environment?[br][br]For the cases examined in this environment[br][br]•Vertical sliding of f(x) leads to f(x) + a. [What happens if a>0, a<0][br][br]•Horizontal sliding of f(x) leads to f(x+a). [What happens if a>0, a<0][br][br]•Vertical stretching & squeezing of f(x) leads to af(x). [What happens if a<0, 01][br][br]•Horizontal stretching & squeezing of f(x) leads to f(x/a). [What happens if a<0, 01][br][br]Do you believe these statements are true for any function of one variable? If so, can you prove it? If not, can you find a counterexample?[br][color=#ff0000][i][b][br]What questions could/would you pose to your students based on this applet?[/b][/i][/color]
– Judah L Schwartz

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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4.

...with functions & their derivatives

  • 1. f(x) f'(x) puzzle
  • 2. f(x) / f'(x) puzzle
  • 3. f(x) + f'(x) puzzle

4.1.

f(x) f'(x) puzzle

Enter a trial function f(x) that can depend on up to two parameters a and b.[br][br]The goal is to find a function such that the product of the function and its derivative is constant for all x greater than some x[sub]0[/sub].[br][br]If both x and f(x) are quantities rather than numbers [[b][size=85]see "[i]Semantic Aspects of Quantity[/i]" on the [i]mathMINDhabits[/i] website[/size][/b]] explain how and why this constant can be adjusted.
– Judah L Schwartz

4.2.

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4.3.

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5.

... with 1 variable equations & inequalities

  • 1. UNsolving Linear Equations & Inequalities
  • 2. UNsolving Quadratic Equations & Inequalities
  • 3. UNsolving Cubic Equations
  • 4. UNsolving linear equations - two views

5.1.

UNsolving Linear Equations & Inequalities

The solution of a linear equation is a number - let's call it a. The solution of a linear inequality is a range of numbers, say all the numbers less than a, or all the numbers greater than a.[br][br]To UNsolve a linear equation or inequality, drag the GOLD dot in this panel to set the solution. The other panel will show you a linear equation or inequality that has that solution.[br][br]You can drag the WHITE dots in the right hand panel to see other equations or inequalities that have the same solution set. Each of the large WHITE dots control one function - the smaller WHITE dot controls both functions. Why is it permissible to change only one function in an equation or inequality that is a comparison of two functions? [br][br]How many solutions are there? How do you know? Can you prove it?[br]What happens to the inequality when the sign of the scale factor changes? Why?[br][br]Challenge - Make up an equation [i.e., find values for a, b, c, and d] of the form ax + b = cx + d with solution x = 7 and a, b, c, d <>0[br][br]Could any other value of x other than x = 7 satisfy your equation? Why or why not?[br][br]How many such equations can one construct [i.e., how many sets {a,b,c,d} are there]? How do you know?
– Judah L Schwartz

5.2.

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5.3.

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5.4.

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6.

...with 2 variable equations & inequalities

  • 1. UNsolve 2 variable linear equations & inequalities
  • 2. UNsolve 2 variable quadratic equations & inequalities

6.1.

UNsolve 2 variable linear equations & inequalities

The solution of a linear equation in 2 variables, f(x,y) = 0, is a straight line in the x,y plane.[br]The solution of a linear inequality is EITHER all the points above that line or below it.[br][br]To UNsolve a linear equation or inequality, use the a, b and c sliders in the[br]rightmost panel to fix the solution set. The left panel will show the solution set. The center[br]panel will show you a linear equation or inequality that has that solution set.[br][br]You can use the c slider in the rightmost panel to see other linear[br]equations or inequalities that have the same solution set.[br][br]For a given solution set, how many equations or inequalities are there that[br]have that solution set? How do you know? Can you prove it?
– Judah L Schwartz

6.2.

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