IAE Sandbox for Bob





==Algebraic Alakazams: Products of Proper Factors of Natural Numbers

Everybody knows about the sum of the proper factors of a natural number n:

If the sum of the proper factors of n is less than n, then n is a deficient number.

If the sum of the proper factors of n is equal to n, then n is a perfect number.

If the sum of the proper factors of n is greater than n, then n is an abundant number.

Bob woke up in the middle of the night a few moons ago wondering about the product of proper factors of a natural number. Couldn't get back to sleep, jumped up, searched the Internet, didn't find anything. Is the product of the proper factors of a natural number unexplored territory? If yes, let' explore it. We will begin with a table (we love tables)of natural numbers, proper factors of the natural numbers, number of proper factors of the natural number, and the product of the proper factors of the natural number.

First, though, Let's talk about the proper factors of prime numbers.

The one and only proper factor of a prime number is 1. If a product requires two or more factors multiplied, alas, there is no product of proper factors of a prime number. So, aha, and tra la, let's focus on composite numbers. A composite number has three or more factors and two or more proper factors.

The table below provides evidence that we will use to make conjectures.

@@@ BOB STOPPED HERE 2014-05-22 02:16 @@@

Product of proper factors of n = f(n, number of proper factors)

Algebraic Alakazams: MathNEXUS Problem 2013-09-20

Here is the first of several emails about a MathNEXUS problem.

Ahoy Mathemagicians,

Thanks again to Jerry Johnson for MathNEXUS. We browse it every week and always find something interesting. Last Sunday's Problem of the Week is copied below:

Consider this sequence ... can you decipher a pattern?

sqrt(1 + 1*2*3*4) = 5

sqrt(1 + 2*3*4*5) = 11

sqrt(1 + 3*4*5*6) = 19

sqrt(1 + 4*5*6*7) = 29

Predict the value of these roots before calculating them:

sqrt(1 + 8*9*10*11) = ?

sqrt(1 + 9*10*11*12) = ?

sqrt(1 + 50*51*52*53) = ?

Can you express the pattern for the general case?

---

We think we can -- with the help of our TI-84 Graphing Calculator and some algebraic alakazams.

[Our email allows us to use superscripts. We hope that your email shows superscripts because there are many superscripts in the stuff that follows.]

We defined f(n) = sqrt(1 + n(n + 1)(n + 2)(n + 3)) and calculated first and second differences of f(n):

n   f(n)    1st diff     2nd diff

1      5

6

2    11                           2

8

3    19                           2

10

4    29                           2

12

5    41                           2

14

6    55

Conjecture: 2nd differences all equal 2, a constant. f(n) is a polynomial function of degree 2, also known as a quadratic function.

We quick-drew our TI-84 from its holster and used the STAT editor to enter values of n and f(n) for n = 1, 2, 3, and 4 into lists L1 and L2.

L1    L2

1      5

2    11

3    19

4    29

Then we used STAT CALC 5:QuadReg to calculate quadratic regression using lists L1 and L2.

Alakazam! Here is the result we saw on the TI-84's display:

y = ax2 + bx + c

a = 1

b = 3

c = 1

So f(n) = n2 + 3n + 1. We checked it for the given data:

n   f(n) = n2 + 3n + 1

1               5

2             11

3             19

4             29

AOK! Next we cranked out the requested answers

n      f(n) = n2 + 3n + 1

8          89

9        109

50      2651

NOTE: You can use any three of the four data points to calculate the coefficients of the quadratic.

Ha! That's one way. Here is another way.

Assuming that f(n) is a quadratic function, we write: f(n) = an2 + bn + c.

n = 1: f(1) = a(12) + b(1) + c = 5  --->    f(1) =   a +   b + c =   5

n = 2: f(2) = a(22) + b(2) + c = 11  --->  f(2) = 4a + 2b + c = 11

n = 3: f(3) = a(32) + b(3) + c = 19  --->  f(3) = 9a + 3b + c = 19

We then used the TI-84's MATRIX operations to solve the system of 3 equations in the three variable a, b, and c.

Yep: a = 1, b = 3, and c = 1. f(n) = an2 + bn + c = n2 + 3n + 1.

Just for the fun of it, we define g(n) = [f(n)]2

n     f(n)      g(n)   1st diff   2nd diff    3rd diff    4th diff

1        5         25

96

2      11       121                       144

240                          96

3      19       361                       240                           24

480                        120

4      29       841                       360                           24

840                        144

5      41     1681                       504                          24

1344                        168

6      55     3025                       672

2016

7      71     5041

Aha! As we expected, 4th differences all equal 24, a constant.

Conjecture: g(n) is a polynomial function of degree 4, also known as a quartic function.

g(n) = an4 + bn3 + cn2 + dn + e

We entered values of n and g(n) for n = 1, 2, 3, 4, and 5 in TI-84's lists L1 and L2:

L1       L2

1        25

2      121

3      361

4      841

5    1681

Then we used the TI-84's STAT CALC 7:QuartReg to calculate the values of a, b, c, d, and e. The TI-84's display shows:

y = ax4 + bx3 + ... + e

a = 1

b = 6

c = 11

d = 5.999999999 [We rounded it to 6]

e = 1.000000001 [We rounded it to 1]

So, tra la, tra la, g(n) = n4 + 6n3 + 11n2 + 6n + 1

We defined g(n) as the square of f(n), so it would be good if the following is true:

n4 + 6n3 + 11n2 + 6n + 1 = [n2 + 3n + 1]2

It is! Joy is everywhere, funiculi, funicula!

We trust that you will not take our word for this happy equalness, but will verify it yourself, or -- better -- have your students do it.

For f(n), we showed two ways to get the function, using 1) quadratic regression and 2) matrix algebra. So let's do the matrix algebra solution for g(n).

g(n) = an4 + bn3 + cn2 + dn + e

n = 1: g(1) = a(1)4 + b(1)3 + c(1)2 + d(1) + e =    25

n = 2: g(2) = a(2)4 + b(2)3 + c(2)2 + d(2) + e =  121

n = 3: g(3) = a(3)4 + b(3)3 + c(3)2 + d(3) + e =  361

n = 4: g(4) = a(4)4 + b(4)3 + c(3)2 + d(4) + e =  841

n = 5: g(5) = a(5)4 + b(5)3 + c(3)2 + d(5) + e = 1681

Simplify:

a +      b +      c +   d + e =     25

16a +    8b +    4c + 2d + e =   121

81a +  27b +    9c + 3d + e =   361

256a +  64b + 16c + 4d + e =   841

625a + 125b + 25c + 5d + e = 1681

Abracadabra! TI-84 MATRIX solution: a = 1, b = 6, c = 11, d = 6, e = 1

g(n) = an4 + bn3 + cn2 + dn + e = n4 + 6n3 + 11n2 + 6n + 1

May dragons of good fortune dance on your keyboard.

Dave, There several more emails. I have developed some teacher tools to generalize this type of problem.

Bob & George

bob@geekclan.com

Practice Arithmetic with Whole Numbers at Math Magician
Home page; http://resources.oswego.org/games/mathmagician/cathymath.html

Practice addition at http://resources.oswego.org/games/mathmagician/mathsadd.html

Practice subtraction at http://resources.oswego.org/games/mathmagician/mathssub.html

Practice multiplication at http://www.oswego.org/ocsd-web/games/Mathmagician/mathsmulti.html Practice division at http://www.oswego.org/ocsd-web/games/Mathmagician/mathsdiv.html

Practice Integer Arithmetic at AAAMath
Alas, alack, and oh heck, many students have difficulty doing arithmetic with positive and negative integers. This is a big problem in pre-algebra, algebra, geometry, and beyond.

Practice integer arithmetic at AAAMath http://www.aaamath.com. Become a wiz at adding, subtracting, multiplying, and dividing integers. Serendipity! Homework becomes easier, she or he does better on tests, et cetera, et cetera. Here are AAAMath integer arithmetic pages that we recommend to our tutees.

Addition Table of Contents http://www.aaastudy.com/add.htm


 * One Digit Addition of Integers http://www.aaastudy.com/add65_x2.htm
 * One Digit Integer Equations http://www.aaastudy.com/add65_x4.htm
 * Add Three Integers http://www.aaastudy.com/add65_x1.htm

Subtraction Table of Contents  http://www.321know.com/sub.htm


 * Subtract Negative and Positive Integers http://www.321know.com/subint1.htm
 * Subtraction Equation – 1 Digit Integers http://www.321know.com/sub65_x2.htm

Multiplication Table of Contents http://www.aaastudy.com/mul.htm


 * Multiplication Sentences I http://www.aaastudy.com/mulequ1.htm

Division Table of Contents  http://www.aaastudy.com/div.htm


 * Division Sentences I http://www.aaastudy.com/divequ1.htm

Practice Arithmetic with Fractions at AAAMath
You can practice arithmetic with fractions at AAAMath ( http://www.aaamath.com ). You can become a wiz at adding, subtracting, multiplying, and dividing fractions. Serendipity! Homework becomes easier, you do better on tests, et cetera, et cetera.

AAAMath Fractions – Table of Contents http://www.aaastudy.com/fra.htm

We have selected items from the table of contents that we especially recommend to tutees.

Equivalent fractions http://www.aaastudy.com/fra42ax2.htm

Simplify fractions http://www.aaastudy.com/fra66hx2.htm

Fractions and Mixed Numbers


 * Convert improper fractions to mixed numbers http://www.aaastudy.com/fra57cx3.htm
 * Convert mixed numbers to improper fractions http://www.aaastudy.com/fra57cx2.htm

Add Fractions


 * Add fractions with the same denominator http://www.aaastudy.com/fra57ax2.htm
 * Add fractions with different denominators http://www.aaastudy.com/fra66kx2.htm
 * Add mixed numbers http://www.aaastudy.com/fra66dx2.htm

Subtract Fractions


 * Subtract fractions with the same denominators http://www.aaastudy.com/fra57bx2.htm
 * Subtract fractions with different denominators http://www.aaastudy.com/fra66lx2.htm
 * Subtract mixed numbers http://www.aaastudy.com/fra66ex2.htm

Multiply Fractions


 * Multiply fractions http://www.aaastudy.com/fra66mx2.htm
 * Multiply fractions by whole numbers http://www.aaastudy.com/fra66nx2.htm
 * Multiply mixed numbers http://www.aaastudy.com/fra-mul-mixed.htm

Divide Fractions


 * Divide fractions by fractions http://www.aaastudy.com/fra66px2.htm
 * Divide fractions by whole numbers http://www.aaastudy.com/fra66ox2.htm
 * Divide mixed numbers http://www.aaastudy.com/fra-div-mixed.htm

Math Playground Fraction Games
At Math Playground http://mathplayground.com you can learn math by playing games and solving puzzles.

In the menu on the left side of the page, click on Math Games and you will go to http://mathplayground.com/games.html. This page has picture links to many games. We recommend these to our tutees for practicing fraction skills.


 * Decention – a game of fractions, decimals, and percents http://mathplayground.com/Decention/Decention.html. At the Intergalactic Space Games (ISG), teammates wear uniforms with matching number values expressed as fractions, decimals, and percents. Your job, as ISG director, is to make sure that all members of each team go to the teams starting place.
 * Bridge Builders http://www.mathplayground.com/FractionGame/FractionGame.html. Gertie Gecko is on the road to somewhere, but sections of the road are missing. Your task is build spans over the gaps in the road so that Gertie can keep on truckin’ down the road to somewhere.

We suggest that you play these games before suggesting them to your tutee, then observe while your tutee plays the game, and be ready to answer a question, provide a gentle hint, or otherwise enhance your tutee’s game-playing experience.

NCTM Illuminations Fraction Activities
The National Council of Teachers of Mathematics Illuminations site presents more than 100 interactive alakazams for learning math.

Go to http://illuminations.nctm.org/.

Click on Activities and go to http://illuminations.nctm.org/ActivitySearch.aspx. Click on K-2 or 3-4 or 5-6 or 7-8 or 9-12 to select activities for that grade level.

Or click on Select All to select all 107 activities.

We clicked on Select All, tried a bunch of fraction activities, and liked them. Here are direct links to NCTM Illuminations fraction activities.

Equivalent Fractions (3-5) http://illuminations.nctm.org/ActivityDetail.aspx?ID=80.

Create equivalent fractions by dividing and shading squares or circles, and match each fraction to its location on the number line.


 * Click on Instructions, read, and then play.
 * After playing a few games, click on Explorations, read, and then explore.

Fraction Models (3-8) http://illuminations.nctm.org/ActivityDetail.aspx?ID=11 Explore different representations for fractions including improper fractions, mixed numbers, decimals, and percentages. Additionally, there are length, area, region, and set models. Adjust numerators and denominators to see how they alter the representations and models. Use the table to keep track of interesting fractions.


 * Click on Instructions, read, and then play.
 * After playing a few games, click on Explorations, read, and then explore.

Fraction Game http://illuminations.nctm.org/ActivityDetail.aspx?ID=18

The object of the game is to get all of the “racers” to the right side of the game board, using as few fraction cards as possible.


 * Click on Instructions, read, and then play.
 * After playing a few games, click on Explorations, read, and then explore.

Free Ride http://illuminations.nctm.org/ActivityDetail.aspx?ID=178

Vary the gear ratio of a bike. The distance traveled by a half-pedal is determined by the ratio of gears. Can you capture all five flags on a course?


 * Click on Instructions, read, and then play.
 * After playing a few games, click on Explorations, read, and then explore.

Visual Fractions Games at http://visualfractions.com/Games.htm
Very elementary fraction games.

Find Grampy http://visualfractions.com/FindGrampy/findgrampy.html

Grampy hides behind a hedge whose length is given as n equal parts. You can see the top of Grampy’s head at his hiding place behind the hedge. Your task is to supply the numerator of the fraction that names Grampy’s hiding place. It is OK if the fraction is not in lowest terms.

Find Grampy – Strict http://visualfractions.com/GrampStrict/findgrampystrict.html

Grampy hides behind a hedge whose length is given as n equal parts. You can see the top of Grampy’s head at his hiding place behind the hedge. Your task is to supply the numerator of the fraction that names Grampy’s hiding place. The fraction must be in lowest terms.

Cookies for Grampy http://visualfractions.com/CookiesF.html Assemble cookies for Grampy by dragging fractional pieces of cookies to the whole cookie place. Look at the available pieces carefully and plan ahead.

Find Grammy http://visualfractions.com/FindGrammy/findgrammy.html Grammy hides behind a hedge whose length is given as n equal parts. You cannot see the top of her head at her hiding place behind the hedge. You must watch her hide and remember where she hid. Your task is to supply the numerator of the fraction that names Grammy’s hiding place. It is OK if the fraction is not in lowest terms.