The X•Y Chart: Multiplication and Coordinates

The X·Y Chart System

Plotting Points (x,y) Plotting Products xy

Presenting Multiplication for Maximum Mathematical Meaning

© Speier Associates 2014

The X·Y Chart System Mission

To help students master math through a picture that unites multiplication with the Cartesian coordinate system.



In this process students will acquire:

•mastery of multiplication and confidence with the related operations of division, factoring, and exponentiation•skill in graphing with the rectangular coordinate system•a visual understanding of different representations of multiplication•confidence using letters ( x and y) to represent numbers•no math phobia

Multiplication and GraphingCritical Math Skills

With multiplication and graphing mastered, the remaining math is easier.

Consider the necessity of multiplication in:

•division•factoring •reducing fractions•finding the greatest common factor•Finding the least common multiple•coefficients without operational signs•finding prime factor sets



To appreciate the importance of multiplication and graphing, examine any algebra or higher math text book. Notice two things:

•the high frequency of multiplication found in the algebraic expressions•the number of graphs that use the Cartesian coordinate system to represent visually the relationship between two numbers

Representing Multiplication

The following slides show the evolution of multiplication tables


A list of the facts An orderly arrangement

1x1=11x2=21x3=31x4=41x5=51x6=61x7=71x8=81x9=91x10=10

2x1=22x2=42x3=62x4=82x5=102x6=122x7=142x8=162x9=182x10=20

3x1=33x2=63x3=93x4=123x5=153x6=183x7=213x8=243x9=273x10=30

4x1=44x2=84x3=124x4=164x5=204x6=244x7=284x8=324x9=364x10=40

5x1=55x2=105x3=155x4=205x5=255x6=305x7=355x8=405x9=455x10=50

6x1=66x2=126x3=186x4=246x5=306x6=366x7=426x8=486x9=546x10=60

7x1=77x2=147x3=217x4=287x5=357x6=427x7=497x8=567x9=637x10=70

8x1=88x2=168x3=248x4=328x5=408x6=488x7=568x8=648x9=728x10=80

9x1=99x2=189x3=279x4=369x5=459x6=549x7=639x8=729x9=819x10=90

10x1=1010x2=2010x3=3010x4=4010x5=5010x6=6010x7=7010x8=8010x9=90

10x10=100

Memorize this!

Multiplication table of rows and columns,but upside down

How we read.


Not Multiplication.A Feelings Table

“We’re bored. Teach us to think and we’ll be happy. We need math.”


A Rectangular Coordinate System

Not a multiplication table

Sample XY Chart

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

X

Y Series1

How we do math. Sky is up.© Speier Associates 2014

Multiplication Table and Coordinate System

The X•Y Chart

Finally!© Speier Associates 2014

The Visual Presentation of Multiplication

An effective multiplication table has– a natural and logical arrangement of the

multiplication facts – The integration in the first quadrant of the

coordinate system, a natural environment for the multiplication facts

– combination of concrete and symbolic representation: area and numerals

– easy navigation– engaging and accurate graphics


What is the The XY Chart? • The XY Chart is a colorful

picture of mathematics.

• It shows the 10 by 10 multiplication facts in their natural mathematical environment, the first quadrant of the rectangular coordinate system.

• Each product is carefully positioned next to the point of intersection of its factors.


What is the The XY Chart? • The large white X and Y letters,

connected by the multiplication dot, signify the important relationship between the X’s, Y’s, multiplication, and algebra. So do the bold X and Y of the widened black X- and Y-axes.

• The perfect squares are colored gray to highlight the symmetry of the 10 by 10 table and the visible commutative law of multiplication. The learner only needs to learn one side of the diagonal and the squares.

• The bright primary colors of the quadrants signal a change of sign.


What is the The XY Chart? • The sections of the quadrants,

which begin at the intersection of the X- and Y-axes, invite the observer to continue beyond the borders of the chart toward the limiting infinities, the outer bounds of the actual math quadrants.

• The XY Chart is designed to help teach multiplication in an easy to use, effective visual manner. Learning to navigate the coordinate system in the process is an added benefit.


Commutative Property of Multiplication.

Multiples of 1, the unity element of multiplication.

Multiples of 10. (Append a zero.)

Multiples of 2. (Double it.)

Multiples of 5.

How many multiplication facts require rote memorization?

Consider the following:

Only 21 facts remain!

Now memorization may begin.

Memorize Facts and Internalize the Coordinate System!

• After understanding concept: repetitive addition.• After practicing with preliminary activities.• Refer to chart.• Flash cards.• Speed test.• Oral quizzes.• Practice, practice, practice.• Connect-the-dots: numbers to pictures.• List coordinates of figures: pictures to numbers.


Interact with the X·Y Chart! Activities #1.

• Use the X·Y Chart to teach beginning counting.• Use its rows and columns for skip counting, i.e. counting

of multiples.• Connect-the-dots drawings to teach the coordinate

system.• Draw figure on graph paper and describe by

coordinates.• Compare graphs of constant functions to rows • Rectangular area • Use as multiplication machine - - over and up • Use as division machine – over, up, and back over• Use to get square roots


Skip Counting

• Counting multiples demonstrates that multiplication is repetitive addition.


Interact with the X·Y Chart! Activities 2.

• 3 way multiplication test– From memory– The X·Y Chart– The calculator

• Study Descartes

• Study Mondrian


Connect the Dots:Numbers to Pictures

• Or try it on the computer http://www.xycharts.com/test/xyctdMM/xyctd.html


http://www.xycharts.com/neworigin.html

http://www.xycharts.com/neworigin.html

Historical and Personal Math: Rene Descartes

• United geometry with algebra to create analytical geometry

• First to use x, y, and z for variables and a, b, and c for constants

• Introduced the use of superscripts for the exponents

• Cartesian Product named after him


A Night With Descartes

• A fly inspired this French math genius to devise the coordinate system.


• A math titan.

Rene Descartes

His development of the methods of analytic geometry paved the way for modern math.


“A fly’s on the ceiling at (23,87). Where’s the fly-swatter?”

The beginning of analytic geometry

His only regret.

“Placing the multiplication table in the first quadrant of my rectangular coordinate system…what a great teaching tool! Why didn’t I think of that?”

Refutation of a Common Fallacy

A philosopher and mathematician in the money.


Piet Mondrian

• The artistic inspiration for the coloring of the chart.

Artistic Inspiration

Graphic Inspiration: Piet MondrianMath and Fine Arts

•


Strong Mental Math ThroughVisualization

• Master multiplication. Know the coordinate system.

Visualize patterns.

Mentally change patterns to numbers (from geometry to algebra).

Mentally change numbers to patterns (from algebra to geometry).


The Truth About Mathematics

• Contrary to popular belief, we were made to enjoy math.

• It’s the most underrated pleasure in our culture


Bring on the calculator. I know my facts.

First day of summer vacationFirst day of summer vacation..

Connect-the-Dots and Multiplication Quizzes

Useful pleasures.

Math Organization

A rith m etic G eom etry A lg eb ra A n a lytic G eom etry C a lcu lu s

M ath em atics

Math is one, but analysts and institutions break it into many. The X•Y Chart™ represents a unified

vision. This one chart spans the math lessons from kindergarten to calculus.


Kindergarten Math Through Calculus

• Counting by 1• Skip-counting• Connect-the-dots• Draw on grid and code coordinates• “Over and up” multiplication • “Over, up, and back” division • Rectangular areas • Arithmetic sequences• Algebraic expression represented by area• Area under the curve or definite integrals


Multiplication loop for multiplier adding multiplicand.

product = 0

for I = 1 to multiplier

product = product + multiplicand;

Multiplication looping for multiplier and multiplicand adding 1.

product = 0

for I = 1 to multiplier

for j = 1 to multiplicand

product = product + 1


Recursive Definition of Multiplication.

Int recmult (int x, int y)

{

if (x == 1)

return y;

else

return y + recmult (x-1,y);

}

X * Y = Y + (X – 1)Y


Multiplication and the Definite Integral

• Consider a fence that is 6 feet high. Designate one end of the fence as the origin. As you walk down the fence from the origin the height of the fence remains 6 feet. A mathematical representation of the fence is the function f(x) = 6. Here the x represents the distance from the end designated as the origin. This is a constant function, ie y’s value doesn’t change with x. Now consider painting that fence for 5 feet from the origin. How many square feet will have to be covered? In arithmetic it’s an obvious opportunity to multiply 5 x 6. In calculus you would be looking for the area under the curve and use the definite integral of f(x) = 7 from 0 to 5.


The Leading Question

• We educators can learn from business marketing.

– Advertising employs the old method of rote learning, i.e. memorization through repetition.

• Businesses invest billions of dollars in “rote memorization”, because it works.


The Leading Question

• How do businesses successfully teach our children to know and to need so many products?– Pokemon: 150+ random names and attributes

learned by heart in a matter of days. • Why is learning the multiplication facts so difficult?

– How many commercial logos have our children already memorized?

• This is “branding” on the brain.

– Our children learn what our entertainment providers sell to them effortlessly, but through repetition.

• Our children are not aware it is happening.


Advertising Awareness

1. Consider a few products or services successfully sold to our

children.

2. How does the advertising work? What techniques of instruction do the marketers employ? “Student centered”, “constructive”, “Socratic method”, “drill and kill”, “rote memorization”?

3. Homework ?

4. Teacher effort?

5. School effort? Standards? Assessments? “Leave no potential customer behind!”

6. Parent effort?


Summary

• Designed to work like an effective advertisement• Requires repetition: at school and at home• Integrates multiplication and graphing in one picture for

all ages of math learners• Complements every math program• Works without a teacher’s explanation • With teacher the learning is deepened• Useful math patterns• Math historical origins: Descartes• Integrates fine arts: Mondrian• Simulated activities: www.xycharts.com


http://www.xycharts.com/

Display it! At School and at Home

• Recognize that successful ad campaigns require frequent viewing. Seen everyday the information can be mastered.

• Children see on average 20,000 TV ads per year. Imagine the benefits to our culture if these ads taught mathematics!

• A strategy is clear: Connect classroom with “homeroom”.

Display it!Classroom walls

School hallsBedroom wallsRefrigerators

Bedroom ceilingsDresser tops

Desktops

– How often do your students see what’s on their walls at home? 20,000 times per year?– Why not display math?


Learning through constant seeing X·Y Chart System

• Large wall chart in each classroom

• Several in school hallway

• 8 ½ by 11 inch for student binders

• Postcard size version on school desk

• In home: bedroom, refrigerator, ceiling


Visual Presentation of Multiplication

• Apply successful methods of visual learning as used in advertising

• Burn meaningful mathematical image into the minds of the young viewers through repetition

• Presentation worthy of valuable knowledge– Good design– Good printing – Best materials– Universally understood– Math is the world’s language


Outline

• The X·Y Chart system mission

• The two major math skills: multiplication and graphing

• Evolution of the multiplication table

• At home and at school

• Memorize facts and internalize the coordinate system

• Interact with it! Some activities

• Historical and personal math: Rene Descartes

• Graphic inspiration: Piet Mondrian - math and fine arts

• Math as pleasure

• Learning from successful advertising


The X•Y Chart: Multiplication and Coordinates

Education

Transcript of The X•Y Chart: Multiplication and Coordinates