Elementary Mathematics (K-6) Explained 1: Counting using the grid plane

We start this course in elementary mathematics suited for a general audience by introducing geometry in the simplest possible way: by using a sheet of graph paper as the basic model, and then doing some simple arithmetic in a geometrical context.

We introduce the horizontal and vertical lines of the grid plane, grid points, segments and rectangles, along with the idea of measuring their sizes. This is a good place to start practicing counting: and we count a number of interesting things associated to a rectangle, including numbers of sub rectangles of a certain size, paths from one corner to another, and tilings by 1x2 or 2x1 dominoes.

Category:

Education

Tags:

• Elementary
• mathematics
• K-6
• explained
• Wildberger
• course
• education
• grid
• plane
• geometry
• arithmetic
• rectangle
• counting
• lines
• points
• dominoes
• paths
• area
• size
• segment
• tilings
• sub-rectangles

[CorFortgens]

You have used the idea of counting rectangles. Did you consider the possibilities of using vectors? Adding vectors along a line is as simple as rectangles. One can evenh see a vector as a series of rectangles with an arrow pointing tot the right (positive number) or to the left (negative number). Making the step to negative numbers then is perhaps more 'natural'.

[njwildberger]

@CorFortgens While adding vectors is natural, and negating them simple, multiplying them is the problem. How do you explain multiplication from this point of view? Nevertheless, you have a good point, and it is an alternative possibility.

[piemanminter]

Dear Mr Wildberger, Thanks for this wonderful resource. I'm a 32 year old plumber who is considering studying an engineering degree, but despite a keen interest mathematics i have always struggled. Hopefully this course will steer me on the right track with some of the elementry mathematics that i am missing and stand me in good stead.

Kind regards piemanminter

[yn30s]

I'm going to recommend this series to my friends who have been frustrated with math. I'm a PhD student & for what it's worth I think Mr.W could easily be pushing boundaries in math with his creative thinking, but instead has dedicated himself to reforming math education from the bottom up - something that desperately needs to be done but rarely gets this sort of dedication from someone so intelligent. Also, if you have a problem following something you should speak up, this guy really does care.

[jehovajah]

You emphasise neatness, systematic, and logical. I hope you will also connect these things to chaos! I feel fundamentally that repetition(iteration) and simplicity must be shown to produce Fractal geometry and dynamical systems of extraordinary beauty at this Elementary level.

Way to go, prof!

[jehovajah]

Simple introduction to combinatorics! great!

[jehovajah]

@Quatermaths

Do not please do this to a perfectly reasonable observation. Analogies are not exact, and in fact the objection does relate to fundamental notions of space and our interactions in it. The issue here is abstraction, and which set of abstractions are most utilitarian. That actually takes centuries or millenia to determine!

[jehovajah]

I like the beginning, but hate the sectarianism. Your approach recombines what human self interest has divided. I know you have to be modern, but i feel the internal tensions are what distract form the study of space, introduced by Pythagoras, redacted by Plato and taught by Euclid.

Good job, but not revolutionary enough!

[gpjelliss]

@QuantumMaths You've obviously never done any practical tile-fitting or paving.

[edd9139]

Very interesting. I've studied a few university mathematics courses now (including one with you last year) although had never thought about how to define a rectangle or counting in that way.

[QuantumMaths]

@gpjelliss

That is one of the most incredibly non-sensical objections I have ever heard to anything in my entire life.