I have been developing this computer software / program / application for some years now, and it is now accessible as a web page, to run in your browser.
It provides basic geometric construction facilities, with lines, points and circles, from which endless possibilities follow.
Just try it out, it’s free.
Click on this or copy and paste for later : www.mathcomesalive.com/geostruct/geostructforbrowser1.html
.Here are some of the basic features, and examples of more advanced constructions, almost all based on straightedge and compass, from “make line pass through a point” to “intersection of two circles”, and dynamic constructions with rolling and rotating circles.
Two lines, with points placed on them
Three random lines with two points of intersection generated
Five free points, three generated circles and a center point
Three free points, connected as point pairs, medians generated
Two free circles and three free points, point pairs and centers generated
GIF showing points of intersection of a line with a circle
Construction for locus of hypocycloid
GIF showing a dilation (stretch) in the horizontal direction
Piston and flywheel
Construction for circle touching two circles
Construction for the locus of a parabola, focus-directrix definition.
CCSSM talks about “the standard algorithm” but doesn’t define it – Oh, how naughty, done on purpose I suspect, since there are varieties even of the “American Standard Algorithm”. Besides, if it is not defined it cannot be tested (one hopes!). I checked some internet teaching stuff on it, and as presented it won’t work on for example 403 – 227 without modification.
Anyway, I was thinking about subtraction the other day (really, have you nothing better to think about?), and concluded that subtraction is easiest if the first number ends in all 9’s or the second number ends in all 0’s. So, fix it then, I thought, change the problem, and here are the results
I am quite sure that some of you can extract the general rule in each case, and see that it works the same in all positions.
While I am going on about this I would like an answer to the following-
“If I understand subtraction, and can explain the ideas to another, and I learn the standard algorithm and how to apply it, and I have faith in it based on experience, WHY THE HELL DO I HAVE TO BE ABLE TO EXPLAIN IT?”
I guess this post counts as a rant!
Let’s have a number line. We can count up or down by moving to the right or the left, or actually up or down if the number line is drawn vertically.
But what else ?????
The following pictures show how points on the number line which represent fractions can be found exactly by simple geometrical construction, and then how results of multiplication and division of fractions can be found exactly as points on the number line (sorry, the numerical values are however not found).
This arose from a statement in the CCSS math document that the fraction 1/6 could be represented by a point one sixth of the way from zero to one, BUT NOWHERE DOES IT SAY HOW TO FIND THAT POINT.
No more borrowing and paying back, or the new alternative
Use addition and then simple subtraction, naturally there are several steps
Example 234 – 187
-187 (should be lined up!)
Now 7 is greater than 4, so we add 3 to both numbers, giving a zero in the units position of the second number
234 -187 = 237 – 190 (must be able to add 3 to 187 !!!)
moving to the 10’s position, and very formally, 90 is greater than 37 so add 10 to both numbers, giving
237 – 187 = 247 – 200, which is now simple, with result 47
Of course, if the lower digit is less than the one above just subtract it from each number
234 – 117 = 237 – 120 = 214 – 100 = 114
It is best to keep the “one above the other” layout, but I have fought with the editor enough today.