Coordinate Grid Pictures, Slope, Distance, and Square Root Estimates and Sir Isaac Newton

I shared a story about Isaac Newton about what can happen during a shutdown. In 1665, Cambridge University closed as the bubonic plague swept across England. Isaac Newton, a 22-year-old student, was forced to retreat to the family farm, Woolsthorpe Manor. Isolated there for more than a year, on his own he revolutionized the scientific and mathematical world. Newton said that this shutdown freed him from the pressures of the curriculum and led to the best intellectual years of his life. Here is what he did:

Optics: He discovered the fundamentals of color. It was already known that sunlight passed through a glass prism produced the rainbow spectrum—bands of red, orange, yellow, and so on. But were these a fundamental aspect of light as opposed to some artifact produced by prisms? To find out, he passed a single-colored beam from the first prism through a second prism and got the same color once again. The glass did not change it.

Gravity: Sitting in the orchard behind his farmhouse, he considered two orbs in his view, the moon and a single apple. Newton wondered if the force that drew the apple to the earth was what held the moon in orbit. These musings led him to later construct the laws of gravity and motion that tied everything in the universe together.

Calculus: He wrote three papers inventing calculus. (Shortly afterward, Gottfried Leibniz came up with the same principles.) Thinking about the rate of change as an object accelerated falling to earth, he realized that one could get an accurate total of the area under a curve by summing the rectangles, down to the infinitely small rectangles, that made up this area.

In 1667, Newton returned to Cambridge, the plague having abated. He presented all this work to his mentor and professor, Sir Isaac Barrow. Two years later, Barrow resigned his chair in favor of Newton.

Martin Seligman of the University of Pennsylvania said of this situation, “Don’t let current circumstances interrupt learning. Do imagine what you might do when freed from conventional routines and requirements. When it comes to curiosity and creativity the mind knows no boundaries.

I asked the Mathletes to think about what they would like to invent during their time at home. One idea was to create a cell phone case that would bounce like a ball if you dropped it; one student wanted to create a light saber that could help to clear your driveway of snow; and my favorite comment came from Andy who said that he wanted more children around the world to be able to take our Virtual Mathletes Classes so children around the world could get better at math and problem solving. Wow! What will you do next?

Now for the lesson:

Coordinate Grid Pictures, Slope, Distance, and Square Root Estimates. 

Coordinate Grid Pictures: I provided the K-2 students with dozens of 1st quadrant coordinate points to create pictures such as a Bald Eagle, MATHLETES, Bart Simpson, Diplodocus, Honest Prez, Father Christmas, Owl, Sailboat, and Super Mario. For students from 3-6th grade, 4 Quadrant pictures such as Lisa Simpson, Superman, Batman, Princess Sally Acorn, Whammy, Miles Tails Prowler, Rouge, Sonic the Hedgehog, Rotor Walrus, and Antoine D’ Coolette. These will keep them busy for months and focused on the geometry and algebraic notation of location in a two dimensional plane. The children should color their pictures and send me their work product (include name please) so I can post on this site (SEE THE FIRST FEW DRAWINGS IN THE ATTACHED PDF).

Slope of a line: As they create each segment, I wanted them to understand slope. This is taught in Algebra and Geometry way too late for it to be second nature. Every time they look at any steepness of a hill, they should be thinking about slope. The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. The slope of a wheelchair ramp is required to be 1/12 in order to allow a person to self wheel up and down safely. This creates an angle of about 4.7 degrees. A slope of 1/5 creates an angle of about 11 degrees, too steep for a wheel chair.

I showed them a picture of a skier in Switzerland skiing down a mountain with a near vertical slope. The children discussed why this is a bad idea due to gravity and lack of friction. In fact this slope is referred to as “undefined” since the concept of infinity is undefined. Calculators will say Error when a vertical rise is divided by a zero horizontal run; you can’t divide by zero.

We then looked at a horizontal line which has a slope of zero because a zero rise divided by any horizontal run is still zero. Cross country skiers like this slope. Not me, I like a negative slope which goes downhill from left to right. Positive slopes go uphill from left to right. The steeper the slope the closer the numbers get to negative infinity or positive infinity. For example, if I am going up a hill of slope 1/4, that is a relatively flat grade compared with a slope of 1 which creates a 45 degree angle with the ground (but a 1/4 slope is way more steep than our wheelchair ramp example of 1/12). As the slope gets steeper, like 2, 4, 8, 16, it begins to look more vertical until you are approaching infinity or undefined slope.

For the 3-6th graders, I had them think of slope as a fraction of rise/run, with rise always a positive vertical movement and the run horizontal. The run determines whether the slope is positive or negative. A run to the right is positive; a run to the left is negative.

It is great practice for the children to name the slope of each line they create in their pictures. When they get to Algebra and Geometry, this will be second nature.

Distance and Square Roots: For some of the 5-6th grade groups, we were able to discuss the length of these segments. Essentially, this 2500 year old mathematical discovery says that the sum of the squares of the perpendicular legs of a right triangle will equal the square of the hypotenuse (the longest side of a right triangle).

The example above shows a famous Pythagorean Triple of 3:4:5, but most distances that are not horizontal or vertical are irrational numbers (square root of the sum of the perpendicular leg squares). 

To understand square roots, first a student must master his/her squares (a number multiplied by itself). For example, 5 squared is 5 x 5 or 25. So the square root of 25 is 5. The square root of 100 is 10. 

If your points are separated by a right triangle with perpendiculars legs of 1 and 3, the distance will be the square root of 1 squared plus 3 squared, or the square root of 1 + 9. The square root of 10 is between the square root of 9 (3) and the square root of 16 (4). Oddly enough, the square root of 10 happens to be close to Pi (3.1415….); coincidence??

I developed an algorithm some years ago for estimating square roots of any whole number (this process is outlined on page 9 of the Print pdf for 3-6th graders). Try this with a calculator so you can show your friends how scarily close your estimate is to the actual square root. Note that your estimate can never be exact since an irrational number can never be expressed as a fraction of whole number numerator and denominator.

I cannot wait to see what the children create over the coming months with their coordinates pictures.