Math 1.0 August 21: Length, Coordinates, Dice and Histograms

Semester introduction

The founder of the Urbana Math Circle Yuliy will be away for a semester, and so I slipped in to try my hand at teaching the older students for a few months.
I am framing the semester around two themes, and will add other bits and pieces.

The first theme is a fundamental law of physics: the harder you push something, the faster it goes. More specifically, that acceleration and speed are determined by how heavy something is and how hard it is pushed,  acceleration is proportional to force over mass a = F/m which is one arrangement of Newton's second law, F=ma. 

We start today reviewing a few basics - understanding length, understanding units, how to draw a graph with x and y axes. Later we will explore position, time, and the relationships between these. We will also learn about functions.

Another topic will be 'expectation'. What do we expect will happen if we roll a dice, flip a coin, pull a card, etc. This will provide some foundations for probability some day when students are ready.

Finally student choice: which of Yuliy's lessons were their favorites? 

Length and Measurement

First, students used a ruler to draw a number line. Twenty squares long. Then they measured this in centimeters. They came up with 12 or 13 cm until we looked into the notches and learned to read the millimeter marks. Then we translated this to inches (20 squares is 5" on a 1/4" grid), a non-standard grid that elicited an ode to 1/2 cm grids (and where to find them) from the professor in the back of the room.

Next, we enumerated the number line. every other square was one unit. This then became the x-axis, and we built a y axis.

For good measure, we plotted a sequence of numbers, starting at x=0 and y = 2:

From here, we thought about what number would go at x=-1 or x=3 ...

Dice (or expectation)

This week we discussed what could happen if you throw a single dice. Students realized that possible outcomes were 1, 2, 3, 4, 5, or 6.  

I asked them to write them down and start filling in a square corresponding to the number of dots each time they rolled, thus:

After we did this for a while I asked each student to say what value they got the most of. I enumerated those. Different students got different answers. Then we set up a second 'trial' and began rolling, to see if the same value won both times:

One student came to what seems to be a very unexpected outcome on the second trial:

He got one one, two twos and so on. Perfectly. We were all amazed. Did this fit in with what we expected?

Student choice: favorite topics

At the end of the meeting I asked students to list their favorite activities from Yuliy's math circle. 

I was amazed that the students came up with 'Roots', 'Vectors', 'Roman Numerals' and 'Algebraic notation', then there were some logic puzzles (remembered as 'truth or mad' ... which of the two twins is lying? homework, below).

Hopscotch came up. An app that lets kids build their own games. Emil built a variant of geometry dash.

Homework:

Bab and Bob are twins. One always lies and one always tells the truth. You meet one, but you can't tell them apart. How do you find out if you are talking to Bab or Bob?