513 Jeffery Hall, Queen's University, Kingston ON, K7L 3N6

math9-12@queensu.ca

(613) 533-2434

Neutrinos follow a straight line through a 10×10 grid of atoms each with radius 0.1. Which atoms do they hit, if any? We then work with a 100×100 grid of atoms each with radius 0.01. The students will work with a spreadsheet and will need a precise geometric condition to decide whether a given atom is hit. This activity involves them in a Pythagoras and similar triangle argument.

Teacher manualTwo tall grain elevators in the distance appear to be the same height though it is known that one (B) is twice the height of the other (A). How can I walk towards them in such a way that they always appear to be the same height.

The students will start by guessing, but what’s really needed is a coordinate system and an equation. We want the locus of a point that is always twice as far from B as from A. It’s a perfect task for grade 10 students who are developing facility with triangle trig and coordinate geometry.

The answer is a circle. Is that a surprise?

Teacher manualThe student is given a diagram in which a transformation *T* (from graphs 1 to 4) is “factored” as a composition of “basic” transformations (these are the
“primes” of the system, rotations, dilations and shears). The objective is to calculate the parameters θ, a, b and h. In phase 1 we use geometric reasoning,
simple Euclidean geometry and triangle trig.

Phase 2, comes towards the end of week 1. Here we tell the students how to find the 2×2 matrix of a transformation and that composition of transformation corresponds to multiplication of matrices. That striking “isomorphism” allows them to solve the problem algebraically.

For the remaining examples, we often let each of them choose the method to use and it’s interesting to see their preference. On the whole, the majority seem to prefer the algebra because it’s more algorithmic––even though it involves the solution of four non-linear equations in the four unknowns!

Teacher manualAt time t = 0 a red ball is projected from the origin and travels along the line y = mx at constant speed 2. At the same moment a blue ball is dropped from the point (4, 4) and travels down the line x = 4 at constant speed 1. What value of m will produce a direct collision of the two balls? Of course the fun part is constructing the animation and deciding what you want to happen when the two balls do collide.

Teacher manualA cylindrical bottle contains 900 ml of water. At t=0 a hole is punched in the bottom, and water begins to flow out. It takes exactly 100 seconds for the tank to empty. Draw the graph of the amount of water z in the tank against time t. Explain the shape of the graph. Now do an experiment and collect and plot the data. Basic principles of physics tell us that the graph should be a parabola. Verify that this is the case. [There is a neat way to change variable so that your points should lie in a straight line.]

Teacher manual