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### Math9-12 Projects

For many years now I have been developing “enrichment activities” for high school working with “motivated” students in a mixture of grades. The focus has been to develop those mathematical thinking skills that I find are weak in most of my first-year students at Queen's. My recent work, and the objective of Math9-12, is to construct some new problems and refocus the best of the old problems to fit particular grade levels in the Ontario curriculum structure.

Thus I have taken the Ontario curriculum documents and slotted each problem in at the right technical level. For example, my Transformations project that works with coordinate geometry and triangle trig was pegged at Grade 10, and my Recursive Thinking project which works with the Fibonacci sequence was slotted into Grade 11. And, along with my graduate students, I took these into various classrooms.

The experience was enlightening. For the most part, the students found the experience interesting but I feel that only about half of them were really “ready” for it. I started to get the feeling, and this was shared by some of the teachers, that maybe some of the problems belonged at the next grade level, that the students are conceptually ready for the problems a year after they have encountered the technical material. I’m still unsure about that––much depends on the problem-solving level of the students.

In any event many of the problems I describe below are tagged at two grade levels, one that fits the curriculum specification at the technical level, and another that might better fit student readiness––this is an interesting version of “cycling.”

Grade 9: Missteaks. Grade 9 students are supposed to practice the manipulation of algebraic expression, solving equations, operating with exponents. Here’s a fun way to get such practice and more. And opportunities arise for friendly competition among groups.

Grade 9: Lines and curves. One of the Specific Expectations in Grade 9 is to “Investigate the relationship between the equation of a relation and the shape of its graph,” but the examples given in the document involve straight lines. I feel that for a proper grasp of rate of change, the students need to access the wider world of curves and indeed I find that they seem to have no trouble interpreting the slope of a tangent to a curve as a rate of change. Here we look at a collection of optimization problems involving the interaction of lines and curves.

Grade 9-10: Neutrinos. Follow a straight line through a lattice network of atoms. Do they hit a nucleus or do they pass right through? In working with the geometry of lines and circles, Pythagoras and similar triangles arise. This is an excellent problem for good Grade 9 students, but it should also work well in Grade 10.

Grade 10: Grain Elevators. Two tall grain elevators in the distance appear to be the same height though it is known that one is twice the height of the other. 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. It’s a perfect task for grade 10 students who are developing facility with triangle trig and coordinate geometry.

Grade 10-11: Transformations. This is our most ambitious project. We designed it as a 3-week project––indeed it spans a block of 2/3 of the curriculum, Analytic Geometry and Trig––but our main grade 10 trial was only two weeks long. The students were certainly interested but most of them found it quite demanding and said they needed more time. The teacher said he’d like to see how it would work in Grade 11.

Grade 11: Piece of wire. The idea here is that the students are given a target graph and a family of related graphs, and they have to find a member of the family that can be picked up (as if it were a piece of wire) moved and flipped (but not bent) and laid exactly over the target graph. This type of algebraic transformation is part of the grade 11 curriculum but done in a very technical way. This activity tries to provide a more constructive, hands-on version of things.

Grade 11: Parabolas and lines. These investigations offer a number of problems intersecting parabolas with translating or rotating lines. The purpose it to build the students’ sense of the connection between the equations and the graph.

Grade 11: Musical magic of 12. This is an extraordinary tale and well worth some play-time. The students will encounter many things they already “half know,” and they will be enchanted at the end. The technical math here is at exactly the right level (working with the exponential function) but of much greater value to the student will be the connection of the math with a world that they are all involved with, principally music, but also, for example, the nature of perception.

Grade 11: Water tank. You should bring a large plastic soda bottle to class for this one. Fill it with coloured water and punch a hole in the bottom. Use a scale marked on the side of the bottle to track the water level z against time t. If the bottle is cylindrical, basic physics tell us that the z-t graph should be a parabola. We test for that and get an interesting surprise at the end.

Grade 11: Zero and asymptotes. We begin with the aim of giving students a qualitative experience with the graphs of polynomial and rational functions focusing on roots and asymptotes and the sign of y. We then move on to the study of a couple of parameterized families of rational functions and make use of the animation capacities of Desmos to help them understand the dynamics of a parameterized graph. Most of the examples here trespass into Grade 12 territory but it is my belief that there are too many new things in Grade 12 and some of the ideas, at least in a preliminary form, should be dropped into Grade 11. More generally students need a wider and more sophisticated range of “works of art” earlier in their studies.

Grade 11-12: Trains. This is a conceptual adventure with some of the remarkable properties of the Fibonacci numbers. We develop a surprising method for constructing proofs for some of these properties, for example that the sum of the squares of two consecutive Fibonacci numbers is always a Fibonacci number. So far I have only tried this out with a grade 12 class and the experience made me feel that although the topic "sequences" is on the grade 11 curriculum, these problems might well work better for grade 12 students with their more advanced thinking skills.

Grade 11-12: 3D graphs. Many of my first-year students have trouble when we come to the calculus of two-variable functions and often the problem is found in visualizing and working with surfaces in 3 dimensions. Indeed, as has been pointed out so many times, it is absurd that high school students do so little mathematics in 3-dimensions. These activities give students an introduction to some simple 2-variable function graphs and their contour diagrams.

Grade 11-12: The parent game. This is a fascinating 2-person asymmetric game. Both players make an investment and each gains a benefit determined by the sum of the two investments and incurs a cost determined by their own investments level. In this version, player A goes first, so that player B knows A’s investment before he had to choose his. What should each player do? We work with both an algebraic and a graphical approach.

Grade 11-12: Jacqueline and the beanstalk. A discrete growth process involving both multiplicative change (beanstalk growth) and additive change (Jacqueline climbing). An unexpected twist reveals that the problem (will she reach the top) is actually equivalent to an annuity problem. This problem was constructed for Grade 11 but my suspicion is that most students won’t be quite ready for it until Grade 12.

Grade 12: Rolling wheel. A wheel of radius 1 rotates without slipping around a fixed circle of radius 2. The task is to use Desmos to construct an animation. This draws on the grade 12 circular trig and involves some preliminary vector ideas. This task is simple enough but requires careful organized thinking. A wonderful feature of this is that you can tell right away whether your calculations are correct––just run the animation!

Grade 12: Inverse quartic. An important notion in Grade 12 is that of the inverse of a function, but the examples that the students are given to work with are often not so interesting. Understanding come from examples that have a rich structure, and the graph
that we work with here seems to have the right level of complexity. Its inverse is not a function graph but it can be broken into four functional pieces that the students must identify both graphically and algebraically.

Grade 12: Optimal driving speed. How fast should we drive on the highway if we simply want to minimize our fuel costs for a fixed distance? Working with a fuel consumption graph (litres/hour vs km/hour) we use a graphical approach to find the optimal speed. We then put a dollar value on our driving time see what difference that makes. Finally we build a simple fuel consumption model for a car and use calculus to solve both of the problems.

Grade 12: Tire pressure.You have a hole in your tire. You pump it up to P=400 kilopascals (kPa) and over the next hour minutes it drops to P = 100. What function describes the P-t graph? Using simple dynamics we build a model to show that the curve is expected to be exponential and then we find a real tire and collect some data. We use a log plot to see if the data are linear.

Grade 12: Exponential dice. The students do many experiments each one starting with 50 dice and rolling repeatedly each time removing all the sixes. There is lots of variation but the average result might well provide an example of exponential decay. We use a log plot to check this out.

Grade 12: Coffee and milk. You buy a large coffee (300 ml) at 80 degrees and four milks (15 ml each) at 4 degrees, and get into your car which stays at a constant temperature of 16 degrees. It will be a 10 minute drive before you can drink the coffee, and you want it to be as hot as possible at that time. Do you put the milk into the coffee mug at the beginning or after the 10 minutes when you’re ready to drink it? Do you mix first and then drive or drive first and then mix? Does your intuition give you an answer? This is a nice example of exponential decay and the principles around heat-loss that we will encounter are ones we meet in our day-to-day lives.

Grade 12: Forage and fly.A bird forages among patches leaving its current patch when it has been well-enough picked over to search for a fresh one. One question is how long the bird should remain in the patch, when the continued foraging shows diminishing returns, and a second related question relates to the speed at which it should travel in the search for a new patch, as higher speeds mean shorter search time, but higher energy costs. This enticing problem can be solved both graphically and algebraically.