At a glance
Pairs · two working periods at the end of the unit · due on the modelling day · one data story: table, graph, equation, two predictions
What you are making
Somewhere near you, a quantity is multiplying: water cooling toward room temperature, a ball losing height with every bounce, a tire losing pressure, a rumour finding ears. Choose one you can actually measure, collect the data yourselves, and model it with an equation of the form .
You finish with a one-page data story: the table, the graph, the fitted equation, a doubling time or half-life in real units, and two predictions — one inside your data, one beyond it — each with a sentence on how much you trust it.
Milestones
- Phenomenon chosen and a measurement plan written — what, how often, with what tool
- Data collected; table and graph drawn before any equation is attempted
- The constant ratio hunted down in the table — this is your , as The Exponential Function explains
- Equation fitted and checked against the graph in Using Desmos
- Predictions made; trust boundary stated; data story assembled
How it is assessed
Growth over polish, per How Marks Work. The measurement plan and the trust boundary carry as much weight as the equation, because they are where the thinking shows — see What Makes a Model Good. Your Math Journal entry on where the model fails is part of the evidence, not an afterthought.
Success criteria
| Quality | What it looks like in your data story |
|---|---|
| Real data | You measured it; the messiness is visible and kept |
| A hunted ratio | The base comes from the table, not from a guess |
| A defended fit | Equation and graph demonstrably agree |
| Real-unit answers | Doubling time or half-life lands in minutes, bounces, or days |
| Honest reach | Predictions carry a stated trust boundary |
If your data will not behave
Real data never sits perfectly on a curve — that is a feature. If no constant ratio appears, check whether your quantity approaches a floor (room temperature, zero pressure) and measure the distance to the floor instead. A model that almost fits, with the gap explained, beats a perfect fit to invented numbers.
Curriculum connection
B3.1
collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data Sample problem: Collect data and graph the cooling curve representing the relationship between temperature and time for hot water cooling in a porcelain mug. Predict the shape of the cooling curve when hot water cools in an insulated mug. Test your prediction.
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B3.2
identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve) Sample problem: Using data from Statistics Canada, investigate to determine if there was a period of time over which the increase in Canada’s national debt could be modelled using an exponential function.
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B3.3
solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations Sample problem: The temperature of a cooling liquid over time can be modelled by the exponential function , where is the temperature, in degrees Celsius, and is the elapsed time, in minutes. Graph the function and determine how long it takes for the temperature to reach 28°C.
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