Representations of Relations

This is going to be my fifth year teaching Algebra 2, this year I am changing schools so it will be called Advanced Algebra. I was doing some curriculum writing with my new team of teachers and the first section we are going to cover as a class is Representing Relations my first reaction was...



At this point with just coming back to school students may not remember what relations are, what functions are, or domain and range.

I thought back to Dan Meyer's talk of headaches and aspirin and why do students need to know there are different ways to represent relations: ordered pairs, tables, graphs, and mapping.

I took the second graph from New York Times: What's Going on in this Graph?  and re-organized the information differently.

Give the students the following information:
This data is organized from by: country (guns per 100 people , mass shooters per 100 million people).
United States (85, 28)
Canada (26, 9)
Afghanistan (2, 20)
Iraq (37, 4)
France (35, 15)
Yemen (55, 40)

Ask the students what they notice? what do they wonder? As the teacher write down everything they say. One question I have is what is the data saying? Is there a different way to represent the data?

Give the students the following information:



What do they notice and wonder now? What has changed? You can show them mapping as well, but eventually you will need to introduce other things, but the last one is the graph from The New York Times.




How do the three representations differ, do they all tell the same story? Do some tell the story better? I'm not sure if this is the way to start the year out, but nothing is perfect. I want students to feel that there is some context to mathematics other than its day 1 therefore we do lesson 1.

"The Grasshopper and the Ant" teach Exponents

What do you notice, what do you wonder when you see this image?

It looks cold, how do you know which one is the ant? Which one is the grasshopper? Which one looks warmer? How do you think the discussion is going?

Next have the students read the fable that is associated with it:

A Grasshopper gay Sang the summer away,
And found herself poor By the winter's first roar.
Of meat or of bread, Not a morsel she had!
So a begging she went, To her neighbour the ant,
For the loan of some wheat, Which would serve her to eat,
Till the season came round. "I will pay you," she saith,
"On an animal's faith, Double weight in the pound
Ere the harvest be bound." The ant is a friend
(And here she might mend) Little given to lend.
"How spent you the summer?" Quoth she, looking shame
At the borrowing dame. "Night and day to each comer
I sang, if you please." "You sang! I'm at ease;
For 'tis plain at a glance, Now, ma'am, you must dance."

What do you notice? What do you wonder now?

Where math comes in to play is the idea of where the grasshopper says, "I will pay you, she saith,

On an animal's faith, Double weight in the ." When you can't afford something say, the full price of a car, how do you afford it?  Most students will talk about saving money or a loan. If the grasshopper wants to survive, she wants a cut of the ants food savings and next season the grasshopper will give double back. Ask your students is this fair? Is this how a bank works?

There is a way to calculate it mathematically, but right now I want students getting use to the idea of exponents. The equation I would have them use is the compound interest equation.

A=P(1+r/n)^(nt)

Where A is the amount
P is the principle
r is the interest rate
n is the number of times it is compounded per year
t is the time in years.

I would have students think about what each of them means and how the rate effects how much the grasshopper would way in the long run.