This lesson is not a feat of great technology. This is not a three act post. This is however something that I am proud of because it was simple, clear, and really really fun. I am in my third year of teaching, and every year I notice misconceptions that I want to get out of students heads. I also want to introduce them to natural understandings of mathematics that transitions students easily to higher level thinking, and higher level mathematics. I love this lesson because I feel as though it did that.
When (plural noun) understand they (adverb) learn!
My struggle has been to get kids to understand that a variable is a number. They see x and they think that it is a letter, but I want them to know that it is a number. However, I want them to know that it is any number. Depending on the number we put in, we get a certain outcome. Rather than tell them this, I start with a favourite past time of mine, mad libs.
Students in my class have a vague concept of variable but I am really driving home that a variable can be anything, so I begin with variables in English. I prepared this spread sheet and mail merge document so that they can see clearly how variables work.
The paragraph first looks like this:
- Once upon a/an «Noun_1», there were three little pigs. The first pig was very «Adjective_1», and he built a house for himself out of«Pl_Noun». The second little pig was «Adjective_2», and he built a house out of «Pl_Noun_2».
And wonderful grade eights turn it into this beauty:
- Once upon a fire breathing llama, there were three little pigs. The first pig was very colourful, and he built a house for himself out of cats. The second little pig was scary, and he built a house out of apes.
I talk a bit about programming here and say, "This is what computers do, they take information and insert it where it is supposed to be. Here wherever I have 'Noun 1' what information is stored there gets displayed." I then talk about we get a certain outcome based on the variables we put in. I ask about the first variable 'noun 1', "What if I put in this variable the word 'time'? How would that change the outcome of the sentence?"
"It wouldn't be funny!" Most students say. At which point we move on to our next session...
Wow, that _______________________ smells really good!
I tell students here that we are going to use ________________________ as a variable. I give them a chart, and I want them to fill in the chart with variables that produce a factual sentence, a nonsensical sentence, and a funny sentence. Here are a few examples....
- Cannibalistic Llamas
- bucket of degrading cat carcasses
At this point you can't help but have wonder and awe at the buy in factor! It goes with out saying that you have to remind students to be appropriate (the cat carcass one is weird, but it made me laugh!), but kids love thinking of these and sharing them.
It is important that you wear a/an X before you jump into a pool!
At this point I tell students that they have used variables for ever in math. They have all done 3+__ = 5. The __ is a variable, you can put any number in there (although only one number makes that statement true!). So in this example we do the same thing, but instead of _____________ we now say X. We discuss again what is factual what is not. We can even begin to talk about one to one correspondence (is "speedo" factual or funny? Hint: the answer is yes).
But wait, are we not in math class?
The jump then becomes ridiculously easy. I have students do the same thing with numbers. There tables now read even and odd (instead of funny and factual), and they insert values of x that produce even and odd. They do three functions: x+3, 3x, and 2x+1.
Now all you wiley math teachers out there are saying
Becausewith 2x+1 you get some interesting results. Kids start to X out the even section. Here's where the messing with your kids in a totally awesome way part comes in. Ask them, "you can't find a value that produces an even number?" They say "no it is impossible." You say, "Yes it is." I let the class know that there is a number that works, they just have to 'open their mind' (but not in a hippy way). They rack their brains, and finally asking, "can we use decimals?" to which you respond "is it a number?" because remember a variable means any number.
Going even further!
That's right, there's more!
Why stop at odd and even? This I feel is the homerun to this lesson. I gave them first 4x+1>5 and then 2x+11=7. They fill in a true and false table for each of these, and what they notice is fantastic! In the first example students ask, "What about when x=1?" To which I reply, "Is it greater than 5?" "No" "Then is it true?" I had students writing down 1- (-infinity) in the false section. This wasn't 1 minus negative infinity, this was 1 TO negative infinity! In one lesson I had students figuring out inequalities, variables functions, and equations.
At that point all we needed to do was show a little notation, and they were totally on board for inequalities and equations. Awesome! I am going to keep giving them these right up until I formally teach them equations (inequalities aren't even touched on in grade 8, but hey, get them ahead!). Because once they get bored of filling these in, we can say, "How can we speed this process up?"
Trick them into learning... yes...
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This is the worksheet I whipped up. Take it and do WHATEVER you want with it!