This is what I told them to do, and how I demo'd it for them on the board:
Me: Next work out how much x + y is for your ordered pair, and write the equation to express the result like this, but of course put your own result:
Me: Next work out how much x - y is for your ordered pair, and write the equation that expresses that result:
Me: Ready? Okay, Susie, tell us how much your x + y and your x - y were:
Me: How can we figure out what Susie's ordered pair was?
Class: Guessing? Asking Susie? We can't?
They didn't see this as a system of linear equations, which is weird because that's what we were doing yesterday.....it seems that since they each had a secret ordered pair and equations of their own making, it all looked different than it had yesterday.
So I gave them a hint:
Me: What happens if we add the two equations together?
Class: Oh! We get 2x = 18..... so x = 9?
Me: I don't know......Susie is that right? Was your x = 9?
Class: Whoah!!!!!!!!!! For real?
I swear it was as if I had just levitated. Like I had David Blain-ed right there and then.
Now that the cat was out of the bag, it was a simple matter for them to find Susie's y coordinate. Next I asked Joe to give us his equations, and for everyone to direct their solutions to him. Now Joe got to have that feeling of satisfaction that we (sometimes) get - to say to someone yes, you're right, you got it!
All I said next was okay who wants to go next and my board lit up like a geogebra Christmas tree. Everyone wanted someone to figure out their secret ordered pair.
Then I told them that this is how teachers make up questions for them - starting with the ordered pair, then working out equations using it, but of course the equations aren't usually as simple as x + y = ... or x - y = ....
Take-aways for me:
- Getting them to SEE something clearly is so much harder, but so much better, than getting them to DO something correctly
- The initial steps were simple enough to bring everyone in, unsuspecting of the learning that awaited them
- The system was just easy enough to solve quickly, so good to start with that, but next time I'll have them make up more complex linear equations afterward
- Each person made up their own unique numbers, so they kind of owned them, which made it more important that someone else get the right answer than if I had made it up
- I have to hand it to me, I shut up really well, and let the owner of the solution give the nod each time
- Nice lead-in to the next thing we're doing, which is solving word problems using systems. I'm hoping it'll help them make the connection between the variable and the quantity it represents.
Weird how the best inspirations seem to hit me at the least convenient time, like on my way into class, but it happens often enough that it can't be a coincidence. Desperate need truly is the mother of invention!
I really love this lesson. I did something similar in Pre-Algebra when we were learning how to solve equations with variables on both sides. I wanted the students to see where one solution, no solution, or all numbers solution come from so I showed them how to "build up" an equation by starting with x = 2, x = x or 2 = 3. I knew one students had an aha moment when he asked "Is this how you write problems for us?" The lesson, or class the next day, would have been much improved by having the students create the different kinds of equations and give problems to each other like you did in your class. Thanks for sharing your last minute pedagogical inspiration.ReplyDelete
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Darn - I posted a comment, got an idea for a better one, and tried to edit the original, but could only delete....anyway I originally responded to your comment with this:ReplyDelete
That's a great idea for teaching equations! I guess as long as we both remember these improvements, we'll get even more students aha'ing the next time around. Thanks for your comment! I would love to read more about your lesson - why not post about it at Math to the Seventh Power?
Then I realized that the next time I do this lesson, I will get them to build systems that, as you did, have one solution, many solutions, and no solutions! Awesome idea! Thanks again!
Something I can make use of as I return to the classroom next week.
If you do, let me know how it goes, Michael, I'd love feedback. Good luck at Traf! And thanks for the follow :)ReplyDelete
I am going to try this tomorrow to teach elimination! I'm thinking about adding different layers to it. Having some students volunteer their point to the equations (x+y= and x-y=) and (2x+y= and -x+y=)and possibly another. Let them choose between the different sets of equations to plug their point in and have a discussion about solving systems using elimination.ReplyDelete
Also, I did something similar to introduce standard form. I asked the students to come in and create a point that when you add the x and y you get 5. We then graphed the points and realized that they created a linear line. We then found the equation of the line using the x and y intercepts as points to find the slope and the y-intercept. I then asked them to figure out the equation they used to make their point (x+y=6). We then talked about how this is standard form and how we can solve the equation for "y" to get to slope intercept form and how easy it is to find x and y intercepts by plugging "0" in for either x or y.
Thanks for the post! I cannot wait to do this activity!
Hi Michael! I'm so glad you found it a good idea, and I really like yours too. I can just imagine their shock when all the points started to line up! If I were teaching this course again, which unfortunately I am not, I would definitely be adding layers like you suggested, and also one like the first comment above about working in some "no solution" systems.ReplyDelete
Please let me know how it works out, or if you have a blog, let the world know, we're all in this together!
I love this idea... Going to try it tomorrow! Thanks for sharing.ReplyDelete
Glad to hear - let me know how it went!Delete
I love this! I am actually incorporating into my lesson for my FIRST interview for my FIRST teaching job! I could not have come across this at a better time!ReplyDelete
I'm so glad to hear that Angie! Good luck on your interview, let me know how it goes, and make sure you tell them that you are part of the culture of teachers who share ideas on the web!Delete
Awesome activity. Thanks for sharing.ReplyDelete
I'm a year late in replying, but so glad you liked it!Delete
Good idea, thanks.ReplyDelete