Throughout the year, I have felt uneasy teaching students to work with problems involving parentheses. If I gave students something like:
Solve: 3(x + 4) – 8 = 16
A large number of students would subtract 4 at the incorrect time in the problem. Over the last two years of teaching I haven’t come across a way to address this problem… Until today.
We were working with solving quadratics by square roots. After a bit of scaffolding, I posed students with this problem.
Quite a few of them wanted to subtract 13 first. I put on the breaks and asked them “If we knew what x was, say 5, how would you simplify the expression?”
We wrote a list of the order to simplify things:
- Add 13 (Parentheses)
- Square that number (Exponents)
- Divide by 2
We then talked about how solving equations starts at the bottom of this list and uses inverse operations to undo whatever is happening. So to solve for x:
- Multiply by 2
- Square root
- Subtract 13
They seemed to catch on after this point. It seems like a simple strategy, but I had never really thought it this way. I always talked about solving equations as a process of working your way from the outside in.
Today was rough.
Algebra is just getting into solving quadratic equations. We looked first looked at solving linear equations like 3x + 4 = 10 and talked about how these were solved and what was happening to x in equation.
After a while I threw up x² + 4x = -3.
Students tried to solve it by moving the 4x over and were frustrated that they couldn’t get the equation down to just one x. They whole group divided and conquered on different values of x, and eventually they reached two solutions.This took a lot of time, time students don’t have, they asked for a better way.
Then I messed everything up.
No matter how I modified the lesson between periods, students struggle with the process of setting the equation equal to 0, graphing the related function, heading back to the equation to remember they were looking for x-values that produced y-values of 0, then using the graph to help them.
I had blank stares all day, something is wrong on my end. Maybe I haven’t emphasized WHY graphs are useful, but, we talked a lot about how graphs generate y-values based on chosen x-values. This is almost the same thing.
I also probably shouldn’t have made them suffer by graphing each equation by hand…
Let me know below how you navigate into solving equations by graphing.
To get students into the swing of comparing 2-4 quadratic functions I had them pick up an equation and graph on their way in today. The equations were of the form y = ax² + c, and either had different a values or c.
They put ’em up by colors and had a quick galley walk, discussing the similarities and differences.
Desmos and the ability to add sliders, made everyone’s lives a whole lot easier when it came down to comparing functions like:
f(x) = x²
g(x) = -¾x² – 12.
Students seemed to pick up on transformations pretty quick and did well in describing the differences between functions.
Algebra is getting into quadratics. They know a bit about vertex and how to determine what direction a parabola will open. To keep momentum after introducing quadratics, I pulled a great acitivty from Dan Meyer’s Blog again.
Last year I spent an entire day on guessing and tracing where the ball would end it. It was a bit of overkill. This year I spent about 20 minutes on it, which was a good amount.
Students came up and traced the path they thought the ball would follow.
I threw the pictures into desmos and fitted a curve to them.
We watched the end of the video.
After a couple, students picked up that the ball followed the same path on the way up as it did on the way down. They are pretty comfortable with quadratics now and finding the max,min,vertex, and axis of symmetry.
“Over two out two from the origin both ways, then go beneath the x-axis, over 5 down 5, then back above the x-axis above 10 and 10 over”
Any ideas what the graph this student was trying to describe looks like?
Algebra is starting quadratics and with that comes a mountain of vocab; upwards, downwards, vertex, maximum, minimum… ect. Instead of having students read/take notes or listen to a lecture on the important of all the vocab I have them do the following:
- Break into pairs
- Each pair needs a single whiteboard, marker, and eraser
- Move desks around so students are facing towards each other; one needs to be looking at the smartboard while the other has their back to it.
- Throw a graph up on the board (Thanks Desmos)
- Without drawing in the air, pointing, listing off ordered pairs or anything like that the student facing the smartboard describes the graph as best they can to their partner (who can’t see it).
- Students struggle.
- Students get it.
- Partners switch positions and repeat.
Some of the best conversations about math happened today. Listen to a student describe to another what an exponential curve looks like for the first time ever was priceless. Students developed strategies and realized which points were critical in their partners success. They also saw the need for some mathematical framework, which was laid in place only after they encountered some tough ones.
Here are a few other graphs I threw up
Can you guess which one the student was trying to describe?