Log 5 X 5

If you have yet to read Sara VanDerWerf's post about her 5 X 5 game go here and read it now. It sounds awesome, doesn't it? I didn't read it until I was already on break, but I look forward to playing it with my students at some point down the road.

At a very early hour this morning, I thought that this game could be modified to help students practice working with logarithms. Instead of having the numbers 1-10 in their grid, they could have logs. They would still place numbers of equal value next to each other, but some would look like log₂8, others like log₃27 and others like 3. They would also still need to find the sum of squares that match. I would have them write the sum of each set of matching expressions as a single logarithm so that they would need to use their log rules.

Clearly, I need to actually work through this, but I wanted to get it down before I forgot. I think you could also do this with trig expressions (sin π/6, cos 5π/3, etc.), but have thought that through even less than the log idea.

If you have feedback, please let me know. Ideas to build on or telling me why this is a bad idea are both welcome :)

MPM2D - Day 70

Today is our last day before the break. This generally means treats, music and movies throughout the school, with very little, if any, work being done. However, 11 of my students opted to rewrite the skills portion of the quadratics test from Monday. I didn't even have 11 students show up to my grade 12 class first period! Their dedication to learning and improving is remarkable. I feel very privileged to be their teacher.

Happy holidays to all!

MPM2D - Days 68 & 69: Test (part II) and Equation of a Circle

Day 68
Yesterday was day 2 of the test. I don't really want to talk about it...

Day 69
We changed gears completely today and looked at the equation of a circle. This is one of those weird topics in our curriculum that doesn't really fit in. I do it as an application of the distance formula, but we only look at circles centered at the origin so I don't even get to make connections to the transformations we have done. Well, it's not part of the curriculum, but I usually have them explore circles that are not centered at the origin.

I started off with a little group activity. They had to plot 12 points and calculate the distance between each point and the origin. This is what it looked like:

They easily saw the pattern (all the answers were 5!) and that the points formed a circle. So then I asked them to define a circle. Here is what they said:

Pretty good, right? We went on to develop the equation of a circle.

Then they filled in a big table giving them different information (equation, radius, intercepts, graph).

Next, we looked at this question:

We hopped onto Desmos to see what this looked like. They could easily see if a point was inside, outside or on the circle, so they talked in their groups about how we could figure this out without the picture.

In our discussions, we talked more about radius that radius squared, so that is how we looked at the example that follows.

And then class ended early so students could go clean up their lockers. They were not sad about not getting homework.

MPM2D - Day 67: Review, Part II

Today, students had the opportunity to work on review questions (quadratics) and ask for help. I returned yesterday's test and went over the questions where they had to solve a quadratic. We also spent a little bit of time talking about how to determine the number of real roots of a quadratic. I like having a day in between test days as it gives students a chance to refocus and ask questions about skills which with they are struggling.

Completely off topic, if you haven't checked out Desmos' latest activity, Marbleslides, go do that now! Here is the link to the quadratic one.

MPM2D - Day 66: Test (part I)

Not much to report today. Test day is always a combination of feelings. Satisfaction that some students have really understood the material, combined with sadness that some really have not. I have to push down that voice that wants to say "But we did this!" as the evidence clearly shows that some just didn't get it. More work tomorrow to prepare for part II of the test on Wednesday.

MPM2D - Day 65: Cycle 3 Review

I collected some great quadratic summaries today and asked four students if I could scan their work and post it on Edmodo. I like students to be able to see each other's work.

They spent the period working on review questions. The cycle 3 test will be over two days. The first day will cover mixture-type linear system questions, shortest distance between a point and a line and some of the skills of quadratics (factoring, completing the square...). The second day will be all about quadratics where they have to choose the right tool to answer the question and interpret answers as necessary.

What's left in the course? Sine and cosine law, comparing y = 2^x and y = x^2 and equations of circles centered at the origin. I think we will look at y = 2^x (and negative exponents) next Thursday and do Penny Circle on Friday and save the rest for January.

MPM2D - Day 64: More Quadratic Problem Solving

We started today with a couple of leftover examples from yesterday.

We talked about the fact that the second answer is not double the first as the diver is speeding up.

They had done problems similar to the next one for homework, but I wanted to make sure that they were all solid on how to approach this type of question.

Then we moved on to making frames, inspired by (stolen from) Fawn Nguyen. Here is her post about it.

Each pair got one picture (I use black & white pictures as they look better photocopied) and four frames (on thicker paper) to work with as I told them their first attempt would likely be unsuccessful, but that they should learn from it.

Eventually they wanted help. They didn't quite beg for it but they did ask for the math that would help them. That was good. I gave them this and let them have a little more time to work it out.

Then we went over the solution together.

Here is one group's finished product!

After that they worked on one more problem.

Time was tight so we looked at the solution in a Desmos kind of way. I explained how to set up the equation - that the variable was the number of decreases in price. They usually find these questions tough.

And their homework for today:

I left it very open-ended as I want them to make something that is meaningful for them.

MPM2D - Day 63: Problem Solving with Quadratics

I struggled with the title of this lesson and blog post as all the "problems" are "math land" questions - we wrap a fake context around what we want our students to show us and ignore real world constraints. However, here is what we did today.

 

We focused a lot on what we were trying to find and which tool would get us there, with stops along the way for those students that were lost and confused. Here is another one:

And a third example:

Here is today's homework.

MPM2D - Day 62: The Quadratic Formula

Several times over the past few weeks we have talked about the fact that we could only find the zeros of a quadratic if we could factor it. Today we found ways of dealing with the case when it does not factor.

As a class they were struggling with this so I opened up Desmos and asked for some values. One student gave me an equation with decimal values for 'b' and 'c' which gave zeros that would not easily have been found algebraically. This, once again, set up the need to find another way to solve quadratics.

We worked through a couple of examples where the equation was in vertex form. I told them that one of the big challenges was knowing when to expand. In these cases, solving would be easier if we did not expand.

This proved to be a great opportunity to review the effect of the 'a' value when graphing. We went back to our pattern of "from the vertex, go right/left 1, up 1; right/left  2, up 4; right/left 3, up 9" to graph each of these parabolas. The algebraic solution, although new, seemed to make sense.

Okay - so now given an equation in standard form, they could complete the square to get it in vertex form, then solve as we did above. I told them we could generalize the process and then I did. The curriculum says that students should be able to follow the development of the quadratic formula, not replicate it, so it was all pencils/pens down as we worked through a case with numbers alongside the general equation.

As is often the case, students were generally unimpressed with this "ugly" formula. I asked if they would rather complete the square then solve each time they could not factor or simply substitute values into the formula. Some were sold.

Then we practiced with three particular questions.

It took a little questioning to get them all to see that what was under the square root was the determining factor in the number of solutions. I had a Desmos file ready to go, but felt that they understood how the discriminant showed the number of roots so I skipped it. We did a couple of examples to be sure.

Then we started on a more interesting question. The actual calculations are not difficult but choosing what tool to use is. 

Here is yesterday's homework and here is today's.

MPM2D - Day 61: Completing the Square, Day 2

As many of my students were away on a field trip on Friday I told them to each find someone who was in class, and have them explain completing the square. They spent about 15 minutes on this. I thought it would be a good way to bring those who had missed class up to speed, but would also be of benefit to those who were there as they had to explain the concept well enough for their classmates to understand.

What we worked on today was really more of the same.

I illustrated the point that we could not make a square with two x^2 tiles. I asked if we could if we have four... they thought a bit and many said yes. What about three? No. Nine? Yes.

I showed them that we divide up the x^2 tiles and create a square for each one, dividing the x tiles evenly among them.

We translated this into a chart method and repeated the process algebraically, too.

We continued with more examples, relying less on the tiles each time yet always tying the process back to them - "Why are we dividing by 2? Why are we squaring?".

When we looked at the next example, using tiles or the chart became less meaningful as it's hard to think of having -3 of each square. I think they had enough experience and a solid enough understanding of why we were doing what we were doing to move to the algebraic form.

I will post today's homework tomorrow as DropBox is not cooperating right now.

MPM2D - Day 60: Completing the Square, Day 1

We started today by looking for patterns in perfect square trinomials.

They noticed the pattern which we consolidated:

Then we looked at it another way. In groups of four, they each answered one of these questions and then wrote the sum of the four answers in the middle box. This allows me to quickly see if they are correct and, if they are not, they have to work together to figure out which question(s) are wrong.

We talked about how we can find the vertex of a parabola. Factor the quadratic, take the average of the zeros, then substitute that value back in the equation. But what if you can't factor the quadratic? One student said he could always find the zeros... "Desmos!", he said :) Then the algebra tiles came out and we starting completing the square.

The idea of making a square is not difficult when they work with the tiles. We kept the 7 unit tiles off to the side and then added one positive unit tile to fill in the square. This meant that we also needed to add one negative unit tile to ensure that we weren't changing the value of expression. Writing the equation in vertex form was quite straightforward, as was stating the vertex.

I gave them the steps - this may be useful for those students who were away today.

Then we practiced some more.

It was time to start to move toward an algebraic solution so we started by noticing what is happening with the numbers, and then repeated one of the previous examples without tiles.

We did a few more examples.

Along the way we talked about why we needed to move away from the tiles. What if the number of x-tiles was not even? But we did a simple case together with tiles - not the actual tiles though, as I do not want them split in half! 

Today's homework was to go back over any old homework that they had either not completed or done incorrectly. Next class we will look at cases where the a-value is not 1.