Cheerio Stacking Race
Today in grade 9 we were working on graphing linear equations, remembering y=mx+b, and next looking for intersection points and interpreting their meaning.
We started with x+y=5 and talked about how sometimes we just know some points that are on the line because they work. E.g. (2,3) works because 2+3=5. We made a list of points and graphed them. We talked about the special points which are on the axes, and named them.
We next looked at an equation that was more complicated 2x-y=4
We found the x and y intercepts and plotted them to make a line. We noticed that the graph had a rise of 4 and a run of 2, and started with a y intercept of -4, so we could make an equation. We also determined the y=mx+b form of the line by isolating y. We realized quickly that the slope of 4/2 is the same as the slope of 2/1 or just 2.
Next we looked at the graph of both the lines, and noticed that there was a point where they crossed. That is the only point that is on both lines.
To give ourselves a context for what intersection points are, we participated in a cheerio stacking race.
Students each determined their cheerio stacking speed, in cheerios per minute. We used non dominant hands just to make things interesting. Once they knew their cheerio per minute speed, we tried to set up a photo finish between the slowest in the group and the fastest in the group with both having a full skewer of cheerios. To do that, the slowest person needed a head start of a certain number of cheerios.
We used graphs to help us figure out the head starts. We knew that we wanted both people to have a full skewer at the same time as the fastest stacker. We used the fastest stacker’s data to draw the red line. We know a full skewer is 54 cheerios, so we want that to be our intersection points. We know the slow stacker’s speed, so we can use their slope to count backwards from that intersection point, and we can extend the line to the axis to determine the head start needed.
Some groups had more luck than others when testing their head start numbers. Some people had gotten better with practice, or they’d calmed down so hands weren’t as shaky. Some had maybe been cheating a bit the first time, so they had misrepresented their speed. Most of them ended up with a pretty close photo finish even if it wasn’t exact.
I was particularly proud of how they were willing to try something new, work together, figure out what information they needed, and how to figure out a head start. Also at the end of the class materials were cleaned up and put away without issue which is not always the case when it comes to cheerios! Thanks grade 9s 🙂
Desmos and Inequalities
Grade 9s are using desmos to explore inequalities and equations on the cartesian plane. We started by making a list of all the possible numbers we know that add up to something less than or equal to 10.
After we compiled the list, we noticed that there was a region that had dots, and a region that didn’t. We then found out that you could type inequality symbols in desmos and it will show the region coloured in, bound by a dotted or solid line depending on if there’s an “or equal to” in the symbol.
We got to review skills of isolating y, remembering what slope and rise and run are, what’s the y intercept. We also looked at how to calculate the x and y intercepts because sometimes that’s a faster approach. We substituted 0 in for x, and then for y and calculated the intercept values.
We’ll keep working on more graphing tomorrow and solving systems graphically. There’s so much to review before the exam and EQAO. I’m very glad we spiralled through material so it all is kind of building on past skills and review.
Arithmetic Series
Today grade 11s were working on making rules for adding up arithmetic sequences. We started off with adding all the counting numbers from 1 to 1000. The challenge was interesting for me to watch. Some groups started adding, looking for patterns in the list. Some got overwhelmed by the 1000 and said it’ll be big, and stopped there. I redirected the class to try adding the numbers from 1 to 10 as a more simple task and then to see id they could find a strategy that might help them.
This group got right into it. They grouped numbers to make tens, they used the 1+9, then the 2+8 etc. The 5 is in the middle and dealt with at the end. The 10 was already a 10 so they didn’t change it. They knew that adding 1-10 made 55, so they then looked at adding 11 to 20 and they knew that it’d be 55 (because you are adding 1×10 again) but you’re also adding 10 more 10s, so they generalized that you’d add another 100 on top of the 55 for each group of 10 that we go up. I had not anticipated this approach at all!
Some groups added first and last, then second and second last and made groups of 11, then realized that there were 5 groups needed.
We tried some more challenges like adding 5+6+…+15, or adding 6+7+8+…+16. This was different because the “a” value changed, but not much else.
We started to see patterns like adding the first and last terms is important, then you need (n/2) groups of that sum.
We tried this with different sequences, with different common differences next, to show that our conjectures were still true.
By the end of class we were comfortable with arithmetic series, and had developed a few working formulae.
Fibonacci Numbers in Plants
In grade 11 we watched these videos and then explored the Fibonacci sequence. We drew spirals, and then explored how Fibonacci numbers are found in unusual places. It’s even hiding in Pascal’s/Kayaam’s/Yang Hui’s triangle.
Next we explored some other benefits of this triangle, and how it can help us with binomial expansion.
They don’t know it yet, but this is a foundational skill for math next year!
Distance Time Graphs
Grade 9s are working on various relationships, and today we looked at slopes on a distance time graph and how that relates to speed. There’s a fantastic game which I enjoy watching students play on the big screen each year.
They click and drag on a stick figure and see what distance/time graph gets made. The goal is to match the given graph.
I asked students to be brave volunteers to either do a good job, or a really bad job at matching the graph. Sometimes allowing it to be done badly and have it be no big deal will encourage more people to come try. There are 12 different scenarios, and they can be replayed if students want to do some head to head competition.
Before each attempt the class puts into words what the stick figure should do. They make a plan and end up saying things like, “one space per second” or “2 spaces per second” or “in 3 seconds it needs to end up at 3 spaces”. Some connected the positive and negative and zero slopes to the direction of movement. It’s a fun way to explore a concept before doing some more individual practice.
Exponent challenge question
At OAME I learned several new problems which I have been sharing with my classes, colleagues, and friends. One that’s been particularly fun with grade 9s has been an exponent question.
The question is: how many digits are in the answer to this question.
We quickly notice that calculators can’t handle this problem (phones can, so I’ll need to change it for the future). It’s fun to tackle problems that are too big for calculators, but our brains can figure them out.
This question illustrated so many misconceptions my students have about exponents. It was good to see them exposed so we could tackle them together and build more number sense.
The first misconception was that the base and the exponent would both be changed when raised to a new power. This makes me think we’ve done too much practice with algebraic bases where we’ve ignored the base because it’s a letter. They knew that the power of a power multiplies the exponents together, but they missed the idea that it’s that many 2s or 5s being multiplied together.
The next misconception is that when multiplying, we add the exponents and multiply the different bases. They are used to adding exponents together when the bases are the same, and maybe thought that you do that all the time. They are working on remembering rules, and not understanding the concepts.
We brought it down to a smaller scale to see if that would help. We tried 2 squared x 5 squared etc,
Students were making connections more quickly now, seeing that all the answers were powers of 10. We’d just done scientific notation so this was a nice connection to see.
Some spent a lot of time exploring various powers of 2 and 5 and seeing how they grew. We practiced how to calculate this (where the exponent buttons are) and how to interpret scientific notation on our calculators.
While consolidating the lesson we had time to talk about many aspects of the math that we’ve learned. We reviewed what exponents mean, and what the power of a power does, and why the base stays the same.
We reviewed 2 different ways to multiply 16×625, using area model or by doubling 4 times.
We looked at powers of 10 as well and made the connection to the scientific notation that we’ve practiced.
We looked at how a power of a product is the product of the power as well.
All in all we have enjoyed working with this problem in several classes now. It helps to show what skills are solid, and where we need some more practice.
Fractions
I’ve been working with some grade 8 students, supporting their understanding of fractions. We have been using various representations and manipulatives to help. Today I brought Cuisinaire rods, and we used concept circles.
We used 12 as our “1” and represented a variety of fractions based on that. I wrote the fractions and the students needed to place the blocks.
The second time we cleared the values in each segment, put different blocks in each section, and then the students needed to write down the fraction.
Finally we tried it again using the 10 rod as our “1” which pushed their thinking. By changing what the “whole” is, each block takes on a different meaning. When the whole is 12, each white block is 1/12 since it takes 12 white blocks ro build the whole. When the whole is 10 each white block is 1/10, since it takes 10 white blocks to build the whole.
Students are starting to engage more, and respond well to the manipulatives. We are working on conceptual understanding and filling in some gaps in both skills and confidence. It’s been a fun challenge!
How Tall is the Novelis Sign?
We have a very tall sign visible from the park beside our school. I’ve always wondered how tall it is, so decided we should try as a class to figure it out. We can’t actually walk to the sign, so we are trying to calculate the height using 2 different angles of elevation separated by a known distance.
We have clinometers that we are learning to use, and we tried our best to get good measurements, but it was surprisingly windy on the field, so even with classmates becoming windbreaks it was hard to get accurate measurements. We calculated the height to be anywhere from 6m (which is very much too small) to 40m which seems too large.
It was a lovely day to go outside, and get our hands on the measuring tools and build some sense around how we might apply our trigonometry skills to solve real problems. Next time I’ll check for wind in the forecast as well as for rain!
Cookie Towers
In grade 9 we built cookie towers to generate some single variable data to use for our statistics and box and whisker plots.
Each group made 12 towers and kept track of how many cookies were in each. From there, they calculated the mean, median, mode and range of their data.
Then each group made box and whisker plots for their data. We learned about how each quartile will hold the same number of data points, and the graph will give us lots of information about how the data is spread out. Students used the idea of density to talk about which quartiles had data that was more dense and less dense.
I apparently forgot to capture pictures of their box and whisker plots! It’s been a busy time.
this activity was inspired by “math equals love” blog years ago, but I can’t seem to find the post.
What’s been happening in Grade 11
Lately grade 11s have been working on understanding periodic and sinusoidal functions. After our linguine math day we started off by making tables of values for both f(x)=sinx and f(x)=cosx.
We made connections between the CAST rule and the quadrants we’ve become familiar with, and these graphs. We learned a lot of vocabulary like amplitude, period, cycle, and axis.
Next we started to do some modelling, thinking about ferris wheels, or pebbles stuck in tires, and the height over time, or height over distance in each of those situations. Next we got into some transformations, a good chance to review mapping statements once more.
Today we explored some fun problems like where does sinx=cosx? We saw tables of values, graphs, quadrants and the cast rule and special triangles, and also algebra where we divided both sides by cosx and found ourself solving for where tanx=1. (I should have taken pictures, it was really neat work).
further to that, we looked at graphing tanx, a bit of an extension, but touches on periodic functions and reviews rational functions and identities which is always good to revisit. Again, there were so many interesting approaches, and strategies explored to end up with the graph for f(x)=tanx.
We’re going to work backwards next, starting with graphs and making equations. New challenges to explore tomorrow!
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