Volume Relationships
Today in grade 9, we took a few minutes break from our review to explore the volume relationships between prisms and the pyramids with the same base and height.
These transparent solids are great for modelling the relationship. We can pour water using the pyramid and try to fill up the prism. The class took some guesses and most landed on 2, or maybe 1.5 as the guess for how many times the pyramid will be filled and dumped into the prism to fill it.
Here’s the picture after 1 time, 2 times, and after 3 times. We noticed that for each prism it takes 3 of the pyramid to fill it up. That means the volume of the pyramid is 1/3 of the volume of the prism. We’ll be using this on our next cycle after tomorrow’s test.
Linguine Math
My grade 11s are starting to learn periodic functions today, and we used the activity 8.5.2 Freddy the Frog rides the Mill Wheel, listed here
We started off by drawing 15 degree increments around the wheel, and then we measured the height of each wedge using linguine, which we snapped to length and glued onto the graph paper.
Students were able to make predictions about the repetitive nature of the graph, and felt that they didn’t need to snap and glue more pasta. The top curve of the graph shows Freddy’s height above the water, and the bottom curve of the graph shows Freddy’s height below the surface.
We consolidated with an introduction of new words like period, and amplitude, and axis, crest and trough. We looked at how we can do a sinusoidal regression on desmos that will give us a periodic graph. We also made connections to the quadrants and CAST rule, where we know that from 0-90 degrees, quadrant 1 sine is positive (just like on our graph) same for quadrant 2, from 90-180 degrees sine is positive, but from 180-270 quadrant 3 sine is negative, and also from 270-360 quadrant 4 sine is negative.
We’ll be looking at cosine graphs and sine graphs in more detail tomorrow.
Building rectangular prisms
Today we built rectangular prisms with a volume of 300 cubic centimetres. We used file folders as our building material, and connected them up with tape.
Groups had different approaches to building. Some made nets and folded them up, and others made 6 individual faces to connect with tape.
groups were encouraged to be creative with their dimensions so we’d have lots to discuss at the end.
Groups then calculated the surface area for their prism, and wrote that on the surface. Then we collaborated to put them in order according to surface area.
We noticed that the biggest surface area was the ridiculously long prism, and then the smallest area was the cube. We talked about how to find the side length for a cube using the cube root calculator button.
We talked a bit about how to minimize the cost of packaging, by making objects more cubelike if possible. We also talked about why maybe making everything a cube is not the most realistic, there are ergonomic concerns with a cube shape box of breakfast cereal or a cube shaped juice box.
Next week we will be building square based pyramids with the same volume of 300 cubic centimetres.
Popcorn Picker
Today in grade 9 we did a task that connects what we’ve been working on lately. We have been exploring volume calculations, and how changing a radius or changing a height will affect the volume of a cylinder.
We watched the video of popcorn picker act 1
Students were wondering about how much popcorn would fit in the tubes made by curling a letter sized piece of paper one way or the other.
They got the dimensions 8.5×11 inches, and then set off to calculate the volume of the resulting cylinders. Usually in a class someone will hypothesize that they are the same volume, but nobody in this class thought that. They had a good intuitive sense that one would be larger than the other.
There was a big misconception that rose to the surface. Some groups decided that the radius of the cylinder would be half of the circumference, instead of half of the diameter. other groups thought that the diameter would be half of the circumference. We cleared up the relationship between circumference and diameter, and circumference and radius.
I was impressed that students were trying things, and showing their work. I was very glad to see them using their strategies like multiplying decimals using the area model.
This was one of those days where it’s nice to have an air popper and some popcorn kernels! We made popcorn and filled tubes and then counted the result to verify that our answer was correct.
The taller tube had 222 popcorn pieces and the wider tube had 263 popcorn pieces.
The big takeaway of today is that a change in the radius will bring a bigger change to the volume because the radius is squared in the volume calculation. A change in height will change the volume but by a lesser amount.
Trig Introduction
Today in grade 10 applied, we introduced trigonometry and reviewed our measurement skills.
The sequence of the task was as follows: Everyone needs to draw a right angle triangle with one angle that is 30 degrees.
We quickly noticed that students were struggling with protractors and rulers, and even the vocabulary of right angle.
I took a step back and tried to connect to prior knowledge. I cut out a triangle and labelled the angles then ripped out the angles and recombined them to form a line.
We knew that half of a circle is 180 degrees since a whole circle is 360 degrees. We know that all the angles in the triangle add up to 180 degrees. We know that our protractor also is 180 degrees. We talked about different types of angles, and different triangle words we knew, and then we got back into trying the task.
We needed quite a bit of help to make sure that our angles were accurately drawn, but eventually we got there!
Step 2 was to measure the 3rd angle.
Step 3 was to measure all of the 3 sides, using decimal numbers to be accurate.
Step 4 was to put all of the data from our triangles in the table on the whiteboard.
These triangles we made are all kind of the same, since their angles are all the same. The vocabulary of similar triangles was introduced and can be built on later.
I told the students that I’m magic and I can tell by looking at the table which triangles were perfectly accurate, and which were close, and which were not so close. I’m not sure if they believe me or not, but some wanted to try to learn my trick!
It didn’t take long for someone to notice that the long side is double the short side for the triangles that were deemed to be perfect.
We introduced new vocabulary of adjacent, opposite, and remembered hypotenuse, and then introduced the mnemonic device SOH CAH TOA
For an introductory day we covered a lot of skills, and reviewed several skills from earlier years. Looking forward to getting out the clinometers and doing some measurements next week.
Solving in 3 Dimensions
In Grade 11 we are working on solving trigonometry problems in 3 dimensions. The biggest challenge can be understanding the situation and creating a drawing, or interpreting the drawing that’s given.
We’re working on calculating sides that are shared between triangles so that the information can be used to calculate the value of unknowns. We’re getting better at knowing when to do sine law and cosine law and when we can use primary trig ratios.
Sometimes it helps to break a 3 dimensional picture down into flat triangles that are separate, but we still need a way to relate sides that are the same from one drawing to the next.
Changing Dimensions
Grade 9s are working on calculating area and volume and perimeter, but then also seeing how these will change if dimensions are doubled or tripled or multiplied by other factors.
We’re expanding on what we learned with algebra and toothpicks when we built models of (x)^2 and (3x)^2, when we triple the side length the area is multiplied by 9.
We connect this to our volume calculations as well like converting 1 cubic yard into cubic feet. We know there are 3 feet in a yard, so 1 cubic yard is (3feet)^3 which is 27 cubic feet.
These ideas blend so many of our skills, it’s important to spend a bit of time connecting all the concepts and representations that we know.
OAME 2026
I was so pleased to be selected to present at this year’s provincial math conference in London Ontario this past week. I attended some excellent workshops, met good math friends, and learned a lot of new strategies and problems to bring back to my classes.
We had a workshop based on a task which we played with and modelled (or tried to model) and then the workshop main topic of consolidation was introduced.
We were split into groups and tasked with organizing how we would consolidate the work if we were in a class. We talked about the approaches and communication that we would highlight, and in what order.
A big part of teaching through tasks is having good tasks that take some time and effort to work through, which can lead to a bit of a messy time consolidating, as not all groups will have had a similar journey through the tasks, and not all communication is clear and easy to follow. It is a good skill to work on: watching the groups, keeping an eye on all of the boards and having a good plan about how to debrief at the end in a way that validates the efforts of all, and the processes that were explored, and highlights the skills we are working on developing.
I led a workshop on cup stacking and the algebraic modelling that can be explored through these hands on activities. My slides are in the professional development section of this site.
I went to several problem solving sessions, with a goal of getting problems that I can use with my classes and with our grown up math club. One session was in French, and it was so nice to get resources in French for my immersion classes. It was a nice small group so we all worked together to explore the problems.
Another session I attended was all about the historical ways to calculate pi. It was all a bit over my head, as I’ve never thought about how someone might have done this way back before writing implements existed. One of the neat approaches was using rings of circles and ratios. Someone in the room had just bought some counters so we tried to build a model to help us understand.
I’m glad to have heard so many good keynote addresses, and attended such well organized workshop sessions. I’m looking forward to attending in Ottawa in 2028.
Gas Cost
I’m heading to OAME in London, and my grade 9s helped me determine how much I will spend on gas for the trip. We looked up on a map that it will be 440km from Kingston to London. Gas today costs $1.64/L and then we needed to find how much gas it takes to drive my car for a certain distance, the fuel efficiency. Looking up online we saw that my car has an efficiency of 7.5L/100km.
Groups used different strategies to do calculations, and it’s interesting to see how they approached the task.
Some started by multiplying 7.5×4 which is 30, which makes sense because it’s 7.5L per 100km and there are 4 groups of 100km in 440km (the total distance). Next we need to multiply 7.5 by 0.4 since there’s a total of 4.4 groups of 100 km in 440km. To do this multiplication some groups divided 7.5 by 10 which is 0.75 to find out the value for 1 tenth and then multiplied by 4 to get the value for 4 tenths (which is 3). The final value for 7.5×4.4 is then 33. All of this can be done without a calculator. This tells us how many litres are needed for a one way trip. Since we’re going both ways we need twice that, so 66L. Next we multiply by the price per L.
Other groups started multiplying 1.64 by 7.5 which gives the price for the gas that will take you 100km. Then you multiply by 4.4 to get the price for 440km, then double it for the entire journey.
Some groups wrote it all down like fractions and proportions, and others wrote out their steps one at a time.
I’m glad to see a variety of strategies. Students are using methods that make sense to them, not just memorizing one way to solve a problem. I’m glad to help them develop a more deep connection and understanding of their numeracy skills.
Solving Ratios and Proportions
Grade 9s have been working on fraction skills lately. Today we worked on solving questions with ratios and proportions.
We used a few strategies: for some questions we looked at creating equivalent fractions by multiplying the numerator and denominator by the same values. If we multiply the denominator 5 by 6 to create 30, we need to multiply the numerator x by 6 to create 2. We can then solve for x.
We also found a strategy of making common numerators, or common denominators. We know that if 2 fractions are equal, if we can make the numerators equal, then we know the denominators will be equal. In this example above, we make the numerators both 70. This leaves us with denominators of 5x and 126 which are both equal as well. We can then solve the equation 5x=126.
We also saw how this skill can help us solve proportions in similar triangles. It was a busy day with lots of solving and working together.
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