Area Model
In Grade 11 today we were doing grade 10 review, and looked at many topics but spent some time on multiplying with an area model.
One lovely thing we can observe when we use an area model is that the diagonals in the box have the same product. It’s always true! This relationship helps us when we come to factor. We can work backwards and use logic to help fill in the box as we factor.
We’re growing beans!
Today I was invited to do a lesson in a colleague’s grade 10 applied math class. We’re going to do a long term activity that starts off with germinating some beans. I used to do this task all the time at KCVI, and packed up the beans when we moved to KSS, but due to windowsill limitations, I’ve not embarked on the task again, until today.
We wondered a bit as a class about how to grow beans in the classroom, and if different beans grow differently. We have the old beans from 2014, but I was nervous that they might not germinate, so I also got some new beans last night.
Each group of 3 students made a team name, and chose either old beans or new beans. We counted the beans, wrapped them in paper towel, dunked them in water and then put them in a baggie on the windowsill. We also estimated what fraction or percent of each type of bean will germinate. We will compare this to our actual results which we’ll explore next week. We’ll plant the beans and water them and measure them as they grow, and graph them and make some predictions. It’ll be great! Stay tuned for updates.
After starting our bean project we did some dot talks and explored the idea of multiplication as an array, or as a “groups of” model.
We next got out the base 10 blocks and worked on representing multiplication.
Here is 2×7 as 2 groups of 7 or a 2×7 array.
Above we did 3×5 as 3 groups of 5, 5 groups of 3, or an array that is 3×5.
Here is 9×10, not quite a square.
Here is 10x11done with a 10×10 and 1 bar of 10.
Here’s 12×13, we can see it as a width of (10+3) and a length of (10+2). We can count this as 100+30+20+6 which is 156. Area models are so important. It was nice to practice with manipulatives.
Welcome to grade 9
We started off the term with skyscraper puzzles.
In random groups of 4 or 5 we worked on solving these puzzles. Each row and column contain a skyscraper of height 1,2,3,4. Also, each edge has a number which indicates how many buildings are visible from that point of view. After a few false starts we were all on task and striving to complete the puzzles.
Some groups even mastered a 5×5 puzzle!
We debriefed the task together talking about how important it is to work as a team to solve these puzzles because each person has their own perspective and can see different information from each position around the puzzle board.
We also talked about perseverance and how to keep working even when frustrated. We talked about strategies to manage our emotions, like taking a break, taking a breath, getting some water, looking from a new direction, doing some pushups, asking a friend or working together with someone who knows more. All of these skills are very transferrable to math problems, where we need to lean in and enjoy the productive struggle.
We are off to a great start in grade 9!
Welcome to Grade 11
Today was the start of semester 2, and we hit the ground running in grade 11. We’re sitting in random groups, and working at the walls in smaller random groups.
Today the goal was to review grade 9 and 10 math. We did some work on fractions, prime factoring, patterning and modelling relations with tables, equations and graphs. We saw the very first function notation to calculate figure 10, or f(10) for each equation. We also practiced adding and subtracting polynomials, and solving equations with and without fractions.
We have such a big class we’re working on walls AND windows! It was an impressive start to the term, shaking off the cobwebs and working on our collaboration. Looking forward to a great semester together.
Pathways P.D.
A colleague joined me to do a math professional development session at Pathways to Education for their staff and volunteers.
We used concept circles to explore base 10 blocks and saw how to use them to practice place value and integers.
We worked on multiplication, like 6×23 and how to represent that as 6 groups of 23 (an additive approach) and also how those 6 groups can be made into a rectangle with length 23 and width 6. The area is the product. We can then visually split up the area model into 6×20=120 and 6×3=18 so 6×23=138.
We looked at algebra tiles and how we can represent multiplication with the area model using tiles, and also how to simplify and add polynomials. Finally we used tiles to solve equations.
I hope we’ll do more sessions over the term to build on these skills.
Sphere Day!
Today we calculated the surface area and the volume of spheres.
We explored the volume with a displacement tank. A tennis ball just happens to fit perfectly in a juice concentrate can, so I made a cylinder that is perfectly suited for the tennis ball by cutting the cylinder down so the height of the can is the same as the height of the ball.
We put the tennis ball into the displacement tank and had to press it down since it floats. We caught the displaced water in the cylinder. It filled it 2/3 of the way.
To prove that it was filled 2/3 of the way, we used the cone and sphere from the solids set.
We saw that half a sphere has the same volume as a cone. We already know that a cone has the same volume as 1/3 of the cylinder, so a whole sphere (2 cones worth) is 2/3 of the cylinder.
We went through the derivation using some intense algebra and substitution. We needed to substitute thar h=2r, and then we could get the formula for volume of a sphere.
Surface area was next on the list. We used oranges to help us, since they can be peeled. The peel is the surface of the sphere. We drew circles with the same diameter as the orange, then we peeled the orange and fit the peels into the circles.
We can fill 4 circles perfectly with the peels. This corresponds to the formula of the sphere’s area.
This was our last new learning, and we’ll be preparing for our exams which take place next week.
Popcorn Pandemonium
Today I was invited to work with a grade 9 class with the goal of exploring systems of equations. I love the 3 act task popcorn pandemonium, so that’s was the game plan to start.
We watched the video and noticed and wondered. This class has come a long way since September with their ability to dive into a task and try to figure things out.
Next we watched the act 2 video
we made use of the extra information in the video to model the popcorn eaten, and to figure out who would eat more after 1 minute (when the timer elapsed)
This class had a few different ways to approach time, time elapsed, or time remaining on the counter, so some saw the graphs come together in a different way, looking at what the values of popcorn would be when the timer was at 0 (the y intercept). This is why some groups had negative slopes and some had positive. We didn’t correct/clarify as they worked, because the problem could be solved by either interpretation.
Here’s the graph that we made
From the graph we looked at who would eat the most popcorn at the 1 minute mark. Next we looked to see if we wanted Don to win, how long would they need to eat for. This brings inequalities into the discussion. For t>100 Don wins. We looked at the point of intersection as the “tie” and then for t<35 Jon wins, and for 35<t<100 Tim wins. We could see lots of points of intersection to explore all the different “ties” possible.
Next we looked at what would happen if Don ate slower, or if Tim had a different head start, and how that would affect the graphs. If 2 ate at the same rate, but one had a head start, they’d never have an intersection because their lines would be parallel etc.
We could explore a lot through this task. It was a nice way to get them ready for EQAO and the thinking questions they’ll have to tackle then.
We didn’t get to watch the 3rd act because we were so deep into debriefing the act 2 math, but here’s the video anyway.
Beading
One of our grade 9 classes is participating in a beading workshop right now (similar to what other classes did this fall). It is so nice to see the creativity that they have, and chat with the students about the math that they see in their patterns.
Many thanks to our Indigenous program team for facilitating these educational experiences for our students.
Ball Roll Race Finale
Today was all about analysis and making predictions, and then we did the final race.
We spent some time at the start of class making graphs by hand and with Desmos in our groups, and then using them to predict what we thought would be the end results.
We looked at some composite graphs from all the groups, and since class I made one with ALL of the data here
We can see a non linear trend when we have lots of data together, but most groups assumed that their 4 points of data could be fit with a line.
Once we had locked in our choices, we went downstairs to the ramp to do our final trial. We lifted the meter stick and the balls rolled down the ramp.
They didn’t roll in straight lines which may impact results slightly. Our order of finishers was: metal bocce, plastic bocce, basketball, ball hockey ball, tennis ball. Nobody in the class guessed the correct order!
We also noticed that the times that we got from our linear predictions were off. We had predicted that it would take more time than it did for the balls to finish the race, so that’s more proof that the graphs are non-linear.
Many thanks for everyone helping to make this a smooth activity! Students worked really hard.
Heat Loss House day 3
The houses were running all night, with their little furnaces keeping them at a temperature of 30 degrees. We came together today to learn the results, and to see where there are leaks by using a thermal imaging camera.
We learned about how insulation and windows affect houses in the real world, and how we might use a thermal imaging camera to find leaks in our own house. We learned about how a water leak by a windowsill could lead to fibreglass insulation compacting and sliding down inside the wall, and causing a big cold spot.
We learned about how any watt saved is good because we are burning fossil fuels to produce much of our energy and that contributes to the greenhouse effect, and climate change. We also saw how this house project has been on the road to Indigenous communities last year and hopefully is helping raise awareness of insulation and helping communities make good choices about home repair options.
There were many houses that were very well insulated and performed well, but one group won the prize. Good job Red Team!
We took apart the houses in the end, so they can be used at another school.
Many thanks to the Heat Loss House Project for sharing their knowledge with us as we went through this 3 day task.
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