How Tall is the Atrium?
Welcome to our atrium.
Today the challenge for my colleague’s grade 12 college math class is to determine how tall the ceiling is in the blue section of the atrium.
We started off estimating and justifying our estimates. Guesses ranged from 20-50 feet. Ideas were justified by imagining a 7 foot basement stacked up on itself, or by estimating the height of the classroom using the door frame height or a person’s height, then doubling it since the atrium covers 2 storeys of the school. Interestingly enough the students preferred to estimate in feet, but defaulted to measuring in metric to do our data collection.
To collect data we stood at the back of the atrium, just at where the overhang ended. We marked our position, then measured with the clinometer. We walked forward by a measured distance, then repeated the process. These 3 measurements, along with the height of the clinometer operator’s eyes were enough to allow the calculation of the atrium height.
The calculations include some supplementary angle calculations, and sine law and sine. We got part way through the calculations and there are some pinky swears given that the calculations will be finished for tomorrow. We’ll see. I’m quite curious to see what the height is, and how that compares to the school height that my grade 10s calculated.
Popcorn!
This question was inspired by the popcorn picker 3 act task. I showed act 1 yesterday and asked the class to think about which holds the most popcorn, an 8.5”x11” paper made into a tube in portrait orientation (we called this hotdog) or in landscape orientation (we called this hamburger).
The students asked if we’d have popcorn the next day…so I agreed.
Today my popcorn popper was in full gear as the students worked through the task.
Some decided to work in cm and others in inches. It was great to see students flock to the formula sheets and use their skills with algebra and equation solving to isolate and solve for different variables. They knew how to calculate the radius when given the circumference.
They knew to calculate the volume, and did a great job communicating their work and checking it over.
To begin with we had people in 3 distinct camps: the tubes will hold the same amount, the tall one will hold more, the short one will hold more.
In the end, we all agreed that the short tube would hold more based on the math. It was time to test it out.
We added popcorn to the tubes. Then dumped them out and counted. There were 50 more pieces in the shorter tube.
The reason this works is that the radius is really important. The radius is squared. Squaring a bigger number makes it grow a lot, so this has a big impact on the volume. Height in the volume calculation is not squared, so an increase in height does not cause as big of an increase in volume.
Building Pyramids A Lesson in Productive Struggle
Today we tackled a challenge. This challenge had some constraints, but was open enough to allow each group to do something different if they chose to. The task is to build a square based pyramid with a volume of 300 cubic centimetres. This followed up on us building rectangular prisms with a volume of 300 cubic centimetres.
It took some reminders with water and our solid forms to get that a pyramid with a volume of 300 cubic centimetres has the same length width and height as a prism that has a volume of 900 cubic centimetres.
Many groups chose dimensions of 10 by 10 and a height of 9.
Making the square base was the easy part. There was some trial and error when it came time for the triangles (and this is where the productive struggles began). Some groups made triangles with sides of 10, 9 and 9. That made a pyramid that was too short. Some made triangles with a base of 10 and a height of 9, which when inclined to form a pyramid left the height too short.
Some groups decided the triangles needed to be taller than 10 so that when inclined the pyramid height would be 9. They guessed and checked and ended up pretty close to accurate. Other groups used the pythagorean theorem to calculate the height that they needed for the triangles.
Another group decided to just make their own pyramid and to not care about the constraints! I guess we all choose an entry point to the task.
Next time I might give different entry points, and provide different extensions.
mild version: Make a pyramid, calculate the volume and surface area.
medium version: make a pyramid with a specific volume, calculate the surface area if there’s time (there will be iterations needed to get the volume correct).
spicy version (for those who need more extensions): make a pyramid with a specific volume, calculate the surface area and the angle of elevation of the pyramid walls.
How Tall Is The School?
We had an interesting time estimating how tall the school is. Guesses ranged from 100m to 20 times the height of a student (which would be 120ft), to 40-60 ft, and we decided 15 ft would be too small. We were working on narrowing in on what would be reasonable before we went outside to do the measurements and calculations.
We had groups of 3: a measurer, a recorder, and a clinometer operator. Each person had responsibility for bringing a piece of equipment downstairs and back up. Clinometer, meter stick, paper and pencil.
Groups looked through the tube on the clinometer to see the roof of the school. We measured where the string fell which tells us the number of degrees in the angle of elevation.
We measured to the height of our clinometer operator’s eyes, and we measured the distance to the school.
From there we had enough to calculate the height of the school. We went back inside and worked away at our trigonometry. With practice we’re getting better. Hopefully with this context we can make some connections to what the calculations mean and when they could be useful.
Math Workshop
A colleague and I were invited to present a workshop after school all about fostering confident math learners. The audience was a group of elementary educators from our feeder schools.
We introduced our “graph our math life” task which we often do with our students. We thought back to how we felt as math learners and teachers over time. There is a big connection between feeling good about ourselves as math learners and how we will be able to progress through the productive struggles along the way. This task is based off of the work of Liesl McConchie.
We can use this as a springboard for conversations about what the stories we hear about math at home. Some of us grew up with positive experiences and messaging around math and challenge, and others have hit roadblocks where we didn’t have such positive feelings or messages. We know that students are hearing lots of messages at home or in pop culture. We want to be able to refute some of the negative messages they hear, and help them replace some negative self talk with more constructive messaging. Instead of I can’t do math, or math is hard, or I’m not a math person, they can say “math is tough but so am I” or “i can do hard things” or “i’m working on my math skills” or “i’m learning and asking for help”.
Our next task was to make some art with the patterning blocks. We looked at all the math we could see: rotational symmetry, lines of symmetry, fractions and percent of blocks that are of a certain colour, parallel lines, angles that are complementary, angles inside a polygon, perimeter, area…so many ideas. We also used the first letter of each colour and made an algebraic expression for each picture e.g. 1y+6r+1g+1b=flower
Next we did a thinking classroom style problem at the walls. The prompts were to break up 25 into pieces. All the pieces will add to 25. Next change the addition to multiplication and look for the biggest product.
It was neat to see how teachers of various grades tackled the question in different ways. Different groups imposed different frameworks and constraints. We haven’t yet got to the largest product, but some are still working away at it.
It was a great opportunity to share some of what we are doing at KSS with educators in our feeder schools. We are glad of the invitation and hope to go again in the spring and share some more tasks.
Similarity and Proportions
We looked at similar triangles today in grade 9 as a way to practice ratios and scale factors. We did a few examples, and made a pseudocode to calculate the proportionality constant between the similar triangles, and then students had some questions to solve in groups up at the walls.
We’re trying to work in some pseudocode as we go this year so it’s not a shock when we look at EQAO preparation.
It has been helpful because students can look at the algorithm in code and then use it to help them with the process.
We’re getting better at expressing our steps and solving equations with proportions.
Fractions
For our fraction warm up today we counted by 1/4 around our table groups. To make it more interesting or a bigger challenge we added clapping on the whole numbers and then stomping on the half numbers. We then tried it as a whole class with 16 people present. The next challenge was to put on a number line where we’d end up if we went 3 times around the room, or 5 times around the room counting by 1/4. The final thought experiment was, if we counted by 1/5 how many times around the room would we need to get to a number between 20 and 30? It was a good introduction and got us thinking about unit fractions.
Next we got up to the boards and tried to put price tags on these pieces of cake. A very creative baker decided to cut cake in some interesting ways. Each cake costs 60$.
Groups had different approaches to their calculating. Some looked at fractions, some divided the 60 up, and some did both! Groups were drawing extra lines to subdivide pieces and help them figure things out.
We consolidated all the thinking together before moving on to look at ratios and fractions.
We know that 1:7 is comparing 1 to 7, and we can write it as a fraction 1/7. We can also say that 1 and 7 are both parts, so the whole would be 8, and we can make fractions of 1/8 and 7/8 as well. This is helpful when we are solving ratio problems.
The question we had next was to make a bag of trail mix that is in a ratio of 2:3:4 sunflower seeds: raisins: peanuts. If we want 600 grams total how many grams of each ingredient do we need to mix?
It was great to see their different ways of presenting their work to show their understanding, and how they verified their work to check that it made sense and was reasonable.
Bucket of Zeros
This morning I had the pleasure of working with some of our grade 7/8 teachers as they practiced using the bucket of zeros as a way of visualizing and making integer work concrete.
It was great to see the intentionality of the sequence that they are using, and how they plan to integrate bucket of zeros, and number lines, group practice and individual practice, and games and problem solving. It opened my eyes to what happens in grade 7/8! I’m thankful to have had the opportunity to sit in and work with them this morning.
Fractions Decimals Percents
We added to our wall of fractions and decimals today. We now can slot percentages into the mix.
We took some time to verify the order of all of the cards, and showed some equivalences written in other forms. We noticed some challenges with small percents for example 0.2% is 0.2/100 which is 2/1000 or 1/500. In decimal form it is 0.002 (we divide percents by 100 to write them as decimals.
We also learned that fractions with denominator of 9 have a neat property. 4/9=0.44444444, 5/9=0.5555555 7/9=0.7777777
We looked at some financial implications (calculating sales tax, or looking at inflation percentages, and looking at percentage appreciation ans depreciation).
impressive collaboration today to get all these cards sorted out!
Fractions
Today in grade 9 we had a look at some fraction fluency. We practiced counting by “one eighths” and we clapped when we got to a whole number. We also worked on representing fractions, and what the numerators (parts) and denominators (total number of pieces) mean. We coloured in 1/2 in multiple ways. We know that 2/4=1/2 and 4/8=1/2 and 8/16=1/2.
Next our challenge was a card sorting task as a class. In 3 small groups students sorted through cards with fractions or decimal values on them. Each group ordered their stack of cards. We had some interesting conversations about equivalent expressions, or improper fractions and mixed numbers, or what does 0.3 with the line mean compared to 0.3 without a line over the 3.
Each group chose someone to take the smallest value that they had over to one side of the whiteboard, and another volunteer to take their largest value to the other side of the whiteboard. They had a chat to decide on the smallest and largest values in the room. We then had the edges of our number line, and worked together as a class to put all of the values in order.
There was a lot of rearranging and sorting things out as all groups merged their lists.
Next we added some benchmark values and tried to get our spacing accurate.
Tomorrow we’ll put some percentages in the mix and see if we can manage the new challenge!
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