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!
Surface Area and Volume
Grade 10s have been working on surface area and volume calculations, but it seems to be overwhelming everyone a bit. For some it’s knowing the difference between surface area and volume, for others it’s that the formula sheet has so many letters and equations, and for others they don’t know how to substitute in values and solve.
We gathered everyone together and did a few examples together and they wrote down an example solution for them to follow.
We also had a look at some different nets and how they come together to make prisms and pyramids.
Hopefully we can remember and use our example next week when we do more of these questions.
Fractions
In MTH1W we were working on doing BEDMAS questions with fractions, and my students had had enough halfway through the period. They asked to do something else, or to have a “fun Friday”.
I pulled the question from our math club for grownups.
We extended the pattern, and wrote the fractions in improper form.
One group got excited about exploring these fractions as percents, and others were keen to use my prompt of “use the numerator and denominator as legs of right triangles” to explore and practice Pythagorean Theorem calculations.
I’m glad they knew to ask for something different to work on. This was a much more fun way to end out the week.
Math Club For Grown Ups
We had our 2nd meeting of the math club for grown ups. The first meeting was in June last year. We’re getting together as educators/admin/board office staff/EAs/teacher candidates (hence the all encompassing word “grown ups”) and doing math together.
We did several questions.
We worked on the questions in random groups and got pretty close to solving them. Part of the fun is deciding what steps to take to solve them.
It was such fun to work with colleagues from 4 different high schools, and 2 of our feeder schools, and to have our principal and several superintendents doing math all together. We look forward to another meeting in December.
The questions for this meeting came from Peter Liljedahl’s old website of good problems for teachers, and from Al Overwijk’s site slamdunkmath. Have a look there for the sequence and ideas pulled from the sequence of fractions.
Similarity and Area
I’ve been working with a colleague brainstorming ideas for exploring similarity using pattern blocks. We can make similar shapes quite nicely, and prove they are similar because the angles are the same and the side lengths are proportional. We can see that when the side length changes by a factor of 3 the area changes by a factor of 9 etc. My colleague took pictures from his class to show what they did. Students noticed that there was a quadratic pattern in the area, since the change was changing by a consistent amount (the 2nd differences are the same).
We can do this with other shapes too. This was from my brainstorming.
and this was from my colleagues class. They found creative ways to grow the shapes.
I’m thinking now about how these same pictures could be used later when fractions are discussed.
Students said they enjoyed the task and liked working with the blocks ro show their learning.
Exploring Volume and Surface Area
Today in MFM2P we had a challenge. I gave each table group the task of working as a design team to create a rectangular prism to meet my specifications. It needed to be made of card stock (old file folders) and tape, and it needed to have the volume of 300 cubic centimetres.
Groups had access to rulers, protractors, calculators, scissors, folders and tape.
As groups they had some choices to make about what the dimensions should be. We had talked about volume earlier. We know that the volume is the insides of a 3D shape. We know that for prisms it is calculated by the area of the base multiplied by the height, or specifically for rectangular prisms volume=(length)(width)(height). It took a while to find dimensions that would work, and then the struggle was with how to cut out the pieces to build it, and what pieces we’d need.
Some drew each side separately and taped them all together. Some made a foldable net.
When we got them assembled the goal was to check to see if the volume met my criteria of 300 cubic centimetres, and then I asked them to report on the surface area of paper required to build them.
Some groups had time to make several different boxes. And some realized after they made the box, that it wasn’t quite right, so they had to be redone. This small one has a volume of 90 cubic centimetres. It’s pretty cute though.
one group got really creative and persevered to make a box that was 100cm x 3cm x 1cm.
We will keep working on this tomorrow and consolidate with talking about how prisms with the same volume don’t always have the same surface area. We’ll arrange them to see any patterns, and connect the idea to surface area to volume ratio and how that’s important in packaging, and also in biology.
This task was pretty open ended, which caused some struggles getting started. We managed emotions like frustration pretty well, and we are continuing to work on our collaboration skills and working as a team to get tasks accomplished. We’re making headway on our clean-up skills which is allowing us to take more risks with hands on tasks.
I was quite intrigued to see this box being built at the end of class. We’ll have lots to chat about when we come back to this tomorrow.
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