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.
Introduction to Fractions
in grade 9 we started working on some fraction fluency tasks. Today we got out the big bin of fraction strips and explored the connections we saw.
We noticed that the denominator showed how many equal sized parts that the 1 is split up into. We also noticed that when there are more parts each part is smaller.
Then we looked at how to make some equivalent fractions. We noticed for all of the things equal to 1/2 they had even number denominators. For all of the ways to make 1/3 the denominators were all multiples of 3 etc.
Then we started to get creative. We can see here that 1/2 can be written as 1/6+1/6+1/6 or 3/6. But it could also be written as 1/3 since 1/6+1/6=2/6 which is equivalent to 1/3. So we know 1/3+1/6 is another way to write 1/2.
Next we looked at this statement. 1/4=1/12+1/6. To make sense of this students saw that the 1/6 is also equal to 1/12+1/12 so we could write 1/4 as 1/12+1/12+1/12 which is 3/12. If we divide both the numerator and denominator by 3 we get 1/4. We know that if we split 1/4 up into 3 parts each one will be 1/12.
This could be written as 1/4=1/12+2/12. We saw that this could also be a subtraction. 1/4-1/6=1/12 as well.
We showed that we could start with the question written in fractions, model it, build it in the same sized pieces and then get the answer. 3/4+1/8 was the question. We built it as written. Students then saw that we needed to make each 1/4 into 2/8. Then we can count the number of 8ths.
next we tried adding 1/4+1/2+1/3+1/12. We decided that it made sense to write it as 12ths.
3/12+6/12+4/12+1/12=14/12 which is more than 1. We wrote it as 12/12+2/12 which is 1 and 2/12 which can be 1 and 1/6, which is exactly what we got with our blocks.
We are going to keep working on fractions for the next little while, building up skills and getting more confident with the concept and the different representations and manipulatives.
The Most Special Triangles
I had the opportunity to teach a lesson in a colleague’s MCR3U class, a course that I have never taught yet, so it was fun to explore a sequence of steps to learn about special triangles.
Here’s my sequence of prompts.
We noticed that all of the hypotenuse calculations resulted in the side length multiplied by root 2. We noticed that the sine cos and tan of 45 all simplified to be the same, regardless of the side length of the initial square.
We worked through a similar task to develop our next special triangles.
We calculated that the height is root 3. And we see that sin30 and cos60 are the same, and sin60 and cos30 are the same because when the reference angle changes the opposite and adjacent sides now refer to different sides. The adjacent to the 30 degree angle is root 3, and that is the opposite to the 60 degree angle.
Next step was to do a task. This challenge came from Peter Liljedahl’s old website of good problems for teachers.
Students worked hard to represent their thinking. We had some physical representation, some calculations of angles and sides, and some use of the new special triangle learning.
It was neat to see which direction various groups took, and whether we all arrived to the same place in the end.
Math test today
Grade 9s had a test today. We decided to make it fun and students could choose where they sat. We have a small class so I let them have free rein. It was a little chaotic. Some chose to sit so they faced the window and could look out. Some faced the whiteboard so they wouldn’t be distracted. One embraced and encouraged chaos and sat right behind the door, which would have been problematic had anyone needed to come in.
We also used Howie Hua’s test talk routine, where the first 5 minutes of the test is time to talk…pencils down, students read the test over and can talk with friends (without writing) to go over anything they’re confused about. This helps calm everyone’s nerves about the test, and encourages students to read things over and strategize about which questions to answer first.
Our tests have some questions that everyone answers, and some where there’s a choice of “mild” “medium” or “spicy” questions to do. The questions have more complexity, more steps, more thinking to do. Answering mild questions is the bare minimum, and earns a level 1, medium done well can earn level 3, spicy done well can earn level 4+.
We make sure everyone has what they need, the same manipulatives as we use in class are available for the test. This test some students used algebra tiles, and linking cubes to help model the algebra and measurement questions. There’s also a little “roche dans la poche” (a smooth stone) that’s kind of like a worry rock that students have to help keep us grounded and focused.
3 act task: mowing the lawn
We worked on composite area and pythagorean theorem today in grade 9. There’s a good 3 act task called mowing the lawn. It starts with a lawnmowing video to pique the interest, and get questions flowing.
Next, another video, with a timer, and also dimensions given for the mower and for the lawn
I usually wait until they ask how wide the mower is. Often we get so bogged down with area calculations that we forget the mower has a width too!
We then worked really hard for quite a while to calculate the area of the lawn and how long it would take to cut.
Some common errors I’ve noticed were in applying pythagorean theorem to calculate a triangle leg vs hypotenuse. Also some students struggled with 18 inches being 1 foot 6 inches which is 1.5 feet (not 1.6 feet). Others struggled with how to use unit rates that would be helpful like how many seconds it takes to cut one square foot of lawn. That can then be scaled up to figure out the time to cut the entire area of the lawn. There were good conversations about how many pieces to break the yard up into, and if it mattered if he mowed in a spiral pattern or back and forth in rows.
I’m thankful that our meter sticks are also yard sticks, so we can work in inches and feet when we need to.
In the end we watched the whole video to see the answer. Watch for motion sickness!
We did a lot of calculating and reasoning today. We need to work on our communicating…that will be our next step.
Area of Regular Polygons
Today in MFM2P we looked at area calculations, to review and activate prior knowledge and then apply it to some new challenges.
Groups started with a sheet that led them through some calculations. They worked collaboratively in pairs or groups of 3 to go through these early questions. It’s interesting to me what ideas stick (area=basexheight seemed to be in their memory, but they had forgotten the word parallelogram)
We talked about how you could find the area of the triangle pieces and the rectangle piece and add them up, but ALSO how you could slide the triangle piece over and create a rectangle. The formula for area is still base times height, but we have to remember that height is perpendicular to base always.
Next in the sequence was to look at a right angle triangle and relate that formula to what we know about rectangles. This is half a rectangle, so the area should be (1/2)(base)(height).
The bottom triangle has the same formula for the same reasons. We can draw a rectangle around the triangle. Looking at each side, half of the rectangle is in the triangle, and half is not.
Next we branched out to regular polygons. This was a bit of a leap, it is so visually different that some students had a moment of giving up. It’s neat to see how we can make a visually different figure appear to be a parallelogram by using scissors!
We drew lines from the corners to the middle, then cut out along those lines. We could fit them together upright and inverted to build a parallelogram. If we cut one of the triangles in half and slide it over we would have a rectangle. The base would be half of the perimeter of the shape. For this dodecagon we can see that there are 6 of the “edge segments” that make up the top and bottom sides of the parallelogram. The height of the shape is the same as the height of one of the triangles. This distance is called the apothem (not an important word, but it’s like the radius of the non-circular shape).
We can make any regular polygon into a rectangle to calculate its area. We can explore how this helps us understand circle areas, but we’re not there yet. This also sets the stage to calculate volume of any regular polygon based prims (and later pyramids).
M&Ms task
Today my grade 9 class tackled the M&Ms task.
Students counted, dumped, removed all the “M” side up candies, then counted, dumped and repeated the process. They added to a data table as they went, and noted that there was a big drop at first, and less and less of a drop as it went on. Some saw the connection to the drop being about half each time, which was neat to see. Again, we noticed that 8-12 candies had no M printed at all, so we never ended up eliminating all of them.
Groups created graphs and saw that the pattern was not linear, but looked like a decreasing curve. They made curves of best fit to model the data.
We talked about how to make an equation to generalize what we saw happening. We know that to find half of a number we can divide by 2 or multiply by 1/2. If we find half of THAT number then we’d have to multiply by 1/2 again. We showed that in our table of values for an example case of starting with 160 candies.
We could use our exponent knowledge to help us build the equation. This type of modelling will be important when we look at compound interest later on.
Finally we modelled the class data on desmos and looked at how to do a regression. We followed the same model as we created together, but left the initial value as “a”, the base of the exponent as “b” and the vertical shift for the horizontal asymptote as “c”. We talked about the R squared value and how it’s a really good fit. We compare this to the linear regression that Desmos does. We noticed that the linear regression has an R squared of 0.59 which is not as good a fit.
This was a nice way to spend a Hallowe’en Friday, eating some candy and making some graphs, and learning some pretty sophisticated modelling skills.
Ms. Bearse's Site 