Collecting Data in Grade 9
We’ve started collecting some data in grade 9, and we’ll be collecting more data each school day, for a total of 10 days.
Each student is going to be doing a “five minute frenzy” multiplication challenge.
After the challenge is done, we tally up how many we got right, and how many errors we made, and how many we left blank, and how long it took. These are our dependent variables. We will be exploring how they correlate with the date that the challenge was completed.
Each student will then get to select data that tells a story. They will graph the data, interpolate and extrapolate with a line/curve of best fit, and discuss the trends of the graph.
Along the way, it is an excellent springboard to discussing multiplication strategies, and to try them out the next day.
Today, we looked at using the distributive property to split up big numbers like 12 into friendlier numbers like 10 and 2, and using that to help us multiply by 12.
We also explored how this can help us multiply big numbers together.
We looked at multiplying by 6 as well, and how it relates to doubling the result of multiplying by 3. Here we used the distributive property and the associative property to illustrate this multiplication.
We looked at multiplying by 4 and by 8 using the associative property as well.
Students are keeping track of their data each day, and some are already noticing some big changes as we practice and learn new strategies.
Bucket of Zeros
Today in grade 9 I was in a class where students were working on integers and using the bucket of zeros to show that a negative times a negative will be a positive number. Many times we have been told the rule, but it is important to be able to show why it makes sense.
Students started out by making a bucket of zeros with many zero pairs on their desk.
The next step is to decode the question. This question was -7(-4), which we can be viewed as removing 7 groups of (-4).
Once the 7 groups of (-4) are removed, we look at what’s left.
We have some zero pairs, and also 28 red tiles. So we have proven that -7(-4)=28.
To consolidate, students were doing questions on the board, showing their work with the bucket of zeros.
They then consolidated with a 4 quadrant meaningful note.
Volume of Pyramids and Cones
Today we looked at the volume of pyramids and cones, and developed the formulae using water.
By using the geometric solids we discussed the properties of prisms and pyramids, and then guessed how many cones it would take to fill a cylinder with the same base and height. Many students guessed 2, 2.5, 3, 4 as possibilities. After the cone was filled once, and dumped into the cylinder, students were prompted to look at it and refine their guess if needed. Most converged on an idea of 2.5 or 3ish.
Finally, everyone could see that it took 3 cones to exactly fill the cylinder to the top. The same process was repeated for square based pyramids and a square based prism, and then again for a triangle based prism and triangle based pyramid.
In the end, students saw that it took exactly 3 pyramids to fill a prism with the same base and height. (Note: if you are doing this activity be very careful not to overfill the pyramids since a meniscus would affect the volume).
Next, the idea of working backwards from cylinder to cone means that we multiply by 1/3. A cone contains 1/3 of the water that the cylinder did.
From there, students were up at the boards in small groups working on calculating the volume of pyramids and cones, and cylinders and prisms.
Building Prisms in Grade 10
This morning in MFM2P we were building rectangular prisms with a volume of 300cm^2. It was an interesting challenge for the class, to find dimensions that would work, and then to actually construct the prism.
We used old file folders donated from our main office, and we needed rulers, tape and scissors. Students were in groups of 2 or 3.
some groups made a net, and then had a much easier time assembling the prisms. The groups that cut out 6 individual faces had some frustration putting it all together.
There was some great learning that happened when assembling the boxes. Some people learned that if sides fold to join, they should be the same length. Something that you realize pretty quickly when your prism has a hole in it! Others realized part way through that they had chosen numbers that would not actually multiply to 300. I recommend not choosing a side length of 17 if you are working without a calculator!
Another group made a side 30cm instead of 3cm, so they have a much larger volume than everyone else!
We have a cube that was made by our fearless leader. There was some discussion about if that counts as a rectangular prism. Some students are still not convinced!
At the end we calculated the surface area of the prisms, and we will look at them again tomorrow and put them in order from smallest surface area to biggest, and notice any patterns. It is a neat way to experience how prisms may have the same volume, but that doesn’t mean they have the same surface area.
Great work everyone!
Day 2: we organized the prisms from biggest to smallest surface area, to look for trends.
we noticed the cube is the one with the smallest surface, and the ones that are longer or flatter have bigger surface areas.
we talked about surface area in terms of doing drywall, or flooring, and we noticed the floor tiles are a square foot. We talked about packaging items, and why choices are made to minimize packaging, or not. We talked about how surface area and volume are important to biology, with cells dividing to maintain an optimal surface area to volume ratio, or how worms can breathe through their skin, and how our lungs by design, increase the surface area for gas exchange.
we talked about how many groups had integer dimensions, and these values are all factors of 300. Some groups had side lengths that were decimals, so they needed to use division or trial and error to get the correct side lengths. The cube led to an interesting discussion about how to get the sides correct. We need to take the cube root to undo the effect of cubing a number. The cube root of 300 will give us the side lengths of a cube with volume 300.
Friday Photos
I was in several classes on Friday and took some photos of the neat tasks and activities that were going on.
One class was working on adding and subtracting positive and negative numbers, with an emphasis on zero pairs.
They also had a neat quiz, based on the max/min dice game
To start the quiz a dice is rolled 4 times, and students place the 4 numbers into the boxes, then evaluate each expression.
Another class was doing this slow reveal data talk. A series of images are shown, and students notice and wonder more and more as more information is revealed.
Another was consolidating the R2D2 3 act task, looking at 2 different ways to solve the task, by a ratio of areas, or by converting the dimension in inches into a dimension in “post it notes”. Consolidation ideas included making sure the number sentences were all written in a way for others to follow clearly.
Finally, in my class we were working on order of operations. We were working on the 4 4s task where we use four 4s to create a math expression that equals 0,1,2,3,4,5,6,7,8,9,10.
e.g. 4+4-4-4=0
It was impressive to see the different approaches. We practiced using brackets, and exponents, and found out that since square and square root are opposite operations, they have the same level in BEDMAS. Groups looked to other groups for inspiration, and students were working hard to check and double check their work.
It’s been a fun week in the math department!
Dot Talks
Today several classes were working on dot talks as a way to encourage students to express themselves and explain how they perceive different patterns of dots.
An image is projected for several seconds, and students are asked how many dots there were on the screen. Students then take turns explaining how they know how many dots there are.
As they explain, the teacher groups the dots and shows the different ways that the students suggest. It’s always impressive to see how many different ways we can see things to get to an answer. It’s a great activity to encourage participation, and to celebrate different approaches and viewpoints.
some sources of dot talks are: numbertalkimages, and youcubed.
Working at the Walls
Today in grade 10 we did some review in groups of 3 at the walls. We worked on simplifying algebraic expressions, and solving equations including ones with fractions.
It was great to see the students working so well with each other and collaborating on day 2. Having lots of practice in grade 9 has been very helpful in building the expectations of collaboration and the ability for groups to ask for help within and amongst the groups. Our class has 32 students, so I am not able to be the sole source of information, nor the sole checker of “is it right?”. I was impressed by the different strategies students used to solve equations with fractions, and how they communicated their work.
We have some work to do with fractions and integers coming up tomorrow, which will hopefully work to increase accuracy of solutions. Many groups worked through questions a few times, finding out where they made errors, and checked their work with each other. I was so lost in the moment with them that I didn’t take any action shots, just photos at the end of class while I cleaned up the room. It was an impressive whirlwind of activity last period, working right to the bell!
Teamwork in Grade 9
Today Grade 9s tried the 1-100 task, where they circle numbers in order.
The first time through, as individuals, and we timed it…it took about 8-11 minutes. I took photos as they worked to debrief the class later.
Next, working in groups of 4 (each student has a different colour marker).
If students follow the rules, and take turns circling numbers one at a time in order, they will see some neat patterns emerge. On the left, the rules were followed. On the right is what you see if there’s a group of more/less than 4, or students aren’t taking turns.
this round took 4-6 minutes. It emphasizes that when we work together we can accomplish things quicker, and help each other. Many students naturally started looking ahead for their number, or pointed out numbers for peers, or were counting out loud to help themselves keep track.
For the final round groups strategized, and moved themselves, or furniture, and made a plan before attempting the task again.
This group decided to work around the desk so everyone saw the numbers right side up. They drew a grid to help narrow down their search areas, they spoke out loud, and helped each other find the numbers. This round took under 2 minutes for some groups.
At the end of the lesson I showed the pictures I took, and we discussed the ways that they worked well as a team.
The final stage of the task was to brainstorm in groups what good teamwork looks like, sounds like, and feels like. These ideas will be combined to make a list of group norms for teamwork.
At the end of this task we had some time left, so we did an intro sheet which starts with graphing their math life. This was inspired by a workshop by Liesl McConchie (@lieslmcconchie on Twitter).
Students think back to how they felt about math over the years, and represent that on the graph. While they worked I circulated, and had some discussions with students about the peaks and valleys on their graphs and learned a lot about them, and how they view their journey through math class.
Productive Struggle on Day 1
This year I am supporting staff and students in the math department as a coach/facilitator. Today I was helping in grade 10 applied math, and brought a non-curricular activity to challenge everyone on day 1. It is based on this activity(spoilers if you scroll to the end of the linked page).
The goal was to use a half piece of paper to recreate this shape. Students couldn’t touch the pink page, but could come up and examine it. Some brought rulers and took detailed notes.
In groups students worked to recreate the shape. It is not an easy task, so they needed to try a few different approaches. It was neat to see how different groups approached the same task.
While they worked, I documented some of their commentary. There was evidence of frustration, and exasperation, and also evidence of some students perseverance.
We talked about how we will experience struggle in math class, and in life, and how we can lean in to the struggle, and enjoy the challenge and work together to solve problems in groups, and helping each other.
The class didn’t solve the challenge yet, but some will likely keep working at home. We did have a successful vice principal who visited, and managed to solve the challenge!
We’ll keep working on leaning in to the productive struggle this term.
Welcome back Grade 10s!
Grade 10s worked on visual patterns today. Many of them did this kind of task with me in grade 9 last year, so we were remembering what we knew, and making connections to quadratic patterning which is new this year.
Students worked on remembering their modelling skills from last year. We had so many representations!
we worked on going directly to an equation from the pattern, since the table of values told us it was not linear, we needed a new approach.
We did quite a few examples to practice our new skills
our final task of the day was to tackle this question, to find out for which figure number the perimeters are the same.
We had lots of students making tables and comparing, but then we used our new tools (magnetic graphing grids) to draw on the whiteboards with more precision.
this is a nice introduction to solving systems graphically, which we will be doing a lot over the next few weeks.
We’ve got a lot of enthusiasm and energy on day 1 which bodes well for the future.
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