Building Exponents
My grade 9s built exponent models today. We were working with toothpicks (representing x) and plasticine to construct models of x^2 and x^3.
We used the concept circle framework where we can show our knowledge by building a lot of different expressions, and then compare them. Students were told the list to build: 2x, x^2, x^3, 3x^2, (3x)^2, 2x^3, (2x)^3, x^3+x^2+x.
We always have an interesting conversation about what the brackets mean. Here we can see (3x)^2 is a square with side lengths of 3x. The side length is always the base of the exponent (what the exponent touches). We noticed that it is equivalent to 9x^2 visually, but also in algebra since (3x)(3x)=9x^2.
The same is apparent with (2x)^3 which is a cube with side lengths of 2x. We can see that it is equivalent to 8x^3 visually, but also algebraically (2x)(2x)(2x)=8x^3.
We extended to another more “spicy” concept circle with fractions. We can show 1/2x^2 and (1/2x)^2 and see the difference. For (1/2x)^2 we make a square with side lengths of half a toothpick, which ends up being 1/4x^2 since (1/2x)(1/2x)=1/4x^2.
We’re just starting on our exploration of exponents, but this task helps explain lots of things: how x and x^2 and x^3 are physically different so we cannot add or subtract them. It also shows how volume corresponds to cube units and area corresponds to square units. It also helps get at the idea of doubling or tripling the side lengths and how that affects the volume and area.
We’ll be learning more exponent representations and exponent laws in the coming days.
Bean Finale
The grade 10 class that I have been working with are wrapping up their bean growth experiment before March Break. Today we took all of the growth data that we’d collected and we made a graph to help us calculate the growth rate.
We made scatter plots, and discussed the importance of good communication like scales and titles. We talked about linear and non linear trends, and made lines and curves of best fit. For the lines we calculated the growth rate.
We found 2 points on our line of best fit, and then determined the rise and run between them. We simplified that fraction or made it a decimal. That is the growth rate. To find the height of the plant on any day, we needed to add the starting height to that rate. Some groups measured from the table and some from the soil level, so the initial heights were sometimes bigger than others.
It’s been a good project. Students can now being home their beans for the holiday and watch the flowers form and bloom.
A word about Test Return
In my classes we’ve done our first test now, and are now working through some corrections.
It’s so important for students to take a minute to revisit their work and use this as a learning opportunity to improve skills. I have been using an error analysis page for years, originally inspired by mathequalslove. The types of errors are: inattention (focus/concentration/not reading the instructions/all the “whoopsie” mistakes that you could fix if you were paying attention). Computation is a calculation (zero pairs, integers, order of operations, fractions issue), precision is communication (then”let x represent” statements, the final statements, the units, the clarity of the work, handwriting sometimes causes numbers to migrate from one to another), problem solving is the most serious kind of error, those ones you got stumped and couldn’t start, or couldn’t continue and you likely need help from someone else to sort through those ones. Test taking strategy is an error like spending a lot of time on page 1 and never getting to page 4.
As we go through the course skills build on each other, and if we can get a handle on the errors we make now, we’ll be in better shape by the end. I like to spiral through courses as well, so no learning cycle is truly final. We will see the skills later, so confronting the errors now is important.
There are many emotions around getting tests back. Excitement, confusion, disappointment, judgement, but we can make this a moment for connection and mutual support as well. We can make study groups, help each other out with problems, encourage and support our classmates as we navigate the challenges together.
I ask for corrections ro be done, and submitted. I’ll keep returning them until they’re fully and completely done. Some people submit them 3 or 4 times but eventually everyone has a full solution set for their records.
Experiment with Sound
Today in Grade 9 we went outside. I’ve been waiting for a nice day, and we sure lucked out today. The snow is almost all melted, and we were ok with no coats.
The task today was all about data collection and graphing. Each group had a bottle (with straight sides) full of water, and a device with a frequency measuring app. The one I’ve always used is apparently not in any app store anymore, but there are a bunch of free apps that work. Students had been tasked with learning how to make a noice by blowing over a bottle’s opening for homework. Some had success! Today we are looking for the relationship between the height of the air column and the frequency of sound produced when blowing over the mouth of the bottle.
We started with an almost full bottle, measured the height of the water, measured the height of the air, measured the frequency, then repeated to fill in a data table.
We got partway through making graphs and answering follow up questions, and will need to keep working on the analysis tomorrow. It was such a nice day for an outdoor experiment!
Some keen students want to try to play a song. We’ll have to do some tuning tomorrow to see what’s possible.
Modelling Data in Grade 10
I was back helping in a MFM2P class today. We checked in on our beans. Some are getting quite big.
The next task we did was a “fun Friday” task where students worked in pairs to decide how many cups would be needed to make a stack as tall as their teacher. (5 foot 7). Students used unit conversions, some choosing to work in inches, and others in centimetres. They made ratio tables, or used linear tables of values to determine how many cups they’d need.
They experimented with different ways to stack the cups as well to add a little bit more height, so they could be precise. Their teacher had a prize on the line. Chocolate bars for the group that got closest to his height.
Each group had 4 cups to test out their plan, then they needed to lock in their total number of cups that they’d need, and show their thinking. Most groups had a good grip on proportions. If 4 cups would stack to a certain height 8 cups would stack to double that height, and they’d double again to know what 16 cups would stack to.
Some groups used weights to help balance their stacks, and others decided that making a triangle stack would be the best approach to make the base wider and more stable. The triangle stack quickly evolved to needing all the cups in the room.
It fell down once or twice, but dedicated students built it back up!
Many groups got their towers to be the desired height, but only one group did it with the number of cups they had predicted they’d need.
Lots to debrief from this task in the coming days. Linear vs quadratic growth, tables of values and graphs, proportional reasoning, and unit conversions. Other alternative approaches are to start stacks on the desk, introducing a “b” value to mx+b, and also we can note how the graphs would produce parallel lines starting from the floor and the desk.
It was a great Friday challenge. The groups worked very hard!
Back up at the Boards
In Grade 11 we are done our first test, and started learning new material. As a warm-up we reviewed modelling from a visual pattern, and finding domain and range.
It was so interesting to see how many strategies popped up to model the same pattern.
Some worked from a table of values, using the 2nd differences to calculate “a”, and figure 0 to calculate “c” and substituted a point to calculate “b”. Others saw the pattern as half of a rectangle, filled in the other part of the rectangle, made an equation using area model then cut it in half. Others saw that they could move some of the dots to make rectangles, and then modelled in our usual visual manner, identifying squares of x, and groups of x.
Some found the vertex by completing the square, others by finding the zeros then the axis of symmetry, then the vertex, and others used -b/2a to find the axis of symmetry from standard form, and got the vertex from that.
I’m thrilled that groups have the flexibility to approach problems in different and creative ways. We are thinking, and not memorizing.
Next we looked at new material: how to simplify and work with fractions. We confronted a few misconceptions along the way, when we need a common denominator, when we can “cancel” out terms that divide to make 1, and if it’s possible to solve for x or not. Some students started applying equation solving skills of “doing the same thing on both sides” to the expressions, before realizing that there are not 2 sides.
I’m sure we will continue to confront more misconceptions as we dig into operations with fractions.
Finally as a fun factoring review we watched this excellent song.
Height vs Wingspan
Today my class explored how to make a scatter plot with a spreadsheet. We collected data of our height and our wingspan.
Next we put our data into a spreadsheet, shared it with the class, and everyone practiced making a scatter plot of the data.
We noticed how if you change the scale on the axes the data can look quite different. We looked at how the R^2 value will tell us how strong the correlation is. We looked at how to make a line of best fit, and how to use the graph to interpolate and extrapolate, and also how we could use the trendline equation to help as well.
We will be making use of these skills over the coming days. We are going to be graphing out multiplication data.
We’ve been collecting multiplication data for the last few weeks. Each day we try a 5 minute frenzy multiplication challenge, and each day we do number talks about different multiplication strategies. With practice, skill building, and repetition we notice that in general we can improve how many we get right, or how fast we can complete the grid. We track many variables in a big table of values, and in the end we will each make a choice about which variable is the most interesting to graph. We’ll make scatter plots by hand and with a spreadsheet.
Math Club for Grown Ups
We had another great meeting to try some challenging problems together.
Many thanks to those who came, and those who brought problems to present. Looking forward to our next meeting in April.
Experiments in Grade 8
I’m working with a grade 8 class for the next little while, once a week. Today they were doing experiments and comparing theoretical and experimental probability.
It was interesting to work with the students and watch them work through different tasks involving dice and also marbles. Looking forward to getting to know the students better over the coming weeks. Thankful for the opportunities!
Distributive Property
Today we looked at how to multiply polynomials. We modelled several questions, and showed how the “groups of” model and the “area model” work together. Example 2(3) is 2 groups of 3, but also a rectangle wth dimensions 2 by 3. We applied that to multiplying polynomials. Here’s 3(x-2) shown in 2 ways.
We practiced several similar examples and noticed that we can multiply the number in front by each term within the brackets. Now we have 2 ways to approach the questions.
When we have (x)(x+1) we run into a problem with the “groups of” method. We don’t know how to represent x groups of something. We need to shift to the area model, or distributive property where we multiply x by each term in the brackets.
We had some breakthroughs with the visual representation, and we talked through how (5)(5) is 5 squared so (x)(x) is x squared. It’ll take some time to solidify the skill, but to practice we did a puzzle in small groups.
I love the puzzles from this site. There are so many neat ones.
we’ll keep working on more algebra next week.
Ms. Bearse's Site 