Thin Slicing Questions to Learn Exponent Laws
Today we spent grade 9 math up at the boards in random groups to work on representing exponents, and worked all the way to learning 2 exponent laws.
We began with questions like “represent 2 squared”, and got all the notations on the boards (expanded form, a 2×2 square, 4, 2^2). After some repetition we looked at some cubes, and students represented and expressed them e.g. 7 cubed=7x7x7, 7^3, and a 7x7x7 drawing of a cube. They evaluated what that would be without a calculator as well since we are working on our fluency strategies whenever we can. We looked at what fractions would be when raised to an exponent as well, again with the fluency strategies.
Next we looked at what x squared and x cubed would be, connecting to yesterday’s toothpick task. We then revisited what (2x)^3 would be. We’re working on remembering to always multiply the coefficients and the variables together and always writing down exponents when needed. We looked at multiplying variables by each other, and what to do if there are different variables and also coefficients in the mix. We then did some power of a power examples.
At the start of the class students were fidgeting and uncertain about exponents, and some were hesitant to contribute. Pacing a lesson like this is tricky because I want to make sure the majority of the class is understanding, but also give enough challenge so that those who do understand aren’t totally bored. We had a quiz/exit card at the end of the lesson to see what concepts stuck and what ones we need to go over again tomorrow. Pretty proud of the progress today!
Fluency Task
Today I did a fluency task with a colleague’s grade 9 math class. Students are working on their multiplication skills, and today’s task was to use base 10 blocks to build a rectangle. They used 6 of the tens and 12 of the unit blocks.
Groups had to try to build all of the different possible rectangles. We then had a good conversation about how all the rectangles had the same area of 72, but the dimensions are different. We know area is length times width, so we know that each rectangle has a different multiplication represented.
1×72 and 2×36 and 3×24 and 6×12 were all possibilities.
we looked at doubling and halving when exploring the list. 1×72, double the 1, halve the 72 and you get 2×36.
Some students asked about if we could make a square. I removed the constraint of using the tens, so with 72 unit cubes we tried to make a square. The closest we could get was 8×9. Some knew that 72 was not a square number, so it was impossible.
One did know that with 72 you could make 2 squares of 36 though.
we looked at how doubling and halving can help us make multiplying easier. Instead of multiplying 15×60 we could double the 15 and halve the 60 to make 30×30 which is 900, the same answer as 15×60.
We tried another question as well, 18×7 we halved the 18 and doubled the 7 to make an equivalent multiplication of 9×14. We know 9 is close to 10, so we can do an overshoot and return to get to the answer of 140-10-4=126.
I was interested to see that many students were attempting to skip count by 18s or by 7s. Hopefully we can show different strategies that can be more efficient than skip counting, so students move into more multiplicative thinking and proportional reasoning.
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.
Ms. Bearse's Site 