How many times can you fold a piece of paper in half?
Today grade 11s were working on modelling exponential growth and decay. We folded paper in half repeatedly and modelled the number of sections that we get after more and more folds.
We decided this was exponential growth and the equation is y=2^x
We also looked at the page area, and how that changes as we folded more and more. Some called the initial page area 1 (measuring with a “page” as a non standard unit of measure), and then we saw that the area of each folded section decreased each time (the base is 1/2). The exponential decrease function is y=(1/2)^x. Others who measured the page and had the area in cm^2 then would have that as their “a” value for their equation.
Tomorrow we will be working on modelling word problems, so this is a good introduction to those skills.
PS. We had papers folded 7 or 8 times before it got to be too challenging to fold any further. We tried with larger pages too without any luck. Here’s a group that managed to use toilet paper and fold it 13 times.
Beading Begins
Today my grade 9s started their beading projects that we planned yesterday. They are all looking so good!
Each row has 11 beads, which all need to be strung in order then threaded again to lock them in place on the warp threads on the loom.
Some are designed with geometric patterns, or representing various things like flowers, ducks, music notes, or books on a shelf. I look forward to seeing what other images will be revealed.
Students store their work in giant ziploc bags with their name on it so it will all be organized and ready to go tomorrow when we pick up where we left off.
It is impressive to see what students are able to come up with, and how quickly the beadwork comes together once the initial lesson is complete.
Many thanks to our guest from the school board who is sharing their expertise with us and helping when there are challenges. We are remembering the teachings and working on keeping a good mind while we work. We are being sure to take a break and ask for help when we are needing to.
More updates to come tomorrow.
Split 25 with grade 8s
Today I had the opportunity to lead a grade 8 class through the split 25 question “thinking classroom style”.
we got into random groups of 3 using my team shake app, and then working at the boards I gave this prompt. Split 25 up into pieces. Example: 20+5 or 1+1+1+1+1+…+1 and then multiply all the pieces together, so we’d have 20×5=100 in the first case and 1x1x1x1x1x…x1=1 in the second case. We want to split up 25 in a way that maximizes the product of the pieces.
Groups knew the rules of 1 marker per group, and the person with the marker writes the thoughts of the other members.
We had some interesting approaches. Some groups used big pieces, some used small pieces, some used equal sized decimal pieces, and one group wanted to start off thinking about negative pieces (something I didn’t expect at all).
With a bit of experimenting and some calculators, we got right into the task.
Some groups got really big values for the product, but still are searching for the biggest.
I consolidated the task by talking about thinking creatively and responding well to challenges, and also we talked about how to write their repeated multiplications with exponents, so it could be a more streamlined process of they continue calculating at home.
I’m proud of them for diving in and giving it a good try. I’ll have to bring another good challenge to them next week.
Beading in Grade 9
Grade 9s learned about loom beading today from a member of the Indigenous program team at the school board. We have a project we are working on this week: to learn about patterning, symmetry, ratio, percent and area while designing a beaded bracelet or key chain.
We learned about the importance and significance of beading as part of Indigenous culture, and saw many examples of beadwork. We learned about how beads were made, and saw several examples of wampum and learned about the agreements they represented.
We learned about how it is very special to be learning this artform, as there were efforts over the years to prohibit Indigenous people from doing traditional cultural practices. We are thankful to be learning this new skill, and exploring math through a different lens.
We started to plan out our designs. We have 11 beads across in our pattern, so we draw designs on a template, and will start the loom beading tomorrow.
Modelling growth in grade 12
Today I was in with my colleague’s grade 12 college math class. We were working on modelling various patterns yesterday, and continued today.
We made pictures, then tables, and noticed that they double and double and double. We made graphs, and explored what would happen if we went to figure -1 and figure -2 and figure -3, we’d be dividing by 2 as we move up the table.
Next we explored making patterns of our own to correspond to pattern rules (equations). Some groups had linear patterns, others had quadratic, and one had exponential.
Once they represented their pattern, they got a graph paper, and transferred their tiles to make the graph.
Groups got another set of rules, built the patterns, then the graphs. Some had time to do a few patterns. In the end we consolidated their work with a gallery walk and introduction of new vocabulary.
We could see how same slopes, and same y intercepts impact graphs. We could see the difference between x^2 and 2^x as well. I’ve done this task with lines often, but I am enjoying how many functions we could explore with this method.
M&Ms Task
Grade 11s are learning about exponential growth and decay, and to get us started we did a task with M&Ms.
With each pour, they removed the m&ms with the m side up. The number of m&ms that remained got smaller and smaller until all that were left was the ones with no “m” at all.
Groups made graphs, and we looked at features of the graph, then tried to make an equation. We used our reasoning skills about how we were losing half each time, so we’d be multiplying repeatedly by (1/2), which led to the creation of an exponential equation. We noticed that in our situation there was a horizontal asymptote that was not 0, there were a few m-less m&ms. We played around with transformations to see if we could fit the data better.
Finally we did a desmos regression to see how the program would do it. Using the the exponential regression that’s built in forces the asymptote to be at y=0.
It was a fun task. Looking forward to doing more exploration tasks as we move forward.
Taco Cart
Grade 9s are working on pythagorean theorem, and we worked on the 3 act task: taco cart.
We watched act 1 (available here)
and we noticed and wondered about who would get to the taco cart first. The students immediately knew that they needed more information and to my surprise asked for both the speeds and the lengths of the paths.
Here is the information to start act 2:
Dimensions and Speed information
We had a good time working with unit rates, and pythagorean theorem, and then converting time to minutes and seconds to see how much time separated the two people at the end. I had a hard time locating the final video in the moment (websites were down), but here it is!
Patterning in Grade 12
I’ve been invited to come work in a grade 12 class for a few days to work on patterning, reviewing linear and quadratic, and introducing exponential.
We did a few warm up patterns, and then tried these questions: are the patterns growing linearly or not.
Students got into it, drawing the missing figure, determining pattern rules, making tables and deciding if it’s linear or not. There was some debate about the growth in the 2nd pattern. There were some excellent conversations happening around the room.
Groups got into exploring the tables to see what happened when the x values are negative. They used their 2nd differences to help fill it in, or used their equations to calculate the y values for negative x.
Next we looked at this pattern that grows as a rectangle. We looked at patterns in the dimensions, and also how we can see 2 squares of the figure number, and 1 group of the figure number, so the equation is 2x^2+x.
Tomorrow we’ll introduce some exponential patterns then do an activity with blocks to show some graphical connections.
From Patterns to Algebra
Today my grade 9s worked on solidifying their understanding of linear patterns, equations and graphs.
We started off today with each group getting a pattern rule (equation). They needed to use the 1 inch square tiles to build figure 0, figure 1, figure 2, figure 3, figure 4 of their pattern. Some needed to use 2 colours to represent positives and negatives.
The next step was to disassemble each figure, one at a time, and recreate it in a stack on a graph. We have 1 inch grid chart paper which is beautiful, because each square tiles fits one grid space. We line up figure 0, then put a dot at the top LEFT (along the vertical axis). Next we line up figure 1 tiles in a stack beside figure 0, and pu a dot at the top left. We keep going and noticed a pretty regular pattern.
Each group then took the tiles off their pages, and connected the dots to make their graphs complete. We repeated the process with another pattern rule or two all on the same grid. I planned ahead to make graphs that are interesting to compare, and that would bring forth a lot of vocabulary and concepts. We put all of the papers up on the whiteboards and did a group consolidation. Here’s a gallery of what we looked at. Some groups had asked for some spicy challenges!
We explored slope, steepness, parallel, positive vs negative, we looked at y intercepts, what the equation looked like if there was no “b” value (passed through the origin), or if the b values of 2 lines were the same, that’s their intersection point. We found other intersections, some by estimating. We showed what it means for a point of intersection to satisfy both linear equations. Some groups showed really good rise and run. We had some groups working on fractional slopes and non-linear patterns too by the end of the class. I hope this helps connect some of the patterning and graphing we’ve been working on this week. Tomorrow we will explore some more with desmos.
Cheerio Stacking Race
I’ve been working with a colleague this term to bring more data collection tasks and hands on activities to his Grade 10 applied math class. We sure had an interesting time with this one!
I’ve run this activity a few times, and it’s different with each group that tries it. This time, there was a lot of initial competition, and potentially some falsified data which affected what we could do in the end.
To start with each partnership had 1 skewer and a blob of plasticine to stick it upright on the desk. Each partnership took turns to see how many cheerios they could stack on the skewer in one minute. This established our stacking speed. (If I were to do this again, I’d stipulate that they got one try to do this, and they couldn’t keep redoing it to get a better time)
Once we had everyone’s stacking speed data we made a table to show how many Cheerios you’d be able to stack in 2 minutes or 3 minutes based on the initial data. We then made graphs to show both partners data and compare their speeds. (Because so many students had practiced, their speeds that they recorded were pretty close to the same so the graphs didn’t really show major slope differences).
The next part of the challenge was to see if we could create a photo finish between ourselves and the fastest stacker in the room (who may have inflated his stacking speed..which ended up being problematic). We wanted to create a photo finish with a full skewer between each of us and the fastest stacker, so we all needed a head start of a certain number of cheerios.
We counted the number of cheerios that could fit on a skewer, then groups used their graphs or their understanding of ratios and proportions, to figure out how many cheerios of head start they’d need. Below are the calculations for the time it’d take for the fastest stacker to fill their skewer. Several students were keen on calculating precisely, in minutes and seconds. It’s great to see their enthusiasm!
In the end we got people to prepare their head start skewers and then we all started stacking at the same time, and raced to make a photo finish.
It didn’t work because our “fastest stacker” had claimed a speed that was not repeatable, so everyone ended up finished before him.
We had major issues with cheerios on the floor, and others who wanted only to stack and stack and stack, and avoid all analysis because they were busy stacking. I’m hopeful that my colleague can use this task as a springboard to further discussions about points of intersection, rates, “head starts” (y intercepts) and more as they continue their work with solving systems of linear equations.
Even though the lesson didn’t go as imagined, I think that some students made connections, and thought about the problem. They did an experiment, collected and represented their data, and made predictions and compared their rates with others.
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