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.
Visual Patterning
Grade 9s are learning about patterning, and linear and non-linear growth. We started by using patterns from visualpatterns.org to explore how something grows.
We look at how the tree sequence grows. There are always 2 more each time. We can also see 2 vertical columns of “x” for each figure, and one lonely tree that’s always there (that’s the constant) and it will be the only thing for figure 0. This is the pattern y=2x+1.
We next made some patterns of our own, given figure 2 only.
It was neat to see how many different patterns they made. We put pictures up on the screen and analyzed the pattern rules of each.
Next we looked at whether patterns are linear or not. We used visual clues to build the missing term, and then looked at the first differences to see the growth. We made equations for each of the rules, even the non-linear one!
a good start for patterning indeed!
Negative Exponents
I’m pretty pleased with the sequence that we did for introducing negative exponents today. We’ve been doing patterning for ages, so students are used to the routine of modelling visual patterns.
This one was a bit different than the usual linear and quadratic patterns. We were modelling the number of dots or terminal branches.
Students had great success with making tables and spotting that the number doubled and doubled and doubled as we got higher figure numbers. It took a minute for some to connect that with writing repeated (2)(2)(2)(2) and making it 2^x. Students graphed for a positive domain quite readily, but needed a push to think about if the domain of this would be positive x only. The visual patterning breaks down, but the equation continues, and we got there by using the tables. As you go down the table (for larger x values) we multiplied by 2 each time. To go up the table (for smaller values of x) we divide by 2 each time.
We had light bulb moments as we saw the numbers getting smaller and smaller and smaller, as we divided by 2 each time. They made connections to graphical features saying there’s an asymptote, and telling that the range is y>0 but not equal to 0.
We debriefed about how 2^-3 would be 1/(2^3) etc. It’ll take some practice to get the skills solid, but it was a nice introduction. We worked on expressing different bases to different negative powers.
Inequalities
Today in Grade 9 we worked up at the boards in random groups and we explored a new topic to many of us. We were representing inequalities on a number line. We worked from questions presented out loud e.g. “show me what x is bigger than 3 looks like on a number line”, and also in print e.g. show me x<-4, so we could work on connecting a conceptual understanding of the number line and also of the symbols and notation.
We now know that if a point is not included we draw an empty circle around it, and an arrow to one side or both, and if a point is included we fill in the circle.
We saw how we can show a region as well, here x is greater than or equal to -1 and also less than 2.
We did a lot of practice, and then levelled up to solving some simple inequalities and representing them on number lines too.
We also wrote our 6th weekly email home, and took up some questions that were tricky from the homework and quiz that we had done earlier. A good way to wrap up before March Break.
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!
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