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!
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.
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