Visual Patterns
Today I was in a colleague’s grade 10 applied class to work with them on modelling linear and quadratic patterns.
We remembered from grade 9 how to continue a pattern moving forward and backward. We made a table of values and a graph, and we looked at how to make equations. To get to an equation my new strategy is to have students think about figure 10. In the case of the trees, we can think of it 2 ways:
- Start with 1 tree (figure 0) and then add 2 10 times. f(10)=1+2(10)
- notice that in each figure, there are 2 columns of that value, and one extra solitary tree, so f(10)=2(10)+1
From there we can find figure x by swapping out the 10 for x. f(x)=2x+1. Introducing function notation works here, and I use it for grade 9s and 10s. The 10 applied class seemed ok with it too.
The next pattern we did in parters at the whiteboards.
Finally we did some practice with quadratic patterns. We noticed they grew differently, not going up by the same each time, but that the “change” changed by the same amount each time. We named that the first and second difference, and saw how we can find squares of the figure number, and groups of the figure number, and the constant visible in these patterns.
Again, we look at figure 10, there’d be 2 squares of 10 by 10, and then 2 left over as the constant. So the equation is f(10)=2(10)^2+2
and f(x)=2x^2+2
Students were starting to get the hang of it after a few examples.
For more examples go to visualpatterns.org
Cookie Towers
I was working with a colleague’s grade 9 class today, and the goal was to recall mean median mode and range, and to start looking at box and whisker plots.
I love to get kids hooked on data collection, and what’s more fun than a cookie tower competition?! We made rules of stacking one cookie on another, one at a time, using one hand only. Each group members took turns stacking towers until we had 12 data points in their charts.
Groups were getting pretty keen, and a little competitive. Some started experimenting with how to stack the cookies, or even “sanding” down the cookies to make them flat.
After some frustrations, some regrouping, practicing and strategizing, several students became very skilled!
Groups had their data tables complete, then we had to decide who wins. Each group found their max, min, range, mode, mean, and median.
We were able to see which group had the highest average tower, and which group had the tallest short tower, and which group had the highest median tower, and which group has the highest repeated tower height. Some groups won on several criteria, and other groups won on other criteria.
The next step we took was to introduce the idea of a box and whisker plot. We had the data in order to calculate the median, so it was a quick process to calculate the quartiles (the middle of the bottom half, and the middle of the top half).
We noticed how easy it is to read the values from the diagrams. If we had 100 data points, searching a list for the maximum, minimum, and determining the median would be tedious. If we were looking at the box and whisker plot we could quickly determine the values at a glance. We can then compare data sets with ease. It was a fun class, and students worked very well together to collect and present their data, and calculations.
Fractions Decimals and Percents
Today grade 9s looked at how to compare fractions, decimals and percents. After a brief introduction and brainstorming session about what we remembered from earlier years, each group got a stack of cards with fractions, decimals and percents on them. Groups were tasked to put them in order.
There were good conversations at each table about equivalencies, and how to make comparisons. Some groups looked at numerator and denominator sizes, and others made everything into decimals and compared.
All groups next had the mission of sorting all of the cards from all of the groups into one big number line. To start this, one member from each group brought their group’s smallest value to one end of our whiteboard and decided amongst themselves which was the smallest smallest value. Another member of each group brought their largest value to the other end of our whiteboard and decided amongst themselves which was the largest largest value. From there we started to put all of the cards in order.
We had some excellent conversations and some leaders emerged to place the benchmark values of 1/2, 1, 1.5, 2, 2.5 in place, then order the others within the sections.
Unfortunately today was super humid, and the tape was not cooperating, so cards were falling off the wall as we worked, and as I was consolidating.
We learned about how to determine the decimal value for ninths, e.g. 4/9=0.4444 repeating. We also saw how 0.9999 repeating equals exactly 1, because it is 9/9 which is 1. A bit mindblowing.
we saw how some fractions are improper, the numerator is higher than the denominator. We looked at how to write them as mixed numbers as well, and we saw how various representations can have the same value, e.g. 8/4, 2, 6/3, 200%
It was a great way to end a short week. P.A. Day tomorrow 🙂
Incredible Shrinking Dollar 3 Act Task
Today in grade 12 we explored exponential decay with the 3 act task of the incredible shrinking dollar by Dan Meyer. (Available here).
We watched the Act 1 video, noticed he was copying money but shrinking it on the copier. They immediately started to make models. Most were based on percent like this: A(n)=100(0.75)^n
We then started to wonder if the photocopier shrinks the area of the page to 75% of the original, or if it shrinks the length to 75% and the width to 75%. A quick google search confirmed that it is indeed a linear shrinking of length and width both.
we had a good conversation about whether this is linear decrease or exponential decrease. Some groups thought that you’d lose 25%, so after 4 decreases there’d be nothing left since 100-4(25)=0. Others claimed that it’d never disappear just get smaller and smaller and smaller since it would be losing 25% of the current length each time, so it’s losing less and less each time.
Finally someone asked what the dimensions of the original bill were, and we unlocked the “act 2” data where the dimensions are shared.
Groups worked on making tables, graphs and equations for the area decrease, and they looked at domain and range within context of the question.
Others worked on modelling how the linear dimensions would change, and how the dollar would shrink by length and width each time.
Groups got a lot of practice with a skill that is challenging, sorting through information given, like they may see in a word problem, and making equations that make sense which can model the problems.
We’re heading towards our 3rd test soon, and we’re getting lots of practice in each day.
Numeracy Challenge
Today I worked with a grade 8 class. We’ve been working over the last few weeks on developing some confidence to start and persist while problem solving.
We had such intense focus and good spirits of competition and challenge today. There were high fives and cheers, and students told me that they liked the problems we did.
The lesson sequence is from Peter Liljedahl. I didn’t post anything, but said it all out loud. Groups are given the answers to 5 equations, and they need to make the equations to follow the rules. Once a group was done with one set of numbers, I came around to check, and then give them the next set to work on. (See below)
In 45 minutes we had some groups get through 6 of the 7 sets. Most groups got through 5 of them.
Groups not only practiced their operations, but they also needed to use logic to determine which number is made with the fewest possibilities, e.g. in that first number set, 21 can only be made by 3×7 so we know that 3 and 7 cannot be used in any other equation. Students also practiced checking their work for small details like how many repeated operations they used, and if they repeated any of the values 1-10 in their work.
Proud of them for their excellent attitude and effort!
Using Fractions
Today we built on the fraction skills we started learning/remembering yesterday. We started off with multiplying a fraction by a whole number like 2(1/3) is 2/3 and 4(3/4)=12/4=3, and 5(2/5)=10/5=2. We started noticing that sometimes we can predict when the result will be a whole number and when it will be a fraction.
We practiced a bit, in table groups on mini whiteboards, using our fraction manipulatives to help us.
By the end of class we were getting comfortable with combining some of our skills: area and perimeter calculations but now with fractions.
Groups collaborated to solve these problems together, and there were some celebratory high fives and cheers when they got them correct. Some students are confronting their past views of themselves as someone who can’t do fractions. We’ve worked on building the skills bit by bit, and we’ll keep practicing some fraction fluency moving forward, so hopefully they’ll be less surprised that they got it right! I’m so glad that we are building skills and confidence with fractions, as they are key for many skills coming up in grade 9 and beyond.
Stacking Tower Game
My grade 11s are working on problem solving, and today we started with a game.
There are 5 discs of different sizes stacked on one peg. There are 2 empty pegs. Students need to find the minimum number of moves to get all discs to another peg. The rules are: one disc moved at a time. No disc can be put on top of a disc that is smaller.
The next challenge is to extend it to the number of moves needed to move 10 discs, and if a group says the minimum number of moves was 3267, how many discs did they use?
It was fun to watch students wrestle with the problem, and how they supported each other through the frustrations. Groups developed different ways to tracking moves, and modelling the situation.
One neat justification for the hypothesis that it is exponential growth is that it took 15 moves to get one stack of 4 transferred to a new peg. If we are considering moving a stack of 5, we’d have to first move the stack of 4, then move the bottom one, so that’s 15+1, then we’d need to move the stack of 4 again to get it on top of the 5th disc, so that’d be 15+1+15, which is 31 moves. If we’re working on moving a stack of 6, we’d move the stack of 5, then move the 6th, then move the stack of 5 again. That’d be 31+1+31, which is 63. We can see that the growth is close to doubling each time.
It was a fun modelling task which led to a fruitful discussion about whether the claim of 3267 being a minimum number of moves, and about whether we should consider a fractional number of discs.
Fractions Introduction
Grade 9s are working on exploring, and enjoying fractions. We got out the unit fraction strips and played a bit. We organized them, and noticed that when the denominators get big, the pieces get small. We noticed that it takes 4 of the “one fourth” pieces to make 1, and 10 of the “one tenth” pieces to make 1 etc.
Next we started exploring other ways to make things that are equal. Some made equivalences like 1/4=1/8+1/8, and 1/3=1/12+1/12+1/12+1/12. We saw how we can quickly add unit fractions in a similar way that we add x terms in algebra. 1/4+1/4 is 2/4, just like 1x+1x=2x. When the denominators are the same, we are able to add those similar unit fractions together.
We had some interesting combinations where we needed to make common denominators before we could add unit fractions.
One neat example was given where it looked like there was equality, but we explored it further and saw that it wasn’t really equal.
We made common denominators with each side separately, then simplified each side separately, and then eventually made common denominators with both sides so we can compare, and we saw that they were NOT equal, off by 1/120 which is tiny, but not insignificant.
Getting hands on experience with fractions and manipulatives can help us explore and communicate with fractions. It takes away some of the mystery about what fractions are, and lets us build to understand and connect to the operations.
From there we connected our understanding of area models for multiplication to multiplying fractions.
We looked at how to multiply unit fractions, non-unit fractions, mixed numbers, and then later how to divide fractions. We will keep practicing these skills and building our fluency and reasoning skills with fractions.
Math Club For Grown-Ups
We had another fun evening with Dr. Taylor and educators (and future educators) from our region. We looked at 3 rich problems, one involving quadratics, another involving exponent laws, and another involving exploring Napoleon’s Theorem and a neat demonstration of Desmos Geometry.
Many thanks to QSLMA for sponsoring our snacks! We looked forward to another meeting in June.
Word Problems
In grade 11 we are working on developing our critical thinking skills by tackling word problems where we don’t always have an equation. We are getting good at making tables and modelling, and also making equations and solving.
We are able to identify exponential growth or decay depending on if the base of the exponential is greater than or less than 1. We know that the rate of increase or decrease is related to what is added or subtracted from 1.
It takes time to build up our skills and confidence.
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