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.
Experiments in Grade 8
I’m working with a grade 8 class for the next little while, once a week. Today they were doing experiments and comparing theoretical and experimental probability.
It was interesting to work with the students and watch them work through different tasks involving dice and also marbles. Looking forward to getting to know the students better over the coming weeks. Thankful for the opportunities!
Distributive Property
Today we looked at how to multiply polynomials. We modelled several questions, and showed how the “groups of” model and the “area model” work together. Example 2(3) is 2 groups of 3, but also a rectangle wth dimensions 2 by 3. We applied that to multiplying polynomials. Here’s 3(x-2) shown in 2 ways.
We practiced several similar examples and noticed that we can multiply the number in front by each term within the brackets. Now we have 2 ways to approach the questions.
When we have (x)(x+1) we run into a problem with the “groups of” method. We don’t know how to represent x groups of something. We need to shift to the area model, or distributive property where we multiply x by each term in the brackets.
We had some breakthroughs with the visual representation, and we talked through how (5)(5) is 5 squared so (x)(x) is x squared. It’ll take some time to solidify the skill, but to practice we did a puzzle in small groups.
I love the puzzles from this site. There are so many neat ones.
we’ll keep working on more algebra next week.
Update: Beans and Ropes in Grade 10
The grade 10 class I’m working with has been growing beans and measuring them each day.
some groups have a lot of beans that are sprouting, and others are still waiting and watering. We are measuring the height each day and will do some graphs of our data after a few more days of measurements.
The other task we worked on last week was a knot tying experiment. Today we consolidated that task and worked on graphing, making equations, and interpreting the rate, initial value, and point of intersection.
We co-constructed an exemplar of a graph, and each group made progress on interpreting their data. We made connections between the thickness of the rope and the amount of rope consumed per knot.
We found the average rate of change by looking at the length change for 10 knots, and then divided that number by 10 to get the average rate of change per knot.
Next week we’ll be working on some new data collecting tasks and introducing algebraic methods of solving.
Snow Day!
Today we had some fun with the small group of students that were at school on the snow day.
We did an impromptu trigonometry lesson to introduce how to use clinometers.
We calculated how high our atrium wall is. We got pretty accurate numbers despite the many opportunities for human error.
next we did some testing with a few elastics, and then used proportional reasoning to estimate the number of elastics we’d need for a safe but thrilling bungee jump.
Finally we bungeed several figures off the edge. We had 1 crash on the floor, 1 was thrilling, and 1 was a really boring bungee.
students had fun, teachers had fun, we did some math and learned some new skills, and went home with a story to tell from a very exciting snow day.
Introduction to Algebra Tiles
I was working in a colleague’s grade 9 class today and was introducing algebra tiles, an excellent way to visualize and conceptualize algebra when students are learning.
one of my first steps in introducing algebra and the tiles is to make some art. Students get very creative!
Our next step is to remove all the zero pairs to simplify their polynomial art to the simplest forms.
I posted some pictures of student work on the whiteboard, and one by one students practiced being brave and coming up to the board to cross off the zero pairs until we got it down to a monomial, binomial or trinomial.
We practiced vocabulary and what constants, variables, coefficients and exponent are.
Next, we made 2 trinomials (x^2-2x-3) and (-x^2+3x+3) and then we added them up, and simplified by removing zero pairs. We ended up with a monomial, 1x
The challenge problem that followed was to create 2 binomials that add up to a trinomial, and 2 trinomials that add up to make a binomial.
We had so many possible answers.
It was refreshing to hear students making connections between the tiles and algebra, and realizing that they could always use them if they need them. Some students were encouraging their peers and helping them to understand.
best quote of the day: “The tiles really help bro! Give them a try.”-grade 9
Inverse Functions
Today my grade 11s were working on inverses of functions. The activity requires a single hole punch, and some creative folding. Here it is to download if interested. I was inspired by a post on social media which I have now misplaced.
Step 1 was to draw the line y=x
Step 2 was to fold along the line y=x
Step 3 was to use the hole punches to make holes along the given function.
Step 4 was to join up the new dots to make a different line/curve.
We were able to debrief a lot of things through this task. We saw that the lines have coordinates thar have reversed values, e.g. (1,3) will correspond to (3,1) on the new graph. The two are also reflected across the line y=x. We can also see that the first line has a slope of 3 and a constant of 0 so the output is input times 3. For the 2nd graph the slope is 1/3 and the constant is 0 so the output is input times 1/3 or input divided by 3.
We can see how the function and its inverse will undo each other. The operations of times by 3 and divide by 3 are inverses. If the output of the function is used as the input of the inverse you’d get the initial input back again.
We needed some creative folding to get the hole punch to reach the dots sometimes.
This one has a function of y=1/2x-5 and the inverse is y=2x+10
we were able to switch x and y in the function and then by using opposite operations we could isolate the y. Y=2(x+5) which is the same as the inverse if we expand.
We saw that some functions are their own inverses. We showed how that works with algebra.
We also saw how this function is not invertible. If we follow the process we don’t get a function in the end. The original function needs to pass the horizontal line test for it to be invertible. We can restrict the domain of the function to make it invertible.
I think we have laid the groundwork conceptually, visually, and algebraically and we will be able to practice more as the course continues.
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