Linear Systems
Grade 10s are working on remembering lots of what they learned in grade 9 which will help as we move forward in grade 10.
We’ve done lots of patterning, and tables and graphs and equations. Today I gave groups a pattern rule to model with blocks, then to line up the blocks from each figure vertically, and place a dot at the top left corner. Once groups had one pattern modelled, I gave another pattern to do on the same axes.
We then put all the graphs up on he boards and consolidated vocabulary and connected graphical features to parts of the equations.
We talked about parallel lines never crossing because they have the same slope. We compared steep and shallow slopes, we looked at rise and run, and positive and negative slopes. We saw points of intersection, some we need to estimate. We looked at y intercepts. We also had one group that wanted a challenge to do one that’s not a line.
We’ll start to solve systems by graphing next.
inverses
We’re working on understanding inverses in grade 12 math. We’re exploring what operations are inverses of each other. Inverses undo what the original function does. It takes the output of the function as the input of the inverse, and will return as output of the inverse what was the input of the function. Basically we swap inputs and outputs. All input related (x related) things for the function, including domain restrictions, will show up as output related (y related) things for the inverse, including range restrictions.
We talked about how we sometimes need to restrict the domain of the function so that it will be invertible. It must pass the horizontal line test to be invertible. Some notation confusion cropped up while we were talking about how we’d need to restrict the domain of the sine function to make it invertible. We’d choose from -90 to 90 so it passes the vertical line test. The inverse of sine x is sin^-1x, which some students thought was the same as cscx. The misconception and somewhat confusing notation of the exponent of -1 sometimes representing inverse, and when placed on numbers represents the reciprocal. What adds extra complication is that in French (some students are immersion) the word reciprocal is translated as “l’inverse”. We’ll have to keep working on this!
To be sure of the differences we plotted all 3 graphs together on desmos.
Making Predictions
I was working with a grade 9 class today as they explored a neat activity that allowed them to make and test their predictions, explore some geometry, and get their hands on some math.
Step 1: make 2 strips of paper into loops. Tape the loops together so that the loops are at 90 degrees to each other.
Step 2 is to predict what will happen when each loop is cut down the middle of each strip. Groups talked, and came up with several different possible outcomes before actually getting out the scissors.
Step 3: cut the loops, and unfold to see what happens. (Spoilers below)
When one loop is cut and unfolded this is what we see.
This might be enough to see what will happen when the next loop gets cut.
That was quite a surprise! Groups were pretty astonished to see that it’s a square that is formed. We looked at the characteristics of a square, and how we know it’s a square.
Students were then prompted to see what changes to the initial conditions would create a rectangle that was not a square. It was interesting to see what they chose to change. Many had a good idea, and then tested the hypothesis.
The next challenge was to see how to change the initial conditions to end up with a parallelogram that was not a rectangle. Several groups succeeded with this challenge.
We had a good conversation about the properties of parallelograms, rectangles and squares, and will continue to debrief tomorrow on the topic.
This task encourages 3D spatial sense, making predictions, and deciding how changing initial conditions can help create different results. Through the task we could address properties of quadrilaterals, access many important vocabulary words, and lay the ground work for many more lessons and conversations.
Conjectures in Grade 9
Today I was working with a grade 9 class as they are starting up their number sense activities. We talked about multiplication as repeated addition and as the area model. We introduced the idea of commutative operations, and looked at how we can use doubling and halving to help us understand how to use our multiplying by 10 skills to be able to multiply by 5
Students tried to fill in as many multiplications they could do in 5 minutes, then we debriefed some of the patterns we saw.
Students were curious about the patterns along the diagonal. They saw that the blue diagonal went odd, even, odd even, odd, even. The adjacent diagonals are all even, then the next adjacent are alternating even and odd again.
They made some conjectures about what’s going on.
next we tried to see if it’s true. We used some big numbers to check if it’s still valid for numbers off our grid.
students chose 112×1011. We used area model to multiply the numbers.
we see that oddxeven=even yet again!
we also had a conjecture that oddxodd=odd.
Finally we noticed that when we draw all the shaded numbers like an area model the rectangles are squares.
After a brief BEDMAS review we ended with a round of max/min dice game, where we start with a framework, then madlibs style we fill in the numbers from a 10 sided dice. We were aiming for the biggest number we could get.
Exams!
We made it to exams. Best of luck to all who are writing. Thanks for joining us on our math adventures this semester.
Ambiguous Case Desmos
My class was working on exam review and came across an ambiguous case question. Here it is:
Angle A is 30 degrees, side b is 40 cm. Find the side length of BC that will allow for 2 triangles to be formed.
I’ve done some fiddling with desmos geometry and I think I have a pretty good illustration for this question
https://www.desmos.com/geometry/ac2yadxuui
The bottom point, on the circle can move, and that will show the red sides of the 2 triangles created. The black side is the case where there is 1 triangle, the right triangle, when the side length is 20. There will be 2 triangles until the side length is 40 and from there there will only be one again. It’s neat to be able to build and show these models to explore relationships in geometry.
Fractions with Cuisinaire Rods
I was working with grade 8s again this week and we’re building our fraction number sense. This time we worked through a sequence of steps from this document. The goal is to work with fractions greater than 1 and less than 1, and to understand that quantities are fractions of other quantities. We would give different rods different fractional identities and then solve using other rods.
For #3 if the orange rod and red rod together is one whole, make 1/2, 2/3, 3/4 and then #4 make 3/2, 4/3, 5/4 and 7/6, we could wrestle with the understanding of the numerator and denominator, seeing that in #3 the numerator is 1 less than the denominator, and in #4 the numerator is 1 greater. Then we use the learning to help us solve #5 which can’t be done with the rods.
We needed to first figure out which block showed thirds, then find a block as long as 2 of them, which would show 2/3.
We did something similar for 3/4. We found out which block represents quarters, then found a block that was the length of 3 of them.
Here’s the compilation of all of the fractions greater than 1 to help make some inferences about what makes them bigger or smaller. Some guessed that 7/6 is bigger than 3/2 because the numbers are bigger. The rods really help show that 3/2 is bigger because the unit fraction 1/2 is larger than the other unit fractions since the “one” is split into fewer pieces.
Next we assigned some rods different fractional values, and use that to solve problems. Here the yellow is 5/4, and out job was to find the rod that is 1 whole.
It was a good sequence to work through. We had some moments of struggle as we got used to the blocks having different names sometimes. The red could be 1/2 in one question or 1/4 in another, because the “whole” was changing. Hopefully associating the fractions with concrete objects help with the understanding, and that it can endure over the summer!
Fractions with Pattern Blocks
We did some work with fractions today in grade 8. I’ve been helping a small group of students build their skills this term. Today we worked with pattern blocks and concept circles. This first one had 1 yellow hexagon as 1, then I placed blocks in each segment and students needed to write the fraction. I made sure that we had some that were greater than 1, and some that could be simplified.
Next I put fractions in the segments and the students needed to build them. We branched out and did some adding as well. It was neat to see how quickly they were putting together the various fractions to make hexagons. We talked about how we can represent everything in triangles and get to the same answer using a common piece (common denominator).
We got partway through this last one, which was a bit tricky since we renamed 1 to be 1 red trapezoid.
Because we were used to building hexagons, it took a minute to get reoriented, and count trapezoids instead. It was a good challenge to change what “1” was. Using the manipulatives and combining them with fractions can help take away the fear of the fractions in their abstract state. Sometimes seeing fractions and knowing that they can be built, or drawn, or represented in another way can add meaning, but also help keep us calm when using them.
Problem Solving with Guests
Today we had the pleasure of hosting several mathy people from Queen’s. They came with some rich tasks to present to a grade 11 class and a grade 10 class
The first one was a question about maximizing area of a fenced field located against a long barn.
Students worked on different ways to model the problem, with a picture, and variables, and equations. They had different strategies for finding the vertex, some completed the square, some found the axis of symmetry, some used the quadratic formula and found roots and then the axis of symmetry.
It was great to see how actively groups collaborated together to solve the problem. The next challenge was a bit trickier. We needed to find the rectangle inscribed within a certain parabola with the maximum perimeter.
Students had challenges knowing where to start, and how to proceed. Many of them struggled with calling the length and width of the rectangle x and y, so that brought in 2 conflicting definitions of x. By labelling the width of the rectangle as “w” we were able to build an equation for w in terms of x and use that in the other equations.
Here’s some examples of works in progress. It was neat to see some groups base their perimeter calculations on different things: some counted x as the distance from the origin. Others counted the horizontal distance from the axis of symmetry to the edge of the rectangle.
We had a lot of struggle, but kept working through the struggle and made some progress before the end of class.
Making Mistakes
Today was a day that made me glad that I had made a mistake. We all make mistakes, and the response my students had today made me so happy. It was a small moment, but made me realize that my messaging around mistakes is being heard and adopted by my students.
Here was the question that we were doing. Students were working in small groups up at the whiteboards. We’ve done lots of work with sequences and series, and now we are practicing reading problems and interpreting what we need to do to solve them.
After working for a while, one group was huddled by my laptop. The projector screen was frozen, and the students had advanced my slideshow to see the solution to the problem to check their work. I went over to see what they were doing, and they alerted me to the fact that their answer and my answer didn’t match. They had looked through my answer and noted where I had gone wrong. They were discussing how a 710 in one line had changed to a 720 in the next line. I had a look, realized that it was a typo, and agreed that they were right, and mine needed a correction. They told me their answers so I could update the slide, and then they told me that it was ok, it wasn’t a conceptual error, just a typo. They told me that my process was correct and this wasn’t a big deal of a mistake.
We’ve worked a lot this term on learning from our mistakes. We classify them as inattention (loss of focus or concentration…an “oops”), computation (mistake with integers, order of operations, exponents etc), precision (sloppy work that’s hard to follow/read, incomplete communication for introducing variables or doing final statements, missing units etc), and problem solving (conceptual issues, getting stuck, not having a plan). Each test gets handed back with an error analysis sheet for students to engage with their errors, identify what type of error it was and to write the correct solution. Those pages get handed back to me to get checked, returned, and revised if needed.
Today my students identified the error, classified the error, realized it was an “oops” and carried on. They were polite and respectful while letting me know about the whole thing.
The conversations that followed were pretty rich. We talked about how this question might be marked, and how many marks the question should be assigned. We agreed that a small error like this should not be a large impact on the mark, that this answer would be a level 4 answer, or a 4+, just not perfect, but so close. We’d want to have maybe 5 or 6 points total so that a loss of 0.5 would still leave a high mark for the question.
I’m glad that I made an accidental mistake, and allowed for some time and discussion about it. I’m thankful my students were able to share with me that they are understanding that not all errors are the same, and that they know which ones to worry about more. I hope they will give themselves the same grace they extended to me!
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