Ramp Race Day 2
Today we got to work with our ramps, finishing up all of the data collection and averages.
We looked at the data to see if it seemed linear or non-linear, and tried to model the data.
We used Desmos to help with our models. From the data for most groups a linear model seemed to work.
There are a few challenges we’ve noticed. Some of our boards are not straight, so for some, the tennis ball will stop part way down the ramp, because the ramp momentarily goes up. For next time we’d try to get thick boards that wont curve, and to check that they are as straight and flat as possible. We also noticed that the timing can cause some challenges. Since the ramps are short, a small error in timing will be a big percentage of the time value that we get, which is likely why our data is looking linear-ish. Having only 4 points for each ball also might not be enough to see a real trend. In any case, we’re going with the idea that they might be close enough to be linear.
Groups are using Desmos to get lines of best fit, and then Monday they are going to make predictions about how long it will take for the ball to go down the 8m ramp. We’ll hopefully be able to get the final race done on Monday as well, and we can use the final results to evaluate our models to see if the real world race is modelled by linear equations, or whether it might be more complex.
Many thanks to Pathways to Education for sending a grad connector to help with some of the groups. This is a task that takes a lot of coordination and checking in. We are much more on task now than we were in September though which is a great development.
Ramp Race Day 1
Today we are starting our culminating task inspired by Al Overwijk (blog)
We have a 3 degree ramp at our school that is 8 meters long. We are going to race 5 different balls down the ramp. To predict which one will win, we are modelling the ramp in our classroom with 6 different lengths of board.
We had to calculate how high up the rise should be. We had to figure out how to elevate the board, and keep it stable.
Once the ramp was made, we chose 4 locations along the board to start rolling the ball. Each ball will be rolled 3 times from each location and averaged.
We’re busy filling data into our table of values.
We’ll keep working tomorrow, to get through testing all 5 balls at each location, then we’ll do some analysis to figure out how to model the relationships.
Heat Loss House Day 2
Day 2 of the project includes building the houses and using the insulation choices from day 1. Insulation choices were fibreglass, rockwool, or foam, and window choices were single, double or triple pane.
Groups worked together to get the houses built. The kits included sensors and a “furnace” walls, doors and windows which all clipped together with 3D printed parts.
In the end, the houses were set up and turned on at the same time. They will run all night with a calculator tracking the sensor output.
We look forward to seeing the results tomorrow.
Many thanks to St. Lawrence College for providing this opportunity for our students.
Barbie Bungee
Today we bungeed Barbies! This is a task I’ve been wanting to do for ages, but never did. I went to the thrift store and dollar store and got a variety of characters to bungee.
We started off with each team getting 7 elastics and then we started to test the stretch to figure out how many elastics would be needed to bungee down the 537cm from the top of the atrium stairs down to the floor. The goal was to give the character a thrilling and death defying ride.
We had groups try different strategies. Many did trial and error with different number of elastics, checking how far the bungee would go.
groups put data in tables and used proportionality to find their answers.
Groups used desmos to find averages, and also to solve the problem of how many elastics are needed.
Finally we bungeed the Barbies!
We had quite the audience for some of the plunges. It was a great success, with only one ending up on the floor.
We had a lot of fun on the snow day and working through some math!
Substitution
Today in grade 10 we worked on solving a system of equations by substitution. We started off with a tug of war puzzle.
4 acrobats vs 5 grandmas are a tie
1 dog vs 2 grandmas and an acrobat is a tie
we need to figure out who will win with 3 grandmas and a dog vs 4 acrobats.
Students worked out a few ways to solve the problem. I wish I had photographed them!
One method had them assign a value for the strength of each grandma and each acrobat. They decided that if a grandma had a strength of 20, then 5 grandmas have a strength of 100, and if they are tied with the 4 acrobats, each acrobat would have a strength of 25. They figured out the strength of the dog by substituting in the values into an equation: dog=2(grandma)+acrobat to calculate that a dog has the strength of 65. Then they could calculate the final situation to see that the side with the grandmas and the dog will win.
Another method was to make equations. The idea of the tie means that both sides are equal, so:
5G=4A
1D=2G+1A
when we get to the final step:
3G+1D, we can replace the dog with (2G+A)
3G+(2G+A)
5G+A
but we know before that 5G=4A so we can replace that too
4A+A
which is 5A, so now we have a final show down with the equivalent of 5A on one side, vs 4A on the other side, so we know that the 5A side (the one that originally had the dog) wins!
We then did some substitution without a context, to practice.
We made the connection to x=-1 and y=-1 being the intersection of the 2 lines using desmos.
Next we tried an interesting case:
this case led to a discussion of what happens when the final line is 2=2 and what does that mean? It means “yes” because it’s true. We are looking for the intersection of 2 lines and math says yes.
these two lines are identical and intersect at all points, so all points on one line are on the other!
We did a lot, and had a fire drill too!
Heat Loss House Day 1
We are so lucky to have a group from St. Lawrence College to come and run a project with our grade 12 college math class. Day 1 involved making some choices about how to insulate a house on a budget. We also did calculations about mortgages for the land and construction loans.
Later we will build the houses to the specifications chosen and test them overnight to see which ones lose the least heat.
Fraction Brainstorming
I worked with a colleague today to look at various approaches to conceptualizing fractions, and adding & subtracting, multiplying and dividing them.
We talked about area model and how to visually represent adding and multiplying fractions, but then we decided to dig a bit deeper and look at using concrete tools like cuisinaire rods to help us.
The plan we came up with is to start with a concept circle with the rods in place and students need to determine how to write down the value of each rod/combo. Using 12/12 as 1 is helpful for this task.
Next we’d do a concept circle with fractions written down, using quarters, thirds, halves and sixths, and the central 12/12 as 1 and have students build the fractions with the rods.
Next we thought that we could build equivalences with the rods, making combos that are the same length, and then writing them down in math. We know that if they are the same length they are equal, and if it’s a combo you can add up the various blocks. You could get at subtracting with an overshoot and return strategy using the rods as well.
After that we started talking about how to best show fraction division. I shared my recent new learning that fraction division can be done straight across just like multiplication can, and if we get a common denominator first then it simplifies beautifully.
e.g. 3/4 divided by 5/7
is the same as
21/28 divided by 20/28
dividing the numerators gets us 21/20 which would be over the quotient of 28/28 (which is 1) so the answer is 21/20. Now this isn’t a way to use visuals or manipulatives, but it sure is cool.
I was recalling the mini feud between Howie Hua and Mr G (videos are on tiktok and instagram, not sure how to link them here) about various fraction division methods.
This is something fun to dive into as a team. So glad colleagues are keen to ask and try and push me and my thinking in new directions!
I’m glad to have spent some quality time with the book rethinking fractions this summer. It has been very helpful.
EQAO practice
It was a snow day today and grade 9s were busy trying practice tests for EQAO
We worked through test 1 of practice questions. We’ll have to do a bit more practice and a bit more learning before we take the test for real later in the month.
Snow Day Math
Today there’s a storm and busses are cancelled so we had fewer than normal students. In grade 10 we worked on a task about rectangles. Each group needed to choose a number between 30 and 70 and that would be the perimeter of each rectangle made. They needed to make a list of all the possible rectangles, and then make the rectangles and calculate the area.
Next we cut out the rectangles and used them to graph the relationship between side length and area.
It was neat to see the quadratic relationship show up, and how the optimal area would be the square, which for a perimeter of 42 would have side length 10.5
It was a pretty fun task to do on a snow day, first day back from holiday.
Group Graphing
This lesson was inspired by reading this post on Al Overwijk’s blog.
We have been working on some quadratics on this spiral of grade 10 applied math. We have used algebra tiles and expanded and factored, and talked about the area model, and then explored the graphs on desmos. In our earlier spiral we talked about linear and non linear data and tables of values and trends that we see in the tables and graphs.
This task was a lot to take on for a Friday in December, but we did our best! I was thankful to have another adult in the room to support and keep the groups on task and moving forward. With a big busy class (24 students with a lot of needs) it would be a challenge to do solo.
As students entered the class and got situated into their random groups, each student was given a card with a decimal number from -3 to +3 and we started out by ordering ourselves in a row, in order. Next each student got tape and we made a well spaced out number line on our board. We talked about using integers as our ruler, and that the 0.5 is in the middle of 0 and 1, and the 0.25 is between 0 and 0.5. We are good at spacing out halves, quarters and 3 quarters which is what was needed for the functions I chose to graph.
We next got split up into 7 groups. Each group got assigned an x value to plug into each of the following equations. We used x values of -3, -2, -1, 0, 1, 2, 3
It took groups several tries to substitute and simplify using order of operations. The biggest challenge for some groups was correctly copying down the equation (many minus signs went missing), and then the issue of squaring negative numbers, or adding before multiplying caused challenges as well. It was very helpful for me and my colleague to have an answer key to help make checking work go quickly.
We’re working on showing steps and communicating our thinking. It was helpful for me to see the misconceptions so I know where to go from here next week when we consolidate the task.
Part 2 involved number lines. Each group got a number line with zero marked out. I cut strips from grid chart paper to make the number lines. They marked their points, then got the right colour marker and drew coloured dots on their number line.
The next step was to assemble the strips together. I taped them back onto a sheet of grid chart paper to help with spacing. I’m rethinking the scale of this, as it will be tricky to consolidate from as it is kind of small. Perhaps I’ll work from a projection of the image of the group graph to consolidate.
It looks a bit messy at first glance, but if you look carefully at one colour dot at a time, you can see the lines and curves take shape. If I were doing it again I’d have the students count each grid line as 0.5 to help us spread out the graph a bit more.
We will next explore the graphs, and use the tables of values and equations to identify key points of each equation, and features of the graph (slope, direction of opening etc).
We were mostly all engaged for most of the period, and even had some visitors in the room (grade 8s on tour, 2 v.ps and a colleague who popped by because it looked interesting). Many thanks to Al for writing his initial lesson study. I’m going to try this task with grade 9s and linear equations in various forms next!
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