Tarsia Calculus Review
We had a fun time in calculus today refreshing all of the derivatives we know.
This puzzle has a lot of different derivatives to do, and then evaluate at x=1.
There’s one issue with this puzzle. There is a repetition of the value 2. One student has offered to make some changes to the puzzle. There was also the suggestion of making the puzzle form a 3D object in the end if the pieces are taped together. I’m intrigued to see what they come up with! I’d like to use it next time I teach the course.
Intersection of 3 Planes
Today in Calculus and Vectors we were working on understanding the different ways 3 planes can intersect. We explored several situations as a class, and then each student was given a set of 3 planes to explore.
Students used matrices or elimination to decide if the planes intersected in a point, or a line, or didn’t intersect all together because they were either like an H (2 parallel and 1 not) or a triangular prism.
When they had decided, they cut out the 3 pieces of paper and slid them together to form a physical copy of their planes intersecting.
Having a different colour of card stock for each situation helped students find others with the same case, and they could trouble shoot their intersection calculations.
It was a nice way to end the week!
Math is Everywhere!
We are working towards our culminating task in grade 9, which involves exploring the math that we see in an image, or the math that the image inspires.
Today we debriefed the example that we tried on our quiz.
We talked about volume, area calculations and comparison. We talked about questions like “if you know the volumes are the same, and one radius is 3 times the other, how are the heights related”, we looked at classifying shapes by number of faces and showing that as a ratio, fraction, decimal, percent. We talked about how tall a tower we could build. We looked at Euler’s formula “faces + vertices=edges+2” and examined which solids fit that rule.
As a class, we then split up into groups and brainstormed questions we could ask about these images.
We had very creative ideas! Many students wanted to calculate volume of ice cream, or surface area of the cone, or look at the angles of the palm trees. Some wanted to fill the ferris wheel with ice cream and wondered about how many cones it would take. That led to another group wanting to fill the slide with ice cream! Lots of creative math to tackle.
We’ll keep working on creating mild/medium/spicy questions as we head toward our culminating task at the end of term.
Algebra Review
Grade 9s are deep into algebra review today. We dug out some math Tarsia Puzzles and had some fun putting them together.
We had 2 different puzzles to do, which kept most students actively involved and discussing math for most of the period.
Puzzle had fractions and order of operations work, vocabulary, exponent laws, different forms of linear equations, distributive property and equation solving.
I made these tarsia puzzles in a word processor, but in the olden days I made them by hand, copied them on card stock, and cut them out. Some groups tried one of my puzzles from years ago. They had fun putting it together.
I was impressed at how they were working hard, and having good discussions with peers, and reminding each other about what we’ve learned this term.
Here are the .pdf copies of the puzzles.
Remember to print them single sided, and if doing more than one puzzle in a class, having different colours for different puzzles is helpful. When I am very organized I’ll give groups sandwich bags, and have them put their puzzle pieces in the bag to either bring home to practice, or for me to keep and use with future classes.
More about Matrices
We dove in to matrices today and solved a system of 3 equations in 3 unknowns.
We are getting good at making the bottom left corner into zeros, and now we know how that will help us solve the system.
Once we have reduced our matrix to this point, we can jump back into equations to arrive at the final solution.
Something new this year that I’ve been exploring is the desmos 3D graphing. It is powerful to see that this solution is actually the intersection point of the 3 planes we started with.
here is a link to the desmos graph so you can interact with it and try it out.
Understanding Bias in Algorithms
Grade 9s looked at several articles today and we practiced marking up text and summarizing. We talked about strategies good readers use to interact with the text, and then we broke off into smaller groups jigsaw style to read and learn from several articles.
One article talked about pulse oximeters registering different values depending on the skin’s melanin content (because melanin is interpreted as higher hemoglobin counts and higher oxygen content of blood), so algorithms and devices have a built in bias. Another article talked about how AI can be a big help, but if it is trained on biased materials, it will reinforce and amplify the bias. Another talked about how data security is important, and how people should consider what health information is entered onto apps, using the example of period tracking apps in the United States where abortion is becoming more illegal in certain states. Another talked about data security of apps like tiktok where foreign countries can have influence and access.
We had some interesting discussions, and some students are rethinking which apps they are using, and who may have access to their information. We also recognize the importance of having a diverse group of people represented at the coding table!
Math Is Everywhere!
In MTH1W we are working toward our exam and culminating task. Our exam will be similar to our tests, a menu of mild medium and spicy questions covering the many strands of the course. Our culminating task will be quite different from that. It is something we explored 8 years or so ago, but are going to try again, and streamline it differently this time.
To prepare for our culminating task we are practicing our skills of noticing and wondering.
Today my class looked at this image, and brainstormed the different ways that they can see math in this image, or the different math questions this image makes them wonder about. We also prompted them to think of the information that they’d need to solve the questions that they are wondering about.
It was interesting to hear the students’ ideas. Many were interested in the circumference and area of the circle. Some were curious about the proportion of the image that was green compared to red. Some were noticing angles in the circle from the people, or their shadows. Others were curious about the basketball and how to calculate things about it, or how it compared to the circle on the ground.
As we approach our culminating task we will practice the skills of asking good questions about the math that we see in various images.
Our culminating task is inspired from an OAME presentation years ago, where students are presented with an image, and they are tasked with creating good questions about the image. The questions should fit within several key strands of the curriculum. For us this year it will be measurement/geometry, numeracy, and financial math.
Day 1: Students will be in groups to examine the image, and create a mild/medium/spicy question for each of the 3 strands. They will also decide on what information they’d like to ask us for. While in groups they can brainstorm ways to solve the questions that they are creating. Groups will submit their questions, and their information request at the end of the class.
Day 2: Students will be working independently. They will receive a copy of their group’s questions, along with the answers to the information requests that they submitted. Students will choose to answer the mild/medium/spicy question for each strand, and submit their work at the end of the class.
Students will be evaluated on the depth of question that they asked as well as the answers that they determine using the provided information.
We are trying to value the group work that we do all the time in class, and the practice that we have been using frequently of “noticing” and “wondering”, and asking for more information. We hope that this culminating task will allow students to show skills that they can’t show on an exam.
Welcome to the Matrix
In Calculus and Vectors we are working on solving systems of equations to tell us points of intersection of lines and planes. We will need to solve a system of 3 equations in 3 unknowns. To get us geared up for that challenge we looked at the 3 act task “tech weigh in” (sequel).
For those new to 3 act tasks, there is usually an intro video to get us thinking. We watch the video and notice things and wonder things.
Act 2 brings more information, often at the request of students. They know that the first video didn’t have enough information in it, so the next videos provide more details that can be used to calculate something. In this case, it is the mass of an iPad, a Chromebook and an iPad mini.
My students knew that for 3 unknowns they needed a lot of information. They knew to keep asking for more scenarios!
3 unknowns need 3 equations, so here is the 3rd scenario.
Now that we have 3 situations, we can create 3 different equations to solve for our 3 unknowns.
Some groups went directly to elimination. This is a great task to use as an intro to elimination as the equations seem to invite that solution method, and students intuitively take that path.
Others use substitution, but in the end we all get to the answers no matter the method. We look to streamline our communication though, and we notice that some methods are more direct than others.
In the end we can watch the 3rd act to verify our answers.
We are working towards solving systems of 3 equations and 3 unknowns because we will be looking for the intersection of planes in the near future, using the Cartesian equations in the form Ax+By+Cz+D=0.
At the end of the lesson today we started to look at what a matrix is, and how to write them. We will practice using them in the coming days to keep our calculations organized so we don’t go around in circles like some of us experienced today!
Catapults Day 2
We spent time today calculating the x intercepts and the vertex for our catapult trajectories.
We made graphs and made predictions about where to place the catapult if we were going to aim for a target on the floor, or if we were aiming for a target on a chair.
One group even tackled the situation where we’re catapulting a pompom onto an angled surface.
By using their understanding of slopes and angles and trigonometry they used the tan ratio to solve their problem, and then it continues to be a quadratic problem which we’re getting really good at solving.
Our individual portion of the culminating task is coming up next week, where we will do similar analysis using a new set of data.
Finishing our beading
We paused our beading progress for a while, but were working hard today to finish them up, and add glue and leather. We needed to weave the thread back and forth for about a centimetre which secures the threads and gives a surface to attach the leather end pieces and ties.
We glue the threads to keep them stabilized, and then we glue the leather on the ends.
We’re putting together a document to share our photos and explain the math that we see in the designs.
We have learned a lot through doing this project, both about beading and about patterning and about fractions and percents, and symmetry. We also learned about managing our time and emotions and persisting with a task over a period of time.
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