Welcome Back to Grade 10
We’re back in action at KSS today! We started off with some introductions, and got to work. In grade 10 applied math we filled in an introductory page with a twist. Students need to graph how they felt towards math over the years. This was inspired by work from Liesl McConchie who posts here on Twitter.
I can get an idea of graphing skills and also see how a student’s math journey has gone. It’s a spring board for future conversations. It’s followed by some questions about their math experiences and how to help them do their best.
Tonight I can read and learn more about my students, and be more responsive tomorrow based on this information.
From what I saw as they were graphing their experiences, I made a choice to start my grade 10 applied class with some dot talks to have an entry point for all students, and a chance to see how we all can think about problems in unique ways and we can learn from each other if we share our ideas.
Dot images from Number Talk Images are flashed on the screen for 3 seconds, and then students are asked how many dots there were and how they knew. For this one, some students used “fast counting” since there weren’t so many dots. Others saw the dots in groups and showed how the groups helped their counting. We had fun annotating the picture using all the colours, and representing our thinking with math as well.
Students engaged with the task, and were keen to share their ideas.
We next looked at a which one doesn’t belong thought routine, where we need to brainstorm reasons why each quadrant might not belong with the others. We used lots of vocabulary about measuring sides, and angles, and counting angles, and colour of the shape, and if it was sitting on a horizontal side, or on a corner.
we then had a look at some patterning from grade 9, where we drew figure 0 and figure 4 and then noticed how the pattern was changing each time.
We determined that this pattern grows by 2 each time, and it starts at 1 for figure 0. We used this information to figure out how many trees are in figure 10, figure 100 and figure 1000. We’re going to do more patterning tomorrow.
The highlight for some students was our game at the end of class. We stood in a circle and counted around, with each student saying a number in order. When we got to a multiple of 5 though, we had to say buzz instead of the number. Anytime we missed saying buzz, or said buzz instead of the number, or said the wrong number, that person had to sit down. We’re pretty good at identifying multiples of 5 because they end in 0 and 5. This game can evolve (and will) over time to include a few pieces of information to keep track of, but for today we started with just one thing.
It was a fun start today! We’ve got a big busy group, and we’ll make good progress this term.
Meeting Grade 9s
Today Grade 9s came to school and met their classmates and student leaders and walked through timetables meeting teachers and learning more school skills and hearing about opportunities available.
My session was the how/when to write an email session, which was pretty fast! Students sent me emails telling me about their favourite number and why it is their favourite. After that we had time to do some math.
Glad I have some problems in my back pocket. We looked at the 4 4s problem, figuring out how to use 4 4s, and the operations we know (including any exponent), and making expressions that equal 1,2,3,4,5,6,7,8,9,10
We got several expressions figured out. We were working in small groups, practicing some of the norms in our classroom.
I had not anticipated having time to get into any math at all today, so this was a lovely start to the term. We start for real with students next Wednesday.
Summer Institute
Today I had the pleasure of leading a session at our summer institute. We did some activities together and looked at some strategies to build brave spaces in your math classroom. Here’s a copy of the slides.
We looked at a few tasks which had easy entry points but allowed for discussion and extensions. We did a dot talk, which one doesn’t belong, and then got up to the boards to try some tasks together.
The Unusual Baker is a problem that works on understanding of fractions, and different ways to partition a square. We looked at how adding a few extra lines to the images can really help us understand the pieces better. We talked about how doing this with students, they could model the cakes with construction paper and cut out the different representations of one fourth, and show how they can be cut up and rearranged to be equivalent.
This image was a springboard for a lot of discussion. People talked about how they could see rectangles and squares, and how there was a pattern that the regions were divided in half each time. Groups took different directions, some looking at the fibonacci numbers and golden ratio and spiral, others looked at how the fractions had denominators that were all 2 to the power of something. Another group made a connection to binary numbers being base 2, and another explored the sum of the fractions will all add up to 1 whole square. Several groups were upset that the drawing was incomplete and knew that the middle rectangle needed more subdividing…but where to stop?
It’s neat to see how many different directions a task can take!
The final task we looked at was stacking Cheerios, which is fun on many levels. We get data collection, an understanding and comparison of rate, solving a linear system, exploring translations of lines by increasing the y intercept, and also we get a bit of head to head competition.
The activity is here, so you can print it off and try it yourself. Different groups will need more or less scaffolding.
Many thanks to the 30 teachers who took time in the summer to come and do math together. It is great to see such enthusiasm at the start of term.
Review
I was helping in a grade 10 class and saw a neat strategy for review. Several topics were written as titles on the boards and students circulated adding in their strategies for achieving the goals.
how to factor
how to find the roots/x intercepts/zeros of a quadratic
how to find the vertex
how to go from factored form to vertex form
how to go from standard form to vertex form
how to go from standard form to factored form
This was a work in progress as I snapped these photos. Students were collaborating, and adding their strategies to those listed, and editing what was written if something was missed.
Students were then working on a review package with a great deal of support material on the walls to help. This was a nice way to reactivate prior knowledge.
Exams are coming soon, it’s a good time to be reviewing!
Calculus Review
Today we did a calculus review puzzle. Here is the file.
It’s a nice review of all the derivatives we did a while ago. We’ve got a few days to review before our exam on Tuesday. Good to see so many smiling faces working hard until the very last day!
Problem Solving Workshop for Teachers
Today we had a great time solving math problems after school with teachers from various schools in our region.
Many thanks to Dr. Peter Taylor from Queen’s University, and his students for bringing us some great problems to explore.
The first problem was about 2 concentric circles that had a special property. The segment AB which is a cord to the outer circle, and tangent to the inner circle has length 10. We needed to find the area of the outer ring.
We had lots of ideas, and approaches. Some used trig, others substitution and pythagorean theorem. Some made assumptions at the beginning, others didn’t. It was very neat to see the various strategies all get us to the same answer in different ways.
The next problem was about mirrors. 2 mirrors are placed with a 50 degree angle between them. If you sit so your eyes are 1 m from where the mirrors meet you can see 6 images. The goal is to determine how far from each of your images you are.
We set up the mirrors, got to measuring, and then had a lot of fun mapping out light rays.
We needed our knowledge of reflection from science, and triangles from math. We didn’t get to the answer for this problem, but we have lots to consider and think about. Sometimes we think that math needs to happen in 75 minutes, but for some problems they need more time. Time to make connections with other math/science, time to think of different approaches, time to do the calculations, or time to make a good plan, or a generalization of patterns that you see. We need to give ourself the time to be playful with the math, and enjoy the collaboration and the journey to the solution.
Often teachers are caught up in the busy seasons of marking and setting exams, and we lose the spark of excitement that comes from working together and doing the math. Today was a lovely refreshing evening of collaboration which I hope we can do again in the fall.
Many thanks to all who came, and to Dr. Taylor for bringing us some great problems to try.
Paper Planes
We looked at intersection of planes this week in calculus and vectors. To test our knowledge groups got a system of 3 planes and used their skills to determine if the system was consistent (there was a point/line/plane of intersection) or whether the system was inconsistent (no point/line/plane of intersection) and how.
The next step was to cut out their system, and slide the pieces together to see how the planes intersect.
We had 4 groups and 4 different results.
It was a fun way to visualize what our matrices told us!
Problem Solving Review
My class of grade 10s has come a long way since February, in terms of math skills, confidence as problem solvers, and our sense of community in our classroom.
Today we started looking at some interesting problems that mix a lot of ideas together. It’s a nice way to end the term, and show them how far they’ve come. We also get a good exam review by doing problems that involve many aspects of our course.
To start the class we looked at what types of math we’d learned, and what are some of the key words or equations or strategies they might need.
After listing all this we started to work on our first question.
We needed to do a bit of brainstorming with our groups. What type of question is this? We noticed key words, and units. The word angle made us think of trigonometry. The squared units made us think of area and quadratics. The 2 equations for perimeter and area made us think of substitution. We know we need to make a diagonal line, and we know diagonals join opposite corners. We need a good picture and some variables and some equations!
what happened next was magical. Groups got to work. Independently, and collaboratively. There was such a hum of activity in the room.
It’s a pleasure to be able to stand back and watch them work, but also it’s great to interact and see and hear what each group is thinking. There are good questions being asked, and students are helping each other to know what the next step is. They are checking to see if they are on the right track, and reassuring each other. They are checking to see if their work is reasonable, and what’s even better is that they are having fun with each other and also with the math.
In this particular question many students questioned if we are looking for x intercepts or for the vertex. There was good conversation about what to do with 2 positive x values as roots to their equation. We talked about if we could use sine cosine or tangent or if sine law or cosine law were better.
we moved on to look at some questions with systems of equations and percents
Here’s a group mid trouble-shooting…they are figuring out why they ended up with a negative value. They figured it out…it all had to do with how they set up their equations to start with. Hopefully by working through these situations now, if it happens on a test they will know what to check for and how to correct it.
The next problem we did was one involving rates and trigonometry.
I added an extension for groups who finished early, asking how far the boats would be from each other after 45 minutes. This allowed us to enter into a nice discussion about similar triangles.
We had almost everyone engaged in problem solving for almost all of the class today. We’ve built up this stamina over time. It’s a process, but at the end of term it’s great to see how far we’ve all come.
It felt good today. Hopefully this can continue into tomorrow’s review as well.
Exam review begins
We’re blending our final rest review into our exam review. This is one of the joys of spiralling the curriculum material. At the end of the year we are doing exam review for quite a while, and everything has been covered recently, so there’s less of an issue with forgetting major concepts.
Today we looked at solving some word problems.
We are working on connecting our knowledge of quadratics to the context of word problems. We’re remembering lots about different forms of quadratics, and what we can read quickly from equations, and what needs a bit of calculating.
It’s great to hear the students working together and helping each other. We have so much expertise in the room!
We looked at a trigonometry word problem too where we needed to draw our own picture.
We’re getting better with translating words into pictures, and making a plan to solve.
We’ve come a long way since February!
Catapult Results
Today we looked at our catapult results for the entire class.
It was interesting to see how varied the results are. Some groups had to work in the atrium to have a bigger height, since they kept hitting the ceiling. It had the farthest range, and the highest vertex.
Other groups with closer x intercepts had much more repeatable experiments.
We have practiced doing lots of calculations to be ready for our individual portion of the task on Monday.
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