Bucket of Zeros
Today I had the chance to work with a grade 9 class on integer adding, subtracting and multiplying.
The bucket of zeros includes understanding of zero pairs.
When adding, the tile are all combined, and zero pairs identified, and the answer will be any unpaired tiles at the end.
Subtracting involves using the bucket a bit more. There are animations in my slideshow which show how groups are taken away, including from the bucket, and how by using zero pairs the result is simplified.
The bucket can also be used for questions like -3(2), which can also be written as 0-3(2), which would mean we start with zero, then remove 3 groups of 2. The addition of the 0 in the front of the questions helps to ground us with our original bucket of zero.
It is with this type of task that I can explain to people why a negative times a negative is a positive, or why a negative times a positive is a negative. It brings conceptual understanding rather than just memorizing some rules.
It was great to see students engaging with manipulative tiles at their desks and building confidence with this new thinking routine.
Locker Task
Today in grade 9 we challenged ourselves to solve a problem. We experienced productive struggle and persevered for the entire 70 minutes of class time on this one problem.
Here is the prompt:
There are 100 lockers and 100 people. The first person walks past and opens each locker. The 2nd person walks past and closes every 2nd locker. The 3rd person walks past and changes the state (open/closed) for every 3rd locker. This continues with the 4th person changing the state (open/closed) for every 4th locker etc. When the 100 people have walked past the lockers which lockers will be open?
Students worked on representing the problem in a manageable way.
some used 1 and 0 to represent open and closed. Others used mark and no mark to represent open and closed, and other groups used colours to track the open and closed lockers.
It took us a long time to go through the task without making mistakes. Some groups decided to shrink the problem and work on a case of 30 lockers and 30 people.
We eventually landed on the following open lockers:1,4,9,16
From there, some students saw that the numbers increased by increasing amounts in a predictable way. Others saw that these numbers are all square numbers.
The next question is: why do these lockers stay open?
We talked a bit about how we used our multiples to open and close lockers, but also we saw that the factors of the numbers represent the people who touch the lockers to either open or close them.
All of the square numbers have an odd number of factors (since the middle factor represents the square), so the lockers are open, closed then open again.
I was impressed by the stamina that was shown in the class, and how they were keen to explore the problem in a few ways, and to make use of their skills in factors and multiples in the problem solving.
Group Work in Grade 9
A great way to start a conversation on group norms is through Sarah Vanderwerf’s 1-100 task. Students are given a 1-100 page with numbers all over it. We estimated how long it would take to circle all the numbers in order. Guesses were from 1 minute to 6 minutes. We put on a timer and started to circle! It took the fastest student about 8.5 minutes, and some were still working at 13 minutes.
Groups of 4 students each with a different marker colour tried the task again. They anticipated it would take less time, but they were pretty surprised when they saw the pattern that emerged. Each person’s numbers were concentrated in a corner, which became visible when the different colours are used.
Groups finished in about 5 minutes this time. Students helped each other find their numbers, and they were able to look ahead and plan for the next number to circle while others took their turn.
We tried the challenge once more, now that the pattern was visible. Groups had the opportunity to strategize, to move furniture, move themselves, make a plan, and it sure worked!
Groups drew outlines on their search quadrant, they were speaking numbers out loud, helping to point out what comes next, and of course, as usually is the case, someone just decided to circle all their numbers NOT in order to be fast. This task allows all kinds of behaviours to be observed! It took most groups 2.5 minutes to complete the task this 3rd time.
We talked through all of the positive group work skills that were demonstrated, and hope to see all term.
First day of Grade 9
We started off Grade 9 with a fun challenge. We’ve done this for many years now, but this group seemed to be particularly good at solving the task and being resilient when confronted with challenges.
The challenges are Sky Scraper Puzzles. They are kind of like sudoku puzzles. You need a building that is 4 floors tall, 3 floors tall, 2 floors tall and 1 floor tall in each row and column. Also, there are numbers on the side of the grids. These numbers indicate the number of buildings that are visible from that location.
There are 4×4 grids which are generally quite solvable once you find a good strategy, and there are 5×5 grids which in the past have been very challenging. This year most groups succeeded. Some solved 3 of the 5×5 puzzles!
We got to experience frustration, elation, teamwork, strategizing, starting over after making mistakes. There were many conversations about how looking at a task from different directions is so important, and trusting your group members to catch some mistakes, and trusting yourself enough to engage in the task with effort.
After we worked through these challenges we celebrated our very first Vennsday! (We celebrate on Wednesdays)
Our Venn for the day was one circle that were divisible by 3, another divisible by 5 and the third was greater than 100. Teams worked together to put as many numbers as they could in the different sections. We had some good chats about the logic used to approach the task, and also some of the characteristics of each section on the diagram, like if something is divisible by 5 and by 3 it will be divisible by 15. We noticed that there is a limitation to how many values are in that particular section. The centre section could actually have many more values, since numbers bigger than 100 keep going forever. We also got to talk about how 100 is not greater than 100, it is equal to 100. Hopefully this type of task will also help when we explore inequalities later on.
We enjoy these weekly moments to talk about numeracy, and to keep bringing up different number sets and their properties, and densities.
Grade 9s did such a good job cleaning up, that we had time for a game of “buzz” like we played in grade 10 (the previous post). Since this class is in French, there are added challenges with being able to say numbers larger than 60. It’s a good thing to practice!
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.
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