Linear Matching

A while back, Pam Wilson shared an old linear matching activity. It had students match up a graph, two points, a slope and two forms of a linear equation to form a set. I really liked it, but it used old calculator screen captures for the graphs. I cleaned it up and ran it with my students. It went well, but I learned that it works better if each type of card (graph, slope, equation, etc.) is printed on one colour of paper so that students have a complete set when they have one card of each colour.

Here is the .docx file and here is the .pdf file. I'd love to hear if you use it and how we could make it better!

MPM2D - Day 10: Stacking Cup Systems

Before beginning today's activity I wanted to clear up some issues that I had seen in Friday's homework around multiplying binomials that had a constant in front. I asked them each to expand and simplify the one shown below and then we looked at a number of ways of tackling it.

Then, on to today's activity: stacking cup systems! On their desks were 10 of each type of cup and this handout, which I copied on ledger paper.

Then I gave them their task for the day: given Styrofoam cups starting on the ground and red cups starting on the desk, determine what equal number of both types of cups would produce the same height.

They started collecting data, some groups more precisely than others. It was interesting to see that some added 2 cups, measured, added another 2 cups, measured, etc., while others added 1 cup at a time but did not use all 10 cups.

I circulated and helped them ensure that they were measuring vertical height, not along the side of the cup. My conversations with several groups about their data meant that they had to start collecting data again. For example:

S: "Each cup makes the height go up by 1 cm, so at 10 cups, the height is 17 cm." 
Me, measuring 10 cups: "Is that 17 cm?"
S: "No, that's 19.5 cm..."

There were a lot of conversations about what the numbers all meant. Rate of change was interesting to talk about, especially for groups that had added 2 cups at a time. The fact that the model says that there is a y-intercept when in reality we know that 0 cups have a height of 0 cm was another avenue for discussion. One group got as far as graphing, but they had neglected the fact that the Styrofoam cups needed to start on the ground so they had to look at their models again.

Tomorrow we will have a quiz. Here is the homework I gave them today. After the quiz, they will keep working on their cup stacking systems so that they can then test their results!

MPM2D - Day 9: Linear & Quadratic Matching

We finished up the last example from yesterday's work to start today's class. We addressed a few issues, including how to evaluate -(4)(-4). I asked them all to evaluate (2)(3)(5). It was cool that some had multiplied 2 & 3 first, and others 3 & 5, but the point was that they all got 30. No one multiplied both the 3 and the 5 by 2. We tied this back to -(4)(-4) which I wrote as (-1)(4)(-4) and contrasted with (-1)(2x + 3) where the distributive property applies. With that misconception (hopefully) cleared up, we moved on to one of my favourite activities. It is from Shell and can be found here with the full pdf at the bottom of the page. 

Students have to match up linear and quadratic expressions in words, algebraic form, tables and area models. Each pair received the 4 sheets on good, brightly coloured paper and cut out the pieces. My students were having great mathematical conversations, were arguing and really engaged for the entire class. Here are some pictures of their work along the way.

I did warn them that there may not be 4 pieces to each match. In fact, some have 6, another only has 1 (they add the ones that are missing). As I circulated, I had good conversations about concepts like whether 3n^2 was the same as (3n)^2. The area models really help me guide students without giving away the light bulb moment.

This is what the back of my classroom now looks like:

I will leave a note to my future self - most students need a full 75 minutes to complete this activity.

I gave them this "Quadratics - Putting It All Together" handout for homework.

MPM2D - Day 4: Visual Patterns...going quadratic!

Today was awesome! My students were talking about math and wanted to share the patterns they were seeing and there was this great vibe going by the end of class. The bell rang and we all looked up, surprised. One student was not interested in leaving because she wanted to (needed to!) explain to me how she was seeing a pattern. This is how every class, every day, should be.

We jumped right in to our visual patterns, starting with two that we had already looked at. I wanted my students to start seeing them in a different way.

Instead of looking at how many blocks were being added each time, we discovered how the shapes related to the step number. They could see that the "extra" block on each side were constant, as was the width of the rectangle. Finding a pattern, based on the step number, for the length of the rectangle was more challenging. I wrote the values in a table to help them come up with a relationship. I had seen it as double the step number minus 1, which is also how at least one of my students saw it. But another said that it was the step number added to the previous step number. We figured out how to write that and simplified to show that both ways of seeing it produced the same result (2n - 1). It was great to have more than one way of seeing this. I asked how many blocks were in the rectangle at step 3. Although some counted the blocks to get 15, others said that they could multiply the length and width. We did this for our length and width at step n, simplified with a little help from the distributive property, and arrived at 6n -1, which was our result from yesterday.

Next, we took another look at the volume pattern from yesterday.

Again, we found the pattern without having to go back to step 0, instead relating what we could see to the step number.

Then, my favourite pattern and several ways of finding its rule.

There is so much richness in this pattern. The first three ways really reinforced the idea of relating the number of blocks to the step number. To work out the rule based on the area of the large square and the area of the inside square, we needed to step into the world of quadratics. My students have never really worked with quadratics before - the word itself was new to them. I loved that one student had created a table of values AND had calculated the first and second differences. We figured out a way of writing the area of the outside square for step n, but in order to show that the rule did still work out to 4n + 4, we needed to expand. They had never done this before, but I tied it back to the distributive property which seemed to make sense to them (I normally use some kind of area model to introduce multiplying binomials, but did not want to add that to the mix today).

And we kept going:

Then I let them work on the next few in their groups. It was so cool to have the same conversation over and over about the length of the rectangle for pattern 8.

They would tell me that you couldn't get it by adding something to the step number and as soon as I suggested multiplying the step number by something, the wheels started turning and almost immediately the light bulb went on. Visibly! Their faces lit and and they said an enthusiastic "Oh!". And I walked away.

I asked them to try to finish the visual patterns over the weekend (although #11 is hard) and gave them this linear review, as well. This is a modification of something I had created for my grade 10 applied class a few years ago. There is a worked example and then a practice question for each skill. As we haven't really done any formal taking of notes, I thought this would be a good approach.

I was planning on doing Solve Me mobiles next class, but now I think I will stick with quadratics for a while since we landed there today. My planning is always a work in progress!

MPM2D - Day 3: Doing What I Never Do & More Visual Patterns

Many years ago, I stopped taking up homework. I check homework every day in the classes in which I assign it, and I try to write some feedback if students got stuck on a particular question. If the majority of the class could not do a question, I will take that up or do a similar question that will help get them going. But today, after spending a couple of hours (!) last night looking at and giving feedback on the first homework set, I felt the need to take up a good chunk of it (I don't normally spend that much time looking at homework but I really wanted to get an idea of what my students know, how they think and how they communicate their understanding). I wouldn't say that I hated every moment, but I'm not sure that it was the best approach (I did way too much of the talking). Because I am spiralling, I am creating my own homework sets (I will likely take questions from the textbook). It is a very interesting process to think about what I want to include and it is equally interesting to see what made no sense to my students and to think about how I can improve my questions. I clearly need to spend more time planning how to better address homework in class. Today we focussed on algebraically finding an x-intercept and determining the equation of a line given two points.

After dealing with the homework, we got through a few visual patterns. My students are doing really well with the linear patterns. I love how they saw this one growing in different ways:

They got stuck on the next one though where they had to find a rule for the volume and a rule for the surface area. 

I pulled out the linking cubes to help them visualize and see the patterns. We started with volume so I used red blocks to show the growth at step 3 and at step 4. They could see that five blocks were being added, one to each branch. 

We didn't look at other ways of seeing the pattern, so I may start with this tomorrow. I can see it as 5(n - 1) + 1, as each "branch" is 1 less than the step number and there is one central block. 

Next we talked about surface area. Here is a different view of the blocks:

The different colours really helped them see how many faces were showing on each cube and helped them see a pattern.

My students told me that the next one was too easy, so we did not spend a lot of time talking about it.

Then we started on my favourite pattern, but didn't get very far. So I already know that tomorrow will be fun!

MPM2D - Day 2: Testing Our Models & Starting Visual Patterns

We started today by recapping the Crow & the Pitcher activity from yesterday. I started with a glass that was more than half full of water and asked what we needed to do first. They said that we needed to measure the height of the water so I handed over the glass and a ruler to a student. He measured by placing the ruler inside the glass. I asked the rest of the class if it was okay to do this. One student said no because the "0 cm" of the ruler is not actually at the end of the ruler. Another said that by placing the ruler inside the glass, the height of the water would increase. I thanked the student who was measuring for having done that as it led to a great discussion.

Here is the 2nd attempt at measuring:

At first he said "...about 12.2 cm", to which I replied "About? No. We want an exact measurement!". He tried again and here are the results:

I asked my students to use their work from yesterday to determine the number of marbles needed to get the water to the very top of the glass. I asked them to write their answers on the board (without names) and this is what they got:

Quite the range! I collected some data myself and did my own calculations which I went over with them.

From the work they did yesterday, they knew the relationship was linear. Here is my data plotted with Desmos:

I made sure they understood all of the parts of the equation they were using by having them explain each element to the class.

We then talked about which numbers went where and why before calculating the number of marbles my model predicted.

Here it is with Desmos:

Notice that their regression line found the same number of marbles that I had predicted. And then for the actual test... Here is the glass with 94 marbles. So close!

I continued adding marbles until the water was just about to drip over the edge.

My model's prediction was closer than any of my students'. I assume this had a lot to do with their precision when measuring yesterday. It gave me a good opportunity to talk about how to do a good job when collecting data.

I really like this activity. It definitely got my students thinking and talking about math, and working together. Clearly, you need a lot of marbles! I will have to buy more for next time. I would also recommend getting marbles that are all the same. None of my students got through it quickly enough to try with the larger marbles I have, but perhaps another time. You could then extend this activity to linear systems by asking when two glasses would be at the same height with the same number of marbles. You would have to carefully choose the marbles and the starting heights, but this could work.

Next up, visual patterns! I had made this handout which they jumped right into. I wanted them to show me how the patterns were growing, and they did this pretty well. I did not see a lot of tables of values popping up which was really good. We took up the first two in class:

It was interesting to see how difficult some students found it to explain what their numbers meant. They seemed to enjoy these and will continue with more tomorrow.

MPM2D - Day 1: The Crow & the Pitcher

We started the school year in sweltering heat (the school has air conditioning but they are replacing all of the cooling/heating systems - yes, they waited until September to do this). 

I wrote about today's plan here, including the handout. Here is what it looked like as students added marbles to the water and measured the height:

Overall my class worked really well. I circulated and learned all their names and helped out when they were stuck. Most did a good job of collecting and recording data and creating a scatter plot. Some had reversed the independent and dependent variables which prompted some good discussions. I am much more aware of my questioning and am trying not to lead them with my questions. They were very good at realizing that neither the table nor the graph would allow them to determine the number of marbles to reach the top of the glass. There were lots of conversations around the y-intercept and slope which I tried to steer back to the context of number of marbles and height of the water so that those numbers had meaning. One group said they calculated the slope to be 5.4. When I asked "5.4 what?", and they replied cm per marble, they immediately realized that their answer didn't make sense. We didn't get a chance to test their models so will start with that tomorrow.

MPM2D - Day 0

I will be teaching grade 10 academic math this fall (along with grade 10 applied and grade 12 advanced functions) for the bazillionth time, but it will be different as I decided to spiral through the curriculum as I have done in MFM2P for the last two years (click here to learn more about spiralling). I have always incorporated activities in my classes, but have been intentionally adding more over the past couple of years. I will take those activities and add new as I get rid of units and look at this course in a whole new way. There are fewer overlaps between curriculum strands in this course than in the grade 10 applied course, but I think I have found a way of organizing the cycles that makes sense and will help students be more successful. Here are the curriculum expectations (standards) for this course:

CURRICULUM EXPECTATIONS

Analytic Geometry:

• Model and solve problems involving the intersection of two straight lines.
• Solve problems using analytic geometric involving properties of lines and line segments. Verify geometric properties of triangles and quadrilaterals using analytic geometry.

Quadratic Functions:

•   Determine the basic properties of quadratic relations. Relations transformations of the graph of y = x² to the algebraic representation of y = a(x - h)² + k.
• Solve quadratic equations and interpret the solutions with respect to the corresponding relations. Solve problems involving quadratic relations.

Trigonometry:

• Use knowledge or ratio and proportion to investigate similar triangles and solve problems related to similarity.  Solve problems involving right triangles, using the primary trigonometric rations and the Pythagorean Theorem.  Solve problems involving acute triangles, using the sine law and the cosine law.

There are also the mathematical processes to consider:

The mathematical processes that support effective learning in mathematics are as follows:

• Problem solving

• Reasoning and proving

• Reflecting

• Selecting tools and computational strategies

• Connecting

• Representing

• Communicating

I decided that my first cycle would be all about relationships. Students will do activities that allow them to collect and work with data and they will look at patterns to help establish important relationships right away. As always, time is flexible with any of these activities and things never go according to plan! But I have a plan anyway.

Day 0: The Crow and the Pitcher
I stole this from Pam Wilson who blogged about it here. 

I bought some great, tall cylindrical glasses (pictures will be in Tuesday's post) and more marbles and created this handout (update: the handout now includes homework). I have different sized marbles so if a group finishes early they can experiment with larger marbles or combinations of different sizes. By the end of the class, I will put some water in my glass and have each group determine the number of marbles needed to fill it. Then we will see which group came closest by actually adding marbles until the water reaches the very top of the glass. It should be fun.

I love doing activities on the first day, not only because it is a great way for the semester to begin for the students, but also because I get the opportunity to observe my students. I did not teach any grade 9s last year, so I won't know any of them. I will learn their names as I circulate and also learn who the extroverts are, who is quiet, who thinks they can sit back and let the others do the work, who is enthusiastic, who is a perfectionist...

Day 2 & 3: Visual Patterns
I have often professed my love for visual patterns and particularly for Fawn Nguyen's site: visualpatterns.org. I decided to spend at least two days working through linear and quadratic patterns. I made this handout (made = stole pictures and formatted stuff). The first three pages are the patterns students will work on in class and the last two will be their homework. This will help set up the mindset of seeing more than one way to solve a problem and will hopefully challenge some of the high flyers that are used to not having to do too much thinking.

Day 4: Solve Me (Balance Benders)
No handout on this one yet, but we will be spending time here, having fun while solving equations.

That's the plan for my first week. After that we will do ropes, frogs, Desmos, some 26 squares and more. Stay tuned!

Exeter Conference - Day 5

We started with a little 3-act fun from Andrew Stadel. Here is his blog post about Filing Cabinet with all the links to the videos. Here is my blog post from the latest time I have done it with my students. 

I also showed them some of Nathan Kraft's craziness (that's crazy in a good way, of course). Toothpick insanity and Starry Night.

Then we played Polygraph: Parabola from Desmos. It was a lot of fun and everyone saw how the game encouraged the use of correct vocabulary and helped you create better questions by showing you what others had asked. 

We also took a quick look at Central Park.

I probably sound like a broken record, but if you haven't tried Desmos Activities, you really need to check them out!

Time to get up and get moving! Tying Knots was next. The first part of this activity involves determining the relationship between the length of a rope and the number of knots in the rope. I really like this because, unlike most linear data collection activities, this has a negative rate of change. 

We skipped the part involving putting everyone's data together to be able to find the relationship between the diameter of the rope and the rate of change (but it's on the handout) and moved on to figuring out how to get the ropes to be the same length with the same number of knots. 

All but one of the groups got it to work which led to interesting discussions and to me adding what you see below for next time. 

I have blogged about the Tying Knots activity here, here and here and the handouts are here and here.

We took a quick look at some Always-Sometimes-Never statements and discussed how these can be really good warm-up activities that help students think beyond just their initial reaction to the statement.

I then gave a choice of matching activities.Quadratic (credit to the teachers at Sir Wil for that one), rational, right-angle trigonometry or combinations of functions.

A couple of participants then had a quick but lively game of log war. I think I originally got these from Kate Nowak, so I'll give her due credit.

And, finally, I showed off some of my students' parabolic art - art work created entirely with quadratic equations. Here is that activity's blog post and this is one of my favourites: