Median, Altitude, Perpendicular Bisector Warm-Up

My grade 10 academic students have spent the last couple of classes working on finding the equations of medians, altitudes and perpendicular bisectors. Some did a lot of work and some, well, let's just say that some did less. As we move into looking at properties of quadrilaterals, I wanted to ensure that everyone has at least a basic level of comfort with the prior work. So, I came up with a warm-up that I thought I would share.

It incorporates collaboration and finding mistakes (there is no way that all groups will do this correctly on the first try). I suspect some will struggle finding the group with the same question as them, but they will learn to follow instructions better (hopefully).

Here are the questions, linked here, that I will cut into strips for them.

I have 28 students, so 14 pairs which means that I needed 7 questions in order for each question to be repeated. Two groups will eventually find the centroid (via equations of medians), two will find an orthocentre and three will find a circumcentre.

If you can think of ways of improving this, please let me know in the comments!

Centroid of a Triangle

Yesterday, my grade 10 classes learned about triangle centres. They each had a triangle awaiting them on their desk as they walked in (I copy them onto thicker-than-normal paper). 

I gave instructions along these lines:
- cut out the triangle
- figure out how to balance it on your pencil
- make note of the balance point
- figure out how the balance point relates to the coordinates of the vertices of the triangle
- when you think you know, create your own coordinates for a triangle ABC, write them on the board along with their balance point

That last part was new. Its inspiration is Joel Bezaire's Variable Analysis Game which you can learn more about here. It helped some students see the pattern (especially the last line that I added to make it a little more obvious) and kept those who had found it early engaged as they were checking that other student's coordinates and balance points worked.

It was a fun way to start the class so I thought I would share... Thanks, Joel!

Student-Created WODB

Once your students have done a number of Which One Doesn't Belong, they should be able to create their own. I described what I do with my calculus classes here, but thought I should take the time to outline my creating process.

Thanks to Chris Hunter, both when I am vetting or creating a WODB, I use a table like this where there are 4 criteria/characteristics labelled along the top.

I consider it a strong WODB if 3 of the items share a characteristic that is not present in the 4th. For example:

The criteria used for Shape 5 could be:

I find that working through one like this with students is a really important step for them to understand that "It's the only one that's pink" is not the depth that we are looking for (unless you are looking at colours!).

I also strongly encourage you to do some Incomplete Sets before your students create their own. Here is an example:

I love that there are many different options for the missing number here. And students (may) quickly see how helpful the table is to ensure a strong WODB.

What have I missed? Let me know in the comments!

The Box Method for Factoring Trinomials

I love using the box method (area model without appropriately sized side lengths) to help students learn how to multiply polynomials. I love, love, love to use the box method to divide polynomials. But I only started using it to factor non-monic trinomials last year and I did a horrible job of it. Really horrible. I'm sure none of my students understood it because I didn't really get it. I am happy to report that I now LOVE using the box method for factoring non-monic trinomials. I shared this with the other math teachers at my school (I send out a weekly "Math Minute" - a link to a cool activity, a blog post, an idea that is worth sharing... and this was what I sent this week). I tried to colour code it to make it easier to follow and made a second box to better show the steps.

Here is an attempt at an explanation in case the example is not clear. Start by finding two numbers that multiply to the product of 'a' and 'c' (here -120) and add to 'b' (here -2). In this case the number are 10 and -12. The box represents the area (trinomial) and we are looking for the length and width (binomials). I always put the x^2-term top left, the constant term bottom right and the x-terms along the remaining diagonal. The number we found are used as the coefficients of x so 10x and -12x go in the boxes along the diagonal. Then I common factor the first row and the first column (that's where the 2x and 4x come from). This would normally all happen with one box, but now jump down to the second box. Figure out what multiplied by 2x will produce 10x and what multiplied by 4x will produce -12x. Those complete the factors and you can check that it all works out with the constant term (does -3 times 5 equal -15?). There you go. I love this because it is not a trick - it makes sense and has built-in error checking. I find it really fast, too.

I should note that I also show my students how to factor using decomposition and give them the choice of which method to use. So far more are choosing to use the box method. I can't wait to show this crew how to divide polynomials in a couple of years!

Rethinking Factoring Special Quadratics

Have you ever had a day when a lesson you took the time to rethink actually worked noticeably better? Let's be honest - I don't have the time (and sometimes not the motivation either) to rethink all my lessons. "What worked well enough last year is good enough for this year" happens far more frequently than I'd like to admit. I try to make notes if something really doesn't work or if I have a brilliant idea after the fact. And I do my best to act on those notes to my future self. Occasionally, if I teach more than one section of a course, I will make changes on the fly as I teach the second class. But the reality of teaching full-time and raising a family is that every lesson may not be as good as it could be. This is a difficult reality for me.

The change I made to yesterday's lesson was a simple one - I did the opposite of what has been done in the past. Let me back up for a minute (and I apologize if you've heard this all before)... The math teachers at my school all share lessons for all courses. I am the renegade who sometimes does things differently. I have been spiralling my grade 10 applied classes for several years and I spiralled my grade 10 academic class for the first time last year. I put a lot of thought into the order of topics and how each would be approached and blogged daily. This year I am tweaking what I did last year - the biggest change being that I am introducing more quadratics concepts earlier in the course. I am trying to be intentional when I look at past lessons and ask myself whether this is the best way to approach the topic. I looked at the "department lesson" on factoring special quadratics (at this point I have no idea who created it - it could have been me???) and just wasn't happy with it.

The old:

... followed by exclusively difference of squares practice questions. Then:

... followed by exclusively perfect square trinomial practice questions.

The new:

As I wrote above, I got students to do the opposite of the old lesson. Instead of expanding, they factored. This was good practice for them and they could see that there was a shortcut within the patterns. We had a whole-class discussion, talk with the people at your table, test your conjecture(s), come back to whole-class discussion kind of thing going, but we got there. They came up with the patterns (I didn't tell them) and they saw the value in what we were doing (I think). I think the old lesson tended to fall flat because they didn't see a need for more ways to factor - it was just confusing. They didn't see these special cases as being helpful. I hope this year's students do. They also know that they can also successfully factor them as they would any other trinomial if they don't notice that they are dealing with a special case. (Confession: Until I started teaching grade 10 applied, I did not think of a difference of squares being a trinomial where the x-term has a coefficient of 0. Factoring these with algebra tiles was a revelation!)

One of the things I love about spiralling is that it freed me from common test days. When my students need more time on a topic, I give them that time. So tomorrow we are factoring a little more. A few students are really solid with all types of factoring, but most have a more tenuous grasp of what to do when. My room is currently all set up for some factoring speed dating. Tomorrow should be a fun-filled day of factoring!

Distance Between Two Points with Tacos & Zombies

I thought it was about time to dust off my blog. I am spiralling my two grade 10 academic classes this semester (I also have a section of grade 12 Advanced Functions). It is really nice to have last year's plan and homework sets to work from and tweak. I recently saw what Nathan Kraft did to a cool Desmos Activity Builder activity created by Andrew Stadel and knew that I wanted to adapt it for my class. Their activity focused on horizontal and vertical distances between two points. I needed to include the distance between any two points. Due to the size of the activity (so many images!) Desmos struggled to keep up with my changes and I ended up having to hard code the points I used rather than having them referenced with variables. Using variables would have made it easy to move the points, but I will have to save that for another time. I think I'm getting ahead of myself here though. I should back up for a minute and start at the beginning of today's class. We started with Dan Meyer's Taco Cart 3-Act found here. 

After setting the scene with the Act 1 video, I asked what information they needed to know. They requested distances and speeds so they got this:

Next, they worked on big whiteboards in their table groups to determine whether Dan or Ben would get to the taco cart first. We talked a little about the Pythagorean theorem along the way.

Then, I played the Act 3 video to confirm their answers (all groups said that Dan would arrive at the taco cart first).

It was now time for the zombies! I used my Popsicle sticks to select random pairs and handed out Chromebooks. The link to the activity is here.

Despite the activity not running as smoothly as I would have liked (points were not showing up for some students and overall it was very slow), students seemed to learn what I wanted them to. After working out the distance between four sets of points on this screen, they understood the process and were ready to generalize. (Screen 13 was planned as an extension - I really wanted to ensure that everyone completed screen 8.)

We consolidated to close the lesson and did one quick practice question.

Grade 10 Applied Math, February - June 2016

At some point during this semester I said I would blog about what I did with my grade 10 applied class. I have spiralled this course for a number of years now and felt that it was time to change things up a bit. Here is my attempt to remember what I did and why. Firstly though, here is the Google sheet that I used to plan and record what I would be doing during each cycle. There are pictures of the cycles included below.

The first big change I made this time around was to start with trig. I have found that starting with 26-squares did not pose enough of a challenge to some students who decided right away that they didn't need to do any work in this course and behaviour issues ensued. Trig, being new to all students, was a great way to see how they learned and dealt with new content. I also introduced sine, cosine and tangent right away. I used to wait until cycle 3, but found that students tried to pick it up and then forgot how to use the tables. I taught with both trig tables and SOH-CAH-TOA the entire way through the course, and students generally chose one way and stuck with it.

Another change I made had to do with warm-ups. I still did them daily, but I created them as we worked through the course so that they would connect more with the content, either as practice for what we were doing or as lagged practice for a topic that we hadn't looked at for a bit. Here are this semester's warmups. There are no warmups for weeks 6 and 12 because I was testing those weeks.

Here is a look at cycle 2:

Here is a look at cycle 3:

And end-of-year stuff:

I didn't make any other major changes, but did try to spend a bit longer on each topic in cycle 2. I am sure I added a few new things but when you wait several days or weeks to blog about them, you forget. Well, at least I do. If there is anything that you want to know more about, please get in touch!

Things to Remember

Teaching can be a tough job. But it is also the most rewarding. This blog post is something I hope to remember to read on the difficult days.

I find the end of the school year hard. I am so happy to see what amazing people my grade 12 students have become, and for them to go off and make their mark on the world. But this is always accompanied by a sense of loss. I have spent at least 5 months in their company, and for some it has been 4 years, on and off. The connections built over that time are meaningful to me. This is why I write cards to each of my grade 12 students each year - I say something good about each of them and how they have impressed me along with good wishes for the future. Each message is different, as is each student. They also each get a lollipop which is a symbol of achievement in my class - a great result or a marked improvement earns a lollipop on a returned test. So it always fills my heart to receive some thank you cards from my students - I hope they have learned that a small gesture can mean a lot. Their words make me proud of who they are and what we have accomplished together. Here are a few excerpts:

"Thank you for actually making me like math and giving me back some confidence I lost a long time ago about my capabilities in math. You are the nicest teacher I have ever had and I really do appreciate everything you have done for me this year. Thanks for letting me know it's ok to make mistakes and giving me the time and all the effort you have put in to let me grow. I hope no matter what I accomplish in the future, I achieve it by hard work, and with all the kindness, empathy and dignity you have taught me."

"Your style of teaching and attitude towards math is so inspiring to me. It feels like you genuinely care about getting your students interested and getting them to succeed. I am honestly amazed you are able to keep up such a positive attitude and it honestly inspires me so much. ... If I grow up to become half as passionate as you about something, I would be so happy."

"I know I didn't get the greatest marks, however you not giving up on me and continuing to demand my best work will always be something I'm grateful for."

"I want to sincerely thank you for everything you've done for our class this semester. Thank you for being more than a calculus & vectors teacher, thank you for being patient, kind, supportive and always having a smile on your face."

"You are one of those teachers who truly cares about her students, and I'm so thankful for that."

"You have been incredibly inspiring to me and have greatly affected me. I admire your passion for math and especially learning. Your love for math is nerdy and contagious and you are the best teacher I've ever had because of it. ... I wish one day I can be half as lovely and caring as you."

Knowing your content is important. Being passionate about it is far more so. And the relationships we foster in our classrooms help inspire students to be their best selves. We do make a difference and our students notice what is important to us. And every so often we will hear a gem like "Your love for math is nerdy and contagious" - it doesn't get much better than that!

What's Your Best Question?

Yesterday, before class, I tweeted this out:

And, as usual, the #MTBoS came through. Here are just some of the replies I received:

I answered the question thanks to the great replies I got. But it was not until a student asked me how to solve the question that I realized that, despite knowing that many students would struggle with this question, I did not plan out what I would say when they asked for help. My answers ended up being just like those given to me on Twitter - "this is how you start it" which really took some of the fun out of solving this "puzzle". So I am now wondering what a good question would be to help move my students' thinking forward without giving away the solution. I should have at least asked "What do you notice?", but am not sure that would have been enough to get them going. Please tell me if I am wrong! This is the question I came up with in the van ride to take my kids to Jiu-Jitsu:

I am wondering what you would ask - what would your best question be? Please let me know in the comments!

OAME 2016

You know that feeling when you are trying really hard to get caught up on everything, but the "finish line" keeps moving? That's been me this week. I'm woefully behind on replying to emails, apparently have 9 comments awaiting moderation on my blog that I don't even remember getting notifications about (?!?) and I arrived home yesterday to a flooded upstairs bathroom which, of course, had spread to the downstairs bathroom, hallway and bedrooms while I was at school. When it rains, it pours! (sorry - couldn't help myself) 

However, I am attempting to work my way through my to-do list and owe David Petro my slide decks from OAME 2016. So, here they are:

WODB Session

Spiralling Panel Session
(this was with Bruce McLaurin, Jon Orr, Alex Overwijk & Sheri Walker)

Rethinking Grade 10 Math Double Session

Ignite

Gratitude

Last Thursday and Friday I had the privilege of working with some great math teachers from the Niagara region. I was invited to be part of the mini-conference for their OAME chapter (Golden Section) and got to meet so many wonderful teachers. The following morning I worked with a great group of teachers, most of whom have been spiralling through the curriculum this year. I want to thank Elizabeth Pattison and Liisa Suurtamm for giving me this opportunity. I have never before felt like my work has been so valued and cannot adequately express what a great experience this was.

And, as requested, here are my slides.
Keynote
Spiralling Session

Quadratic Visual Patterns

You all know how much I love Fawn Nguyen's Visual Patterns site. I use them a LOT. They have been part of my warm-ups for years now and have been some of the best moments of my class each week. I have been recreating my warm-ups for my grade 10 applied class this semester (no, I can't leave things alone). I decided to do this so they align more with the curriculum expectations we are working on or provide lagged practice for other expectations. The warm-ups have included quadratic visual patterns for a few weeks now and I decided to step it up a little this past week with with a couple of patterns from Michael Fenton. If you haven't tried these ones before, I encourage you to do so before you scroll down.

We didn't actually work with the colour-coding, instead looked at the squares that overlapped by 1 each time. We worked with the number of circles first, established that this is a quadratic relationship and then found the rule by comparing the "side length" of each square to the step number.

I really also wanted to look at this pattern using the colours as a guide so we started over and found that we ended up with the same simplified rule.

I am totally impressed that some of my students can do these as they are not easy, especially for students who have struggled a lot with math and have trouble making connections. They have shown incredible progress and I love how willing they are to try.

Here is the next one we did:

 There is a lot going on with this pattern, but the colours really help show the squares emerging.

I should note that these "warm-ups" took about 45 minutes to work through. It was definitely time well spent. 

Linking Cube Towers

I am not doing a daily blog about my grade 10 applied class this semester. This is not because everything is the same as last year or eve last semester. In fact, I have changed almost everything so far in this first cycle. I have different warm-ups, am doing topics in a different order and have made some new resources, too. If I'm not happy with things, I cannot leave them alone. I have thought about sharing what I have done at the end of each cycle - if that's of interest to you, please let me know.

Last year during a lesson study process we looked at an activity that involved students creating towers out of linking cubes and competing to see who could get the tallest tower. This is what I mean by linking cubes (also known as cube-a-links and unifix cubes):

I believe the students all had cards that told them how many cubes to begin with and how many to add each time. It went fairly well, but once a student was "out" (because their tower fell), they were no longer really engaged. Anyway, that is what inspired today's activity which will be my students first look at solving systems of linear equations. I am trying to have them really understand how the starting value and rate of change will affect their towers (and corresponding graphs). We have done a few visual patterns and solved some equations with a variable on both sides, so I think they will be ready for this.

Here is the file. I would love feedback. I will add a postscript if it's a disaster ;)

The Power of Popsicle Sticks

Content is not important to this post, but it sets the context. We looked at derivatives of exponential functions on Friday, starting with the derivative of y = e^x. We do this by looking at values of f(x) and f'(x) and then calculating the ratio of f'(x) to f(x). Today, we explored finding derivatives of exponential functions from first principles. I therefore didn't think there would be any issues when I asked them all to complete this:

And I might not have known that this was neither clear nor obvious had it not been for Popsicle sticks.

This is my tin of Popsicle sticks for my morning class. I have one for each of my afternoon classes as well. I generally go with the "no hands up except to ask a question" rule which means that the one or two extroverts in the class who have all the answers are not the only voices heard. I choose a name randomly to answer a question and then quite often choose another name to add thoughts to the first response. A response of "I don't know" is okay and is often followed by several other "I don't know"s which tells me that we all need to take a step back.

This morning's Popsicle sticks told me a lot. I went through a lot of sticks - there was a huge pile outside the tin - which means that there were many answers (some right, some not) to my questions and much "What do you think?" from me as I chose another name following each answer. When this happen I have them talk in their groups to see if they can make connections together before trying again. 

The Popsicle sticks help make my classroom a learning space where everyone has a voice and every voice is important. I do my utmost to make it a safe place where making mistakes is not only okay, but important. In addition to that, Popsicle sticks help me be a better teacher. They help me gauge the understanding in the room (I do thumbs up-sideways-down a lot, also) and help adjust the pace and choose what we need to practice in the moment. This is still a work in progress for me, but one that I think is important to help me grow as a teacher and to ensure that my students truly understand what we are doing, not merely mimic completed examples.

I first read about using Popsicle sticks in Dylan Wiliam's Embedded Formative Assessment. Here is a video about this strategy and here is his website.

Log Clothesline - My Post-Activity Post

Yesterday, I ran the log clothesline activity with two of my classes.

What I liked:

• The great math conversations. They had not worked with logs for a while now, so some remembered a lot and some, well, not so much. They were talking to each other about how to get going with tricky expressions at their desks initially and the conversations continued when they were at the clothesline trying to place the cards in the correct order.
• The collaboration. Those who couldn't even remember how to go from logarithmic form to exponential form got help in their groups. They asked each other questions, they checked each other's work and argued about who was right. 
• The struggles. Apart from the one card with a typo (oops!), all of the others worked out. However, I made this a no calculator activity (horror!) so they had to work smarter to reduce the amount of arithmetic required. Those log rules really became useful.

What I would change:

• I found this activity worked much more smoothly with my smaller afternoon class. I only had 13 students in that class yesterday and everyone was engaged and busy. My morning class, which had 24 students, had more "traffic jams" around the clothesline causing other students to step away and no longer be engaged in the activity. I would put up two clotheslines for a larger class next time. 
• I also liked having extra cards in my afternoon class for those who finished quickly, so I would make more cards next time or have two sets for a larger class.
• The change that would have the biggest impact, I believe, is adding more number markers to the line. I had a 0 marker, but the struggle to correctly arrange all the expressions on the line was greater than I had planned. I asked my afternoon class if they thought having more number markers would help, and they unanimously said yes! I loved that some were using the small whiteboards to work out the expressions from other students' cards that were already on the line, but it became overwhelming.

I definitely think this activity was worth doing and that all students got something out of it. Here is the final clothesline:

Log Clothesline

There has been quite the buzz around clothesline activities of late. There is even a clotheslinemath.com website! I first heard about it from Andrew Stadel when he wrote this post. More recently, Jon Orr made a cool one to practice finding slope. Here is a link to his blog post. I thought that it would be great to use a clothesline to practice evaluating logs, but was having trouble finding the time to actually create it. Then we got a snow day (school buses are cancelled, but teachers still have to report to work... all of the students at my school take the bus so we have a day with no students) but I had other things to do (like meetings) and we were allowed to go home early (we got 51.2 cm of snow by the end of the day!). And then, shockingly, they cancelled the buses again today! So here is the log clothesline for you. I plan on actually doing it with my two MCV4U classes on Friday and and will snap some pictures then. I did take a "fake" one of a colleague placing a card on the clothesline.

Here is the Word file, and the PDF version is here. And I'll even give you the answers. I numbered the pages so those correspond to the question numbers, which I have sorted in the Excel file so that I can quickly check for correctness. I printed two pages per sheet and cut them down the middle.

I will give out one card to each student and they will have to evaluate their expression and check at least one from someone else in their group. Once the whole group is confident in their answers, they will go to the clothesline to place their expression in the correct location. I will give them a 0 marker on the number (clothes)line. The trick is that I will tell them that they can't write on the cards so in order to figure out where their card goes, they will have to work out a number of other expressions. Those students who put their cards up first will be responsible for ensuring that all the cards that follow are in the right place.

I will try to write a "how it went" blog post after I run the activity.

Postscript

After hitting publish this morning I started to think more about this activity and started questioning whether it met the spirit of clothesline activities. I worried that as it doesn't really work students' number sense, it might not be an activity worthy of sharing. I sought advice and, though we agreed this would be better suited to an intro to logs activity (more on that in a minute), this practice, with built-in error analysis and collaboration, was worth sharing. 

Back to the intro to logs idea. This is what I envision: students get cards with powers of 2s from 1/64 to 32768 (or something like that, depending on the number of students in the class) and they have to attempt to place them on a clothesline that will have markers of 0 and 1 on it (not as shown below). It would look something like this:

Hopefully it will become clear that there are too many cards between 0 and 1 and that the larger numbers simply cannot fit on the clothesline. So what to do? I'm not sure how to introduce the idea of taking log base 2 of each of the numbers, but that will be the goal. The result will be a clothesline with a logarithmic scale which will allow all the numbers to be seen. I won't actually be able to do this until next fall, so please let me know if you try it out!