Brain Melts and Typos – #MTBoSBlogsplosion Week 4

The theme for Blogsplosion this week is “We All Fall Down”. The times in your classroom where you make mistakes, accidentally teach a mathematical inconsistency, try to do a really cool activity and it’s a disaster…we all know those days.

One particular day immediately came to my mind when I read the theme. Algebra 2 was working on systems of linear equations earlier this year, and Ms. Kormann from next door was filming the class for my model teacher application. Of course filming leads to miniature disasters – why would something go well when you’re recording it?

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The day before, we had been working on an assignment and one of the students called me over, saying, “Miss Mastalio, the solution I got doesn’t work in the equations but I can’t figure out where I made a mistake.” I looked at her work for several minutes and also couldn’t find a mistake in her work. The whole class thought it was pretty funny that I couldn’t figure it out. When the bell was about to ring, I told her to just turn it in and I’d find the mistake later, and I would let her fix her solution then.

So we started this class with me coming in and saying, “Hey, so the mistake that we couldn’t find in [student]’s work yesterday? It turns out that 6+8 isn’t 13, guys. Just in case you didn’t know.” ARITHMETIC, MAN.

We all had a good laugh at our collective student/teacher brain meltdown, and I began the day’s activity.

 

We were completing a Question Stack (one of my favorite forms of practice) to  help them choose a solution method (graphing, substitution, or elimination) and get practice solving systems before their assessment. The nature of Question Stacks means that students don’t begin with the same problem, and they also have an answer bank of all the possible solutions to work from.

So I was circulating, answering questions and clarifying things, when one of my students hit a problem where she kept getting a solution that wasn’t in the answer bank. I came over, quickly glanced over her work, and couldn’t find the mistake. Because of the mistake I had made on the student’s work the day before, I checked all of my arithmetic with a calculator. A few other students noticed me doing this and laughed, then continued working.

When I still couldn’t find the mistake, the student suggested, “what if I tried to solve it with a different method and see what happens?” I agreed that sounded like a great idea, and began circulating again. Soon, another student hit that same system. She picked a different method than the original student was using, because they were sitting next to each other and she realized that there was some issue. So we now had all three possible solution methods going on this problem.

They kept getting the same solution. Other students in the class realized that something weird was happening in our corner – some of them were reaching that system in their work, and some of them decided to skip to it and try it. I called Ms. Kormann over from behind the camera to come check the work as well.

No one could find a mistake. We kept getting the same solution with every solution method. The whole class started a heated discussion of each solution method and the process behind it. Instead of getting mad and giving up, they started getting mad and analyzing their mathematics.

Meanwhile, I was panicking because I could not figure out where I had gone wrong. I grabbed the textbook I had pulled the problems from to double check their solution. In my classroom, I had students checking each other’s work, trying multiple solution methods, and discussing algebra. (To be fair, they were also yelling “WHY” a lot and declaring themselves failures, so that wasn’t good).

 

Anyone want to know what the mistake was?

Because it was awful.

I transposed the x and y in one of the equations from what the textbook problem said.

Just. Switched them.

Of course they weren’t getting the solution in the answer bank.

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That day was the best possible kind of disaster – the disaster where your students stay with you the whole time and persevere in their attempts to solve. Not all of my disasters go that way, believe me. This one was super memorable though. I was so proud of my students. We bonded over brain melts and those days where you just can’t even add.

They still tease me about it, and I’m totally okay with it.

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Not My Words

It’s week 3 of the #MTBoSBlogsplosion 2017!!! I’m really loving this challenge to blog more, and discovering blogs of amazing math teachers through the summary posts.

The challenge for this week is “Read and Share“. Read some blog posts from other teachers, and share their words. I am so excited for this challenge, because it gives me the perfect opportunity to go back through some of my feedly tag folders and read posts I haven’t read in a long time. Click the first word of any quote to read the whole post.

 

These are about the WHY of teaching. The heart wrenching stuff, the hilarious stuff, the moments that keep me coming back every day even on the days I want to quit. I hope they help you remember those parts of the job.

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If you’re not reading One Good Thing, stop right here and head over there. These stories of finding the (at least) one good thing from every day of teaching are often hilarious, sometimes a struggle to find the thing, very real, and make me cry more often than I’d like to admit. I love this one from right before break:

Those are the real victories, my Teacher Loves. Those and so much more that you and only you know because your classroom is a safe place for kids to share their thoughts, their worries, their triumphs, and their fears.

 

 

I’m breaking the challenge rules a tiny bit because Love, Teach is not a math teacher, but I think pretty much everyone knows and loves her blog, and this post is one that I think about. Often. This paragraph felt like she reached straight into my own heart and pulled it out (I also wrote a post in response to Orlando that was very inspired by this one):

I care about all of this so deeply because of you. Teaching has fundamentally changed me, is changing me, and it has to, because I spend hours every week interacting directly with kids who represent a vast array of beliefs, values, and experiences. I love each of you so much that sometimes I think I’m in actual danger of my heart exploding out of my chest, and more than anything I just want all of you to live in a world where you feel safe and strong and valued, because feeling safe and strong and valued makes it easier to be brave and kind and inclusive. And in case you haven’t been paying attention, we need more of that.

 

Jonathan Claydon‘s posts often challenge me to think hard about curriculum and question design, but this one is more of the chills-all-the-way-down-your-spine types. Because you relate to it so hard. And this job is sometimes so hard, emotionally.

This is supposed to be a job? It’s just work?

But also, sometimes his posts just make me laugh. A lot. You all know how important it is to have fun in your classroom, right? Right?

43. Draw a kitty in a spaceship. Yes, you read that correctly. DRAW THE KITTY.

And here are 12 cats piloting spaceships, because I know what you want:

 

This one from Dan Meyer inspired me to do a bottle flipping project in my Sports Statistics class, which the students LOVED:

I didn’t think there was a useful K12 math objective in bottle flipping. My commenters served their usual function of setting me straight.

 

Tracy Zager has some lessons about how often we explain away our students’ interesting ideas because they don’t follow the formal mathematics we’re familiar with:

I drew the squiggly lines and asked who thought they were straight. A few hands went up. Abby raised her hand halfway, then put it down. She said, “I made up a new word for that kind of line. It’s vertiwiggly.”

 

I’m in love with Ben Orlin’s Math With Bad Drawings. I kind of want to wallpaper my classroom with them. Whenever a new one is posted, you never know if it’s going to be hysterical, inspirational, or hard hitting and challenging. Possibly all three. These are some favorites:

Graham’s Number can be defined by human symbols, but never fathomed by human minds. The notation can reach what the mind cannot grasp.

(my students have loved the concept of Graham’s Number when I’ve presented it to them in class)

But I can hate this view, this toxic meme, which I believe is latent in our stereotypes of mathematics: this belief that generating new mathematical ideas is man’s highest calling, while wallowing in old ideas is grunt-work fit only for mules, washouts, and the dim bulbs we call teachers.

I would love to quote the entirety of this post on why we learn math, but…just go read it:

Mathematicsis a safe playground with all the richness of reality.

 

And this post on sharing other people’s words would be incomplete without a nod to Sarah Carter. There are too many posts I could have chosen to include from her, but honestly one of my favorites is this one, where she shared incredibly detailed pictures from her classroom of all the posters. I always find myself looking in the background of people’s pictures from their classrooms, trying to see how they decorate and set things up. Sarah brought this gift to me so that, at least for one other teacher’s classroom, I get to see everything.

Whenever I read other blog posts where people share pictures of their classroom, I’m always left wanting more.  I want to see every single thing hung on the wall.  I guess you could call me nosy.  😉  So, I set out to write the blog post I would want to read.

Honestly, the Math is Secondary

If you had asked me in college, I would have said that the phrase “honestly, the math is secondary” would never come out of my mouth in reference to my classroom. In the last three and a half years of teaching…I’ve said that phrase. Out loud. More than once.

The theme for this week’s #MTBoSBlogsplosion is Soft Skills. Soft Skills are the interpersonal skills, the social behaviors, the habits of learning and communication that aren’t part of your curriculum but are, arguably, just as or more important for your students to learn. Of course the math is still important. I wouldn’t be teaching it if I didn’t think so. Francis Su’s closing remarks as president of the MAA are beautifully moving on the reasons why we should learn and teach mathematics.

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When I was searching for jobs during student teaching, I didn’t really know where I wanted to teach. High school? Junior high? Big school? Small School? I pretty much applied anywhere that had an open position in an area where I might want to live, and went to interviews. At the end of every interview, they ask you if you have any questions for them, and I made my question “Why should I want to work at your school over other schools?” because I thought it might help me make a decision. When I asked this at the interview for the alternative high school I now teach at, the principal started crying as she told me about the students there. About how they come to the school feeling like failures in so many facets of their lives. How they haven’t had many people believe in them. How they don’t know how to succeed because they’ve never done it. I knew in that moment that I’d found my school.

My students often don’t know how to act in social situations. They don’t have many role models in their lives. They’re often angry from experiencing failure after failure, from adults who should never say things like this telling them that they won’t amount to anything, that they can’t succeed.

My first goal with a new group of students every year is to know them. As people, not as math learners. I do surveys on the first day of school that ask their most important values, their favorite things, their previous experiences in math class. Based on their most important values selected from a list, I sort them into a Hogwarts house, which gives me an idea of how they might react to certain types of situations and of ways to push them to be better (Gryffindors thrive on competition, for example). These are students that very much do not want to open up. They don’t trust easily, so I have to find sneaky ways to learn more about them until they let me in. I make them list ten things they love, but give them the out that if they can’t fill the list with ten things, they have to finish the list with things that are the color orange. Many of them can only fill in three things before switching to pumpkins.

Every Friday for the whole school year, instead of a math question for their bell ringer, I post a hypothetical question on the board. “If you had a million dollars, how would you spend it on?”, or “If you were a ghost, what building would you haunt?”, “What would be your three wishes for a genie?”.

They write their answers on their opener sheets. They turn them in. They won’t share them in front of the class…until about December, when some of them start shouting their answers as soon as the bell rings. Every week, I read all of their answers, and I respond. Even if it’s just a smiley face. I start asking them questions in class – about their job, about their grandma who’s sick, about the TV show I know they love.

Slowly, they start to turn the questions back on me. They ask how my weekend was, if I got to see my nephew. They congratulate me when the Packers win. They want to know my answer to the Friday Question (I would haunt Madison Square Garden). As this happens…they do more math. They open up. They learn to trust. They get small successes and want more. They realize that when they ask questions (silly ones or math ones) they get answers.

No one is allowed to judge other people’s answers. They can ask questions to learn more, but they can’t call an answer dumb or boring or weird. They learn to treat themselves and others as people whose opinions matter. That how it has been for them does not have to be how it continues.

When you remember that you’re teaching people, the content becomes easier for them. When they know that they matter to you, they’re willing to ask questions until the Quadratic Formula makes sense.

My email signature contains a quote from Harry Potter: “Every human life is worth the same, and worth saving.” I don’t even begin to believe that I can ‘save’ them all. Not every student passes my class. Some of them leave with the same attitude they came in with. But ‘every human life is worth the same’? That I can show them. That, we can work on.

I have a student who came to us after dropping out for awhile. He barely scraped by in my class last year, rarely speaking, never turning in assignments on time. He sat in my room for a whole afternoon on finals day with me convincing him to keep trying assignments until he left at the end of the day saying “D-? Good enough.”

He won student of the month last month. He earned a B in my class this year. He told me a fifteen minute story about how he watched YouTube tutorials to learn how to fix his phone screen by himself to save money. He finished his assignments on time and then spent his spare time in class working on challenges I had posted. He tells me some of the funniest jokes and stories. He asks me for advice.

Do I think my Friday Questions caused this transformation? No. It was a lot of things, and most of them weren’t even involving me. But through the questions, I learned how much he loves drawing, how important family is to him, and how funny he is. From his intro survey this year, I learned that he’s a Ravenclaw, which helped me to seek out and push his curious and creative sides.

He graduates tomorrow.

Honestly, the math is secondary. (But he learned a lot of math, too.)

My Favorite: Classroom Tool

A thing that I discovered in 2016 was that I love teacher challenges. They give me a concrete thing that I can do to make myself reflect on and improve my teaching, and that other people are doing as well, so I can get inspiration.

We had an observation challenge within our district, I have my #observeme sign outside my classroom which has garnered several ideas for class closure, and I’m doing #teach180 this year.

So, when I saw that #MTBoS (Math Twitter Blogosphere) is doing a blog initiative for 2017, I immediately thought “it’s a challenge, so it will actually get me to blog regularly”. Then I actually read the post and saw that they’re asking the first post to be posted by the end of today and immediately thought “ehhhhhh.”

Then I remembered my 2017 phrase that I just blogged about last week of “action over inaction”, so thanks past me for giving me the motivation to just do this. This is all a longwinded intro to say HERE IS THE FIRST POST OF THE #MTBOS BLOG INITIATIVE 2017.

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I love so many things about my classroom, so when pressed to think of my “favorite” thing, my mind jumped to five different projects, three different games, two forms of assignment, and about twenty seven different concepts that I love teaching. I could only decide what to blog about for “My Favorite” when I reframed the thought process to think about what my students would say is their favorite thing in my class. Then it was easy, because they say it often.

 

My favorite tool to use in my classroom is our tables. As dry erase boards.

I love the ease of saving paper for doing something fast, or for individualizing assignments. The students love feeling like they’re doing something they’re not supposed to do. I also think they love the impermanent feeling of it, knowing that they can easily take the eraser cloth or their sweatshirt sleeve and wipe out a mistake and try again, not having to mess with bad pencil erasers or the fact that they were writing in pen.

When we do question stack activities, or Ghosts in the Graveyard style reviews, or when we’re learning a new concept and I want them to try it before copying a correct solution into their notes, they write on the tables.

It also means that my often lazy students don’t have to make much of an effort to get out supplies, because they are already sitting at the table and I keep the markers and eraser cloths in clear purple Solo cups that sit perfectly in the cord holes that my classroom tables have. Some of my least motivated students will actually complete assignments on the tables because they don’t even have to move to get started.

I also see my students using the tables to try to work out solutions t0 whatever our current classroom challenge is in the last few minutes of a class, or even the few minutes before the bell rings at the start of class. Currently, I have up the $100 word challenge, but we just completed the 5-4-3-2-1 challenge and I’ll be posting the 2017 challenge when this quarter ends next week.

My favorite tool: my tables. So simple, but very effective.

2016 Teaching Top Fives

This is a rundown of what’s happened in my classroom life in 2016. I feel like I have grown to be a much better teacher last calendar year (still so much room to grow!) and wanted to share some moments, resources, etc.

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Classroom Moments

1. Algebra 2 Question Stack Meltdown

This was one of those moments where everything went wrong but it ended up better than if it had gone as planned. We were doing a systems of equations question stack, a format I got from Sarah, to practice their ability to choose the most efficient method of solving. I was also filming the lesson for my application to be a model teacher for my district. One student hit a problem that they could NOT find a solution to amongst the solution choices – they could not get a whole number solution. I looked over the work, and could not find the error. Another student looked over the work, and could not find the error. A third  student looked it over, and could not find the error. The original student redid the problem using a different solution method, and did not get one of the possible solutions. We had Heather, who was helping me film the lesson, check the work, and she couldn’t find the error.

Turns out I had switched x and y in one of the equations when I was typing the Question Stack out. Oops.

But. Every single one of my students in that room ended up involved in discussing the math behind the question. Searching for errors. Analyzing each other’s work. Trying multiple solution methods. And none of them gave up. Even when I was checking the textbook to find the problems I had used to see if I had typed it incorrectly, they were still looking for their mistakes. It was honestly one of my proudest teaching moments as I realized that I’d successfully helped along students who had real perserverance.

2. 2016 May Graduation – First Students

Every single graduation ceremony we have at Mid City makes me cry because I am so enormously proud of these students for succeeding when the world tells them they can’t. Every. Single. One of the 6 I’ve now attended. But last May’s was especially meaningful because of the graduates’ relationship to me. I taught mainly Algebra I my first year teaching, which was mainly sophomores, and some Intermediate Math, which also had some sophomores, so this group of graduates ended up being largely composed of the students who were in my first ever classes. The students who helped me through all my panic and mistakes of my first year teaching and who made me fall in love with this profession forever. Watching them walk across that stage was intensely overwhelming and a very, very proud moment.

3. Interactive Notebooks

After reading about them since I started teaching, I decided to take the leap and start using interactive notebooks in Algebra 2 this year. I started with just one class because I didn’t know what I was doing. My students have responded so well to them and I’ve actually come to enjoy designing the pages or finding them from other teacherston. I’ve never seen a group of students actually use notes so efficiently before.

The real deal-sealer for the INB’s was hearing my students talk to each other about how it helps them to know what the most important main ideas are for each concept – it’s what we put in our notebooks. They realized that if they understood what was in their notebooks, they could figure all the smaller stuff out. I’m planning on implementing them in my other classes next year.

4. UNI Professor visit

Several of my former professors came to visit my classroom for a day as part of a program UNI is doing to try to improve their teacher education program. I got to show off my students and my classroom to some of my biggest mentors. My students got to get praise and attention from “real mathematicians”. They had their minds blown by mobius strips and drawing perfect circles on the board and knot theory and lapped up the attention from the professors. They got to hear stories about me in college. I thought my heart might burst when my professors were asking my students how they could make teachers better and one of them said, “just make more like Miss Mastalio!”

They proved that they are incredibly mature and inquisitive people. They showed how proud they were of their school. One of my students pointed out that I say “sure” the exact same way as one of my former professors when a student is asking a question I don’t quite want to give them an answer to and I’m left wondering if I learned it from him.

5. Student Improvements

I’m thinking of a few specific students here that I have for a second time. Students that I handed grade reports last year that were consistently at D-‘s and that I get to hand B’s to this year because they’ve worked so hard to understand. Students who had breakdowns in front of me because they got so anxious about the math that this year are still struggling but come in for extra help and don’t give up and are proud of their own improvements. Students that I regularly had to yell at for talking during class and being off task who now are the first to volunteer ideas in class discussions.

Having a small staff means I get to see this kind of growth up close, because I often have students for Algebra I and later statistics or Algebra 2 or Intermediate Math. I love it.

 

Twitter Accounts to Follow

1. @midcitymavs

This is our school account and you should follow it to see all the incredible things our students do and all the weird costume days we have and a place that is full of pride and love.

2. @mathequalslove

Sarah has so many downloads and games and ideas that she’s posting every day. It’s incredible. I use a lot of her activities in my classroom and take inspiration to create my own as well.

3. @ProfNoodleArms

This is one of my former UNI professors, TJ Hitchman. He tweets a lot about his experiences teaching with an Inquiry Based Learning style, which really pushes students (like me in his Euclidean Geometry class) to become more like ‘real’ mathematicians when in a math class. He also tweets about 3D models and knot theory and in general I really enjoy staying in touch with him through this medium, but I think others would also find interest in his tweets.

4. @Teach4SpclNeeds

I don’t even remember how I started following Karen but she is one of the most relentlessly positive teachers on Twitter. Plus, it’s fun sometimes for a high school teacher to look through posts by an elementary school teacher and think, “that looks fun and cute. But…not my thing. But fun!”

5. @MrsMeganMorgan

Megan is the Lead Support Teacher of our district and she…leads in support. Literally. She has always enthusiastically supported my ideas and technology experiments and offered resources and words to make me better. She shares a lot of cool things happening around the district and posts things that make me think hard about my teaching.

 

Blogs to Follow

1. MathEqualsLove

Pretty much any math teacher will tell you how much they love Sarah, and this is like the third time I’ve mentioned her in this post, so…just go follow her. She is a wizard with Publisher and creates INB pages, posters, and activities that look gorgeous and gives you all the downloads right in the post. My classroom kind of looks like her blog exploded all over it.

2. Infinite Sums

Jonathan posts a lot of metacognitive things about how his teaching is going. There’s a lot of student work examples that he analyzes, and he’s very transparent about all the changes he makes in his curriculum and teaching style and the thought process behind it. It will make you think “why haven’t I tried it like that?” and sometimes “Why is he doing that?”. He’s trying to change the culture behind advanced math classes at his school and is being very successful at it. Very cool.

3. one good thing

This is a collaborative blog with a few main contributors that write short posts daily about the good things that happen in their classroom. Sometimes the good things are tiny and they write about how hard it was for them to find the good in that day, and sometimes they’re huge. You know how teaching goes. But it helps keep me positive and remember to find the good in my days. Plus sometimes it makes me cry.

4. Math with Bad Drawings

This one is what it says on the tin, basically. An investigation of various concepts in math and math education, accompanied by stick figure drawings. Every post will make you laugh, and more often than not you will be thinking about the message the post is sending for the rest of the day.

5. dy/dan

Dan is a former teacher who is now the Chief Academic Officer at Desmos. He does something called Pseudocontext Saturdays, which calls out the ridiculous context math textbooks try to use to make concepts “real world”. He also takes middle of the road traditional math problems or activities and tries to reformulate them to be more interactive, real world, and have the students tackle the mathematics more head on. All of this is done with constant feedback and tweaks from blog commenters who are teachers.

 

Phrases for 2017

I’ve seen a number of different ways to set the tone for the new year around twitter. One word. Three words.

I’m going with phrases.

I posted to my friends as 2016 was ending that I had three goals for 2017.

  1. See 5 Seconds of Summer in concert more
  2. Not get pneumonia again
  3. Action over inaction

The third one was a throwaway thing my brain thought up because I thought, ‘I need to have something that seems serious on this list’. But the more I’ve thought about it…I’m claiming that as my phrase for 2017.

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Action over inaction.

I’m the type of person that puts things off – especially all the adulty things like making phone calls for appointments, doing laundry, grocery shopping. I wait until I’m stressed out about it and can’t wait any longer. So action over inaction – make the call when you think of it. (I just called to order new contacts this morning, GO ME!)

I’m also the type of person that has, in the past, been very content being a bystander. I’m a member of a pretty privileged demographic (besides being a woman), and I’ve watched as LGBT, as black, as native, as any minority people have been oppressed. I’m going to stop watching. I’m going to say something. Do something.

Over break, my aunt and mom were talking about one of my mom’s cousins who has recently come out as transgender. They kept referring to her with her deadname and with male pronouns as they were talking, and old me was thinking, “it’s fine. They’re not doing it with bad intentions. They aren’t super informed on this issue and it’s okay.” The part of me that had been thinking of this whole action over inaction thing knew I had to say something, so I just said, “You know, you should really try to call her by her new name.” And guess what? Nothing exploded. No one was angry. My mom said something like “It’s hard, because I knew her for so long with the other name, but you’re right, I should try harder.” I know that the reaction probably won’t always be that good, but I’m done sitting by and letting people be treated poorly. I need to use my voice.

New ideas for my classroom? Do them. Don’t wait. Don’t say “maybe when I teach this unit next year” or “maybe this class isn’t ready.” Try it. See what happens.

This is also going to be an ongoing resolution maker of a phrase, because it means that every time I think of something I could do to improve myself…this is telling me to start it right away. Not to wait for some arbitrary start date like the first of the month or the end of the school year or spring break. If I think of it, it needs to get put into action.

(Like this blog post. I have barely blogged all school year, but I thought of this and I’m WRITING IT AND POSTING IT. Action.)

Now, I’m pretty scared of this phrase. Help hold me accountable.

 

The other phrase that I’m using to define this year comes from the fact that 2017 is a prime number.

Prime year, prime life.

It just means…make the most of this year. Find the positives. Celebrate the good things. Why shouldn’t an arbitrary year be the prime of my life? Why shouldn’t every year be the prime of my life? I’ve been attached to prime numbers since I found out what they were, and when I discovered that this year was a prime number, I just got excited. I’m going to try to keep that excitement up throughout 2017. Join me.

Matrices!

I’ve been doing Interactive Notebooks for the first time ever this year in my Algebra 2 classes. The students LOVE this form of notetaking – it helps them to know what the most important concepts are, helps them summarize them, it’s pretty, and their notes are always organized.

The notebooks have been a combo of me borrowing already created pages from others and editing them slightly, and occasionally making my own.

We’ve hit our unit on matrices, and I have not been able to find ANY resources online really for INB pages or even notetaking guides on the skills we need for this unit. This is the first unit I’ve created all the pages entirely on my own, and I thought I’d share the pages if there are other teachers also failing to find resources for this content!

Keep in mind that I am an INB beginner, so most of these pages are similar layouts because they’re the ones I’m comfortable with creating. 🙂

 

The skills we cover in the unit are adding/subtracting matrices, scalar multiplication, matrix multiplication, finding 2×2 inverses by hand, and using matrices to solve systems of equations with inverses.

 

The first page we included covers the basics of matrix terminology on the front, with examples for students to practice adding, subtracting, and scalar multiplication as a whole class on the left inside. The right side of the inside I used as their individual practice for the section. We had to discuss that “matrices” is NOT pronounced as “mattresses”…

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Next, we covered matrix multiplication. I don’t know how much I love the way this foldable is laid out after using it, but it worked well enough. The front breaks down one multiplication problem into each pair of row and column that you need to multiply to get each element of the solution matrix.  The inside has example problems that we did as a class. I think my students may have benefited from having some sort of written out instructions as well as the illustration on the front, so that’s probably what I will add for next year. (start by pairing the first row of matrix 1 with the first column of matrix 2, etc.)

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The inverses foldable has 3 sections: how to check if two matrices are inverses of each other, how to find the determinant of a 2×2 matrix, and how to calculate the inverse matrix using the determinant. I should have had one example that were inverses and one that were not inverses. I should have had matrices with determinants other than 1 or -1 for the examples of finding inverses. Otherwise, this page worked well for the students – we just had to talk about what to do when 1/det(A) was a decimal while they were practicing individually!

I also might have them add words next time I do this to the Inverse Matrix section saying “switch the a and d values, change the signs of the b and c values” because in practice, they kept switching BOTH pairs and getting confused, so having it in words as well as symbolically may help with that.

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For systems I just did a basic flowchart of steps with one example, along with information on how to input matrices into a graphing calculator and use it to find inverses. Our standards only require them to find 2×2 inverses by hand, so I’m just doing 3×3 and larger using graphing calculators. We did many more examples as a class that just didn’t go in their notebooks.

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After we did inverses, I created a problem set where they decoded a message by using inverses and matrix multiplication. I think this was great because they got to practice two skills at once! Also, it was nice to get a positive message out to my students sneakily hidden in their math work. I could have organized the boxes where they wrote their decoded message better so the words weren’t on separate lines (one of my students got confused because it looks like it says ‘edible’ at the end, oops), but I was trying to finish this up pretty last minute so that…didn’t happen.

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All the files are included here, PDF and Publisher versions, including the decoding problem set. The actual matrices are mostly screenshots from Word, so if you wanted to change those you would have to go into Word and take your own screenshots.