Distributive Property

Earlier in the year my 6th graders talk about the distributive property without variables. Partway through this post I shared how I introduce that idea to students.

Later on in the year we start talking about the distributive property with variables.  I started by reviewing how they used the distributive property earlier in the year without variables.  I was so impressed with how many different ways my students came up with to use the distributive property to multiply 7×48.

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This year I used Illustrative Math Unit 6 Lessons 10 and 11 to introduce the distributive property with variables.

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Illustrative Math Grade 6 Unit 6 Lesson 10

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Illustrative Math Grade 6 Unit 6 Lesson 10

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Illustrative Math Grade 6 Unit 6 Lesson 11

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Illustrative Math Grade 6 Unit 6 Lesson 11

As we were working through the resources from Illustrative Math, I loved how they incorporated the idea of factoring, without explicitly calling it that.  I had done a little bit of that in the past with this puzzle from Open Middle.

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Again, I was super impressed with all the different solutions they came up with.  I didn’t quite use the “rule” of the Open Middle problem and allowed students to use fractions and decimals.

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After going through that, students worked on this distributive property puzzle.  When students finished that, they started working on some Yohaku style puzzles I created.  This was my first time creating my own puzzles like this, so I had no idea how it would go over with students.  When I made the puzzles, I found two solutions for each.

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This activity went over SO much better than I even imagined, and my students found solutions that were much more creative than the ones I had found!


When I was explaining how these puzzles worked to students I told them that if I did it correctly when I made them, each one should have at least two solutions.  One student asked, “But what if you did it wrong?”  I told them that very well could have happened. I’m human, and it’s May.  I’m tired.  😉

After the first group found two solutions for the same puzzle, one student told me, “You did it right!  You didn’t make a mistake.”

My students were so engaged while working on these puzzles.  They were so persistent.  I loved seeing all the eraser marks on their paper as evidence of them trying again and again and again until they found something that worked.  Students were cheering when they found a solution.  I wish I had recorded them working on these.  It was fantastic.

After the bell rang one student said, “Could you make some more of these for next week? Maybe nobody else liked them, but I thought they were fun.”

I also am looking forward to have a conversation with this student about the right column.


I completely understand the student’s thinking.  This is the same student who came up with this solution earlier in the week.

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Here is the link to the puzzles I created.

If you create more, I would love to see what you come up with.  After sharing a picture of the puzzles on Twitter, Yohaku created a few similar.  You can find them here.


One Good Thing: Volume 4

I’ll admit, my expectations for today weren’t too high.  I had a final for my grad class last night until almost 8:00, so yesterday was a long day.  Teachers have inservice tomorrow, so my students have a 3 day weekend.  It’s May.  The weather is finally nice.  I felt like I had so much going against me today, but man, today was one of the best days I’ve had in a long time.

My day started with one of my 6th graders working on some Yohaku style puzzles I created.  I hadn’t used these before, so I had no idea what to expect.  Would it be too hard?  Too easy?  I didn’t know.  It was the perfect level of challenge for my students.  I wish the eraser marks showed up better in the picture below.  My students were SO persistent when solving these.  There was cheering when a solution was found, and when they found a solution, I challenged them to find another solution.  And then another using fractions.  My students were engaged in these puzzles for well over 20 minutes.  It was fantastic!  I plan on writing more about this portion of my unit soon.

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Class was over and a student said, “Could you make more of these for next week?  Maybe nobody else thought they were fun, but I thought they were really fun.”


My 8th graders are working on solving systems of equations.  We were getting into systems that have no solution and infinitely many solutions.  My lesson plan wasn’t anything special, at all.  Some of my 8th graders have been struggling to focus lately, but both classes worked SO well, the entire hour.


My other 6th grade classes were testing today.  One student left this note on her test, and then left little notes/jokes throughout her test.  She most definitely made my day.



My 7th grade class was shortened today, and over a third of my class was gone for track and many others were leaving partway through the hour for softball.  While my students were taking a short quiz, I this thought, “Maybe today would be a good time to try Dan Meyer’s Taco Cart 3-Act…”  I’d never done a 3-Act before, but I’d been wanting to try this one for over a year.  I decided now was as good a time as any, so we did.  I let my students know that I’d made a last minute decision on what to do.  That I hadn’t done this one before.  I could be great.  I might not be, but they were going to be my guinea pigs.  One student’s response, “Yay!  I love being a guinea pig.”  Students are so awesome.

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While the math involved for this 3-act was from a prior unit, the conversations students had while doing this task were so much better than the lesson I had planned.


My final for my grad class was yesterday.  That means I didn’t spend today after school working on next week’s homework for that class.  Definitely a good thing.



VNPS Variation

(VNPS: Vertical Non-permanent surfaces)


This is a different time we did VNPS, but my classroom looks like this nearly every time we do this.  Why don’t I do this more often??

Last year I created a “Two Truths and a Lie” worksheet on exponent worksheets.  Students simplify three problems.  Two of the answers are the same, and the third answer is similar but not quite the same.  I was really happy with how the worksheet turned out and how it went last year.

The day I had this worksheet in my lesson plan, I realized I wasn’t really looking forward to another worksheet with my 8th graders.  I try to mix things up, but lately they’ve had worksheets pretty consistently.  I was thinking through other options and was trying to come up with a way to change things up less than 2 hours before I had students in my room.

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In the past when students worked at the whiteboards, I had them do one problem at a time and get it checked by me.  This time, I had students do all three problems in each set before I would check their answers.  They knew that two of the answers were going to be the same and one would be slightly different, so they already were doing some self-checking as they went along!  When students called me over to check their answers, I would tell them how many problems they had wrong but wouldn’t tell them which one.  I really liked the small change in how I had students do these problems.  This is something I do a lot when they’re working in Desmos or with other activities, but I hadn’t done it in this situation before and loved how it went.

I had more students engaged for more of the hour by doing the problems this way, and my students were talking through the problems more than they would have if they were working on the worksheet.  I actually had students cheer when I said we were going to the whiteboards.  🙂 It was a good reminder that I don’t have to always come up with some fancy activity to switch things up.  Something as simple as taking problems from a worksheet and having students complete them in a different way is enough sometimes.

Here’s the link to download the worksheet and others on exponents.  Here is where I shared other things I did in this unit last year.

I Spy

Last year when I was teaching ratios, I had an idea in my head, but I didn’t quite know where to go with it.  I envisioned an “I Spy” type picture where students had to “spy” the ratios in the picture.  I ended up not doing anything with the idea at the time, and I sort of thought that if the idea had potential, surely someone more creative than me would have come up with it already.

Fast forward to this year, and the same idea was still in the back of my head, so I decided to see what would happen if I tried something with it.  I came across a picture online, added the ratios to go with it, and ended up giving it a try with my students.


What happened next was great!

I first asked students to find those ratios in the picture, and then I told them they were going to create their own pictures using Google Slides or Google Drawings.  I gave them a few ratios they had to have in their picture and had them pick the last one.

Here are a few examples of what they came up with.

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ALL of my students were SUPER engaged during this.  They loved deciding which pictures they were going to use, and I loved seeing the different directions students went with this project.  I also had them fill out a half sheet telling me what their ratios were.  This was really good for students because it made them check their own work.

I would love to find a way to make this project more challenging for students in the future so that they have to use more higher level thinking.  If you have any ideas, please send them my way!

In the future, I would also like to share the pictures with the entire class and have students try to spy the ratios in their classmates’ pictures.  I meant to do that this year, but I never got around to it.


8th Grade Unit 3: Functions (Part 3 – Point Slope Form & Standard Form)

Here’s part 1 and part 2 of unit 3.

I used this warm-up the first day after our test on slope-intercept form to get students thinking about equations and graphs again.

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Then I do notice/wonder with this.

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I heard things like:

  • There are two x’s and two y’s.
  • There are little 1’s by one of the x’s and one of the y’s.
  • There are parenthesis.
  • There’s an m (slope).
  • There’s no y-intercept
  • Is it another form of a linear equation?

It leads nicely into discussing point-slope form and students realize that it isn’t as scary as it may look at first because they recognize the similarities between slope-intercept form.

When going over point-slope form, I make a point to emphasize to students why it’s named point-slope form -we can see the coordinates of a point and the slope from the equation.  I remind them that this is similar to slope-intercept form where we saw the slope and the y-intercept.

Then we go over a few examples of writing equations in point-slope form before doing an activity similar to what Sarah shared here.  I didn’t have big foam die like Sarah used, but I do have double dice, which students always think are fun.  Students rolled the dice to create two ordered pairs and wrote an equation in point-slope form of the line between those two points.  Then they checked their answers using Desmos.  Having students check with Desmos was key to helping them see what they were doing when writing the equation of the line.

I also modified this Desmos marbleslides activity to rearrange the equation so that they looked like what my students were used to seeing.  My modified version can be found here.

Then after some more practice using point-slope form, students are introduced to standard form.

I use notice/wonder again to get students thinking about this.

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Again, students came up with the following things:

  • There’s 2’s in all of them.
  • The two is always by the x.
  • One of the equations is in slope-intercept form.
  • One of the equations is in point-slope form.
  • In the purple one, the x and y are on the same side.

We also talk about how, unlike slope-intercept form and point-slope form, we don’t see the slope, the y-intercept, or a point.

Of the three parts to this unit, this one is takes up the fewest number of class periods.  Writing up this post made me realize that I could probably use a few more activities on these concepts.  If you have any ideas for me, please share!

8th Grade Unit 3: Functions (Part 2 – Slope & Slope Intercept Form)

After talking about functions vs relations and linear vs nonlinear, we get into slope and slope intercept form in 8th grade.


To intro this I start by using a Desmos Polygraph.

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Then this year I tried this Desmos activity on steepness.  I liked we were able to move from talking about something students were familiar with, steepness, to a word students may not be familiar with in slope.

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Then I was able to use use this awesome activity Sarah shared here with some of my students, since they were a couple days behind the other class.  It was such a simple idea, but I absolutely LOVED how students figured out on their own how to find the slope of a line.  They also were able to explain when slope would be positive versus negative.  For next year, I might change some of the lines to include slopes such as 2/3 or 3/4.

And what would be a lesson on slope without Slope Dude?  I blogged about another activity I shamelessly stole from Sarah here.sd2

I used this Desmos activity I found online to have students practice finding the slope of lines.  After that, we talk about the slope of horizontal and vertical lines.  Students already know from playing Slope Dude that horizontal lines have a slope of zero and vertical lines have an undefined slope, but now we talk about why.

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Slope Intercept Form

A while ago on Twitter Mickie asked for ideas on introducing Slope Intercept Form to students.

I shared with her a worksheet I created a few years ago where students use Desmos to figure out slope-intercept form.  She had some great ideas of how to improve what I had created.  (Twitter and #MTBoS are so great!)  She shared with me what she and a colleague came up with.  I loved her addition of the table.  Here‘s the editable version of the worksheet she shared with me -I did make a few minor changes.

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When I’m teaching slope-intercept form, I try to make a big deal about how the name for this form of an equation makes sense based on the formula itself.  When we’re given an equation in this form, we can easily see the slope and the y-intercept, hence the name slope intercept form.  This is something that is obvious to me as the teacher, but I found that students don’t always make this connection.  Because of that, it’s important for me to help students make that connection.  This also helps later on when we talk about point-slope form.

One change I made this year to how I teach this is that I had students check their answers after graphing.  I decided to do this for a couple reasons.  One, I hoped that this would slow students down and help them catch mistakes they made when doing the slope of the line.  Two, I hoped that this would help students make the connection between the graph and the equation.  In the past, I don’t know that I have done a good job of helping students make this connection.

After students are comfortable with equations in slope intercept form, we go over writing an equation from a graph, writing equations for horizontal and vertical lines, as well as needing to get y by itself before graphing.

Here is the link to a Desmos activity I used to have students practice going from graph to equation.  I took the images from somewhere online, but I can’t remember where.  Sorry!  If anyone recognizes them, please share so that I can give whoever created them credit.

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You can download a worksheet and the test review here.

And last but certainly not least:  Desmos Marbleslides.  This is one of my absolute favorite activities from Desmos.

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Small Change

Do you ever make a small change to something you do every day that ends up making a HUGE difference and think to yourself, “Why didn’t I think of this years ago?!” or “Why don’t I remember to do this more often?”

This happened today, and it ended up to be pure magic.

Today in one of my classes I was introducing point-slope form.  I put the formula up on the board and asked students what they noticed and wondered about it.  Then we talked a bit more about it and did a few examples.

Then I gave students the coordinates of two points and said that we want an equation in point-slope form that goes through those to points.  Their task was to figure out how to do that.  They found a partner and got to work.  After a couple minutes, the chatter started dying down, and I could tell many groups were stuck.

This is where the magic happened.

I stopped the class and told them that for about 30 seconds they were going to find a new partner and talk about what they’ve been trying with the problem.  Then I was going to stop them again, and they would return to their original partners and keep working.

The room instantly came alive again with students talking about what they had tried.

Many groups were stuck at different places or had tried different things so my students had a lot to talk about with their new partner.  When they went back to their original partners, they had new ideas and were able to continue working.

I had done something similar to this last year, or maybe even longer ago than that, and why it took me this long to do it again, I don’t know.  I think my fear when I do things like this is that my students will be more off task; however, today the opposite happened.  The conversations within my groups were more on task for longer periods of time than they typically are in that class.

I want to incorporate this more into my classes, but at the same time be careful not to overuse it and be intentional about when I use it as I definitely think there are times something like this would be more effective than others.