6th Grade Unit 6: Angles and Triangles

Angle Pairs

I used this Which One Doesn’t Belong? to start our unit on angles and triangles.  I love how starting with something like this gives me insight on where students are at with this topic based on their answers and the vocabulary they are using.

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After taking some notes on different vocab words we came back to the same image at the end of the day, and I asked students to use the new vocabulary to describe the images.

I also had students do this Desmos Polygraph several times throughout the unit as they learned new vocab words.

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Sum of Angles of a Triangle

In one of my classes, I had students cut out a triangle, rip off the angles, and put those pieces together to form a line.  It didn’t go quite as I hoped with that class, so in the other classes I was the only one who cut the triangle.  I would like to figure out a way so that more students see what I want them to see as they’re cutting the triangles and putting the angles together.

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Jo Morgan shared several good Angle Chase activities in this Math Gems post.

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Interior Angles of Polygons

After talking about the sum of the angles in a triangle, I have students Notice/Wonder with polygons divided into triangles for them to figure out the sum of the interior angles of polygons.

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After they’d done some practice with that we made a table to come up with the general formula.

 

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6th Grade Unit 5: Percents

We start our unit on percentages by talking about converting between fractions, decimals, and percents.

I start with this Which One Doesn’t Belong? to get students thinking about percents and for me to see where my students are at in their understanding of this.  Then I ask them to brainstorm everything they know about percentages.

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I created matching cards for converting between decimals and percents years ago.  I intentionally picked numbers with lots of 2s and 4s in them so students can’t just say, “These are the only two cards with a 5 and a 6, so they have to match”.  You can download the file here.

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I spend several days letting students practice converting between fractions, decimals, and percents with different puzzles I’ve found over the years.  If I remember where I’ve found them, I’ll link to them here.

This is one puzzle I like for fractions and percents.

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From Chris Smith‘s newsletter via Jo Morgan’s blog.

Here is an Open Middle problem too.

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And yet another good Open Middle problem on percents.

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Then we get into applications of percents:  finding tip, tax, and discount.  I think this was the first year that I didn’t have a student do a discount problem with an answer greater than the original cost of the item.

One of my students favorite things to do during this part of our unit is for me to pull up a store’s website, find an item, and then calculate tax, discount, or tip.  (Side note:  Little Caesar’s website was super nice for adding things students wanted to the cart and finding the price.)

We used this loop activity for practice.

6th Grade Unit 4: Ratios

To introduce our unit on ratios this year, I started with the following picture and asked students to notice/wonder about it and if they could figure out what was meant by the word “ratio”

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I also had this one ready to follow up with if I needed to.

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Then I used a couple of Desmos activities.  This is one that I modified from something I found from Andrew Stadel.  Then I also created this card sort.  There are multiple correct options for the card sort I created.

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Then I used an I Spy activity.  I blogged about it here.

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I like this Which One Doesn’t Belong? around this time in the unit.

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Then we get into unit cost and finding the better buy.  Some years I have students look up items online and find the unit cost of the items, but I’m finding that more and more websites already give the unit cost on them.

Students always enjoy math fails, and they work great in this unit.  Sara shares a ton of them on her blog here, here, and here!

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I used Robert Kaplinsky’s “Which Ticket Option is a Better Deal?”  I definitely want to spend more time on this one next year and really focus on question 4.

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Don Steward also has some great ratio puzzles on his blog here and here.

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As I was going through my stuff when writing this post, I also came across this video.  I always forget about it and have never actually used it in my classroom.

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!

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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.

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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.


Solving Equations

Then we start solving equations using the distributive property.

I gave them a couple review problems prior to starting this.  The problems were similar to the following.

  1. 3(x + 4)
  2. 3x + 12 = 24

Then I told them we were going to use both of those ideas today and put the following problem up:  3(x + 4) = 24.

As students were working on this one student goes, “Oh Ms. Bergman, you are so smart.”  Another example of a student noticing that I am intentional about the problems I put in front of them, and I love it.

(Also, yes I know we don’t need to use the distributive property to solve 3(x + 4) = 24.  We talk about that too.)

I made an Add Em Up activity for this.  You can download the file here.  Add Em Up is an activity I got from Sara Van Der Werf.  You can read her post on this activity here and a post I wrote here.  Students are always super engaged when doing this!  We also spent some time doing Vertical Non-Permanent Surfaces with these problems and students were also super engaged in the math they were doing.  Here are some of the problems we used for VNPS.

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6th Grade Unit 2: Intro to Algebra (Part 3 -Equations)

You can read more about our Intro to Algebra unit here (part 1) and here (part 2).  In this portion of the unit we get into solving equations.

There are several things I do prior to actually solving equations to get students thinking algebraically.

I absolutely LOVE puzzles like this.

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One of my students caught something in this puzzle that I had missed the first time I solved it.

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In one of my classes we had some extra time, so I had my students create their own.  It was super fun to watch them get excited over this and to see their creativity in what they used in their puzzles.

 

 

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Then we used one of my online puzzles –Solve Me Mobiles, and one of my favorite movement activities, Balance Points.  I blogged about both of those activities here.

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When we play Balance Points I put an equation up on the board and with a partner, whatever the answer is students need to have that many body parts touching the ground.

 

 


When we get into solving equations, I stole Julie’s idea found here.  If you teach middle school and haven’t read that blog post yet, you need to stop and do that right now.  I’m not even going to say anything more about it to force you to go read it.  Most of my students will only work on one-step equations.  In some classes they’re ready for multi-step equations or some students in the class are.  Here is the link to download a couple worksheets I use and an add-em-up activity.

Open Middle also has some great one-step equation problems.

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Exponent Unit

This was the first time I’ve taught exponents without explicitly telling students the “rules” at some point within the unit.  Many students still said things like, “Oh, so when you divide, you subtract the exponents.”  I have mixed feelings over this.  Yes, I want my students to notice patterns, but not at the expense of understanding the math they are doing.  This is one of the things I struggle ensuring as a teacher -that after my students have noticed patterns, they still understand what is actually happening.

I started the unit with a modified version of Andrew Stadel’s exponent mistakes worksheet.  (I know I found someone else’s version of this worksheet that I modified, but I can’t remember where I got it.)  This was something we came back to periodically throughout the unit.  On one of the last days of the unit, we went over the correct answers as a class for the first time.  After going over the sheet, I asked my students to think back to their reaction when I first gave them the worksheet.  Many sort of freaked out and several others were convinced that some of the problems were actually correct.  It was fun for me to see them realize they had learned something throughout the unit because they could now correctly do all of the problems.

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The rest of the first day we focused on identifying the base and writing things in expanded form.  The next several days I spent at least one full day on the product rule, power rule, and quotient rule.  The link for the worksheets I used is at the end of this post.  Again, I know I modified those worksheets from ones I found somewhere online at one point, but I can’t remember where I found them.

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I used this Which One Doesn’t Belong? as a warm-up one day.  I’ve really been loving using these as warm-ups this year.  I love how much vocab students use while doing these.

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About this point in the unit, I was not in my morning class a few days in a row due to state testing with my 6th graders.  I was looking for self-checking practice for students on exponent problems.  The challenge for me was we hadn’t talked about the zero power yet or negative exponents.  Most everything I was finding online included those types of problems.  Here’s what I came up with.

I modified Kate Nowak’s row game to work for where my students were at.

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I don’t know if “Two Truths and a Lie” is the correct name for the next worksheet I created, but I couldn’t think of another name and was running out of time, so I went with it.  Basically, students were to simplify 3 different problems.  Two of the problems would have the same answer (the two truths) and the other problem had a different answer (the lie).

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I also had a sheet of Yohaku puzzles ready which I LOVED, but I didn’t end up using it then.  I did, however, use it later in a few of my classes.  I love that there are so many different solutions to these puzzles.  I definitely want to look at the other puzzles on that site for future use.

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When I was finally back with all classes after state testing, we reviewed using this Desmos activity I created.

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absolutely LOVE this Desmos activity from Mathy Cathy for an introduction to zero and negative exponents.

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We ended the unit with some more practice combining all different types of problems.


Here is the link to download the worksheets from this unit.

One Problem Lesson Plans

I’ve heard of people who spend an entire class period on one problem.  One problem!  With middle schoolers!  Most days, getting middle schoolers to focus on anything for 40 minutes, let alone a math problem, is an insurmountable task.  (Side note:  I’ve been working on a grad paper recently aka trying to make myself sound formal by using big words like insurmountable that you would likely never hear come out of my mouth if I were to ever have a conversation with you in person.)

I couldn’t wrap my head around finding a problem that would engage middle schoolers for 40+ minutes.   I didn’t have a clue what that type of problem would look like.  I didn’t know where to begin with a lesson like that.

I worried that my students wouldn’t “learn” as much by spending so much time on one problem compared to multiple problems on a worksheet or some other form of practice.

Enter this problem:

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I learned I was wrong.  So very wrong.


Several weeks ago, I gave my students the Open Middle problem above.

Oh. my. goodness.

I was not prepared for the awesomeness that took place that day.  I still smile thinking about it.  I anticipated the problem taking 5-10 minutes, maybe.  Some students worked on it for the entire 40 minutes!

The concept of the problem was simple.  Students knew how to write equivalent ratios.  They understood they needed the digits 1-9 and knew they could only use each digit once.  But the answer?  That wasn’t quite as easy to find.  They were hooked.

And so was I.  I wanted to find other problems to re-create that atmosphere in my classroom.  I completely underestimated the rich conversation that could take place from what I considered a simple task.

By the end of that class period, I knew I needed to do more of this in my class, but what sealed the deal for me was listening as students tried to figure out how to continue working together on the task as a group after class.  They asked me if they could do a group chat with each other that night so they could keep working on it.  Then I overheard, “If I figured it out, I’ll email you! And if you figure it out, email me!”

They were excited over solving a problem in a way that I hadn’t seen from them before.


This week I used the problem below in the same class.

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From Chris Smith‘s newsletter via Jo Morgan’s blog.

I was actually nervous for this lesson after how well the lesson with the Open Middle problem went.  I tried not to hope for the same results I got the first time but was worried it would flop.  It didn’t, and again, I was amazed at the conversation that resulted from this one task.


This week I used the problem below from 1to9 Puzzle with my other two sections of 6th graders.  I thought it would take about 5 minutes.

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I quickly realized it would likely take longer than I anticipated and saw the opportunity for those students to experience what my other class had.  While it didn’t relate to the content we were covering like the other two problems, I decided to deviate from the lesson plan and give students more time on this.  It was well worth it.


In the few times I’ve done problems like this, a couple things stand out to me.

  1. I am amazed at how many students don’t know how to guess, check, and adjust their answers on problems like these.   Some students could not wrap their head around the idea of just picking numbers as a starting point and going from there.  It was an eye opener for me, and I realized I need to continue to incorporate more situations where students need to do this.
  2. My doubts about whether students would “learn” as much from doing one problem like this rather than another practice activity were erased.  The conversations amongst students while doing problems of this nature still amaze me.
  3. One of my absolute favorite parts of doing these are watching students’ reactions when they finally find a solution.  They are SO proud of themselves.  This past summer I had the privilege of spending a lot of time learning from Sara Van Der Werf.  One of the things I heard from her over and over again was how one of her goals in her classroom is to get kids addicted to the cycle of being puzzled and becoming unpuzzled.  I was able to physically see this in my students more while doing these types of problems than possibly anything else I’ve done so far this year.

Do I have the “one problem lesson plan” down pat?  Absolutely not.  Is it even close to great?  No.  So far it’s really been pretty unintentional.  I’ve pretty much just been lucky and stumbled upon problems that have turned into great lessons.  I need to get better at bringing everyone back together to close the lesson after doing a task like this.  I’ve added finding more tasks like these to my summer to-do list.


UPDATES:  I’ll add more of “one problem lesson plans” below as I try them in my classroom.

This problem found here was another winner with students.  After students found a solution, they continued to work to find other solutions with me telling them to.  Sigh.  I needed that little reminder that week that we were in fact making progress.

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