6th Grade Unit 3: Fractions (Part 1 -Greatest Common Factor & Least Common Multiple

I start our unit on fractions by going over Greatest Common Factor and Least Common Multiple.  It seems that no matter how many times we go over what factors of a number are and what multiples of a number are and break apart what it means to find the “greatest common factor” and “least common multiple”, some students still get these mixed up.  I try to be super intentional about not saying “GCF” or “LCM” with students very often because I know there is likely at least one student who doesn’t know what those letters stand for -I try to use the vocabulary as much as possible with students instead of acronyms.  I want them to know what GCF and LCM mean because they will likely see that other places, but if I say “GCF” I always pair it with “Greatest Common Factor”.

We start with greatest common factor and first review what factors are, and then discuss what it means to find the greatest common factor of two numbers.  Once students get a quick review of this, they usually remember doing this in 5th grade.  I found this year that the methods my students used as 5th graders to find the greatest common factor were the “rainbow method” and T-charts.

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We do a few examples where students use one of those two methods before I put up an example with larger numbers like 192 and 320.  Students usually give me the reaction I’m looking for by groaning, and I ask why they’re groaning.  They tell me the numbers are big and there’s a lot of factors to list.  I then ask if they’d like a more efficient method of finding the greatest common factor and introduce students to what I call the “ladder method” to find the greatest common factor.  I think I first heard about it from Sarah here.  The high school teachers in my building use this method with polynomials, so I want to introduce it to students.

For practice, I have students write a number on a slip of paper that has many factors.  Then students pair up and find the greatest common factor of their numbers.  I have them check their answers with Desmos.  (Did you know that Desmos can find the greatest common factor of numbers?  It will also find the least common multiple.)  Then once they’ve checked their answers, they find a new partner.

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Then we get into finding the least common multiple of numbers.  Again, we review multiples, and how they are different from factors.



(We review the above nearly every day in this unit, so you would think I’ve got how I want to display it on the board for students down pat.  This is what I ended up liking the best, although I’m still not sold on how I showed factors.)

Then we talk about what it means to find the least common multiple of two numbers.  Students usually start to remember doing this as 5th graders.  We do a few examples of this before we talk about how we could use the ladder method to find the least common multiple.

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Next comes word problems.  I’ve modified this worksheet for practice for students.

Then I have a couple different loop activities that I use depending on when they best fit into the schedule.  The first has greatest common factor, least common multiple, and prime factorization, while the second loop activity has greatest common factor, least common multiple, and word problems.

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Each activity has 6 problems, which is nice because most groups can finish within a class period.  However, it’s not nice for keeping groups small if I want one group per problem. I ended up printing two copies of the activity on different colored paper, and this worked great.  Half of my students rotated through one color set while the other half rotated through the other color set.

You can download the files for both activities here.


6th Grade Unit 2: Intro to Algebra (Part 4 -Inequalities)

The last part of our Introduction to Algebra unit in 6th grade is inequalities.  I also wrote about part 1, part 2, and part 3.

I don’t spend a ton of time with this.  I start with a few Desmos activities.  Polygraph is the first thing we do, and then I come back to it a day or so later after students have done more with inequalities.  Then we review the inequality symbols before students work on this Desmos activity.  It is a basic introduction to graphing, and it is usually the first time that my students see the greater than or equal to symbols and less than or equal to symbols.

The next day to review we start with this Which One Doesn’t Belong.

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Then we work a bit on graphing inequalities when given a situation.  This Desmos activity is one of the first things we do to practice this.

The last way we review is Mathketball.  My students LOVE this game.  They put their desks in a circle around the room.  I put a garbage can in the middle of the room.  I put a problem up on the board, and if students get it correct they get to try to shoot their paper in the basket.  Students would play this for every concept if I let them, but I try to save this for problems that I anticipate all students completing in the same amount of time compared to problems that have multiple steps to get the answer.


I want to find more pictures like these that students can write inequalities for.

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The day before the test a student asked how many problems were going to be on the quiz.  I said, “8 problems tops.”  Another student said, “You should make us write an inequality for that.”  I just love these kiddos so much.

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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Math Telephone

I used Sarah Carter’s Representations of a Relation Telephone activity last year.  You can read all about the activity in her blog post.  In short, it works just like the game of “telephone”.  A student starts by writing six ordered pairs on the bottom, passes the paper to the next person who creates the graph, who passes it on the the next person who creates the table from looking at just the graph.  This continues until you get to the top of the paper with the ordered pairs again.  If done correctly, the ordered pairs should be the same.

I really liked it, but I knew there were some tweaks I wanted to make the second time around, and it went much better this time.  Part of it could have been that I had already explained the activity once, so I did a better job of explaining it and anticipating where students would struggle.  The first year I did this I actually had one group start whispering ordered pairs into each other’s ears.  I don’t know if they just weren’t listening or if my directions were that bad…probably a combination of both.  Thankfully that didn’t happen this year!

Sarah also shared an updated version of her activity here which also helped as she’s included more instructions on the sheet itself.

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The first time I did this I made the mistake of having one piece of paper per group of 4-5 students.  What was I thinking?!  That meant for 10+ minutes about 6 students in my class were working while the others were supposed to wait patiently?!  Not my brightest move ever.  This year all students had their own paper so everyone was doing something at all times.  This went MUCH better.


As I watched the activity with my first class of the day, I noticed that it was taking students FOREVER to just fold the paper, and one student commented, “Folding the paper was the hardest part of this!”  I found that rather than folding it in an accordion at first it worked better for my students if each person just folded the bottom representation under before passing their paper on to the next person -these instructions are also on Sarah’s updated version.


Overall it was a big success.  I loved watching students look at where the mistakes were made when they were done, and the second time they did it several students were thinking ahead when they created their ordered pairs to try to make it “easier” or linear -which will lead perfectly into what we’re getting to later in this unit.  I even had a student who doesn’t usually get too excited about much of anything say, “This is actually fun.”

I was thinking today what other concepts this could be used for and remembered one I made last year for my 6th graders on exponents.  Since last year, I lost the editable version of the document I made, so I recreated it and added some of Sarah’s instructions to it.  You can download the file here.

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What other concepts could this idea be used for?

6th Grade Unit 2: Intro to Algebra (Part 2 -Evaluating Expressions)

I blogged about the first part of our Intro to Algebra unit in 6th grade here.  In this part of the unit, we finally get into the algebra stuff.

Word Phrases

Before we start evaluating expressions, we talk about what variables are and what the purpose of them is.  We review word phrases.  I started this year by putting these words up on the board and having students classify them based on their operation.

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After we have done a bit of large group practice with that, I have students do a card sort/matching activity.  Once I have checked their answers, they can play memory.  Some students have also played Go Fish with the cards as well.

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I also use this Desmos card sort on another day to review this concept.

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Evaluate Expressions

Then we get into evaluating expressions.  This year I made a small tweak to how I introduce this to students.  In the past,  I would stand up in front of the class and tell students what you do when you’re asked to evaluate an expression.  I’m really trying to get away from being the teller of information in my classroom and instead be the asker of questions to get students to explain the mathematical concepts to each other.  Here’s the small change I made this year.

I put “x + 2″ on the board and asked if we could come up with a number answer for this.  I saw several heads shaking no, and when I asked why, they told me, “We don’t know what x is.”  Then I added to the board “x = 7″ and asked if we were able to come up with a number answer for this now.  They told me we could and that the answer was 9 and then explained how they got that for an answer.  I could tell that not all of my students had caught on yet or weren’t fully paying attention, so instead of me rephrasing what the student had just said, I asked, “Can someone else explain to the class how ‘Sue’ got 9 for an answer?”   Then I put something like 4x + 3 up and repeated the same process.

It was a small change, but it felt SO much better than standing up in front of the class telling them a process to follow.

The first activity I do is something I created several years ago, and I realized last year when I did this that it’s pretty similar to Sara’s Add-Em-Up activity.  I created 4 different sets of 5 cards.  Each set of cards is printed off on a different color paper so that I can tell which set students are working on.  Below is a picture of the first page of the document for this activity which has 4 copies (each column) of the first set.  I have 8-10 copies of each set.  Because all 8-10 sets are the same color, before I laminated the cards, I wrote a number on the back of each card in a set.  This way, when I find a random card on the floor I can ask, “Who has the red 3s?” and easily figure out where the card goes.

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Students are put in groups of 2-3 and in their group they work together to evaluate each of the cards.  When they are done, they add up their 5 answers and come tell me what they got.  If they are correct, I will give them the next set.  If they are incorrect, I don’t tell them which one they got wrong, and they go back to their group and work to figure out what they did wrong.  I’ve tried to level the cards so that each set gets increasingly difficult.  Set 1 has one to two operations on each card.  Set 2 has three steps to each problem.  The next set incorporates decimal operations and the final set has the variable in the problem more than once.

One of the reason I like having a different color for each of the sets is that it is easy for me to see where students are when we are doing this activity.  If I look around my room and see one group on the red set (the first set for me) and every other group is on orange or green (the 2nd and 3rd set for me), I know I need to check in with the group working on the red cards.

Below is an example of each of the four sets.




As I was looking through my stuff to find the file for that activity, I came across another activity I created and had forgotten about.  For this activity, I put students in groups and have them start at a problem.  They can choose to solve either problem on the card.  I encourage all students to try at least one of the more challenging problems.  Once students solve the problem, they look for the answer on another card that I have hanging around the room and then solve either problem on that card.  Eventually, they will loop through to all of the problems and end back where they started.

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I also used this Desmos card sort towards the end of the unit to review with students.

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Find the Flub warm-ups are great for evaluating expressions.

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In the past, I haven’t done much with tables in any of my classes, and I know that this is something I should do more of.  I added this Desmos activity to this unit, and overall I was pleased with how it went.  It was challenging for some of my students, which was my goal when creating it.




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Here is the link to download the activities from this post.

6th Grade Unit 2: Intro to Algebra (Part 1 -Exponents, Prime Factorization, Properties, and Order of Operations)

Unit 2 in 6th grade is an introduction to algebra.  This is one of my favorite units.  I love order of operations, and I love introducing students to solving equations.  I break the unit up into multiple parts.  Here is part 1.


We start the unit talking about exponents so that students can use exponents when we get to prime factorization and order of operations.  I typically spend about a day on this and use Kahoot for practice.  I also incorporate this throughout the unit in brain breaks.  “Ok everyone stand up!  2 to the 3rd power.  (Then I give them time to think about what the answer is.)  Do 2 to the 3rd power jumping jacks.”

Prime Factorization

Then we review prime and composite numbers before getting into prime factorization.

(I incorporate prime/composite into brain breaks as well.  “Think of a prime number.  Do that many sit-ups or push ups.”)

I also incorporate a brain break called Factor Hop into this part of the unit as well.  I put four numbers in the corners of my room.  Students go stand next to a number.  I pick a number and if that number is a factor of the number students are standing by they have to move to a different corner, but they are not allowed to walk.  Some students really get into it and have a lot of fun with coming up with other ways to move to a different number.

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Which one doesn’t belong? works great as a warm-up a few days after going over prime and composite numbers to review this vocab.  Students will also usually bring up factors in our conversation.


Since students typically have already learned how to do prime factorization using the factor tree method, I do a couple examples of that before introducing them a method similar to the birthday cake method I found on Sarah’s blog.


I’ve started using this method because for a couple reasons.  In my opinion it’s more organized than the factor tree method, and I like that it can be applied to other concepts such as greatest common factor as well as with variables.  The high school teachers in my district also use it.


Then we get into properties of numbers.  We start with the associative property, identity property, and commutative property.  I co-taught with a teacher a couple years ago who was a huge help when it came to teaching properties.  She did a great job of helping students see the connection between what the word actually means and what is happening in the property.

Commutative Property:  You see the word “commute” so the numbers “commute” or change places.

Associative Property:  You see the word “associate”.  For example, you may associate with certain people at basketball practice, and you associate with other people at church.  In the associative property we see numbers “associating” with different numbers.

Identity Property:  Identity is who you are, so in the identity property the number wants to keep it’s identity.  It wants to stay the same.  After we talk about that, I introduce this property by saying, “I’m a 5.  We’re adding.  I want to stay the same.  I want to keep my identity.  What do I need to do?”  Then, “Ok, now we’re multiplying.  I’m a 5, and I want to keep my identity.  What do I need to do this time?”

Then for practice, we use this Desmos activity from Cathy Yenca.  I edited her version to not include the Distributive property, since we hadn’t covered that one yet.

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Then I used Sarah Carter’s Two truths and a Lie activity.  My students really enjoyed this. You can download the template from here blog post here.

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I loved this one from one of my students.  I read it too fast the first couple times and missed their mistake.

For a few days leading up to teaching students the distributive property we do math talks, and this has made teaching the distributive property go SO much better for me.  In almost every class, I will have a student who will use the distributive property in the math talk so we can talk about so-and-so’s method of multiplying and then I’ll later introduce the term distributive property.



Then for practice, I came up with this Desmos activity.

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I color coded the cards, and I usually go over this with students before they start the activity so they don’t become overwhelmed when they start.

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Order of Operations

I’ve started introducing order of operations by having the following up on my SMART board along with an example problem on the whiteboard and having students do a stand and talk to talk about which things need to be done before others.

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I’ve liked this change.  I enjoy listening to their conversations as they talk, and it also gives me insight into where they are at in their understanding of order of operations as well as how they were taught this as 5th graders.

In every class a student usually brings up PEMDAS, and then we discuss what I don’t like about that acronym.  I love that students are able to tell me things like the “P” stands for parentheses and there are other grouping symbols besides that, and “it looks like you have to do multiplication before division, but you don’t.  They’re on the same level and you read it like a book going from left to right.”  It was also music to my ears when a student said, “PEMDAS?  What’s that?  I’ve never heard that before.”  To which I replied, “Great!  You don’t need to know what it means!”

This has also become one of my favorite warm-ups of all time.

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Over the years, I’ve built up a quite a collection of order of operations activities, and I’ll pick a few of those for practice.

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  • Espresso Puzzles from Greg Tang Math (scroll through this page to find the Espresso Puzzles)

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6th Grade Unit 1: Area & Decimal Operations

6th Grade Unit 1: Area & Decimal Operations

In 6th grade we start the year with a little bit of geometry and decimal operations.  In Minnesota, students add and subtract decimals in 5th grade and in 6th they are introduced to multiplying and dividing decimals.  I made some improvements to this unit this year, and even though I wish I were better at teaching some of these concepts, I can’t ignore the fact that I did make improvements from last year.  If I keep working each year to make it better, eventually I will get closer to where I want to be when it comes to teaching these things.

Our first unit is broken down into 4 parts:  the coordinate plane, area, multiplying and dividing decimals, area on the coordinate plane.

The Coordinate Plane

I read Tom’s post on creating a need for the coordinate plane a couple of years ago and knew I had to try it.  This is the second year that I’ve used it, and I think it is a really good way to introduce the coordinate plane to students.

After we have talked about the parts of the coordinate plane and plotting order pairs, students do this Desmos card sort.

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Then we do this Desmos activity from Nathan Kraft.  This is usually students’ first time using Desmos, and I love watching them go from being somewhat confused with how Desmos works at the start to absolutely LOVING Desmos about 2 slides later.  🙂


Then we start working on finding the area of figures.  I use a lot of Notice/Wonder with GIFs to talk about finding the area of various shapes.




For students to get some practice finding the area of shapes, I use this worksheet.  It is similar to Sara Van Der Werf’s Add ‘Em Up activity, but in worksheet form.  I use this before we do area of trapezoids.

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For practice on finding the area of trapezoids, I don’t do anything fancy.  I have pictures of trapezoids that I tape around my room and have students walk around in groups solving the problems.

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I do a few other things with compound area and estimating decimals before we start multiplying and dividing decimals.

Multiply Decimals

I tried something new this year to introduce multiplying decimals.  I started by putting a decimal multiplication problem on the board and had students estimate it.  Then I had students multiply the two numbers and told them to forget about the decimal until the end and to use their estimation to figure out where to put the decimal.  After doing one problem together, I had students do several problems in a group and told them to look for a pattern regarding where they put the decimal at the end.  For the most part, students were able to see the pattern and tell me the “rule” for multiplying decimals.  I realize this isn’t perfect, but it’s better than what I had been doing in the past, so I was happy about that.

Open Middle has a couple great problems for multiplying decimals.

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Dividing Decimals

I used the following image to start our conversation about dividing decimals.

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Generally students notice that the answer stays the same.  Sometimes students think that the answer of 4 each time is wrong, so we have a conversation about that.  Someone usually says that you add a zero to the divisor and the dividend each time, and usually someone else in the class knows that both numbers are being multiplied by 10.

In my experience, this has lead nicely into dividing decimals, and as we continue to work on that, I reiterate that when you multiply both the divisor and dividend by 10 (or multiples of 10) the quotient remains the same.

The reason I put decimal operations in the unit with area was so that after doing both concepts, I can put them together to review both at once.

Estimate Area on the Coordinate Plane

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I have always struggled to find good practice problems for students on the types of problems in the standard above.  This year, I found a few pictures online and turned it into a Desmos activity.

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If you use anything in your classroom that you feel would fit with this unit, I would LOVE to hear about it!