6th Grade Unit 2: Intro to Algebra (part 1)

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.


Exponents

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.

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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.


Properties

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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8th Grade Unit 1: Solving Equations (Part 1)

The first unit we do in 8th grade is on equations.  I start by reviewing order of operations, evaluating expressions, and simplifying expressions.  Then we get into solving more basic equations.  Here is a semi-brief overview of the first part of this unit.


Order of Operations

We start off with order of operations.  I use the following Notice/Wonder to lead into our discussion/review of order of operations.

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We also review absolute value as well as square roots as part of our order of operations practice.  These are great problems for vertical nonpermanent surfaces (#VNPS)

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This Desmos activity from Cathy Yenca is also a great review of squares and square roots.

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After a couple days of absolute value problems and square root problems, students work on a worksheet similar to the one below.  You can download it here.  I’ve thought about changing up this worksheet since it doesn’t include square roots or absolute value, but it is a good challenge for students, since students are only allowed to use the numbers 0 through 9 once, and I like that about it.

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Students also see their first Find the Flub warm-up in this unit.

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Evaluate Expressions

Then we spend a little bit of time on evaluating expressions.  I use the worksheet below as practice for students.  I blogged about this type of worksheet here.  You can find the link to download it in that post.

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Simplifying Expressions

Both years I’ve taught this, I forget that students aren’t as comfortable simplifying expressions as I expect them to be.  I start by having students simplify expressions that don’t involve the distributive property, and then I add that in a day or so later.  I found a Desmos activity in the Desmos Bank that I modified and uses on one of the first days on this topic.  Here is the link to the activity I modified.

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Then we do a couple days of simplifying expressions with the distributive property.  Again, I use a “One Incorrect” Worksheet.  You can download it in this post.

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The Notice/Wonder I used below was GREAT to discussion some common mistakes I was seeing students make when simplifying expressions.  For example, I had students who would say that 5x² was 25x.  We had a really good discussion about the differences in the expressions below and how that changed things.

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Solving Equations

Then we get start solving equations.  A few years ago, I had a group of students that struggled to plot points on a number line, so when we got to solving equations, I saw that as an opportunity for them to get more practice with that by having them graph the solution to the equation.  They also struggled with order of operations/evaluating expressions, so again,  I decided to have them practice this by checking their answers to the equations.  I’ve never looked back, and now I have students graph and check their answers to nearly every problem they do for me.

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If you’re interested in the worksheet I use, you can download it here.  Below are a couple of warm-ups we use when we’re talking about solving equations.

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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.  🙂


Area

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.

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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!

Like Terms & Simplifying Expressions

To introduce the idea of like terms to my students.  I use this Desmos card sort.  Initially I have students group the card however they choose.   Students will inevitable group some cards that are like terms which leads us into talking about what it means for terms to be “like”.  Then I have students group the cards into groups that are like terms.

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Then we play “Like Terms Uno”.  The version I use I got on Teachers Pay Teachers a while back, and it looks like it’s no longer available.  In a quick Google search of like terms Uno, several other versions came up.  I’m not sure if those versions are uploaded to the internet legally, which is why I haven’t included links to them.  So if you’re interested in a version of this, seriously just Google “Like Terms Uno” and several different options will come up.

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After this we start talking about simplifying expressions.

I start with the video below.  A couple minutes in many of my students are groaning.  I may look at another option for next year, or cutting down the video clip somehow, because watching that video for over four minutes is torture, but it serves it’s point.  I tell students, “You know how your ears hurt after you watched that video for a few minutes? That’s what it’s like for mathematicians every time they see something like 5x + 7y + 3x + x + 8y.  How could we rewrite that so it doesn’t ‘hurt your ears’?”

We practice combining like terms one day, and then the next day we do practice with the distributive property.

Open Middle has some great problems for simplifying expressions.

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I also made a One Incorrect worksheet for these types of problems.  You can download it here.

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Integers

Up until a couple years ago, I had never been the one to introduce students to integer operations.  I definitely have room for improvement when it comes to teaching integer stuff, but then again, when isn’t there room for improvement when it comes to this job?  Below is a snapshot into our unit on integer operations.


Adding and Subtracting Integers

I’ve tried a couple different ways of introducing students to adding and subtracting integers.  I’ve used Sarah’s “Sea of Zeros” and a number line to represent what is happening.  This year, we focused more on number line rather than using the colored counters, and I’m not sure why, other than time.  It seemed like every time we were going to use the colored counters, something else that day took longer, and I didn’t take the time to get them out.

Last year, I saw this Desmos activity, and loved it, but I didn’t use it for whatever reason.  I did create one for multiplication and division based off of this activity that I used.  This year, I used both activities.  What I really like about these activities is that students practice noticing patterns, generalizing patterns, and applying those ideas to new problems -something we talk a lot about the first few days of school, and integers are the first unit with my seventh graders.

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Multiplying and Dividing Integers

The Desmos activity I made to parallel the activity above is primarily what I use for multiplying and dividing integers.  I wrote about that in this blog post.

After students work through that, we also have the conversation about the idea that negatives are opposites.  If we’re trying to find -3(2) students know 3(2) is 6 and we want the opposite of that, or in the case of (-3)(-2), we find 3(2) and then want the opposite of that and then the opposite again.


Practice

Depending on the year I’ve done multiple different combinations of the following activities.

  • Desmos Card Sort-  I know that this is probably a pointless Desmos card sort, but Desmos had just come out with card sorts when I created it, and I wanted to try it out.  Here is what I came up with.

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  • Game-  I don’t have a name for this game.  I first learned about it when I was doing my student teaching for Spanish.  Another Spanish teacher made one for vocab.  The goal is to have the most cards at the end.  To play students are put in groups of 3-4 and one person is the dealer.  I suppose the dealer could rotate so that all students take turns doing the problems on the cards, but I have never done it that way.  When it’s your turn, the dealer flips over the top card and you answer the question on the card.  If you get it right, you get to keep the card and can choose to go again.  As long as you keep getting questions correct, you can keep getting new cards.  At any point in your turn, you can choose to be done and then you are guaranteed that you will get to keep the cards you answered correctly for that round.  If you answer a question wrong, you lose all the cards you had answered correctly in that round.

As an added twist to the game, there are several cards that have either a smiley face or a

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frowny face on them.  If a student gets a smiley face, they get to keep

all the cards they had won up to that point in that round; they can’t lose them.  If a student gets a frowny face, they

lose all their cards from that round.

If students disagree on an answer, I will either check it, or I will have them check it with a calculator.  (The link to download this is at the end of the post.)

  • I Have…Who Has…  I found a couple of these games for free on Teachers Pay Teachers.  I learned the hard way that I like the cards where the question is something like “Who has 12(-3)?” because students end up doing more problems than if the question was “Who has -36?”  Two of the ones I have found are here and here.

Order of Operations

After doing some practice with integer operations, we start doing order of operations with integers.

  • Warm-ups:  One of my favorite warm-ups when we are doing order of operations is “Find the Flub”.  I love that it forces students to think through an already worked out problem to find the mistake.

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  • Witzzle- I first heard about Witzzle from Sarah’s blog. You can read more about it in her posts.  Essentially, students need to use the three numbers in any row, column, or diagonal to make the target number.  The target number can be anywhere from -12 to 36.  I tried this game for a warm-up for the first time this year, and I really liked how it went.  I can see myself using this more often.

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  • One Incorrect Worksheet-  I blogged about these here.  My students sometimes get frustrated with these worksheets, but I see that as a good thing.  They get frustrated because the worksheet forces them to go back and fix their mistakes when they get something other than -13 for the answer to more than one problem.  (You can download this at the end of the post.)

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  • Add ‘Em Up- Sara Van Der Werf first introduced me to this activity.  You can read her post here.  I created one for integer operations that you can download in the link at the end of the post.
  • Review- I wanted the review that students did to be self-checking, so I modified Sara’s Add ‘Em Up activity and made it into a worksheet.  There are two different types on the worksheet.  The first is simply integer operations.

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The second is order of operations with integers as well as some problems including absolute value.

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Here is the link to download the files for the activities I shared in this post.

Finding Mistakes

I love that getting a new group of students reminds me how far the group of students I had the previous year grew over the course of the year -even if it didn’t feel like it at the time!  There are numerous times in those first few weeks (even months) of school that I find myself thinking, “Oh my gosh, that’s right!  I was intentional about teaching them that.”

One of the things that has stood out to me the most with my 8th graders this year is how when looking at a worked out problem that has a mistake in it, they struggle to see the mistake, whether it’s their own or someone else’s work.  Not all students know how to look at a worked out problem and process through what was done to solve it.  If there was a mistake in their own work, they often don’t even try to look for the small mistake, rather they just erase their work and start over.

At first, I didn’t remember this being as much of a struggle for last year’s group.  Then I remembered I was thinking of my 8th graders last spring -not my 8th graders last fall.  That group of 8th graders struggled with this too, but by the end of the year they had improved in this area so much.  It caused me to stop and think about what things I do throughout the year that helps them grow in this area.

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I use Find the Flub pretty regularly for our warm-ups at the start of the year.  This is one of the first experiences students have looking for mistakes in an already worked out problem in my classroom.  I like that it’s low risk in that students aren’t looking for a mistake they made.  It’s someone else’s mistake.  I’ve found that adding numbers to each line of the problem gives students a good way to talk about the mistakes they see.  “From line 1 to 2, …”

My Favorite No

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This is another warm-up activity I use from time to time.  I tend to use My Favorite No more often, but I still like this one as it provides me with information on how each of my students are doing with a concept.  I put a problem up on the board for students to do.  Then I collect all of their answers and pick my favorite mistakes and put that work up on the board and have students figure out what was wrong with it.  I usually combine mistakes from a couple different students.  I originally saw this idea on the teaching channel, and I talk about how I use this in this post.

Spiral Worksheets

Another big way that students get practice looking over already worked out problems for mistakes is by making corrections to spiral worksheets.  This is the only one of the three that I have already helped students find the mistake, but they have to figure out why it is a mistake and fix it.  I wrote about that process in this post.

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I would love to hear other ways that you help your students improve in this area.

Spiral Review

The idea of reviewing concepts throughout the course of the year is nothing new, so nothing is this post will likely be anything new.  I’ve fallen into a routine with how I review concepts throughout the year in my classroom, and overall I’m happy with it, so I thought I’d share what works for me.

Part 1:

The first day of the week students get a Spiral Review Worksheet that has anywhere from  8-12 problems on it.

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At the start of the year these are review problems from the previous grade -things that in my first couple years of teaching I spent time reteaching.  I’ve found that by putting them on these review sheets and encouraging students to Google them if they don’t remember, I don’t have to spend much time, if any, reteaching these concepts, and it gives me an opportunity to help students through the process of Googling things and how to use their resources to figure out a problem.

As the year goes on, the problems are mostly things we’ve covered this year.  When I know students will need a previous concept in an upcoming unit, I will make sure to include it in a few worksheets leading up to that unit.

The worksheets are due on the last day of the week, and I correct them by highlighting their mistakes.  I know I’m not perfect at grading this way.  I miss things, but I’ve found that when I use this method of grading more students actually look at their mistakes in the problem.  The  video in the link above explains what I do.  Overall, I try to highlight the first time a student makes a mistake in the problem.  I only highlight things after the initial mistake if the student made another mistake beyond the initial one.

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When I first started creating the Spiral Review Worksheets, I typically put the problems in order on the worksheet based on when we covered them in class with the newest problems at the end.  However, I’m now intentional about mixing them up so that students have to jump around from various concepts.

Part 2:

I didn’t start doing this part of the spiral review worksheets right away.  It took me a year or so for me to realize that this part of the process was possibly more important than reviewing the problems to begin with and to come up with a system that worked for me.

Then the second week, instead of getting a new worksheet, the assignment is for students to make corrections on the Spiral Worksheet from the previous week.

I have students make these corrections on a separate sheet of paper and turn in the corrections along with the original worksheet.  For the assignment to be counted “complete” in the grade book, students need to get most of the problems they initially got wrong correct.  It takes a little bit for students to get used to this system.  Some of my middle schoolers don’t understand why an assignment shows that it’s “missing” in the grade book when they turned in it, but after a couple times of doing this they understand the process.