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.

A Day Full of “One Good Things”

I love reading the One Good Thing blog.  It always encourages me to find the positive in my day, regardless of what kind of a day/week I’m having.

Yesterday was one of those days that was full of One Good Things.

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I started with the following Which One Doesn’t Belong in my first 6th grade class.  The conversation was SO good, and I loved hearing all the vocabulary my students remembered.  The conversation went equally well in my other two sections.

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Then we used this Desmos activity.  It was the first time this group of students used Desmos.  It was love at first Class Code.  The groans when I paused the activity were music to my ears -I even gave them a 5 second warning.

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In my 8th grade classes I used Vertical Non-Permanent Surfaces (#VNPS) for the first time this year.  Whenever I do this, I ask myself why I don’t do this more often.  I’m always amazed at how much more engaged students are and how much more they participate when we do this compared to seat work.

After grouping students randomly for this, I saw that two students who I struggle to get to do anything on a lot of days ended up partners.  I had my doubts about how their group would function, but the 20 minutes those students worked together was by far the best either of them had worked all year!

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When it came time for my 5th hour class, I logged in to Desmos to find that none of my custom activities or history showed up in my account.  What could have been a disaster (my 30+ minute lesson plan with 6th graders, gone), ended up going just fine.

For whatever reason, my students were crazy patient and quiet while I tried to log in to Desmos and figure out what was going in.  Then I finally realized what happened.  My school email changed over the summer, and I had been trying to change my Desmos account to my new address.  I had contacted Desmos earlier in the week to help with this and saw in an email that they were able to make the change for me just before 5th hour.  The change didn’t quite go as expected, and I needed to change my password and wasn’t able to do this.

We had just set up something else on my students’ Chromebooks, so I had them work in that while I emailed Desmos to try to figure out the issue.  I spent the next 10 minutes or so emailing Denis from Desmos -we may as well have been live chatting for as quick as Denis was to respond to my emails.  (Have I mentioned that Desmos is awesome!  After my initial split second of panic when everything in my account was gone, there was never a doubt in my mind that Desmos would be able to fix the issue.)

The hour ended, and I still wasn’t able to get my stuff back in my account.  However, I tried the same thing I had been trying during class one more time, and it worked!  I was ready to go for 6th and 7th hours.

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After school I was able to go and support a colleague whose family is going through a difficult time.  While the situation isn’t a good thing, I was glad I could be there for her during this difficult time.

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.