# 8th Grade Unit 1: Equations (Part 2)

I shared a bit about the first part of our first unit in 8th grade here.  In the second part of this unit, students start solving equations with square roots, x², and absolute value.  They are also introduced to the idea that not all equations will have one solution.  They learn that equations with absolute value or x² have 2 solutions and then we talk about equations that have no solution or all real numbers as the solution.

One of my goals when I’m teaching them how to solve these new types of equations is to help them understand how it’s similar to solving problems they already know how to solve.  I want them to see the similarities in the problems below.  A recent conversation with a colleague reminded me that these connections are SO important and that I need to continue to work on helping students see the similarities in these problems.

I sort of want them at first to view the square root, x², and absolute value portion of the problem almost as a variable in and of itself.  I want them to understand to use inverse operations to first get that part of the equation by itself.  Once they have done that, I want them to understand how to use inverse operations to undo the square root or exponent and then solve the remaining equation.  In the case of the absolute value equations, I want them to understand why there are two parts to the answer and how to come up with those two parts once the absolute value is isolated.  (This is the one I have the most work to do to improve how I teach it in the future.)

##### Square Root Equations

We start this part of the unit by reviewing inverse operations, and I tell students that we’re going to focus on squaring and square roots as inverse operations.

Solving equations with square roots typically goes pretty smoothly.  Students understand to get the square root by itself, square both sides of the equation.  There are two main mistakes I see students make early on when solving these types of problems.  In the third example below, some students will want to undo the subtraction before undoing the square root.  In the last example, students don’t always recognize that the 2 is being multiplied by the square root, especially if the number being multiplied is negative.

By this point in this unit, I LOVE seeing students applying what they know about solving the types of equations on the left to the equations on the right in the picture below.  Most of my students are at least willing to start the top right equation before we do an example together as a class.

Students do usually struggle with the bottom right equation, and when this happens, we discuss how to solve the same equation without the exponents.  That is usually enough for students to understand what to do.

For practice on solving these types of equations, I use a worksheet I modified from one of Kate Nowak’s Row Games.

##### All Real Numbers/No Solution

This year I used the following to introduce these types of equations to students.

When we start solving equations like this, students all of a sudden forget what they have been doing in middle school up until now and want to put “All Real Numbers” or “No Solution” for every answer.  I always make sure to include some equations that have one number as an answer when students are doing problems like this, especially equations that have zero as an answer or where there are similar numbers on each side of the equation but the negatives are different.  For example -3x + 9 = 3x – 9

We had time for this Open Middle problem in one of my classes.  I loved watching students work through this.

##### Absolute Value Equations

Of the concepts in this portion of the unit, this is the one that I feel I need to improve the most for next year.  It always starts out well.  Students understand in the first equation below that x can be 5 or -5 and can explain why.  The second one goes pretty well too.

We do a few more examples before getting to one like the third example above, but in that problem, students understand to get the absolute value by itself, but then that’s where more of them struggle.  After I taught this lesson this year, I thought that this might be a great topic for smudged math, but I haven’t had time to think through how that would go yet.

When I was almost finished with this portion of the unit, I realized that I got super worksheet heavy.  It’s even more evident to me know as I put this post together.  In the classes that didn’t get to the Open Middle problem, there wasn’t anything other than a worksheet.  I now know what I need to work on for next year!

Here is the link to the worksheets I used in this part of the unit.

# Open Middle Inequality Problems

I started using Open Middle problems a bit more last year and want to continue using them more in my classes.  A while back I saw something Sarah shared about a problem she created.  I really liked this task, but I had just finished this concept when I saw her Tweet.

I remembered this problem recently when I was looking for things for my inequality unit.  I modified Sarah’s original problem slightly to create 3 different Open Middle tasks for inequalities.  The first one I made is actually the last one I will end up using in this unit, but I was too eager to try it out with students, so I ended up making two problems to use sooner.

This is the first one (and only one I have tried with students so far).

When I shared the first one with a colleague to get his feedback on it, he had a great idea from when he has used Open Middle problems in the past.  He suggested to start by letting students use whatever numbers they want, and then after they come up with a solution to restrict them to only using certain numbers.  I thought this was a great idea.  I started by telling students they could use any integers they wanted as long as they didn’t repeat any of the 12 numbers (I did say they could use 2 and 12 if they wanted).  When a student came up with a solution, then I gave them the added challenge of only using integers from -6 to 6.  I really liked how this played out with students.

This is the second problem I plan to use with students, but it’s the one I played around with the most when I was creating it.  Even though in the directions I have that students can only use the numbers 1-12, when I use this with students, I will probably do what I did with the other problem and say that students can use any integer as long as they don’t repeat any numbers.

I plan on discussing absolute value inequalities that have no solution or all real numbers as the answer after students work on this task.  However, I know those ideas may come up as students work on this.  I’ve thought about how the conversation will go if it comes up while students are trying to solve this problem, and I can’t quite picture what will happen.

Here’s the last one I plan on using with students.  Again, my plan is to start by letting them use any integers they want as long as there aren’t any repeats, and then restrict them to using integers between -6 and 6.

If you try any of these with your students, I would love to hear how it goes!

# 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.

##### 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.

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.

##### 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.

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.

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.

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.

##### 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.

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.

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.

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

# 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.

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)

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

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.

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

##### 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.

##### 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.

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.

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.

##### 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.

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.

# 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.

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.

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.

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.

I eventually did turn these problems into a loop activity so it was more self checking for students.  I’ve uploaded that file along with the area worksheet here in case you’re interested in either of them.

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.

##### Dividing Decimals

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

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

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.

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.

Update 10/18

This year instead of using the card sort above, I had students do a stand and talk to introduce this idea.  I really liked how this went, and I’ll probably use the stand and talk in the future versus the Desmos activity.

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.

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.

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

Update 10/18:  I also used this question stack from Sarah Carter.  In the future I would need to do some more work to set this one up than I did this year.  I did not do a good enough job of preparing students for terms with a power greater than 2.)

## Update 1/2021

I recreated some of these Open Middle problems in Jamboard to use with students while in hybrid/distance learning. The links to those are below.

The first time I did this, I made the mistake of not creating a copy of the template I made for each class. It did take a couple times of doing this with students for them to get used to finding the slide for their group and not just starting on the first slide. I had students use the same slide as their breakout room room in Google Meet.

# 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.

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.

(Update 10/18:  This year when I taught this, I spent a lot more time with the integer mats and got out the counting chips on multiple days and let students pick on other days whether or not they wanted to use them.  I need to remember to make more time for this in the future.  Some students really benefited from this.  I overheard one student say, “I guess I am a hands on learner.” so I knew it was beneficial to take the time for these things.  I also used the Desmos activity for subtracting integers, and thought it went well and plan on using it again in the future.

I also added this activity to practice adding integers.  Students really enjoyed this.)

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.

• 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

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.

• 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.)

• 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.

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

(Update 10/18:  I also added this sometimes, always, never activity towards the end of the unit as a review.  This was the first time I had done a sometimes, always, never activity, but I was encouraged to try it after attending Chris and Mattie’s session at TMC18.  I wanted to get my students talking and debating about integers.  This activity worked great for that.)