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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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.

“I got it, and I feel amazing!”

As a teacher, one of my favorite experiences is watching a student struggle with a problem, persist, finally get it, and say something like, “I got it, and I feel amazing!”

That’s what I overheard one of my most challenging students say a couple weeks ago in my classroom.  About a math problem.  I’ll be honest, at first I thought maybe she was being sarcastic, but a little bit later she was telling someone else, “I did it, and it feels great!”  She was truly proud of herself and wanted those around her to know what she did, and it. was. awesome.

Here was the task students were working on.

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The back side of the sheet had problems like this.

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As I think back on this day, something stands out to me.  If this student had been in my classroom last year, she probably wouldn’t have had that experience.  Why?  Because I probably wouldn’t have put that worksheet in front of her, or any other student in my class.

This was a worksheet I created my first year teaching.  I had found a worksheet with similar problems in our textbook resources, loved how it went, and wanted more problems like it.  I was excited to have students work on this task, but that excitement quickly turned to frustration when I found that students struggled with these problems much more than the ones on the worksheet from the textbook.  They were frustrated, and I was frustrated.  I was frustrated that things weren’t going as I had planned and that I didn’t know what to do.  I was disappointed what I thought was a great idea, didn’t turn out so great.  So we moved on, and the worksheet found its way to the back of my filing cabinet.

I let that one bad experience with this impact decisions on what activities I would and wouldn’t do in my classroom for several years.

Rather than try to learn from that day and try again, I opted for more familiar activities where I could pretty much predict how the lesson would go.  I avoided activities where I anticipated a similar outcome and chose to use activities that were comfortable for me because I didn’t know if I was prepared to pick up the pieces of a failed lesson.

And then like Princess Mia in Princess Diaries (1:15), I realized how many stupid times a day I use the word I.

What about my students?  How often do I rob students of their own “I got it, and I feel amazing!” experience because I choose not to use a task I knew was good for fear of how the lesson might go simply because it was less familiar and more uncomfortable for me?

Probably more often than I want to know.  So next time I’m planning a “safe” activity, I hope I will remember to stop and think twice about it and think about my students.  Sometimes safe is ok, but sometimes safe doesn’t lead to “I got it, and I feel awesome!” moments for students, and I want more of those moments in my classroom.


Here is the link to download both the pdf and Word versions of the worksheet.

“One Incorrect” Worksheets

One of the things that’s a constant struggle for me every year is giving students access to answer keys for homework problems.  I want homework to be useful to students.  I want them to be able to check their work to know if they’re doing it right, but I go back on forth with whether I should just give students answers or worked out solutions.  I am also terrible at remembering to upload answer keys to Google Drive for students.  If anyone has a system that works for them, please share!

Last week in two of my classes, I assigned a worksheet like this:

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I was introduced to this type of a worksheet this summer by Sara Van Der Werf.  She had us do the one below at one of her PD sessions this summer.  It was this activity from Don Steward.  All but one of the expressions simplifies to 5n + 3, and you need to find the expression that doesn’t and show that all the others do.  I like it because students know that 7 of the answers will be the expression in the middle.  I don’t have to worry about remembering an answer key for students!

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I created similar versions for evaluating expressions and order of operations with integers that I used last week in two different classes.  It was easier to make 7 problems that all evaluated or simplified to the same answer than I anticipated.  For the problem that doesn’t work, I tried to create a problem that if students make a common error, the expression still equals what’s in the middle.  For example a problem might have -62 and if students say it is 36 the final answer will equal what’s in the center.

So far, I am really liking how well students have been working together on them.  For whatever reason, it seems that students are more engaged and are more willing to go back and find their own mistakes rather than asking me for help right away when they know that 7 of the answers are the same.  I’m not really sure how this is any different than having the answers in the back of the book, but I’ll take it!

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I’m using this one this week for simplifying expressions because I was looking for slightly different types of problems than what I found on Don’s website.

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You can find pdf files for the worksheets here.  I don’t know that the Publisher files would be of any use to anyone since I created the equations in Word and copied them to Publisher and used a random font for the other text, but if you want them let me know.  I can upload those too.

If anyone has created something similar to this or ends up creating similar things, I would love to share resources!

 

Commit and Capture: An Order of Operations Activity

 

This post and my previous post on another order of operations activity may seem sort of random with all of the other start of year blog posts out there, but they are results of me reflecting on what I do in my classroom as I’m trying to be more intentional about the things I put in front of students -more specifically, I’m trying to think big picture.  Being intentional seems to be a reoccurring theme for be this summer.  Today I finally had several pieces come together and feel like I can put into words what I’ve been trying to process in regards to this…but that’s a post for another day.

Commit and Capture is an Open Middle type problem.  Like the last order of operations activity I wrote about, I used it more often my first year teaching than I did the past couple years.  Again, I’m not exactly sure why because it’s great.  It will be making more frequent appearances in my classroom this year.

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Commit and Capture is one of several activities I got when I went to a session by John from Box Cars and One-Eyed Jacks at a conference a few years ago.

To play, I write 3-4 expressions on the board, either from the sheet above or I’ll create my own.  In pairs, students write down the expressions from the board.  I roll a die, usually 10-sided but it could be whatever you want, and call out numbers one at a time.  The goal is to get the greatest answer possible for each expression.  I call out numbers one at a time.  As I’m doing this, students must agree with their partner on where to put the number -they can put the number in any box in any of the expressions.  Once students have a number written down, they cannot erase it.  They also can’t write down all the numbers on the side of their paper and wait until the end to put them in the boxes.

I’ve always had students work in pairs because I love the conversations they have as they decide where to put numbers!  There are so many different things they think about during this activity.  They are looking to get the greatest value for not just one expression, but for several.  They also have to consider their chances with the die and the number of boxes left.  For example, if there are only two boxes left and they’re looking for a high number.  If I roll a 6, do they use that as their high number or take their chances that the final number will be higher?

A heads up if you decide to try this in your classroom, regardless of how many times I tell students they can put the numbers I call out in any of the boxes they want, there’s almost always at least one group that thinks they have to fill in all the boxes in the first expression before moving on to the second expression.  I even do a quick example before hand and model how it works, but no.  There’s still always that one group that misses memo.

I haven’t tried these, but here are a few variations I’ve thought of.

  • Have students try to get the lowest answer rather than the highest.
  • Have students add their answers for each expression to get one number.  (I’ve always kept them separate and had a winner for each expression.)

I couldn’t find the exact handout from the session I went to on their website, but here is a similar one.  I’ve also used Betweeners, Order in the Court, and Balanced Equations all found on that handout.  Balanced Equations is another one of my favorites.  You can find several other handouts here from other conferences they’ve done.

 

Number Muncher: An Order of Operations Game

One of my favorite activities/games I started using my first year teaching is something my students named “Number Muncher”.  It’s an old Discovery Toys game I had from when I was younger.  (Thanks mom for instilling my love of mathy games young!)  I couldn’t remember what it was called but knew it needed a name, so I decided to have my students name it.  I made a list of several of their ideas and had them vote from that list.  Number Muncher it was.  I’ve since learned that it’s actually called Number Jumbler.

I don’t think I fully recognized the value of activity my first year teaching because since then I’ve used it less and less each year, and I don’t know why because it’s SO good.  It’s low floor/high ceiling.  It’s a great way to review order of operations, exponents, properties of multiplication and division, and even fractions.  It allows students of all levels to be successful but also challenges all students.  I’ve mentioned before that I’m working to be more intentional about the tasks I have students do in my classroom, and this one makes the cut.  It will be making a comeback this year!

The goal of the game is to write expressions equal to the middle number using the numbers around the outside.  You don’t have to use all of the numbers, but you can only use the numbers on the outside and can’t use them more than once unless there are duplicates.  (For example, in the picture below, you can only use one 3 but you could use three 6’s.)

Because the numbers are random, some sets of numbers are  much more challenging than others.  I want students to feel success with this game at first, so I try to make sure that the numbers are nicer to work with the first few times we play (i.e. when I write the numbers on the board I change some without students knowing -particularly the middle number.  I think the numbers on that cube are 10, 20, 30, 40, 50, and 60.  I like to keep that one smaller at first.)

I vary how long I give students to work depending on the numbers, but it’s usually around 3-5 minutes.  I usually make it a competition, and whoever comes up with the most correct expressions wins.  Each expression is worth a point, and depending on what math concept I want to encourage students to use, I may give students bonus points for certain expressions.  For example, I may tell them they get 2 points for every expression they come up with using an exponent.

After time is up, we go over several of their answers on the board.  In nearly every one of those conversations, addition and multiplication properties naturally come up as well as whether the parenthesis a student used in an expression were necessary or not and why.

I’ve used this as a warm-up.  Other times I pull it out when we’ve got an extra few minutes at the end of class, or when I’ve got some early finishers.   How else do you think I could use this with students? As I’ve been writing this post, I’ve thought of the following ideas:

  • After students have had individual work time, put them in pairs or groups and give them a few more minutes to work as a group to come up with a collective list.  Then we could have a group winner.
  • Rather than going over students’ solutions on the board as a class, pair students up and have them check each other’s work.  Then maybe as a class we would highlight some of student’s favorite solutions.
  • Play sort of like Scategories and only give points for unique answers.

I know this is similar to some of the other order of operation activities out there, but students love to be the one to roll the purple thing to come up with the numbers.  They think it’s so cool.  I couldn’t find it sold on the Discovery Toy’s website or on Amazon, but I did find a few on eBay if you’re interested in one for your own classroom.  You could also use 7 regular dice and pick one to be the “middle number”.