# 8th Grade Unit 5: Systems of Equations (Part 2 – Elimination & Choosing Method)

I shared here part 1 of our unit on systems of equations -solving by graphing and by substitution.

##### Elimination

I started this unit with the following warm-up. Then I followed it up with this notice/wonder. One of my students this past year noticed that, “These are like the problems we did for warm-up.”  I love when my students notice connections between the warm-up and the lesson for the day and that I’m not just having them do random stuff.

I was really pleased with how this led nicely into the lesson for the day.  Students noticed that in each problem there was a zero.  They were able to tell me why that happened.  When I told them that this is another method for solving systems of equations called elimination, at least someone in each class was able to explain why they thought that elimination was a good name for this method.

It had been a while since we had done Vertical Non-Permanent Surfaces, and these problems work great for that. ##### Choosing Method

The last part of this unit was having students choose the best/most efficient method for solving a system of equations.

I started with this Desmos Activity.  I didn’t have students solve the systems the first day.  I wanted them to just think through what method they would want to use to solve each.  The next day they actually solved some of the systems. I love this Desmos Activity from Paul Jorgens. I’ve uploaded some of the worksheets and notes I used from this unit here.

# 8th Grade Unit 4: Applications of Lines

We start our unit on applications of lines by discussing independent and dependent variables.  I have a note to myself to remember to use the following language next year because it worked well this year.  Nothing earth shattering, I know.

• “(independent variable) causes change to (dependent variable)”
• “(dependent variable) depends on (independent variable)”

I use a lot of Sarah’s resources found here for my notes, and I’m pretty sure that’s where I got the problems for this Desmos activity.  The next day we do Sarah’s Ghosts in the Graveyard activity with independent and dependent variables.  Every time I use that activity I think to myself, “Why don’t I do this more often?  It’s great!”

After students have a pretty solid understanding of defining the dependent and independent variable, writing linear equations from word problems goes a lot better.

Then we get into parallel and perpendicular lines.  I blogged briefly about what I did last year here.

I start with parallel lines and use this Desmos activity.  One of the downfalls of starting with that activity is that when students are asked to solve problems where they need to write the equation of a line parallel to a given line through a specific point, they want to use Desmos to guess and check.  This is a good strategy, but I also want them to know another method.  I start the next day with a couple problems like these. After spending another day or so on parallel lines, we finally get into perpendicular lines.  I start with this Desmos activity. We spend another day or so practicing with perpendicular lines.  I’ve used this activity before and like how it brings back different forms of lines.

We also talk a little bit about parallel and perpendicular lines and quadrilaterals using this Desmos activity.  We do Desmos Polygraph next.  Last year I had a student ask if there were two “loners”, and I will forever think of outliers as loners.

After students do that activity, I put the graphs up on the board and ask students to put them in groups.  They end up describing the different correlations to me. This Which One Doesn’t Belong? is great around this time in the unit. I took a couple tasks from this page and turned them into Desmos activities.  (I know she tweeted out links to the activities at one point, but I couldn’t find them.

Here’s one on correlation. And another on lines of best fit. # 6th Grade Unit 8: Probability

Another new unit and another Which one Doesn’t Belong? to start out. Then we did notice/wonder with tree diagrams. After doing some practice with tree diagrams, I encourage students to start looking for a pattern to figure out the total possible outcomes each time.  Most often students are able to notice the counting principle.  Sometimes students will notice the pattern before I even mention it.  They’ll ask, “Can’t you just…”  I usually nearly scream at them, “Wait! Not yet!  Don’t ruin it for those that haven’t noticed the pattern yet!”

Tree diagrams are good opportunities for students to make up the problems as they go.

One problem I like to give students is “A tree diagram has 16 possible outcomes.  What could the tree diagram be?”

Then we start talking more about probability.  After spending a day on theoretical probability we start talking about experimental probability.  I know there have to be some awesome probability activities for 6th grade, but I haven’t found them yet.  (If you’ve got some, please send them my way!)  What I’ve done the past several years is set up 5 different stations for students to work through:  coin toss, dice, deck of cards, box with different colored cubes in it, and a wheel with different colors on it.  Then students compare their experimental probabilities with the theoretical probability.

Here’s an example of one of the stations for experimental probability. I also use this as an opportunity to review converting between fractions, decimals, and percents.  Another way that I like to spiral concepts in this unit is to give a problem like the following:

The following numbers are written on cards and put into a box:  1, 1, 3, 4, 5, and 8.  What is the probability of randomly picking a prime number?  a factor of 20?  A multiple of 4?

To review we play mathketball.  Students LOVE this simple game.  Students make a circle around the room with their desks, and I put a trash can in the middle of the room.  Students answer a question I put up on the board, and if they get it correct, they get to crumple up their 1/4 sheet of paper and try to make a basket.

Here’s an example problem from that. Here’s a different class playing mathketball, but you get the idea of what it is.  I do try to pick topics for mathketball where the problems shouldn’t take students too long to solve and/or have fewer steps.  I don’t want students to feel rushed, but I also don’t want students who complete problems quickly to be waiting a long time. # 6th Grade Unit 6: Angles and Triangles

##### Angle Pairs

I used this Which One Doesn’t Belong? to start our unit on angles and triangles.  I love how starting with something like this gives me insight on where students are at with this topic based on their answers and the vocabulary they are using. After taking some notes on different vocab words we came back to the same image at the end of the day, and I asked students to use the new vocabulary to describe the images.

I also had students do this Desmos Polygraph several times throughout the unit as they learned new vocab words. ##### Sum of Angles of a Triangle

In one of my classes, I had students cut out a triangle, rip off the angles, and put those pieces together to form a line.  It didn’t go quite as I hoped with that class, so in the other classes I was the only one who cut the triangle.  I would like to figure out a way so that more students see what I want them to see as they’re cutting the triangles and putting the angles together. Jo Morgan shared several good Angle Chase activities in this Math Gems post.  ##### Interior Angles of Polygons

After talking about the sum of the angles in a triangle, I have students Notice/Wonder with polygons divided into triangles for them to figure out the sum of the interior angles of polygons. After they’d done some practice with that we made a table to come up with the general formula.

# 6th Grade Unit 4: Ratios

To introduce our unit on ratios this year, I started with the following picture and asked students to notice/wonder about it and if they could figure out what was meant by the word “ratio”  Then I used a couple of Desmos activities.  This is one that I modified from something I found from Andrew Stadel.  Then I also created this card sort.  There are multiple correct options for the card sort I created. Then I used an I Spy activity.  I blogged about it here. I love this Open Middle problem on equivalent ratios. I like this Which One Doesn’t Belong? around this time in the unit. Then we get into unit cost and finding the better buy.  Some years I have students look up items online and find the unit cost of the items, but I’m finding that more and more websites already give the unit cost on them.

Students always enjoy math fails, and they work great in this unit.  Sara shares a ton of them on her blog here, here, and here! I used Robert Kaplinsky’s “Which Ticket Option is a Better Deal?”  I definitely want to spend more time on this one next year and really focus on question 4. Don Steward also has some great ratio puzzles on his blog here and here.  As I was going through my stuff when writing this post, I also came across this video.  I always forget about it and have never actually used it in my classroom.

# 8th Grade Unit 3: Functions (Part 3 – Point Slope Form & Standard Form)

Here’s part 1 and part 2 of unit 3.

I used this warm-up the first day after our test on slope-intercept form to get students thinking about equations and graphs again. Then I do notice/wonder with this. I heard things like:

• There are two x’s and two y’s.
• There are little 1’s by one of the x’s and one of the y’s.
• There are parenthesis.
• There’s an m (slope).
• There’s no y-intercept
• Is it another form of a linear equation?

It leads nicely into discussing point-slope form and students realize that it isn’t as scary as it may look at first because they recognize the similarities between slope-intercept form.

When going over point-slope form, I make a point to emphasize to students why it’s named point-slope form -we can see the coordinates of a point and the slope from the equation.  I remind them that this is similar to slope-intercept form where we saw the slope and the y-intercept.

Then we go over a few examples of writing equations in point-slope form before doing an activity similar to what Sarah shared here.  I didn’t have big foam die like Sarah used, but I do have double dice, which students always think are fun.  Students rolled the dice to create two ordered pairs and wrote an equation in point-slope form of the line between those two points.  Then they checked their answers using Desmos.  Having students check with Desmos was key to helping them see what they were doing when writing the equation of the line.

I also modified this Desmos marbleslides activity to rearrange the equation so that they looked like what my students were used to seeing.  My modified version can be found here.

Then after some more practice using point-slope form, students are introduced to standard form. Again, students came up with the following things:

• There’s 2’s in all of them.
• The two is always by the x.
• One of the equations is in slope-intercept form.
• One of the equations is in point-slope form.
• In the purple one, the x and y are on the same side.

We also talk about how, unlike slope-intercept form and point-slope form, we don’t see the slope, the y-intercept, or a point.

Of the three parts to this unit, this one is takes up the fewest number of class periods.  Writing up this post made me realize that I could probably use a few more activities on these concepts.  If you have any ideas for me, please share!

Update 2019

Here are a couple other activities I added to this unit.

I used this Desmos card sort.  Students match a graph with 3 equations, one in slope-intercept form, point slope form, and standard form. I also added a loop activity.  The question is on the bottom of the paper, students answer it, and look for the answer on another piece of paper.  That leads them to the next problem.  There are 8 questions, and the last question takes them back the first paper they were at.  I usually print out multiple copies of the activity so that there aren’t as many people at one problem.  If I add more questions to the loop, then not all groups finish in one class period.  I’ve learned to make fewer questions in a loop and just print out 2-3 copies depending on how many students I have.

# 8th Grade Unit 2: Inequalities (Part 2 -Compound Inequalities and Absolute Value Inequalities)

You can read about the first part of our unit on inequalities here.  In the next part of the unit we do some word problems, compound inequalities, and absolute value inequalities.

##### Word Problems

I know the word problems I give students aren’t very “real world” and that this is an area I need to work on -finding/creating better word problems for students and doing a better job of teaching them as well as incorporating them into class.  I don’t have anything fancy I do for these other than a couple examples together as a class and then partner practice. ##### Compound Inequalities

I use Notice/Wonder to start our conversation on compound inequalities.  Then we do this Desmos activity.  I also like this Polygraph activity for compound inequalities.

This year I also realized I could make a connection between “compound inequalities” and “compound words” and “compound sentences” that students are familiar with already from their English classes.  Why did it take me so long to do this? The following day I use this Which One Doesn’t Belong? for a warm-up and then we go back to the image from the notice/wonder the day before.  I put numbers on the graphs and students write inequalities for each. I used one of Sarah Carter’s awesome questions stacks for practice on solving these types of inequalities.  You can download the file she shared here.

One of my classes got to Point Collector.  This was my first time using this activity with students.  It’s SO fun!

One of my students came up with this for the last challenge.  He didn’t quite follow the directions exactly, but I love that he wanted to get the maximum number of points.  I overheard him telling another student about it later on during the class period when they were working on something else.  His friend goes, “Oh, so you cheated the system?” 😉 ##### Absolute Value Inequalities

I tried a couple different ways of introducing absolute value inequalities this year.  In a couple classes I started with Notice/Wonder.  Then in another class I started with an absolute value equation such as 3|x – 1| + 4 = 19 and had students solve that.  Then I changed it to an inequality and asked students what they thought would be the same/different about solving the problem.  Both ways of introducing the topic were good for different reasons.  I think for next year I may try to find a good combination of both. We did some vertical non-permanent surface practice with solving absolute value inequalities at the whiteboards around my room. # 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.  # Notice/Wonder

Last summer I watched Annie Fetter’s Notice/Wonder video at a training led by Sara Van Der Werf.  I instantly knew this was something that I needed to be using in my classroom, and I was able to use the questions “What do you notice?” and “What do you wonder?” while tutoring over the summer to try it out before the school year.  Right away, I was amazed by how those two questions changed things for me while tutoring, but I was unsure how to implement it into a classroom setting compared to a one-on-one tutoring session.

When I first started using it, I wasn’t sure how to do it.  I wondered if there was a “right” or “wrong” way to do it, so I was more hesitant to use it in my classroom.  As I started using it more, it gradually became a common routine, both formal and informal, in my classroom.  I was curious to see the how I used Notice/Wonder over the course of the year, so I went through my files from the year and pulled out the images I was intentional about using with Notice/Wonder.  As I look back on the images from this past year, I would put most of them into one of the following categories.

1. Students noticed/wondered about a problem before beginning to work on it.
2. Students noticed/wondered things about several similar but slightly different images, and then we discussed how these slight differences impacted the math we were talking about.
3. Students noticed/wondered things about a new concept on connected it to what they already knew.
4. Students noticed patterns and/or put new concepts into groups.

The first two usually happen more in the middle of a unit, while the last two were typically how I started a unit.

The routine in my classroom for this is that students work individually for a few minutes writing their answers down.  I typically tell them something like, “Write down 7 things you notice and 5 things you wonder.”  Sometimes when they finish, I do a stand and talk (read about them here– scroll down to #4) before going over their responses as a class, other times we go right into a whole class discussion.

I saw this on Twitter a while back, and I like this idea rather than giving students a specific number of things to notice or wonder.

Here are some things I noticed about how I used this over the course of the year.

• I was surprised by how often students were able to connect the new concepts to what they already know.  This was HUGE for me as a teacher because it helped me to realize I don’t always have to start from ground zero when introducing new concepts.  It also helped my students see how math builds off of itself.
• When students noticed the differences among what we were talking about and made the connections I wanted them to make, it stuck better than if I would have just told them.
• It gave me as the teacher a ton of insight into where my students were at with the math.  Some students would comment on the size or color of what was on the board.  Other students would connect it to what they had learned in the past.  The vocab they used when talking about it also gave me clues as to where I needed to go with what we were talking about.

As the year went on I found myself asking “What do you notice?” much more informally in class.  I would use it when I put a problem slightly different than what students had seen before, or if we were working on a problem and students were hesitant to participate.  Asking them what they noticed was a much safer way for them to participate than jumping into the problem right away.

I also found that I asked students what they noticed MUCH more often than “What do you wonder?”

Is every Notice/Wonder I do awesome?  No.  Are there more effective things I could be doing?  Most likely.  But this is where I’m at right now with this.  It’s made a HUGE impact on myself and my students, and I’ve seen growth in myself when it comes to using this in class compared to the start of the year.  After doing Notice/Wonder more often with students, it became easier for me to find new ways to incorporate it into my lessons, and I saw an improvement in the images I used with my students.

**Update:  I had this post written and saved in my drafts before going to another training by Sara Van Der Werf where she talked a lot about Notice/Wonder and was encouraged to have her validate many of the ways that I’ve been using this in my classroom.   I will try to remember to link Sara’s blog post on this training when she posts it.**

##### Here are some images of how I used Notice/Wonder with my 8th graders.

I used Notice/Wonder the first week of school with Fawn Nguyen’s Noah’s Ark problem to get them to think about the problem before starting.  This idea was completely stolen from Sara, and it was a great way for me to start using this with students. This is another example of a time when I had students Notice/Wonder before they actually started solving the problem.  I was especially hoping they would talk about the exponents. While grading quizzes, I finally saw where the misconceptions were coming from with some of the mistakes my students were making, so I created a notice/wonder to talk about those and the differences between the following terms.  This ended up being one of my favorites from the year. Here are several examples of when I used Notice/Wonder to introduce new concepts.    I used this one to help students see the differences between the various forms of linear equations. Rather than just giving students a new formula like I would have in the past, I used Notice/Wonder.  Students were able to tell me the variables they were familiar with, so the formula didn’t seem completely new to them, and we discussed the variables they hadn’t seen before. In our unit on sequences, I used Notice/Wonder a few times in hopes that students would notice patterns and connect the sequence to the formula.  ##### I felt that I didn’t use Notice/Wonder as well with my 6th graders, but there were a few times throughout the year that it worked well.

I used it at the start of the year to introduce the game Set.

This spring I saw this Tweet.  I’m always encouraged when I see that someone else confirms that something I did in my classroom was a good idea.

After seeing that, I was reminded to use it again to introduce the rules to this Kenken puzzles. I used the following image when introducing the idea that there are 180° in a triangle. A few days later I put up these images to talk about the interior angles of polygons. # 8th Grade Unit 7: Sequences

Arithmetic and geometric sequences was another unit where this was my first time teaching this when I was the one introducing this to students.  There were parts of this unit that I really liked, but there are definitely things I want to improve for next year.  Looking back on the unit, I’m surprised at how many students struggled with arithmetic sequences after having just spent so much time on linear functions.

I started day 1 with this warm-up: After that we did Notice/Wonder using the following image.  Next time, do I keep the sequences color coded but have them random on the page rather than sorted with the line down the middle? We then took notes on arithmetic sequences, common differences, geometric sequences, and common ratios.  I used some of Sarah Carter’s notes found here and made others similar to hers after reading this post.  I also used her half sheet as practice.  The link to my notes is at the bottom of this post. On day 2, we talked about writing the rule for the arithmetic sequences.  We made the connection to common difference and the slope of a line as well as the zero term and the y-intercept. Sara Van Der Werf’s “Sarified” standards have become my Bible for my state standards.  I LOVE how she has combined the vocab allowed along with the examples given for each benchmark.  However, I’ll admit I was a little bit perplexed by the standards and examples for arithmetic sequences, so if you’re a MN teacher reading this and can clarify for me, please do!

Here’s how the standards define an arithmetic sequence:

f(x) = mx b, where x = 0, 1, 2, 3, … But then in this example, unless I’m completely off track here -which could absolutely be the case, -1 would have to be the 1st term based on the answers given. So after debating and overthinking, I decided to give students some examples where the first number given was the zero term and others where the first number given was the first term.  I also told them in the directions which term the first number given was.  Was this the best solution?  I don’t know, but it’s what I did.  If you have thoughts on what I could do next time, please share! Next up was writing rules for geometric sequences.  I didn’t want to just give the formula to students and have them use it, but I couldn’t think of anything I felt was great.  Last minute, I came up with the following and again had students notice and wonder things about it.  Was it the best thing ever?  No, but my students were able to come up with the formula for geometric sequences on their own after looking at this image. Again, we took notes on geometric sequences and did a couple examples together as a class. When students graphed the sequences, we also looked at the graph on Desmos and talked about how because we see a small portion of the graph, it may look linear, but when we look at more of the graph we see that it isn’t linear. When students were working with arithmetic and geometric sequences separately, they did fine, but the wheels started falling off for some students when the sequences were mixed up.  Students mixed up where the zero term went and where the common difference/common ratio went.  I was frustrated that my students were frustrated because, to me, this wasn’t really that much new stuff.  Arithmetic sequences were like linear functions which we had just spent weeks working with.  After school that day I put on my “non-math teacher brain” and tried to look at the formulas from my students’ perspective.

Arithmetic:  f(x) = mx + b  (m = common difference;  b = zero term)

Geometric:  f(x) = ab^x (= zero term;  b = common ratio)

When I wrote the two formulas out like that, I understood why students were frustrated. Although there’s a “b” in both formulas, it doesn’t represent the same thing in both.  In arithmetic sequences it’s the zero term, but in geometric sequences it’s the common ratio.  I could see and understand why they felt the “rules” were changing on them.  Once I understood the confusion, I was able to address this better when working with students.  When students were confused, I pointed out that in both formulas the common ratio/common difference is the number “with the x“.  I don’t know that’s the best way to go about helping students who struggle with that though.

If anyone else has students who struggle with this and have found ways to introduce this so that students don’t get mixed up with this or have ways of explaining this so students understand, please share!

Here is the link to the notes I used in this unit as well as a few worksheets.

Much of the format in the notes is stuff I’ve borrowed or modified from Sarah Hagen.  I don’t do true interactive notebooks (INB).  I upload the notes as a pdf to Google Drive for my students at the start of a unit, and they use an app called Good Reader on their iPads so they can write or type in the notes.  Here is Sarah’s page of INB resources.