6th Grade Unit 2: Intro to Algebra (Part 3 -Equations)

You can read more about our Intro to Algebra unit here (part 1) and here (part 2).  In this portion of the unit we get into solving equations.

There are several things I do prior to actually solving equations to get students thinking algebraically.

I absolutely LOVE puzzles like this.

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One of my students caught something in this puzzle that I had missed the first time I solved it.

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In one of my classes we had some extra time, so I had my students create their own.  It was super fun to watch them get excited over this and to see their creativity in what they used in their puzzles.

 

 

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Then we used one of my online puzzles –Solve Me Mobiles, and one of my favorite movement activities, Balance Points.  I blogged about both of those activities here.

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When we play Balance Points I put an equation up on the board and with a partner, whatever the answer is students need to have that many body parts touching the ground.

 

 


When we get into solving equations, I stole Julie’s idea found here.  If you teach middle school and haven’t read that blog post yet, you need to stop and do that right now.  I’m not even going to say anything more about it to force you to go read it.  Most of my students will only work on one-step equations.  In some classes they’re ready for multi-step equations or some students in the class are.  Here is the link to download a couple worksheets I use and an add-em-up activity.

Open Middle also has some great one-step equation problems.

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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.

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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?

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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.

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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?” 😉

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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.

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We did some vertical non-permanent surface practice with solving absolute value inequalities at the whiteboards around my room.

I also used this Open Middle type problem I made.  You can download the file here.

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Here is the link to download the test review.

8th Grade Unit 2: Inequalities (Part 1)

After solving many different types of equations in 8th grade, inequalities are up next.  We start by reviewing graphing inequalities before getting into solving them.  Then we also work on inequalities that have all real numbers and no solution as answers.


Review of Graphing

Although students have seen inequalities and graphed them in the past, I’ve found that it is worth my time to spend a day or so giving students a quick refresher on this.  There are several great Desmos activities for this.  Here are a few that I’ve used and like.

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Solving

In the past I had an activity I used to get students to discover when the inequality symbol needs to be switched when solving inequalities.  It was sort of lengthy and cumbersome, but I didn’t know how to improve it more than I already had.  Then I saw Sarah Tweet the picture below.  It was EXACTLY what I was looking for!  Thanks Sarah!  Here is the link to download Sarah’s file.

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Then for practice students do a Tarsia puzzle.  I created the puzzle a while ago and don’t know where the file is that I can share.  If you’re unfamiliar with Tarsia puzzles, you can learn more about them here.

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I also have a question stack that I use for these types of problems.  You can read about Sarah Carter’s question stacks here.

 

 

 


All Real Numbers/No Solution

To introduce inequalities that have No Solution or All Real Numbers as the solution, I went back to what students already knew about equations like these.  I had students solve a problem similar to the one below and then asked them what inequality symbol we could replace the equal sign with that would make the inequality have no solution and the same for all real numbers.

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Then for practice, I had students work on this Desmos activity.

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I also tried creating an Open Middle problem for these types of problems after seeing a similar one Sarah created for equations.  I had one of my co-workers take a look at a different Open Middle problem I made, and 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, so that’s what I did.  I started by telling students they could use any integers they wanted as long as they didn’t repeat any of the 12 numbers.  When a student came up with a solution, I said they could only use the integers -6 to 6.

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You can download the files for the Open Middle puzzle here.

Math Telephone

I used Sarah Carter’s Representations of a Relation Telephone activity last year.  You can read all about the activity in her blog post.  In short, it works just like the game of “telephone”.  A student starts by writing six ordered pairs on the bottom, passes the paper to the next person who creates the graph, who passes it on the the next person who creates the table from looking at just the graph.  This continues until you get to the top of the paper with the ordered pairs again.  If done correctly, the ordered pairs should be the same.

I really liked it, but I knew there were some tweaks I wanted to make the second time around, and it went much better this time.  Part of it could have been that I had already explained the activity once, so I did a better job of explaining it and anticipating where students would struggle.  The first year I did this I actually had one group start whispering ordered pairs into each other’s ears.  I don’t know if they just weren’t listening or if my directions were that bad…probably a combination of both.  Thankfully that didn’t happen this year!

Sarah also shared an updated version of her activity here which also helped as she’s included more instructions on the sheet itself.

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The first time I did this I made the mistake of having one piece of paper per group of 4-5 students.  What was I thinking?!  That meant for 10+ minutes about 6 students in my class were working while the others were supposed to wait patiently?!  Not my brightest move ever.  This year all students had their own paper so everyone was doing something at all times.  This went MUCH better.

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As I watched the activity with my first class of the day, I noticed that it was taking students FOREVER to just fold the paper, and one student commented, “Folding the paper was the hardest part of this!”  I found that rather than folding it in an accordion at first it worked better for my students if each person just folded the bottom representation under before passing their paper on to the next person -these instructions are also on Sarah’s updated version.

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Overall it was a big success.  I loved watching students look at where the mistakes were made when they were done, and the second time they did it several students were thinking ahead when they created their ordered pairs to try to make it “easier” or linear -which will lead perfectly into what we’re getting to later in this unit.  I even had a student who doesn’t usually get too excited about much of anything say, “This is actually fun.”

I was thinking today what other concepts this could be used for and remembered one I made last year for my 6th graders on exponents.  Since last year, I lost the editable version of the document I made, so I recreated it and added some of Sarah’s instructions to it.  You can download the file here.

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What other concepts could this idea be used for?

One Good Thing: Volume 3

It’s been a while since I’ve shared my “One Good Things” from the week.  Here’s volume 1 and volume 2.  It’s been a challenging few weeks, so I thought focusing on the good at the end of the week would be a good thing in and of itself.


My first hour students were working on “Number Muncher” the Monday after Thanksgiving break.  When I said it was time to be done, their response was, “What?? Can’t we just stay in math all day?”

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One of my students shared with me about his first Black Friday experience.  He was NOT impressed.  AT ALL.  Even though he got this huge thing of gum balls, he would be perfectly happy to never experience Black Friday shopping again.  Ha!

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We did an activity called Balance Points in 6th grade this week.  It’s always a highlight for the students AND me.  I don’t think there’s a single person in the room who isn’t smiling when we do this.

The second day we did this activity, this grouped asked if we could have an question with an answer of 5 because, “we want to do that one we did yesterday.”  🙂

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A student came to class with this shirt on.  I don’t know why I found it so funny that day, but I did.

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I used Desmos Point Collector for the first time with students.  I’m a fan!  I had a student do this for the last challenge.  Later on during the class period he was telling his friend about it who goes, “Oh, so you cheated the system.”  😉

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High school basketball games started this week.  My dad coaches and many of their games are streamed online so I can watch from home.  It’s the best!  I’m convinced that God made basketball a winter sport to help us get through Minnesota winters.

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A little Mathketball is always a good way to start a Friday.

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We were reviewing for an inequalities quiz and at the end of class someone asked how many questions would be on the test.  I said “8 questions tops”, and a student replied, “You should make us write an inequality for that.”  So we did.

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I was explaining to a group of 8th grade boys that if they do better on the test on a skill than they did on the quiz earlier in the unit, I will replace the quiz score with their score for that portion of the test, but it doesn’t work the other way.  I said, “If you ace the quizzes but bomb the test, I won’t replace your test score with 100%.”  After saying they understood, I walked away but overheard one of the boys say, “I always though ‘bomb’ was a good thing.”  I went back to ask about this and explain what “bombing a test” meant when I was in middle school and that I must be old.  They said it reminded them of a conversation they had with the art teacher this week about the word “dope”.  Did I mention that this art teacher is just a couple years away from retirement?!  I’m not THAT old!

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I finally got to use a couple of Open Middle problems I made and was overall happy with how students did with them.

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6th Grade Unit 2: Intro to Algebra (Part 2 -Evaluating Expressions)

I blogged about the first part of our Intro to Algebra unit in 6th grade here.  In this part of the unit, we finally get into the algebra stuff.


Word Phrases

Before we start evaluating expressions, we talk about what variables are and what the purpose of them is.  We review word phrases.  I started this year by putting these words up on the board and having students classify them based on their operation.

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After we have done a bit of large group practice with that, I have students do a card sort/matching activity.  Once I have checked their answers, they can play memory.  Some students have also played Go Fish with the cards as well.

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I also use this Desmos card sort on another day to review this concept.

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Evaluate Expressions

Then we get into evaluating expressions.  This year I made a small tweak to how I introduce this to students.  In the past,  I would stand up in front of the class and tell students what you do when you’re asked to evaluate an expression.  I’m really trying to get away from being the teller of information in my classroom and instead be the asker of questions to get students to explain the mathematical concepts to each other.  Here’s the small change I made this year.

I put “x + 2″ on the board and asked if we could come up with a number answer for this.  I saw several heads shaking no, and when I asked why, they told me, “We don’t know what x is.”  Then I added to the board “x = 7″ and asked if we were able to come up with a number answer for this now.  They told me we could and that the answer was 9 and then explained how they got that for an answer.  I could tell that not all of my students had caught on yet or weren’t fully paying attention, so instead of me rephrasing what the student had just said, I asked, “Can someone else explain to the class how ‘Sue’ got 9 for an answer?”   Then I put something like 4x + 3 up and repeated the same process.

It was a small change, but it felt SO much better than standing up in front of the class telling them a process to follow.


The first activity I do is something I created several years ago, and I realized last year when I did this that it’s pretty similar to Sara’s Add-Em-Up activity.  I created 4 different sets of 5 cards.  Each set of cards is printed off on a different color paper so that I can tell which set students are working on.  Below is a picture of the first page of the document for this activity which has 4 copies (each column) of the first set.  I have 8-10 copies of each set.  Because all 8-10 sets are the same color, before I laminated the cards, I wrote a number on the back of each card in a set.  This way, when I find a random card on the floor I can ask, “Who has the red 3s?” and easily figure out where the card goes.

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Students are put in groups of 2-3 and in their group they work together to evaluate each of the cards.  When they are done, they add up their 5 answers and come tell me what they got.  If they are correct, I will give them the next set.  If they are incorrect, I don’t tell them which one they got wrong, and they go back to their group and work to figure out what they did wrong.  I’ve tried to level the cards so that each set gets increasingly difficult.  Set 1 has one to two operations on each card.  Set 2 has three steps to each problem.  The next set incorporates decimal operations and the final set has the variable in the problem more than once.

One of the reason I like having a different color for each of the sets is that it is easy for me to see where students are when we are doing this activity.  If I look around my room and see one group on the red set (the first set for me) and every other group is on orange or green (the 2nd and 3rd set for me), I know I need to check in with the group working on the red cards.

Below is an example of each of the four sets.

 

 

 


As I was looking through my stuff to find the file for that activity, I came across another activity I created and had forgotten about.  For this activity, I put students in groups and have them start at a problem.  They can choose to solve either problem on the card.  I encourage all students to try at least one of the more challenging problems.  Once students solve the problem, they look for the answer on another card that I have hanging around the room and then solve either problem on that card.  Eventually, they will loop through to all of the problems and end back where they started.

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I also used this Desmos card sort towards the end of the unit to review with students.

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Find the Flub warm-ups are great for evaluating expressions.

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Tables

In the past, I haven’t done much with tables in any of my classes, and I know that this is something I should do more of.  I added this Desmos activity to this unit, and overall I was pleased with how it went.  It was challenging for some of my students, which was my goal when creating it.

 

 

 

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Here is the link to download the activities from this post.

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.

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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.

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Quadratic Equations

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.

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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.

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All Real Numbers/No Solution

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

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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.

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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.

img_0796-e1511131230283.jpgWe 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.