Transformations

I was able to squeeze a few days of transformations in with one of my 6th grade classes.  These are 7th grade standards in my state, but this is the group of students I will have again next year as 7th graders with the end goal of getting to all the 8th grade standards.


I started with this Which One Doesn’t Belong?

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And followed up with this Desmos Polygraph.

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I was able to borrow notes from a colleague for this unit.  Teaching in a small district this doesn’t happen often as none of us teach the same course as anyone else.  For each different type of transformation, I started with a Desmos activity.

Translations

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Rotations:  Students made a table of values of the pre-image and new image.  They created different images and looked for patterns to predict how to rotate an image 90 degrees clockwise, 180 degrees, and 90 degrees counterclockwise.

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Reflections

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And we ended with Transformation Golf.  I had so hoped to get to Robert Kaplinsky’s Skytypers or Pac-Man, but there just wasn’t enough time.  There’s always next year.

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I was amazed at how engaged students were with the Transformation Golf.  It was the second to last day of school, and we were doing locker clean outs.  I asked students to sit at the tables in the commons and work on this when they were done cleaning out their lockers, and they did!  They were having so much fun with it, it was great!  Desmos saves the day and prevents chaos at the end of the year!  I shouldn’t be surprised by that at all.

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Similarity

I have a third of my 6th graders again next year as 7th graders.  In those two years, my goal is to teach them the 6th, 7th, and 8th grade standards.  It’s been somewhat of a slow process, but I’m making progress.

This year I was able to get a unit in on similarity at the end of the year with my 6th graders.  I definitely pushed them and challenged them in this unit, and my students rose to the challenge.  I was so proud of them for the challenging proportions they were solving.  Some of my students definitely noticed that I was pushing them a bit more and became frustrated that it wasn’t coming as easy to them, which wasn’t a bad thing.  We worked through that.


It had been a while since we had solved proportions, so I started the first day of our unit with a few review problems.  I knew I wanted to use Marcellus the Giant from Desmos at some point in the unit.  As I was planning, I wasn’t quite sure when I wanted to fit this in.  Should I intro with this and have them try it without telling them anything about similarity or scale?  Do I use it after we have talked a bit about both of these?  I ended up deciding to use it right away at the beginning and was glad I did.  Students may have been confused at some points throughout the activity, but by the end they were able to explain what it meant for a giant to be scale or not scale.

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After a couple days of notes on similar polygons and scale, we did this activity around the room.  I had done another Math Lib activity earlier in the year, and my students love it.  I fully admit that I could have created something, but I was end-of-the-year teacher tired and decided it was worth it to get the activity linked above.

The weather in Minnesota was FINALLY nice, so I decided to do this activity outside with students.

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Then I saw this and knew I had to make this happen in my classroom.

Lisa used the game Clue and incorporated scale factors into the activity for students to figure out who was intentionally making math mistakes.  😉  Read Lisa’s post on how she set up the activity with her students.

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I ended up doing it slightly different than how Lisa described in her post.  I printed out Clue boards that Lisa shared and the Clue cards.  I wrote different numbers on all of the cards.

Students started by finding the area of each of the rooms on the clue board.  Once they had done that I gave them one Clue card with a scale factor on it and had them pick which room to use that scale factor for.  After they found the new area, I checked their work.

Nearly every group did what I expected them to do.  They took the area and multiplied it by the scale factor, rather than multiplying by the scale factor squared.  That was one of my main reasons for doing this activity -I wanted students to see how scale affected area since prior to this we had just talked about how scale affected length.IMG_7592

I let students struggle with that for a little bit before giving them a hint to help them out.  I put a grid up on the board and we looked at a 1×1 square.  Then I picked an easy scale factor, I think it was 2, and we talked about what the new dimensions of the square would be.  Then I asked students what the new area of the square would be?  I saw the lightbulbs go off for some students, and then I asked how that would apply to the problem they were working on.  It was enough for all groups to eventually figure out what to do in the Clue activity.

In the future when I do this, I would be more intentional about the numbers that I picked for the scale.  I used some pretty small numbers for some of the cards so after applying the scale factor the areas of the rooms were pretty unrealistic.  Overall though, it was a great activity and I am looking forward to using it again next year.

Distributive Property

Earlier in the year my 6th graders talk about the distributive property without variables. Partway through this post I shared how I introduce that idea to students.

Later on in the year we start talking about the distributive property with variables.  I started by reviewing how they used the distributive property earlier in the year without variables.  I was so impressed with how many different ways my students came up with to use the distributive property to multiply 7×48.

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This year I used Illustrative Math Unit 6 Lessons 10 and 11 to introduce the distributive property with variables.

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Illustrative Math Grade 6 Unit 6 Lesson 10

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Illustrative Math Grade 6 Unit 6 Lesson 10

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Illustrative Math Grade 6 Unit 6 Lesson 11

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Illustrative Math Grade 6 Unit 6 Lesson 11

As we were working through the resources from Illustrative Math, I loved how they incorporated the idea of factoring, without explicitly calling it that.  I had done a little bit of that in the past with this puzzle from Open Middle.

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Again, I was super impressed with all the different solutions they came up with.  I didn’t quite use the “rule” of the Open Middle problem and allowed students to use fractions and decimals.

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After going through that, students worked on this distributive property puzzle.  When students finished that, they started working on some Yohaku style puzzles I created.  This was my first time creating my own puzzles like this, so I had no idea how it would go over with students.  When I made the puzzles, I found two solutions for each.

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This activity went over SO much better than I even imagined, and my students found solutions that were much more creative than the ones I had found!

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When I was explaining how these puzzles worked to students I told them that if I did it correctly when I made them, each one should have at least two solutions.  One student asked, “But what if you did it wrong?”  I told them that very well could have happened. I’m human, and it’s May.  I’m tired.  😉

After the first group found two solutions for the same puzzle, one student told me, “You did it right!  You didn’t make a mistake.”

My students were so engaged while working on these puzzles.  They were so persistent.  I loved seeing all the eraser marks on their paper as evidence of them trying again and again and again until they found something that worked.  Students were cheering when they found a solution.  I wish I had recorded them working on these.  It was fantastic.

After the bell rang one student said, “Could you make some more of these for next week? Maybe nobody else liked them, but I thought they were fun.”

I also am looking forward to have a conversation with this student about the right column.

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I completely understand the student’s thinking.  This is the same student who came up with this solution earlier in the week.

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Here is the link to the puzzles I created.

If you create more, I would love to see what you come up with.  After sharing a picture of the puzzles on Twitter, Yohaku created a few similar.  You can find them here.


Solving Equations

Then we start solving equations using the distributive property.

I gave them a couple review problems prior to starting this.  The problems were similar to the following.

  1. 3(x + 4)
  2. 3x + 12 = 24

Then I told them we were going to use both of those ideas today and put the following problem up:  3(x + 4) = 24.

As students were working on this one student goes, “Oh Ms. Bergman, you are so smart.”  Another example of a student noticing that I am intentional about the problems I put in front of them, and I love it.

(Also, yes I know we don’t need to use the distributive property to solve 3(x + 4) = 24.  We talk about that too.)

I made an Add Em Up activity for this.  You can download the file here.  Add Em Up is an activity I got from Sara Van Der Werf.  You can read her post on this activity here and a post I wrote here.  Students are always super engaged when doing this!  We also spent some time doing Vertical Non-Permanent Surfaces with these problems and students were also super engaged in the math they were doing.  Here are some of the problems we used for VNPS.

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Like Terms & Simplifying Expressions

To introduce the idea of like terms to my students.  I use this Desmos card sort.  Initially I have students group the card however they choose.   Students will inevitable group some cards that are like terms which leads us into talking about what it means for terms to be “like”.  Then I have students group the cards into groups that are like terms.

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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.

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Then we play “Like Terms Uno”.  The version I use I got on Teachers Pay Teachers a while back, and it looks like it’s no longer available.  In a quick Google search of like terms Uno, several other versions came up.  I’m not sure if those versions are uploaded to the internet legally, which is why I haven’t included links to them.  So if you’re interested in a version of this, seriously just Google “Like Terms Uno” and several different options will come up.

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After this we start talking about simplifying expressions.

I start with the video below.  A couple minutes in many of my students are groaning.  I may look at another option for next year, or cutting down the video clip somehow, because watching that video for over four minutes is torture, but it serves it’s point.  I tell students, “You know how your ears hurt after you watched that video for a few minutes? That’s what it’s like for mathematicians every time they see something like 5x + 7y + 3x + x + 8y.  How could we rewrite that so it doesn’t ‘hurt your ears’?”

We practice combining like terms one day, and then the next day we do practice with the distributive property.

Open Middle has some great problems for simplifying expressions.

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

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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.)

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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(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.

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

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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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(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.)

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Here is the link to download the files for the activities I shared in this post.