Good Things: Volume 10

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My students have continued to enjoy our play table.

I got those on Amazon.

Those are Lego knock-offs from Target the dollar section.

I had a couple students who came early every day to class after lunch and each day they would try to balance a larger cube like that.

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I went to a math Ed Camp hosted by MCTM, and it was a great day of connecting with local math teachers!

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I’ve already tried a couple ideas people shared at the Ed Camp.  Jess Strom mentioned that she loves combining Open Middle problems with Vertical Non-Permanent Surfaces.  I tried it, and it was my favorite day of Open Middle problems yet! I found that students were more engaged and on task with the problem than when they do it on paper.

Students were working on this problem and this one.

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I pulled out Jay Chow’s Desmos Breakout activities recently for the first time this year.  Students always enjoy these so much.  You can find all of his activities here.  If you haven’t seen these yet, check them out!  They are awesome!

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The makers of Set came out with a new game, and it’s great!

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One of my students had an awesome way of thinking about converting 5/8 to a percent.

“Well, we already wrote 1/8 as a percent today and that was 12.5%, and I know that 4/8 is 50%.  I added 12.5% and 50% to get 62.5%.”

I use this method quite a bit with other types of problems, especially when finding percents of dollar amounts, but I hadn’t thought to use it in this direction – going from fractions to percents.  I love learning from my students!

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I had a couple of students debating over multiple days about whether 0/0 would be 100% or 0%.  In this class, we had not been covering percents and this debate was completely student initiated.  We did talk about 0/0 being undefined and sort of an irrelevant question, but they were debating that if you HAD to pick 0% or 100%, which would it be.  I loved listening to their arguments for which one they thought it was.

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The last couple years I’ve introduced the Pythagorean Theorem using Notice/Wonder, and I’ve liked how this has gone.  I really like how more of my students continue to think of the what is happening visually after introducing it this way.

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Last year I used Dan Meyer’s Taco Cart for the first time.  It was the first 3 Act I’d ever done and it was a spur of the moment decision to try it when I found out last minute that nearly half my class was gone for the day.  I used it again this year, and I felt like I facilitated it a little better than I did the first time.  I made adjustments to how we did this activity on the fly, and I was proud of myself for that.  Students are turning in their solutions on Monday, and I hope to write a short post on how this went soon.

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We used one of my favorite Polygraphs recently.  I absolutely love this one for angle pairs in 6th grade.

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We had a crazy couple months after Christmas break with several cold days, snow days, and late starts.  We had a brown Christmas, and ended February with normal amounts of  snow for MN.  It’s finally starting to warm up and the snow is melting.  Normally I don’t like this time of year when the snow gets muddy and the grass is brown; however, I’ve never been so happy to see brown grass as I have this year.

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Another idea shared at the MN EdCamp by May Vang was to give students examples of a vocal word and have them come up with the definition.  I tried this with rational and irrational numbers, and it didn’t quite go how I envisioned with my first class, so I changed things up with the second class.  It went better, but still not as well as I had hoped.  So I changed it slightly for the 3rd class, and it went much better.  I added some notice/wonder to the prompt, and it made all the difference.  No surprise there.  I don’t know why it took me three tries to add that to the prompt.

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A student stopped in my room after school one day this week with a baggie of seashells, “I keep forgetting to bring these to class.  I got these shells when we were on vacation, and they’re for you.”  She thought of her math teacher while on vacation.  How did I get so lucky to be apart of these kids’ lives??

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It was a work day in 8th grade this Friday, so I put on some of the NCAA basketball games.  A little math and some March Madness.  I can’t think of a better way to head into the weekend.

Equations and Inequalities with No Solution or Infinite Solutions

I was looking for something a little bit different than what I had done in the past to introduce equations that have no solution or infinite solutions.  I came across this post from Sarah who blogs at Everybody is a Genius, and it was exactly what I was looking for.  I also liked this because when I had these students as 6th graders, I used scales to introduce them to solving equations, so this wasn’t a new idea for them.

I gave this sheet to students and told them to fill in the boxes to keep the scales balanced, and that for each scale, the number in the box must be the same.  Students have done a few different Open Middle problems this year, so some students struggled with the idea that they could no reuse numbers since they are used to not being able to reuse them for those problems, but they eventually understood what to do.

Equation Scales

As I was walking around, exactly what I hoped would happen, happened.  Students got two number 3 and I heard, “What?  This doesn’t make sense.”  “This is impossible.”

As we went over what students came up with, we discussed how in #1 and #4, we could pick any number we wanted, in #2 and #5 only one number works, and in #3, and #6 no numbers work.  Then we took some notes on this.  In the notes sheet I handed out to students, I included a picture of the scale and we wrote out the equation and showed what was happening to the scale as we did the algebra.

I liked that introducing this topic this way to students gave students a visual to help them understand these types of equations.


The next day we did some practice at the whiteboards.  I always include some problems that have one solution (especially ones where x = 0) because some students want to start saying every single equation either has no solution or infinite solutions, even though I stress that this only happens when the variables are eliminated.

Sarah Carter has created a nice Open Middle style problem to go with this topic.  Here students can use the numbers -4 to 4.

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Last week we worked on solving inequalities with infinite solutions and no solution.  I really liked what I did last year for this, so I did something similar this year.  I started the day by having students solve an equation that had no solution.  Then, I asked students which inequalities would make that true and which would make it false.

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We briefly discussed which would make it true and which would make it false, and that was pretty much the only instruction I gave students that day.  They had little to no trouble transferring the idea of equations with no solutions or infinite solutions to inequalities.

I shared at the end of this post a Desmos card sort I use as well as another Open Middle style problem on this topic.

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Overall, I’m really happy with how students are doing with these types of problems.  I think that introducing this idea using the scales really helped my students to see what was going on.

Puzzles, Puzzles, and More Puzzles

One of my projects last summer was going through all of the different places I had resource stored (emails to myself, Twitter likes, Google drive documents, etc.) and compile them into one place.  I created a separate Google Doc for each of my preps and sorted the resources by unit.  It took quite a bit of time to go through all of those things, but I am reaping the benefits already.  As I start a new unit, or a new concept within a unit, I am able to check this document for resources to add to or replace things that I have done in the past.  I’m no longer spending time searching for these things in 4 different places or looking for new resources when I’ve already found things in the past.

Already in our second unit in 6th grade, I tried several new things this year that I’ve loved! We’ve been talking about exponents, prime and composite numbers, factors and multiples.  I found several puzzles that my students have been enjoying and have been SO persistent with.  I had one student at the start of the week tell me he hates puzzles.  I think probably because he’s a student who picks up on things quickly and doesn’t always like that he can’t figure out a puzzle right away.  By the end of the week he was asking for more puzzles.  🙂 Success.

It’s been fun to listen to their conversations as they’ve been working and to see them excited to share the progress they’ve been making on them.


The first task I added to this unit is the following problem from Open Middle.

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Look at the amount of work by one student!! Wow! So proud of him!


Then I used this Open Middle problem from Bryan.  I LOVED this one so much!  It was so challenging, but it really got my students thinking about exponents.

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I know there is at least one mistake in this one, but again, even despite that, look at all the great thinking that took place here!


Then we started getting into prime and composite numbers and multiples and factors.  I came across this puzzle on Twitter.  I love that it incorporates so much vocab into one puzzle.  After using this in one class, I realized that dry erase pockets would work well for this one so students could more easily change the numbers.

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Such great group work on this puzzle!


Lastly, I found this puzzle here and here.  Again, I love how it incorporates so much vocab into one puzzle, and I love the extra challenge of placing the headers versus having them already placed on the puzzle.  I did type up my own version of the puzzle.  You can download it here.  I used this on a Friday in one of my classes and they were so disappointed when it was time to clean up.  I haven’t had any students solve it yet, but several have come so close.

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Again, not everything is perfect in those, but what great thinking by my students!

 

6th Grade Unit 5: Percents

We start our unit on percentages by talking about converting between fractions, decimals, and percents.

I start with this Which One Doesn’t Belong? to get students thinking about percents and for me to see where my students are at in their understanding of this.  Then I ask them to brainstorm everything they know about percentages.

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I created matching cards for converting between decimals and percents years ago.  I intentionally picked numbers with lots of 2s and 4s in them so students can’t just say, “These are the only two cards with a 5 and a 6, so they have to match”.  You can download the file here.

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I spend several days letting students practice converting between fractions, decimals, and percents with different puzzles I’ve found over the years.  If I remember where I’ve found them, I’ll link to them here.

This is one puzzle I like for fractions and percents.

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From Chris Smith‘s newsletter via Jo Morgan’s blog.

Here is an Open Middle problem too.

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And yet another good Open Middle problem on percents.

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Then we get into applications of percents:  finding tip, tax, and discount.  I think this was the first year that I didn’t have a student do a discount problem with an answer greater than the original cost of the item.

One of my students favorite things to do during this part of our unit is for me to pull up a store’s website, find an item, and then calculate tax, discount, or tip.  (Side note:  Little Caesar’s website was super nice for adding things students wanted to the cart and finding the price.)

We used this loop activity for practice.

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”

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I also had this one ready to follow up with if I needed to.

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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.

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Then I used an I Spy activity.  I blogged about it here.

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I love this Open Middle problem on equivalent ratios.Screen Shot 2018-05-28 at 3.05.05 PM

I like this Which One Doesn’t Belong? around this time in the unit.

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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!

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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.

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Don Steward also has some great ratio puzzles on his blog here and here.

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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.

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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8th Grade Unit 3: Functions (Part 1 -function vs relation, function notation, and linear vs nonlinear)

The first test in unit 3 for 8th grade covers the difference between a relation and function, function notation, and determining from a table whether something is linear or nonlinear.

I followed pretty closely to what I did last year.  You can read about that here.

I did use Sarah’s updated version of her representations of relations telephone activity.  I blogged about it here, and you can directly download her version here.

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Once we got into functions versus relations (again read more about what I did last year here), I used this Desmos Polygraph this year.  After I did it with one class, I ended up copying and editing the activity and changed the circle to another graph because my students kept thinking it was funny to pick the circle and have the other person guess on the first try…oh 8th graders!

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Then we started looking at tables and determined from a table whether or not the graph would be linear.  I feel like this portion of the unit is what I need to focus on improving the most for next year.  It went alright, but I didn’t love it.  I started with the following image and asked students what they noticed about the two and what made the green graph a straight line and not the red one.

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Usually someone will say that on the green one, the y’s go up by two’s.  Then I put this image up and ask why that theory doesn’t work in this case.

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We continued talking about what makes the graph linear, and the next day I used this Which One Doesn’t Belong? for a warm-up.

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When it came time to review for the test, I used a function vs relation Kahoot and this great open-middle type problem from Sarah to review functions versus relations.  I used it the same way Sarah did and had students use the numbers -4 to 4 and first had students place the numbers in the boxes so that the three relations were also functions.  Once students completed that, I had students place the numbers so that the three relations were not functions.

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I also have an Add ‘Em Up activity for function notation I created.  You can read about Add ‘Em Up here, and download the activity here.

8th Grade Unit 2: Inequalities (Part 3 -absolute value special cases and word problems)

The last part of our unit on inequalities covers absolute value special cases and word problems.  You can read about part 1 here and part 2 here.


Absolute Value Inequality Special Cases

To introduce solving absolute value inequalities that have no solution or all real numbers as the answer, I tried a couple different things this year.  In a couple classes I put this up on the board and we talked about each of them, which is what I had done in the past.

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In another class I put the following up on the board and asked how we could write an inequality that would never be true.

|x|______________

Then I asked how we could write an inequality that would never be true.  Once students gave me answers for that and understood, we talked about if it mattered what was inside  the absolute value bars.  Both ways worked fine, but I almost think I liked think I liked this new way better.  I liked that students were coming up with the inequalities.

The next day I started with the following warm-up problem.  I always like to include absolute value inequalities that don’t have all real numbers or no solution as the answer so that students don’t forget what they already knew about those types of problems.

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Here is the link to download a worksheet with these types of problems.

I also created an Open Middle type problem for these types of inequalities.  You can download it here.

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Absolute Value Inequality Word Problems

One of the things that my students need to know is how to solve word problems involving absolute value inequalities.  In the past, I just jumped into these types of problems, but this year I took a day to go over setting up the absolute value inequalities given a graph.  The day before this I had each student solve an absolute value inequality to use as this introduction.  You can download the problems I used here.

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I took several of these and had students find the midpoint of each graph and the distance from the midpoint.  Then students looked for a pattern.Screen Shot 2017-12-08 at 7.10.45 PM.png

Then I had students practice writing absolute value inequalities when given a graph and practice finding the midpoint of the graph and distance from the midpoint when given an absolute value inequality.

The next day when we got to solving word problems went so much better than it did last year having spent a day on this ahead of time.

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You can download the test review here.

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.