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.

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

Illustrative Math Grade 6 Unit 6 Lesson 10

Illustrative Math Grade 6 Unit 6 Lesson 10

Illustrative Math Grade 6 Unit 6 Lesson 11

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.

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.

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.

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!

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.

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

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.

6th Grade Unit 2: Intro to Algebra (Part 1 -Exponents, Prime Factorization, Properties, and Order of Operations)

Unit 2 in 6th grade is an introduction to algebra.  This is one of my favorite units.  I love order of operations, and I love introducing students to solving equations.  I break the unit up into multiple parts.  Here is part 1.

Exponents

We start the unit talking about exponents so that students can use exponents when we get to prime factorization and order of operations.  I typically spend about a day on this and use Kahoot for practice.  I also incorporate this throughout the unit in brain breaks.  “Ok everyone stand up!  2 to the 3rd power.  (Then I give them time to think about what the answer is.)  Do 2 to the 3rd power jumping jacks.”

Prime Factorization

Then we review prime and composite numbers before getting into prime factorization.

(I incorporate prime/composite into brain breaks as well.  “Think of a prime number.  Do that many sit-ups or push ups.”)

I also incorporate a brain break called Factor Hop into this part of the unit as well.  I put four numbers in the corners of my room.  Students go stand next to a number.  I pick a number and if that number is a factor of the number students are standing by they have to move to a different corner, but they are not allowed to walk.  Some students really get into it and have a lot of fun with coming up with other ways to move to a different number.

Which one doesn’t belong? works great as a warm-up a few days after going over prime and composite numbers to review this vocab.  Students will also usually bring up factors in our conversation.

Since students typically have already learned how to do prime factorization using the factor tree method, I do a couple examples of that before introducing them a method similar to the birthday cake method I found on Sarah’s blog.

I’ve started using this method because for a couple reasons.  In my opinion it’s more organized than the factor tree method, and I like that it can be applied to other concepts such as greatest common factor as well as with variables.  The high school teachers in my district also use it.

Properties

Then we get into properties of numbers.  We start with the associative property, identity property, and commutative property.  I co-taught with a teacher a couple years ago who was a huge help when it came to teaching properties.  She did a great job of helping students see the connection between what the word actually means and what is happening in the property.

Commutative Property:  You see the word “commute” so the numbers “commute” or change places.

Associative Property:  You see the word “associate”.  For example, you may associate with certain people at basketball practice, and you associate with other people at church.  In the associative property we see numbers “associating” with different numbers.

Identity Property:  Identity is who you are, so in the identity property the number wants to keep it’s identity.  It wants to stay the same.  After we talk about that, I introduce this property by saying, “I’m a 5.  We’re adding.  I want to stay the same.  I want to keep my identity.  What do I need to do?”  Then, “Ok, now we’re multiplying.  I’m a 5, and I want to keep my identity.  What do I need to do this time?”

Then for practice, we use this Desmos activity from Cathy Yenca.  I edited her version to not include the Distributive property, since we hadn’t covered that one yet.

Then I used Sarah Carter’s Two truths and a Lie activity.  My students really enjoyed this. You can download the template from here blog post here.

I loved this one from one of my students.  I read it too fast the first couple times and missed their mistake.

For a few days leading up to teaching students the distributive property we do math talks, and this has made teaching the distributive property go SO much better for me.  In almost every class, I will have a student who will use the distributive property in the math talk so we can talk about so-and-so’s method of multiplying and then I’ll later introduce the term distributive property.

Then for practice, I came up with this Desmos activity.

I color coded the cards, and I usually go over this with students before they start the activity so they don’t become overwhelmed when they start.

Order of Operations

I’ve started introducing order of operations by having the following up on my SMART board along with an example problem on the whiteboard and having students do a stand and talk to talk about which things need to be done before others.

I’ve liked this change.  I enjoy listening to their conversations as they talk, and it also gives me insight into where they are at in their understanding of order of operations as well as how they were taught this as 5th graders.

In every class a student usually brings up PEMDAS, and then we discuss what I don’t like about that acronym.  I love that students are able to tell me things like the “P” stands for parentheses and there are other grouping symbols besides that, and “it looks like you have to do multiplication before division, but you don’t.  They’re on the same level and you read it like a book going from left to right.”  It was also music to my ears when a student said, “PEMDAS?  What’s that?  I’ve never heard that before.”  To which I replied, “Great!  You don’t need to know what it means!”

This has also become one of my favorite warm-ups of all time.

Over the years, I’ve built up a quite a collection of order of operations activities, and I’ll pick a few of those for practice.

• Espresso Puzzles from Greg Tang Math (scroll through this page to find the Espresso Puzzles)