How it Works: Integer Multiplication

This past week, Josh Zagorski forwarded a tweet from John Fritzky, a middle school principal looking for an explanation of how integer rules work for multiplication and division without using rules:

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Graham Fletcher folded me in on the conversation and I haven’t been able to stop thinking about it. I haven’t been able to think of a quick way to sum it all up, either.  So, here we are.

A few people commented on this with some good ideas and I’ll highlight those here.

As I mentioned, I haven’t been able to stop thinking about this.  The more I thought of it, the deeper I dove into it.  I decided to focus on multiplication first and after the past few days of thinking (this is just the beginning), I think I this is involves 2 big ideas:

  1. What are integers and how can I use what I know about them (absolute value, other operations, and the negative symbol “-“, etc.) to make sense of multiplying these numbers?
  2. What does multiplication really mean and how can we use what we know about it to apply it to a new system of numbers?

When I’ve thought about this in the past with my own students, I focused only on the first idea.  But the idea of what multiplication really means plays a huge role in making sense of integer multiplication (and later division).

Integers

The whole numbers and their opposites have several ideas associated with them and all of them really need to be developed deeply and conceptually in order to get to the point of making sense of operating with them.  That said, here are what I see as the big ideas  that stand out for the sake of this discussion (blue bold highlights):

6.NS.5 – Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.6 – Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7.NS.1A – Describe situations in which opposite quantities combine to make 0For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

7.NS.2A – Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

Multiplication

As briefly as possible, here’s what multiplication really means (I’m taking this straight from the standards, to be as transparent as possible here:

3.OA.1 – Interpret products of whole numbers
e.g. Interpret 5×7 as the total number of objects in 5 groups of 7 objects each

So, when reading a multiplication problem out of context:

____ x ____ = ____

really means:

____ groups of ____ things in each group = ____ total things

This idea is important for three reasons:

  1. In the absence of context, students contextualize the “naked math.” Giving meaning to numbers in order to work with them is one piece of the number sense puzzle.
  2. This builds on an idea of grouping and sharing from Kindergarten, so it is accessible to all students.  Building connections!
  3. Representing multiplication with pictures, arrays, and/or number line diagrams makes more sense when students think of multiplication in this way.

Making Sense of Integer Multiplication

 

 

 

 

 

I Didn’t Know What I Didn’t Know…

Let me just start with this.  If you live in Georgia, say within a 2 hour drive to the UGA Griffin campus, seriously consider joining the Masters’ or EdS program.  I’m in my first semester.  It’s amazing!  ‘Nuff said.

Maybe it’s just me… I thought I understood everything I needed to know about fraction equivalence… until this week.  If you get to the end and think, “Oh, I already knew that!” I apologize.  This is post is really for me to reflect a bit.  If it helps anyone else make sense of fractions…well that’s just gravy!

It all started with an assignment for one of my graduate classes. The assignment was to read Chapter 3 from Number Talks Fractions, Decimals, and Percents by Sherry Parrish and reflect on one of the big ideas and the common misconceptions connected to those big ideas.  I chose to reflect on fraction equivalence.

In the section on equivalence, Dr. Parrish talks about how students want to take fractions like 1/4 and multiply by two to get an equivalent fraction of 2/8. This misconception may be fostered by teachers who wish to make equivalent fractions easy for their students to remember. This is never a good idea!  Because really… if you multiply 1/4 by two, that means you have 2 groups of 1/4.  And 2 groups of 1/4 gives you 2/4 and 1/4 can’t be the same as 2/4.

What I learned next came from a phone conversation I had with Graham Fletcher about 15 seconds after I finished reading the chapter.  Sometimes I just think he knows when I’m learning some math and gives me a call.  He had a question about equivalent fractions. Over the course of about 45 minutes talking on the phone, I think we both deepened our understandings about what makes two fractions equivalent.

Take the rule of multiplying the numerator and the denominator both by the same number to make an equivalent fraction.  If we look at 1/4 and multiply the numerator and denominator by two to get 2/8, we get an equivalent fraction, but this isn’t necessarily the whole story.  To really understand fraction equivalence, I had to be asked to dive a little deeper. Graham asked me to dive deeper.  As we talked, multiplying by one came up, then the multiplicative identity.  These ideas definitely strengthened my understanding of fraction equivalence.

I thought I now had a deep understanding of fraction equivalence.  But wait, there’s more.  This is the best part.  I went to class this past Saturday and Dr. Robyn Ovrick gave us this:

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We were asked to fold the paper as many times as we wanted as long as all of the sections were the same size.  Some of us folded once (guilty – I hate folding almost as much as I hate cutting).  We shared our folds and Robyn recorded what several of us did on the smart board.  Then she asked what we noticed.  This is where everything came together for me.  I tried to share my thoughts but I don’t think I was very successful.  I was really excited about this.  Here is my (1 fold) representation of an equivalent fraction for 1/4:

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For my example, someone said the number of pieces doubled, and at this point (my eyes probably almost shot out of my head) I thought, but the size of the pieces are half as big.  I’m usually pretty reserved and quiet, but this was too much.  So, with a lot of help from colleagues in class who know me a bit better than the others it all came clear to me.  We visually made equivalent fractions, but connected the visual to the multiplicative identity and even explained it in the context of paper folding.

Here it is.

The original paper shows 1/4.  When we fold it in half horizontally, we get 2 times as many pieces and the pieces are half the size.  This can be represented here:

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The 1/4 represents the original fraction. The 2 shows that we got twice as many pieces, and the 1/2 shows that each of those pieces is half the size.  With a little multiplication and the commutative property we can get something that looks like this:

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Knowing that two halves is one whole is definitely part of this understanding, but seeing where it can come from in the context of paper folding allows an opportunity for a much deeper understanding. The numerator tells that there are twice as many sections as before and the denominator (really the fraction 1/2) says that the pieces are now half the size.  We looked at another example of how someone folded 1/4 (someone who folded 8 times!) and noticed that it worked the similarly – we got 8 times as many pieces and the pieces were each 1/8 the size of the original.  I don’t think anyone thought it wouldn’t work similarly, but it sure is nice to see your ideas validate something you thought you really understood before waking up that morning!

I’m still thinking about this and I keep making more connections.  This morning, in a place where I think I do my best thinking (the shower!), I realized that this is connected to the strategy of doubling and halving for multiplication.  I’ll leave you with that.  Time for you to chew.

 

 

 

 

Revisiting 3-Act Tasks

My lessons never stay the same.  They’re always evolving.  Recently, I’ve taken a look at some 3-Act Tasks I created and I noticed:

  • Some of the tasks are lacking an act.
  • Others have resources that no students ask for (at least students that I’ve worked with).
  • The quality is low (shaky camera, point of changes, etc.)

So, I finally had a minute (read 2 days) and revisited each.  Below, you’ll see the tasks I’ve chosen to revisit.  An explanation of the original, what I changed, and why I changed it follows.  If you’d like to skip this and get to the revisited tasks, click here.

Revisited #1 – The Candy Bowl

My very first attempt at a 3-act task was the Candy Bowl task.  I was working in an elementary school at the time and Graham Fletcher had created problem to get 2nd and 3rd grade students reasoning about subtraction by removing the numbers from the problem context.  His context involved the lunchroom and numbers of students in three classes.  We talked on the phone about this for a while and though I liked the problem, I wasn’t crazy about the context.  I sat in my room trying to think of a context that would be a bit more engaging for students to think about.  And the Candy Bowl was created.

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It was a good problem, but it really lacked one of the most basic parts of a 3-Act Task… The third act.  The reveal was weak, because it relied on the teacher to give students validation.  The updated version, which had to be done from scratch (apparently whoppers candies are no where to be found anywhere near Valentine’s day), can be found here with all new updated resources for Act 2 and new video including two reveals, depending on which question students decide to tackle.

Revisited #2 – Sweet Tart Hearts

Another one of my early tasks was Sweet Tart Hearts.  I really liked this one from the beginning. There is a huge focus on estimation which allows for students to obtain solutions that are close, but not exact in most cases.  This also allows for the teacher to facilitate a discussion about why answers may not be exact for a variety of reasons.  But again, it really lacked that third act.  The task was good, but the closing of the lesson was weak due to the fact that the students were relying on the “all knowing” teacher to give them affirmation.

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Apparently Sweet Tart Hearts are a hot commodity a few days before Valentine’s day.  I went out the other day for a quick run to pick up a bag.  I had to go to 4 stores and finally found a bag (the last one).  I thought it would take about 10 minutes to do this revisit.  Surely the  numbers for the colors would be similar to the last time.  Not only was that not true, but Sweet Tarts changed the orange hearts to yellow!  But, the revisit is all done and I’m very pleased with the new reveal which allows the video to reveal the answer and the teacher to focus students on the reasonableness of their solutions.

Revisited #3 – The Penny Cube

My final revisit is the Penny Cube.  It is probably my favorite task.  I’ve certainly heard more from teachers about this task than any of the others.  I think I got the reveal right on this one.  The problem I found with this task was that I thought students would ask for things that I would want.  The first time I did this task with students, I guided them to the information I had ready for them.  They didn’t care anything about the dimensions of a penny.  They just wanted some pennies and a ruler.  It’s amazing what you learn when you listen to students, rather than try to tell them everything you think they need to know. So, to all of the students out there, Thank you for making your voices heard!

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So, this was the quickest fix.  I just updated the Penny Cube page (all of the coin specifications are still there – in case anyone wants them).

Note:  In this post I share how I changed my approach to teaching the Penny Cube task. 

So, it took a few days, but I’ve revisited some tasks that have been bugging me for a while and I hope it’s for the best.  I know I’ll probably give these another look in the future.  I’ll just need to start in early January to make sure I get the candy I need.

Happy Accidents

When I was growing up in (rural-ish) central New York, we had one TV.  We received 5 local stations through the antenna on the roof (abc, nbc, occasionally cbs if the wind was blowing just right, then Fox came along, and a pbs station).  This was a time when TV programming on the major networks actually ended at about 1:00 a.m. with a video of the American flag waving in the wind and the national anthem playing.  When that was over, there was nothing on TV but static.  This is something my kids can’t imagine. Not that they watch regular TV that often anyway (YouTube, Vimeo, etc.), but every time they turn it on, there are at least 100 shows to choose from on 4 TVs.

This wasn’t the case for my siblings and me.  Usually, the first person in the living room got dibs on what show was on or there had to be a “discussion” to figure out what everyone would watch.  Sometimes this ended in the TV being turned off by Mom or Dad with a “suggestion” that we go outside and get some fresh air.  Other times, we would decide to figure it out on our own and end up on the local PBS station watching a man with a huge perm (this was the 1980s) paint beautiful scenes in about 25 minutes.

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We (my 5 siblings and I) were all in awe while we watched Bob Ross paint wonderful paintings while talking to us (the viewers) about everything from his pet squirrels to painting techniques.  And at the end of every episode I felt like I could paint just like Bob Ross!  I never tried, but I felt like I could!

Recently, my kids have discovered the talent and wonder of Bob Ross through YouTube and Netflix. They love his words of wisdom:

  • “Just go out and talk to a tree.  Make friends with it.”
  • “There’s nothing wrong with having a tree as a friend.”
  • “How do you make a round circle with a square knife?  That’s your challenge for the day.”
  • “Any time ya learn, ya gain.”
  • “You can do anything you want to do. This is your world.”

And I love that they love these words of wisdom.  You can find more here.

For Christmas this year, my son and I received Bob Ross T-shirts.  Connor’s has just an image, while mine has a quote as well:

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Bob Ross was referring to painting when he said these words; “In painting there are no mistakes, just happy accidents.”  In other words, when you paint your mountain the wrong shape, treat it as a happy accident.  It can still be a mountain, there may just end up being a happy tree or a happy cloud that takes care of your happy accident.

I think it works for math class, too.  Recently, I modeled a Desmos lesson for a 7th grade teacher.  The students had been working with expressions and equations but were struggling with the abstract ideas associated with expressions and equations.  The teacher and I planned for me to model Desmos using Central Park to see how students reacted to the platform (this was their first time using Desmos) and how I managed the class using the teacher dashboard.

During the lesson, there was a lot of productive struggle.  Students were working in pairs and making mistakes happy accidents.  They were happy accidents!  Because students kept going back for more.  At times there was some frustration involved and I stepped in to ask questions like:

  • What are you trying to figure out?
  • Where did the numbers you used in your expression come from?
  • What do each of the numbers you used represent?
  • Before you click the “try it” button, how confident are you that the cars will all park?

The last question was incredibly informative.  Many students who answered this question were not confident at all that their cars would all park, but as they moved through the lesson, their confidence grew.

One of the best take-aways the teacher mentioned during our post-conference was  when she mentioned a certain boy and girl who she paired together so the (high performing) girl could help the (low performing) boy.  The exact opposite happened.  The girl was trying to crunch numbers on screen 5 with little success.  The boy just needed a nudge to think about the image and to go back to some previous screens to settle some ideas in his mind before moving ahead with his idea that the answer is 8.  Then, he got to expain how he knew it was 8 with the picture, conceptually, to his partner.  The teacher’s mistake happy accident was in believing her students would always perform a certain way.  When students are engaged in tasks that are meaningful, they tend to perform differently than when they’re given a worksheet with 30 meaningless problems on it (the norm for this class before Desmos).  Ah-has all around and the “low student” shows that he knows more than the teacher thinks.

The icing on the cake?  Several students walking out of the classroom could be heard saying, “That was cool.” or “That was fun.”

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Let’s treat math mistakes as happy accidents, something to learn from and problem solve our way through.  When students (all humans) make a mistake, synapses fire.  The brain grows (More on this from Jo Boaler here).  What we do as teachers from this point, determines how much more the brain will grow.  If we treat student mistakes as happy accidents, perhaps their brains will grow a bit more than if we continue to treat mistakes in the traditional manner.

Let’s hear it for Bob Ross.  He probably never thought his words of wisdom about painting would be translated to the math classroom.

Now, go make friends with a math problem.

 

 

 

Georgia Math Conference 2016

For the second year in a row, I had the privilege and honor to give an ignite talk at the Georgia Math Conference (Last year’s talk can be found here.)  What makes ignite talk sessions great is that you get a taste of what several speakers are passionate about and you get to walk away with at least one ember of at least one of those talks beginning to burn in you!

Special thanks to Graham Fletcher for putting this all together (in pre and post production!).  Graham is top notch, “for sure” (Must be a little of my inner Canadian there).

The featured speakers this year in the order of their talk:

Me (@mikewiernicki) – I didn’t ask to go first. 🙂


<p><a href=”https://vimeo.com/190360814″>Mike Wiernicki – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Katie Breedlove (@KatieBreedlove)


<p><a href=”https://vimeo.com/190362489″>Katie Breedlove – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Jenise Sexton (@MrsJeniseSexton)


<p><a href=”https://vimeo.com/190364708″>Jenise Sexton – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Karla Cwetna (@KCwetna)


<p><a href=”https://vimeo.com/190381786″>Karla Cwetna – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Carla Bidwell (@carla_bidwell)


<p><a href=”https://vimeo.com/190286621″>Carla Bidwell – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Brian Lack (@DrBrianLack)


<p><a href=”https://vimeo.com/190415942″>Brian Lack – Ignite Talk (GCTM 2016)</a> from <a href=”https://vimeo.com/user21534889″>Graham Fletcher</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p>

Graham Fletcher (@gfletchy) – The great Emcee’s talk is available elsewhere.  I’ll find it and link it asap.

 

 

Student Ideas Matter

Engaging students in math has always been a goal for me.  No… more than a goal…. a passion! And it’s not always easy to do.  For example, I used to hate teaching students how to find the sum of an arithmetic series.  I didn’t hate it because it was difficult to teach or because students had an overwhelming difficulty learning it.  I hated it because I was the only one that saw the beauty in it.  I was the only one who was passionate about it.

This lesson was “fun” (I use the quotes to denote that this was a fun lesson for me – not so much for my students).  But this all changed when I allowed my students the opportunity to think for themselves.

The task was very simple in concept:  Find the sum of the series of the numbers 1-20.

Before going any further, it may be useful to know about the

  • Class norms:
    • Estimate first,
    • the answer is never enough,
    • reasoning, explaining and looking for patterns are all expectations,
    • if you found one way, look again, you may find a more efficient way,
    • get out of your own head and talk about the math with your partner/group while you work

Several started adding 1 + 2 + 3 + 4 + . . .+ 19 + 20.  I noticed this and asked those groups for one word to describe their strategy.  Sample responses:  boring, lame, tedious (actually proud of that one), calculator worthy…

My reply to each of their descriptions:  If your strategy is [insert one: boring, lame, tedious, or just plain calculator worthy] why do you feel the need to use it?

Sometimes students get stuck in their own thinking and just need to be made aware of it. To help nudge students to think in other ways, I had bowls of tiles with the numbers 1-20 written on them available for groups to use.

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It took several minutes before students began to grab tiles and began to notice things like:

  • “Hey, Mr. W., we can make a bunch of 20s.”

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  • “We got a bunch of 21s. 10 of them.  It can’t be that easy, right?”

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  • “We made 10s and 30s.  How did you make 21s?”

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  • “We did the 20s too.  That’s the easiest way for us.”

It was a bit chaotic, and I didn’t know it then, but there was a passion building.  This wasn’t just engaging, these students were ALL IN.  They were more than engaged and wanted to learn more about the strategies they came up with.  They wanted to share. Needed to know.  And the answer was almost irrelevant. The connections between all of their strategies became the focus.

From here, getting to the algebra made sense.  How would you find the sum of the numbers 1-50?  1-90?  1-100?  What about 5-50?  Some saw their ideas with the tiles transfer easily to an algebraic expression and equation.  Others not so much. So, more time to talk and share.  More time to find a strategy that is more convenient to generalize for a series of numbers of any range.  The success of the students’ mathematical ideas gave them power to reach further – to take another chance.

Teaching the lesson this way was a definite improvement on the original. In this version, the students’ ideas matter, so students matter.  In this version, students think for themselves and collaborate with others, and in turn get validation of their thinking, so students matter.  In this version, students built some passion.  They fed off of each other. And the content mattered because of the students’ interaction with it.

Is this lesson the best it can be? I’m not sure.  So, I’ll continue to try to improve on it.

Thoughts and comments welcome.

 

 

 

 

Blogarithm Posts

Last year I had the honor of being asked to write four posts for NCTM’s Math Teaching in the Middle School Blog: Blogarithm (one of the coolest math blog names out there).  They were posted every two weeks from November through the end of December (which just shows that I can post more frequently if someone is reminding me every other week that my next post is due (thanks Clayton).

Pythagorean Decanomial

The four posts are a reflection of a lesson I taught with a 6th grade teacher, in September of last year, who was worried (and rightfully so) that her students didn’t know their multiplication facts.  After a long conference, we decided to teach a lesson together.  I modeled some pedagogical ideas and she supported students by asking questions (certain restrictions may have applied).

Links to the four posts are below.

  1. Building Multiplication Fluency in Middle School
  2. Building Multiplication Fluency in Middle School Part 2
  3. Building Multiplication Fluency in Middle School Part 3
  4. Building Multiplication Fluency in Middle School Part 4

While you’re at the Blogarithm site check out some other guest bloggers’ posts.  Cathy Yenca has some great posts on Formative Feedback, Vertical Value Part 1 and Part 2, and 3-Act Tasks