Making Sense of Invert and Multiply

As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction.  When we do, it’s normally accompanied with Keep-Change-Flip or the saying “Yours is not the reason why, just invert and multiply.”

Both are conceptual cripplers.

I’ve been drafting the 4th installment of the Making Sense Series involving fractions and I’m sharing this post as more of a personal reference should K-C-F make its way round these parts again…and I’m sure it will.

Side note: A while back Fawn and Christopher each shared a post about division of fractions using common denominators. Both posts left lots of math residue and are well worth your time.

Let’s start with a model for 3/5 ÷ 1/4.


Modeling measurement division of fraction by a fraction.


At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.



From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of “1 whole” so they begin to purposefully leave it out.  In doing so, they become more efficient in the procedure of dividing fractions.

Some students begin to eliminate the green and red steps from the above equation because they’re seen as repetitive.  We’ve even had one student that “invented” and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.

It looked something like this…



I keep reminding myself that if fractions are the gatekeeper to algebraic reasoning then I need to slow the process down and conceptually understand what’s happening. This includes K-C-F.

As students understand the power of creating a whole number denominator they begin to search for more efficient ways to get 1 whole.  It’s here that they being to explore with equivalent fraction and the idea of using the reciprocal to create a whole.


What I’ve realized is that aside from complex fractions, the underpinnings of this equation are developed in the elementary grades.

  • 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a÷b = a/b).
  • 4.NF.1-Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b).
  • 3.NF.3.c- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Just like before, students will look to generalize the equation above and find shortcuts. They’ll do so by eliminating repetitive steps which would leave them with…


Invert and multiply.




Posted in 3-5, 6-8, Against the Norm, Fractions, Making Sense Series, Math Progressions, Number Sense, Planning, Strategy Development | Tagged , , , , , , | 8 Comments

The Multiplication Sundae and the Bad Taste of Incentives

Earlier this week I received an email asking for incentive ideas for a school wide fact fluency focus:

Hi Graham,
I need some insight about math facts and incentive programs. I follow your site and have read Not Your Mom’s Flashcards:Conceptual Understanding of Multiplication and watched From Memory to Memorization: There is a Difference. I know facts are about being efficient, accurate and flexible. With that said our school would like to start an incentive program to encourage students to learn their facts. Ex. In the hallway, picture of student and medals earned and a quarterly bingo math party with snacks and prizes. I am all about balance but… I want to make sure we do what is right for all students. My principal would like an incentive program for facts and I want to lead us in the right direction.

If you search the Internet, there are tons of incentives for fact fluency and the Multiplication Sundae is a big seller.  But the problem I have with the sundae is that some kids never even earn the bowl, let alone the ice cream. And the cherry? It doesn’t stand a chance!

In the same week, I was reminded why incentives for fact fluency crush my soul.  I was at my daughter’s award ceremony, she’s a 3rd grade student. During the presentation they awarded all students that had mastered their multiplication facts with an award. There were a handful of students from her class that earned this award. As a dad, I was proud because my daughter received the award but I know she learned her facts the right way.  But what absolutely crushed me is the other 17 students in her class that didn’t receive the award and how they now believe they’re not good at math.  So I’ll ask the question…Is the award worth it?

I really appreciate the email and all the work we do as teachers to motivate our students but now I can’t escape 2 questions:

  1. Is there an incentive idea/program that addresses equity? An idea where EVERY student can be successful?
  2. What role (if any) should incentives play in our schools?

I like to think if we can’t address question #1 with “yes” then question #2 is answered for us…incentives don’t belong.


Posted in Against the Norm | 7 Comments

3-Act Tasks (#49, #50, #51)

They come in waves. It’s both a blessing and a curse.

Seems like everywhere I walk there’s a perplexing question I can’t just let go of. The bug of mathematization has hit my entire family.

My wife is a teacher and for appreciation week she was given a light bulb filled with Skittles. She comes home and says “Hey Gray, I’ve got a 3-Act Task for you!”

And that’s how task #49-Bright Idea came to be.

We love pickles and jalapeños in our house. We have back-ups for the back-ups. My daughter in 3rd grade was analyzing the jars while unpacking the groceries. Then she says, “I wonder how much longer it takes to fill up the little jar compared to the big jar?”

Enter task #50-Dill ‘er Up

I was at a friend’s house and he was slicing some apples for our girls to have as a snack. My youngest daughter (kindergarten) watched intently and ask “Hey daddy, how long do the think the skin will be? Will it be taller than you?”

Voila…task #51-Granny Smith’s Skins


In September 2013 Dan visited my district and said I needed to start sharing and I haven’t stopped since.

51 tasks later and I just want to say thanks.

  • Thanks to my family for putting up with me. Sure it’s fun but I know it’s annoying too.
  • Thanks to Dan for the push to share.  I hope my work has inspired others to share openly and freely as well.
  • Thanks to my friends in the #MTBoS who help push and question me. With late nights and behind the scenes magic… you make me feel normal.
  • Thank YOU for your continued support and encouragement. If you’ve used or shared one task, I appreciate you drinking the Kool-Aid. Thanks for the feedback and for sharing pictures of student work. You inspire me to create more.

“All of us are smarter than one of us”


Posted in Against the Norm | 12 Comments

Becoming a Better Storyteller

I recently had the amazing opportunity to be 1 of 6 speakers at ShadowCon16.  The whole premise of ShadowCon is awesome and full credit goes to Zak Champagne, Mike Flynn, and Dan Meyer for having the vision and bringing this all together.

Conferences quickly become distant memories, however our three friends have found a way to extend the conference experience. Each of the six ShadowCon speakers shared a provocative 10-minute talk and then challenged the audience (YOU) with a Call to Action.

You can find my ShadowCon Talk here, along with my call to action and some extra tidbits. After watching my talk, please take the opportunity to respond to my call to action by posting your thoughts on the message board.

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If my talk didn’t resonate with you and you don’t feel compelled to act…no worries because there 5 more chances to get connected.

Robert Kaplinsky – Empower

Gail Burrill – Math is Awesome: Let’s Teach so Our Students Get It

Kaneka Turner – Extending the Invitation to Be Good at Math

Brian Bushart – Make Your Own Kind of Music

Rochelle Gutierrrez – Stand Up for Students

I have purposefully closed comments for this post in hope that you will continue the conversation on the webpages that NCTM has graciously shared.

I look forward to learning and growing together as all of us are smarter than one of us.



Posted in Against the Norm

Be the Teacher: Moving from Counting to Cardinality

Cardinality…what is it and what does it look like?

If you’re not a kindergarten teacher you might be left shrugging your shoulders if someone asked you to define cardinality. Before students can own the idea of cardinality, they need to have an understanding of one-to-one counting.

Screen Shot 2016-03-05 at 10.52.33 AM.png


So now you know what it is, here’s an example of a 4-year-old working to gain an understanding of cardinality…


If you’re not familiar with cardinality you expect this little guy to say “seven”. Especially after he’s counted and I’ve repeatedly ask him “how many are there?”

But nope, he’s not quite ready.

Be the teacher… 

What activities and/or suggestions can you offer to help me move this little guy from 1-to-1 counting to cardinality?

Please share you thoughts in the comments.  My hope is that by sharing this video parents can begin to look for and promote cardinality at home before students get to kindergarten.


Posted in Against the Norm | 41 Comments

The Progression of Addition and Subtraction

The more I create… the more I learn.  Here is the 3rd installment of this whole Making Sense Series which has truly forced me to be a better teacher.  A more educated teacher.

I can’t stress enough how much I’ve learned from diving into the progressions found here. There are many intricate pieces of learning which are mentioned throughout the progression documents which can’t be included within a 5 minute video.  I strongly recommend finding the little pieces with your team or grade level.

Unlike the multiplication and division videos which were introduced separately, this video is an addition/subtraction mash up…they way they should be explored by students.

Posted in Against the Norm, counting, K-2, Making Math Accessible, Making Sense Series, Math Progressions, Number Sense | 23 Comments

I’m Placing a Hit on the Pseudo-Context

This robs students of everything mathematics should be…


  • The “real world” connection
  • The step-by-step procedure
  • Circle the numbers you need to solve the problem
  • The pseudo-context word problem.
  • Lesson 19.2 infers that this unit is front-end load with procedures and formulas

What a sham!

I can’t help but ask the question FOR students, “When will they ever use this?

As if that wasn’t enough, the “real world” problem is followed up with mind-numbing practice…

Screen Shot 2016-02-14 at 9.25.28 AM.png

Notice that question #7 has been labelled as “H.O.T.”

Instead of going on a rant, I’ll keep my head down and continue to do my best providing alternative opportunities for students and teachers to engage in math.  The Fish Tank and Got Cubes are two examples that push back on pseudo-context questions and volume.

As for Mike’s DVD box, H.O.T question, and curriculum developers, I’ll offer this new task as a means to undo the limited understanding you promote with every edition you pump out.


Question: How are the sugar cubes packed in the box?

There’s no empty space.


Number of Cubes in the Box

I’ll let students play around with the dimensions for a while. Then I’ll share this video of me putting in the last cube.


Now students have two dimensions and know the total number of cubes.


How the sugar is packed

As a potential extension I might ask if this the most cost efficient way to box the 198 sugar cubes. Sure it dives into middle grade standards but it seems like a natural progression.

I doesn’t seem forced and that’s what I’m after.

Posted in 3-5, 3-Act Tasks, 6-8, Against the Norm, Cheese Mover, Intellectual Need, Measurement and Data, Modeling | 20 Comments

The Progression of Division

Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful.  Well, here’s my stab at the progression of division.  I understand that there’s lots of ways to model division and this is only ONE of them.

Understanding the vertical progression of mathematics is really important in the conceptual development of everyone’s understanding.

Hope it helps.

Stay thirsty my friends!

Posted in 3-5, 6-8, Against the Norm, Making Math Accessible, Making Sense Series, Math Progressions, Math Tools, Number Sense, Planning, Strategy Development, Teacher Content | Tagged , , , , | 35 Comments

The Post-it Holds the Mystery Number

There’s no denying the power of the Post-it.  Many brilliant minds within the #MTBoS have harnessed its’ power to help students make sense of mathematics and this week was no different.

Only this time we infused them into a Marilyn Burns must have…

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Coffee stains & post-its.  The signs of a great read.

To start we had students build animals from pattern blocks and had partners guess the animal.


A rabbit???

Some students were not sold on their partner’s creation which lead to some heated discussions.  Score a point for SMP3.

I showed my animal and we decided that it looked most like a dog.


My dog on DAY 1. The tan rhombus was the tail.

I built the dog to show what it looked like on day 2 and 3.


The document camera didn’t want to play so we took it to the floor.


Teacher tip from Mike Wiernicki: if you have foam pattern blocks you can wet them and they’ll stick to your dry erase board…unfortunately these were wood.


The first 3 days


Then we asked… What do you notice?



And we followed that up with, “What will the dog look like on day 4 & 5?”


Students began building the dog for days 4 and 5.


We asked students to organize their thinking in a table and this is what they collectively came up with:

Screen Shot 2016-01-26 at 2.10.14 PM.png

This was a great opportunity to pull out specific vocabulary with students:

  • Students said “The body, head, and feet never change.  They always have the same number of pieces.” The term CONSTANT was introduced.
  • They also noticed that “The dog’s tail grows one rhombus for every day it’s  alive.”  VARIABLE was introduced.

Screen Shot 2016-01-26 at 2.10.56 PM.png

We had students find the total number of pieces for the 11th and 21st day.

Student: Do we have to build it?!?  That will take forever.  What if we don’t have enough pieces?

Me: Can you use what information you already have to figure it out?


The table really helped students recognize vertical AND horizontal patterns which are typically harder to find.  We were really surprised by how quickly students found shortcuts to complete the table.

What they noticed…

  • The head and body column never change so they placed 4s all the way down.
  • The number that went in the tail column matched the number of days.
  • The total number of pieces goes up by one each day it grows. (recursive rule)
  • The total is always 4 more than the day. (explicit rule)

Share your thinking.

For the most part, students did an excellent job completing the table…when the days were identified.  But when we added a Post-it to the day column things went a little sideways.

Students: Hey! That’s not fair. We don’t know what day it is.

Me: I know.  It’s my mystery number.

Screen Shot 2016-01-26 at 2.10.27 PM.png

Big shout out to Mike for introducing me to the Post-It variable.

Student: We can’t finish it without knowing the day.

Me: So give up then!  I guess there’s no way for you to tell me how many pieces there will be on the mystery day.

The students began talking amongst each other because they knew something was up.

Student: Yes!!!  We CAN do it. We know that the feet, body, and head stay the same no matter what day it is.  So put a 4 in that column.

Me: Ok. What else do you know?

Student: Well since we know the tail and day always match, can we put another Post-it in the tail column and put a question mark on it just like the day column? I asked the class to check with their table and decide.  Each group came back and said that’s what they wanted.

Me: What about the total?

Student: That’s easy…to figure out the total we just add 4 to the mystery number.  So we just write 4 + a Post-it with a “?” mark.

Screen Shot 2016-01-26 at 2.10.42 PM.png

total pieces = 4 + day

Me: Oh that’s awesome.  What happens if I ran out of Post-its.  What could we use instead?

Student: Well you could always just draw a square and put a “P” in it for Post-it. It doesn’t matter if we have a “P” or a “?” on it because it’s still gonna be the mystery number.

I could have swore I heard math angels at this very moment.  It was one of those moments I’ll never forget.


Looking back at the lesson I would definitely say it went “much too whole group” towards the end.  I would have liked students to grapple with the idea of the mystery Post-it at their tables before discussing as a class.  I think I just got caught up in the excitement of the moment. (note to self)

I definitely wouldn’t say that we’re Visual Pattern experts but we have till 8th grade to get there.  I’m sure the early start makes Fawn happy.


Posted in 3-5, Algebra, Making Math Accessible, Number Sense, Strategy Development | Tagged , , | 16 Comments

Take it to the Bank

It’s never been a secret that Superman fears Kryptonite. But I think the same can be said for many 2nd grade teachers and the way they feel about teaching money.

There’s no way around it…teaching money to students is hard.  With money comes a level of abstraction that’s really difficult for students to wrap their heads around.

How can a dime be worth more than a nickel if it’s smaller? 

We knew the road ahead was bumpy, but we moved forward nonetheless. Only this time we decided to tackle money with some base-ten understanding up our sleeve.

We started with a quick brainstorming session:


Students identified coins and equivalent value combinations.

Then students were given a Hundreds Grid and asked to create “money” that represented the combinations they identified.  We hoped that by connecting money back to base-ten, future student thinking would be grounded in conceptual understanding.


The value of a quarter made with 2 tens (dimes) and a 5 (nickel).

Lots of mess taking place but also lots of concrete understanding…


Some students wrote the value of coins on the paper without prompting…we liked this!


The fruits of our labor.

After students had finished constructing the manipulative, we had them explore and model combinations for different values. How many different way can you represent 30 cents? 35 cents? 40 cents?

Knowing the paper base ten money wouldn’t stand a chance during the game, the wooden base ten blocks were introduced with coins and the connections continued. 


How many ways can we represent the value of a quarter?

Note: in the past I’ve introduced the wooden blocks without having students create and design the paper money upfront.  Having students create the money was a new wrinkle in the lesson. Our hope was that students would make a conceptual connection when decomposing money in the game.

As students explored the money they were able to make the connection and see how 4 quarters was equivalent to a dollar. (4 x 25 = 100)Screen Shot 2016-01-13 at 10.49.45 AM.png

Once their notice and wonders were fed it was time to build fluency through conceptual understanding. So we played “To the Bank“…

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Two students came to the carpet and the class created a fishbowl as they modeled the game.  From the onset, each student started with 4 quarters.  Player 1 rolled a 4 and needed to pay 15 cents to the bank.

Player 1: Mr. Fletcher I can’t go.

Me: Why not?

Player 1: I don’t have 15 cents to pay the bank. (Sitting with 4 quarters in his hand)

Me: Well then I guess the game is over. You lose. (player 1 sits stunned on the carpet)

Player 2: Mr. Fletcher, he has 15 cents but he can’t pay for it with a quarter.  He has to break apart the quarter to pay the bank.

Me: How can he do that? I don’t have extra money laying around.

Player 1: But he bank does! Can I give the bank this 25 and they give me back 2 tens and a five?

Bingo!  He modeled the fair trade, paid the bank, and they played through a couple more rolls and the class was free to play.

Receiving change is almost as abstract as money itself.  Because I’ve failed numerous times when tackling money I’ve come to find that having students explicitly conduct a “fair trade” before paying the bank works really well.


This seems like a lot of steps and to be completely honest it is!!!  But as students begin to construct their own understanding these steps are really important for students to internalize themselves.   Physically acting out the fair trade and talking through the steps helps students make the transition from concrete to abstract.  There’s a lot of math happening here and students need to see it.

It’s NOT efficient to have students work through this fair trade every time and thankfully, as time passed a couple of students began to find shortcuts. Not every student in the class needed to preform a fair trade to pay the bank but the majority did and that’s ok.


As an option, some students recorded what they spent and modeled it on a hundreds chart.


First player with no money left over won the game.

By the end of class there were 4 students that moved from base ten money to actual coins which was great to see.

Next time we play To the Bank with this class we’re going to start with coins only.  As students are playing we’ll reintroduce the blocks to those students that still have a difficult time understanding the fair trade/making change idea.

There was an alternative board that was introduced to a handful of players.  The board had greater amounts of money owed and students started with three $1 bills instead of 4 quarters.


All in all, it was a successful 75 minutes spent in 2nd grade and money isn’t feeling so scary around these parts.  We still have our work cut out for us but the math residue has been laid and we’ll take that understanding to the bank!

We’re just hoping the bank doesn’t get robbed before we return to the game.


Side note: Even if you don’t have wooden base ten blocks you can create money. Foam base ten blocks work as a nice alternative after some handy scissor work.


Base ten money made from a foam 100.



Posted in K-2, Making Math Accessible, Math Tools, Measurement and Data, Number Sense, Strategy Development, Teacher Content | 12 Comments