A Simpler Solution

I’m guessing this was about 5 years ago. I was at an all-day workshop when a high school math teacher, sitting next to me, asked about the PoW (from mathforum.org) that I assign to my students. I happened to have an extra copy in my backpack and gave it to her.

Dad’s Cookies [Problem #2959]

Dad bakes some cookies. He eats one hot out of the oven and leaves the rest on the counter to cool. He goes outside to read.

Dave comes into the kitchen and finds the cookies. Since he is hungry, he eats half a dozen of them.

Then Kate wanders by, feeling rather hungry as well. She eats half as many as Dave did.

Jim and Eileen walk through next, each of them eats one third of the remaining cookies.

Hollis comes into the kitchen and eats half of the cookies that are left on the counter.

Last of all, Mom eats just one cookie.

Dad comes back inside, ready to pig out. “Hey!” he exclaims, “There is only one cookie left!”

How many cookies did Dad bake in all?

Maybe you’d like to work on this problem before reading on.

The teacher started solving the problem. She was really into it, so much so that I felt she’d ignored much of what our presenter was presenting at the time. She ran out of paper and grabbed some more. She looked up from her papers at one point and said something that I interpreted as I-know-this-problem-is-not-that-hard-but-what-the-fuck.

It was now morning break.


She worked on it some more.


By lunch time, she asked, “Okay, how do you solve this?” I read the problem again and drew some boxes on top of the paper that she’d written on. (Inside the green.)


She knew I’d solved the problem with a few simple sketches because she understood the drawings and what they represented. I just really appreciated her perseverance.

I share this with you because a few nights ago I was at our local Math Teachers’ Circle where Joshua Zucker led us through some fantastic activities with Zome models. We were asked for the volume of various polyhedrons relative to one another. Our group really struggled on one of the shapes. We used formulas and equations only to get completely befuddled, and our work ended up looking like one of the papers above.


Over the years I’ve heard a few students tell me, “Mrs. Nguyen, my uncle is an engineer, and he can’t help me with the PoW.” Substitute uncle with another grown-up family member. Substitute engineer with another profession, including math teacher. I remember getting a note from one of my student’s tutor letting me know that I shouldn’t be giving 6th graders problems that he himself cannot solve. (The student’s parent fired him upon learning this.)

I like to think that my love of problem solving will rub off on my kids. I hope they will love the power of drawing rectangles as much I do. Or just a tiny little bit.

Posted in Problem Solving | Tagged , , , , , | 2 Responses

Multiplication: Finding the Greatest Product

From a set of 1 through 9 playing cards, I draw five cards and get cards showing 8, 4, 2, 7, and 5. I ask my 6th graders to make a 3-digit number and a 2-digit number that would yield the greatest product. I add, “But do not complete the multiplication — meaning do not figure out the answer. I just want you to think about place value and multiplication.”


I ask for volunteers who feel confident about their two numbers to share. This question brings out more than a few confident thinkers — each was so confident that he/she had the greatest product. (I’m noting here that I wasn’t entirely sure what what the largest product would be. After this lesson, I asked some math teachers this question, and I appreciate the three teachers who shared. None of them gave the correct answer.)


I say, “Well, this is quite lovely, but y’all can’t be right.” I ask everyone to look at the seven “confident” submissions and see if they could reason that one yields a greater product than another, then perhaps we might narrow this list down a bit.

Someone sees “easily” that #7 is greater than #6. The class agrees.


Someone says #7 is greater than #1 because of “doubling.” She says, “I know this from our math talk. Doubling and halving. Look at #1. If I take half of 875, I get about 430. If I double 42, I get 84. Both of these numbers [430 and 84] are smaller than what are in #7. So I’m confident #7 is greater than #1.”


Someone else says #5 is greater than #4 because of rounding, “Eight hundred something times 70 is greater than eight hundred something times 50. The effect of multiplying by 800 is much more.”


Someone says, “Number 2 is also greater than #1 because of place value. I mean the top numbers are almost the same, but #2 has twelve more groups of 872.”


But the only one that the class unanimously agrees on to eliminate is #6. Then I ask them to take 30 seconds to quietly examine the remaining six and put a star next to the one that they believe yield the greatest product. These are their votes.

7I tell them that clearly this is a tough thing to think about because we’ve had a lot of discussion yet many possibilities still remain. And that’s okay — that’s why we’re doing this. We’ve been doing enough multiplication of 2-digit by 2-digit during math talks that it’s time we tackle something more challenging. So #3 gets the most votes.

I then punch the numbers into the calculator, and the kids are very excited to see what comes up after each time that I hit the ENTER key. Cheers and groans can be heard from around the room. Turns out #3 does has the greatest product (63,150) out of the ones shown.

Ah, but then someone suggests 752 times 84. I punch it into the calculator and everyone gasps. It has a product of 63,168.

Their little heads are exploding.

I give them a new set of five for homework: 2, 3, 5, 6, and 9. They are to go home and figure out the largest product from 3-digit by 2-digit multiplication. They come back with 652 times 93.

The next day, we try another set: 3, 4, 5, 8, and 9. We get the greatest product by doing 853 times 94. There is a lot — as much if not more than the day before — of sharing and arguing and reasoning about multiplication and place value.

Many of them see a pattern in the arrangement of the digits and are eager to share. They’ve agreed on this placement.


Then we talk about making sure we know we’ve looked at all the possible configurations. They agree that the greatest digit has to either be in the hundreds place of the 3-digit number or in the tens place of the 2-digit number. We try a simple set of numbers 1 through 5, and we agree that there are just 9 possible candidates that we need to test. The same placement holds.


Then we draw generic rectangles to remind us that we’ve just been looking for two dimensions that would give us the largest area.


I remember saying to the class, more than once, that this is tough to think about. To which Harley, sitting in the front row, says, “But it’s like we’re playing a game. It’s fun.” Oh, okay. :)

Posted in Course 1 (6th Grade Math) | Tagged , , , , , , | 3 Responses

Two Pizzas and Five People

I’m thinking a lot about how my 6th graders responded to a pre-lesson task in “Interpreting Multiplication and Division” — a lesson from Mathematics Assessment Project (MAP) .

MAP lessons begin with a set structure:

Before the lesson, students work individually on a task designed to reveal their current levels of understanding. You review their scripts and write questions to help them improve their work.

I gave the students this pre-lesson task for homework, and 57 students completed the task.


I’m sharing students’ responses to question 2 (of 4) only because there’s already a lot here to process. I’m grouping the kids’ calculations and answers based on their diagrams.

Each pizza is cut into fifths.

About 44% (25/57) of the kids split the pizzas into fifths. I think I would have done the same, and my hand-drawn fifths would only be slightly less sloppy than theirs. The answer of 2 must mean 2 slices, and that makes sense when we see 10 slices total. The answer 10 might be a reflection of the completed example in the first row. “P divided by 5 x 2 or 5 divided by P x 2″ suggests that division is commutative, and P here must mean pizza.

 Each pizza is cut into eighths.



Next to cutting a circle into fourths, cutting into eighths is pretty easy and straightforward.

 Each pizza is cut into tenths.



I’m a little bit surprised to see tenths because it’s tedious to sketch them in, but then ten is a nice round number. The answer 20, like before, might be a mimic of the completed example in the first row.

 Each pizza is cut into fourths.



I’m thinking the student sketched the diagram to illustrate that the pizzas get cut into some number of pieces — the fourths are out of convenience.  The larger number 5 divided by the smaller number 2 is not surprising.

 Each pizza is cut into sixths.



It’s easier to divide a circle by hand into even sections, even though the calculations do not show 6 or 12.

 Each pizza is cut into fifths, vertically.



Oy. I need to introduce these 3 students to rectangular pizzas. :)

 Five people? Here, five slices.



Mom and Dad are bigger people, so they should get the larger slices. This seems fair. We just need to examine the commutative property more closely.

 Circles drawn, but uncut.



I’m wondering about the calculation of 5 divided by 2.

 Only one pizza drawn, cut into fifths.



Twenty percent fits with the diagram, if each person is getting one of the five slices. The 100 in the calculation might be due to the student thinking about percentage.

 Only one pizza drawn, cut into tenths, but like this.



I wonder if the student has forgotten what the question is asking for because his/her focus has now shifted to the diagram.

 Each rectangular pizza is cut into fifths.



Three kids after my own heart.

Five portions set out, each with pizza sticks.


I wonder where the 10 comes from in his calculation.

Five plates set out, each plate with pizza slices.



Kids don’t always know what we mean by “draw a picture” or “sketch a diagram.” This student has already portioned out the slices.

What diagrams and calculations would you expect to see for question 3?


There’s important work ahead for us. The kids have been working on matching calculation, diagram, and problem cards. They’re thinking and talking to one another. I have a lot of questions to ask them, and hopefully they’ll come up with questions of their own as they try to make sense of it all. If I were just looking for the answer of 2/5 or 0.4, then only 12 of the 57 papers had this answer. But I saw more “correct” answers that may not necessarily match the key. We starve ourselves of kids’ thinking and reasoning if we only give multiple-choice tests or seek only for the answer.

That’s why Max Ray wants to remind us of why 2 > 4.

Posted in Course 1 (6th Grade Math) | Tagged , , , , , , , , , | 5 Responses


We currently have 145 patterns on VisualPatterns.org. I’ve updated the key — intentionally did not use equation editor as some folks are not able to read it. I color coded it for quicker reference to the function types.

Download: Equations KEY. I’m going to leave the key out in the open for now. What students would visit this space.

I hope you’ll let me buy you a beer or two the next time we meet if you point to an equation mistake I’ve made. I’ll drink it all for you if you don’t like beer and get you bratwurst instead. Guess that’ll be the title of this post. You have to admit it’s better than “The Answer Key.”


Posted in General | Tagged , | 12 Responses


My old math talks site needed to die. It was riddled with missing image icons and plagued by shitty formatting. I had to fix the main blog and do other things, so this could wait. Only 2 or 3 people in the universe were checking it out anyway.

Although traffic to math talks is low to nil, I’m really fond of it. It’s the most important one to me because it’s a collection of my students’ voices, their reasoning, their thinking, their growth —

So the old broken site betrayed how really proud I am of my kids and their mathematical sharing. I just built a new space for it — dot com and dot org were both taken, so I got mathtalks.net.

You see I wanted to write down what the kids were sharing during our number talks and pattern talks. To not write down what they say would be complete disrespect and pointless. I asked everyone in the class to do the same — we scribed to show respect to and to learn from one another.

Then I realized I couldn’t toss the papers that I’d written on into the recycling bin. Not until I recorded the notes somewhere — hence I started the blog for math talks.

Then as I was typing up what the kids had shared for pattern talks, I felt I needed to include a visual too to record their thinking because that was exactly what we did in class. They were circling this piece and that piece of the pattern; they were rearranging the pieces. I wanted to tell as complete a story as I could, so I did the only thing I knew — I drew on the pattern in Paint and attached it to every voice/thinking that described it. (Unfortunately I lost many of these images.)


This site — this small endeavor — was a personal need to record. I still see Daniel’s proud face as he tries to explain where “4n + 3″ comes from. I still hear the excitement in Audrey’s voice as she shares a different way of seeing the pattern. I still hear kids comparing their algebraic expressions and arguing. I still see Blanca shaking her head, having a tough time seeing the general term. I choke up seeing how far Natalie has come along since we started math talks.

So selfishly this site was for me all along.

Posted in General, Teaching | Tagged , , , , , , | 9 Responses

VisualPatterns.org Again

Who needs sleep. It’s been an exciting past couple of months sitting at my desk rebuilding blogs and websites. The only thing more stimulating would be to poke needles in my eyes. I re-created visualpatterns.org so you can now copy and/or save the patterns. It looks pretty much the same otherwise.

Once again, I want to mention all these good people who have contributed the patterns.

  • David Wees
  • John Golden
  • Kate Nowak
  • Sarah Strong
  • Katie N.
  • Avery Pickford
  • Justin Lanier*
  • Chris Hunter*
  • Simon Gregg
  • Jonathan Newman
  • Henri Picciotto*
  • Don Steward
  • Mary Dooms
  • Nik Doran
  • Chris Robinson
  • Mimi Yang
  • Robert Kaplinsky
  • Austin Otero Rodriguez
  • Michael Pershan
  • Megan Schmidt
  • Mike Lawler
  • Matt Owen
  • Jeffrey Hart
  • Matt Vaudrey
  • Dean Adalian
  • Elaine Watson
  • Kasey Clark
  • Math Curmudgeon

*These folks didn’t really voluntarily contribute, I just stole the pattern off their site.

I’m about 40 patterns behind in updating the answer key. If you had requested and received a key from me via email, then I have a copy of your email and will be sure to send you an updated key.

Please share Visual Patterns often and early. Thank you!!

Posted in General | Tagged , , | 2 Responses

First Two Days of School

Tomorrow is the day. I’m excited to meet my new Math 6 students — all 71 of them, two classes of 36 and 35. I should know about half of the 33 Math 8 kids because I taught half of them in 6th grade. Then I get a big break in class size with just 9 students in Geometry. (We’re phasing out tracking, so we do right be these kids and we’re done.)

I have the first two days planned. My mind thinks in terms of conversations, so it’s just easier for me to write in this format.

Day 1

Math 6 (periods 3 and 5)

Hello everyone! Welcome to 6th grade math! I’m Mrs. Win — spelled N-G-U-Y-E-N — I’d be thrilled if you could spell my name correctly because I’ll do the same with your name.

Whatever seat you’re in right now is fine. We’ll have a seating chart by the end of the week; it’s mainly so I can quickly learn your name. Okay, fire drill. It’s the most important procedure I have for first day because you never know, so listen up.

I tell them about the escape route and something about not panicking. Right.

Now I need to take attendance. If I mispronounce your name, I apologize in advance and please correct me. Also, if you go by an entirely different name, please let me know. I once had a student named Anne Marie — her real name — but she went by Bob. Yup, Bob. Reply with a yes or hereNo grunting.

Oh, how many of you have older siblings who had me as their teacher? Yeah? Did they say that I’m really mean? Well, your sister is a liar.

Enough chit chat. Let’s do some math. Please take out a piece of notebook paper. Name and date in upper right hand corner, same as what you’ve been doing for the last 25 years. Write Pattern #2 at the top of your paper.

I show them pattern #2.

You’re seeing the first 4 steps of this pattern. Please draw what you think the next step, step 5, might look like. It doesn’t have to be pretty — just something to show that you know how the pattern is growing.

This might very well be the first time these kids work with a pattern like this. I don’t know. But I’ll guide them through.

Well, what we just did is a warm-up. As we do more, and you get a hang of it, it’ll go faster. By the end of this week, we’ll skip drawing the next step and go right into trying to find the equation. We do a different type of warm-up each day, so on Mondays we do a visual pattern. I’ll tell you what we do on Tuesdays when Tuesday gets here. You’ll need to do tomorrow’s warm-up on that same paper, so let me see you put it away in your binder where you can quickly retrieve it tomorrow. Do it now.

How’s your first day so far?… Good to know. Okay, that’s enough sharing.

I’m passing out a puzzle called Noah’s Ark. I’m willing to bet that in five minutes, it’ll become your favorite puzzle ever. Please follow along as I read it aloud.

Please quietly read it again on your own…

I then ask a few questions to make sure they understand the problem and what it is they are trying to solve.

Okay. I’m going to give you some time to work on this problem quietly by yourself, I think 10 minutes. I’ll set the timer. When the timer goes off, I’ll put you in random small groups of 3 to continue working on the problem. My sincere hope is that I’ve picked the right problem, meaning one that you understand what is given and what is being asked of you, but that it makes you struggle. Oh my God, you have no idea how much I love for you to struggle in this class! You’ll hear me say that word a lot — struggle. It’s all good. You know how after a good workout, your body feels kinda sore? Well, I want your mind to be sore like that.

AND OH!!! This is really really important. Rule #1 for you in my class: DO NOT TELL AN ANSWER. Meaning when you think you have an answer to a problem, please don’t just blurt it out. I know I completely shut down and don’t care to work on the problem any more when somebody does that. So, please, keep your wrong answer to yourself.

I suspect our 55-minute period would end before they even get into small groups. Depends on how long our warm-up takes. (I don’t care if the warm-up takes up the whole period. Initial tasks that are part of the curriculum and norm-building cannot be rushed.)

About 2 minutes before the bell rings…

What’s on this quarter sheet of paper is instruction on how you and your parents can subscribe to the texts that I send out. It’s mostly to remind you of homework. You can only receive texts from me, you cannot reply. You can also receive my “text” message as an email instead. Show it to your parents.

Sadly, there is no homework tonight for math. But you’ll get about 3 hours of math homework tomorrow night, so clear your calendar.  Of course, you’re welcome to continue to work on the Ark problem, but you don’t have to. Remember not to share the answer!

When the bell rings, I need you to be seated and quiet. Please pick up any trash around you and toss it into the trashcan by the door on your way out. Thank you. Have a great day and I’ll see you tomorrow.

Math 8 (period 2)

Having taught half of these kids already in 6th grade will make it easier for me to call them by their name. Same as Math 6, we’ll start immediately with a visual pattern, but pattern #1 for these guys.

Then our math problem is the classic The Proof is in the Pudding.

Geometry (period 1)

I just taught all 9 of these kids last year in Algebra 1. Also, pattern #1. So sure we did this pattern already last year, let’s see what they remember.

They get Using Fibonacci Numbers. I wrote a short blurb here.

Day 2

Math Talks will be our warm-up for Tuesdays. I still need to figure out which of these to ask first — and create a thoughtful sequence. Since I’m only doing one a week, all I need are 40 really good ones.

Then we’ll continue with the problems from wherever we’d left off. I imagine we’ll be in small groups and then on to whole-class sharing and reflecting.

If we have time, I’ll go over everything they need to know about the class — all on one page or I slit my wrist because these are really dry.

image of geometry class


Math 6 and Math 8

So, I think after two days we’ll get these 3 procedures/routines sorta established:

  1. fire drill
  2. warm-up routine
  3. dismissal bell

And I got 1 rule out of the way, even though I’ll remind them of it each and every time we do problem solving:

  1. never tell an answer

Kids forget things you tell them anyway, so I figure it’s better to tell them in context when they need to be doing the stuff.

I stopped having parents sign my rules/procedures handout. It’s not that sacred.

I’ll probably assign textbooks by the end of the week and give them their individual access codes to the online textbook and resources.

I’ve already assigned a Math Forum PoW to each class and will need to give each kid a log in ID and password (it’s rather generic — no one is getting a handout for anything — it’s just “your first initial plus your… plus… “) to allow students access for online submission of their solution.

I’ll update should we end up doing something entirely different like assembling Estes rockets and dissecting fetal pigs. (These were highlights from my science teaching days.)

Don’t forget to tell kids how amazing you are.

Posted in Course 1 (6th Grade Math), Geometry, Math 8 | Tagged , , , , , , | 11 Responses

The Number Sense by Stanislas Dehaene

I heavily skimmed the middle parts of this book when I bought it in May 2012. [Thanks to Christopher Danielson for recommending it.]

photo (9)

I’m re-reading some parts now, and the sub-section Teaching Number Sense [pages 124 - 128] resonates with me, not just in elementary school mathematics, but in K-12 mathematics.

If my hypothesis is correct, innumeracy is with us for a long time, because it reflects one of the fundamental properties of our brain: its modularity, the compartmentalization of mathematical knowledge within multiple partially autonomous circuits… The numerical illiterate performs calculations by reflex, haphazardly and without any deep understanding.

… A good teacher is an alchemist who gives a fundamentally modular human brain the semblance of an interactive network. Unfortunately our schools often do not quite meet this challenge. All too often, far from smoothing out the difficulties raised by mental calculation, our educational system increases them… But our schools are often content with inculcating meaningless and mechanical arithmetical recipes into children.

This state of affairs is all the more regrettable because… most children enter preschool with a well-developed understanding of approximation and counting. In most math courses, this informal baggage is treated as a handicap rather than as an asset. Finger counting is considered a childish activity that a good education will quickly do away with. How many children try to hide when they count on their fingers because “the teacher said not to”?

Despising children’s precocious abilities can have a disastrous effect on their subsequent opinion of mathematics.

… It seems more likely that many of these “mathematically disabled” children are normally abled pupils who got off to a false start in mathematics. Their initial experience unfortunately convinces them that arithmetic is a purely scholastic affair, with no practical goal and no obvious meaning. They rapidly decide that they will never be able to understand a word about it. The already considerable difficulties posed by arithmetic to any normally constituted brain are thus compounded by an emotional component, a growing anxiety or phobia about mathematics.

… We need to help children realize that mathematical operations have an intuitive meaning, which they can represent using their innate sense of numerical quantities. In brief, we must help them build a rich repertoire of “mental models” of arithmetic… The day the teacher introduces negative numbers and asks pupils to compute 3 – 9, a child who only masters the set scheme judges this operation impossible. Taking 9 apples from 3 apples? That’s absurd! Another child who relies exclusively on the distance scheme concludes that 3 – 9 = 6, because indeed the distance from 3 to 9 is 6. If the teacher merely maintains that 3 – 9 equals “minus six,” the two children run the risk of failing to understand the statement. The temperature scheme, however, can provide them with an intuitive picture of negative numbers. Minus six degrees is a concept that even first-graders can grasp.

But let us leave this chapter with a note of optimism… In the United States, the national council of teachers of mathematics is now de-emphasizing the rote learning of facts and procedures and is focusing instead on teaching an intuitive familiarity with numbers… Number sense — indeed, common sense — is making a comeback.

In fact, most children are only too pleased to learn mathematics if only one shows them the playful aspects before the abstract symbolism. Playing snakes and ladders may be all children need to get a head start in arithmetic.

By the way, I don’t read this and put the blame entirely on myself and my colleagues. As long as we place more emphasis on test scores than we do on learning, we are at best hypocrites.

Posted in Teaching | Tagged , , , , , | 1 Response


The word privilege is being spoken and written many times over. Within the past couple of years I’m seeing and hearing privilege used in an undeniably distinct context. (Or it could be that I was unknowingly and partially deaf and blind to this context.) The word is written on cardboard signs, on people’s faces — and no matter what surface it’s on, the impact it has seems unbearable to the canvas that holds it.

I was organizing my classroom yesterday, putting away new supplies, tossing out items that I’d kept for too long. I have enough paperclips to last me two more lifetimes. Elmer’s glue bottles and glue sticks fill up an entire shelf. Same with staples and pattern blocks. And why would I get mad at a kid for not having his pencil — I have a shitload of pencils. I remember feeling a vague sense of shame of not having certain school supplies when I was in grade school. I remember mashing up rice to use as glue.

Kevin was a black teacher-turned administrator at my former school. When his second daughter was born (maybe 15 years ago), he said, “I thank God I have daughters. It’s hard for a black boy to grow up in this country.”

My husband is white. It never occurred to me how white he is until we were walking the streets of southern Vietnam. People looked at us (much more at him than at me) with foreign expressions. I felt safe with him by my side. Although I had no reason to not feel safe. I was back in my homeland — unknown to everyone around me — yet I was thankful to have a personal bodyguard because Hey, I’m with the big white dude.

Graham Smith, age 11. Me, age 11. He said to me, “Go back to your country.” I actually didn’t understand what he said, my Vietnamese girlfriend provided the translation. His expression matched what she said. I’m terrible with remembering even just first names. But I remember Graham Smith.

Michael missed the bus and had no other ride home. I went to the office to see if I may have permission to take Michael home. My vice-principal, Mr. M, reminded me, “Fawn, he just threatened you last week! And no, you may not transport a student.” (Right, when I sent Michael out of my classroom, he said he wished he had a gun.) I said something like, “I don’t think he has a gun on him though. C’mon, I’ll sign whatever papers. The kid needs a ride home.” Realizing that I was ignorant of the teacher handbook, Mr. M got up from behind his desk and approached me, close enough so he could whisper, “A young Asian teacher should not take a black kid home.” I never thought that statement appeared in the teacher handbook, but what struck me was Mr. M, himself a black man, was saying this.

One year the housing committee at my college decided to move our entire floor of student residents to a different building on the other side of campus because it needed our floor for the football players to move in. Upset, I went to the school’s newspaper in hoping they’d give us a louder voice of protest. Near the end of our conversation, the interviewer said to me, “You are very beautiful. For a Vietnamese.”

Poor. Refugee. Gook. Boat people. Foreigner. Young. Asian. Vietnamese.

It’s been a quiet storm for me.

It’s been a violent storm for others.

It was a fatal storm for Michael Brown.

I close my eyes and take your hand. We ride this storm together, and this shall be my privilege.

Posted in Shallow Thoughts | Tagged | 1 Response