Solving an Equation with a Fraction

From CPM:

The Sutton family took a trip to see the mountains in Rocky Mountain National Park.  Linda and her brother, Lee, kept asking, “Are we there yet?”  At one point, their mother answered, “No, but what I can tell you is that we have driven 100 miles and we are about 2/5 of the way there.”

Linda turned to Lee and asked, “How long is this trip, anyway?”  They each started thinking about whether they could determine the length of the trip from the information they were given.

And I like both methods, especially Linda’s.

Without using a visual, we may have students solve for x in the equation (2/5)(x) = 100 by multiplying both sides by 5/2.

But I notice two things: 1) Students don’t always remember why they are multiplying by the reciprocal, and 2) Students have difficulty showing Linda’s method with an equation like (9/2)(x) = 27.

So, I’m having the students think through the problem by answering these two questions:

  1. If we know that nine halves of x is 27, then what is one half of x?
  2. Now that we know what one half of x is, what is a whole x?

As we write the fractions, we can keep our focus on the whole number numerator and treat the denominator as if it were a thing, and that thing is not changing.

Another example,

This helps us go back to finding the unit rate in the first step via division, and then find a multiple of that unit rate via multiplication.

Once students make sense of these two steps and become fluent in solving for a whole x, then they can work on the not-so-friendly equations — such as (5/6)(x) = 4 — because they are more confident and trust the process.

Sure, multiplying by the reciprocal would have solved for x in one step, but there’s something uniquely comforting to students when they can first find just one part of something.

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


Lillian naturally comes to mind when I plan a lesson. She eats it up. She goes above and beyond. She’s thoughtful and appreciative. She shares her thinking with the class in mindful and modest doses. She smiles quietly at my jokes. I wish I had a copy of all her math work — especially her written reflections — because each and every piece holds the joy of my teaching. Here’s one:

At the Continuation ceremony last year, Lillian delivered a succinct and grateful valedictorian speech.

A month ago, on March 11, I got an email from her.

I was looking at old pictures on your Twitter and in my camera roll, and I could totally see how much I loved your class. I was tearing up. I’m moving up to Math 3 Honors next year, yet I’m not sure I’ll ever be as excited about math as I was in your class. My current class is something of speed and prior knowledge… Not my favorite environment for growth, but you live and you learn to deal with it.

To this day, I remember so many little things about your classes. You truly changed the way I saw the world. I think my intense activism and political vocalness is in part your doing. I use my voice because you gave me one. I’m not a shy little sixth grader anymore. I’m beginning to come into my own as a badass bisexual intersectional feminist. I’m learning, and you pushed me to do so. There’s a lot of work for me to do on myself and the world around me. Maybe my first pattern equation wasn’t so far away (You told me “just because her equation is right, yours isn’t any less right”).

I miss being her teacher. I miss watching her persevere and hearing her explain her thinking in number talks.

Then, last week on April 7, late in the evening, I saw this video of Lillian posted on Twitter by her friend, Sam. I asked Sam for a copy and got Lillian’s permission to share it here.

I cried hard. Not because her poem is eloquent and powerful and makes me so goddamn proud, but because her message is all too real and urgent. The expectations placed on students by parents and teachers — on top of self-expectations — can be and are enormous.

We talk a good talk — about respecting the child and letting her learn at the speed of learning, about persevering and playing with mathematics, about nourishing critical and deep thinking in problem-solving, about ensuring access and equity, about cultivating a voice grounded in truth and heart.

But I’ll be the first to admit that I don’t always walk the walk. I’m bound to a system that requires me to issue a grade at the end of the quarter. I have to do this for each child four times a year. Because that’s just how it is.

At what cost?

Posted in Teaching | Tagged , , , , , | 17 Responses

Be Brave or Be Desperate

I had lunch with a former colleague — I”ll call her Laura — whom I hadn’t seen for some 20 plus years. She’s now retired and was in my town for a wedding.

I got into my car, entered the restaurant’s address into my phone, Google Maps said I’d arrive at 12:05 PM, and I was upset for thinking I lived closer. I texted Laura to let her know I’d be 6 minutes late. She texted back, “No worries!” (I hate being late, it’s rude and arrogant.)

I instantly recognized her. Of course, she was wearing an Oregon Ducks sweatshirt. We hugged, and the waitress showed us to an empty booth. Laura reminded me that she still needed to get a pair of TOMS after lunch because of the blisters she got from walking all day yesterday in her new shoes. I then reminded her that 20 years ago, we were in Nordstrom for her to buy new underwear because she was too lazy to do laundry.

She handed me two gifts wrapped in The Sunday Oregonian COMICS — one dated November 8, 2015, the other July 3, 2016.

We both regretted that neither one us thought of the Dammit! Doll. I mean, Jesus, just look at its mishappened head and scraggly yarn hair. I could have made that.

I really wanted to order a thick juicy burger because this place could put together a great thick juicy burger. But Laura said she wanted to order something healthy, so I opted for the turkey sandwich instead. (Who goes out and orders a turkey sandwich when it’s readily available in your own fridge at home?!) Then Laura ordered, lo and behold, a goddamn burger with two strips of bacon! (For a split second, I wanted to tell her that there was a burning car outside right behind her, so when she turned to look, I could steal her bacon.)

When we were colleagues in Oregon, Laura was teaching math, and I was teaching science. We became friends on Facebook just this past year, and she learned from there that I’ve been teaching math and giving talks at conferences. She used the word “brave” to describe my speaking at conferences. She said it at least three times, “You’re so brave.”

I told her I was terrified each and every time that I accepted a keynote or featured speaker assignment. She looked puzzled. I told her that the honor of being invited was always bigger than who I was, so it was hard to say no. And I didn’t want to say no. My father, a math teacher his entire career, would want me to accept. I’d told my own three children that doing the easy stuff ain’t worth their time, so I accepted because I could hear my own voice preaching. I accepted the invitation to speak because I wanted to bring the voice of classroom teachers and students to the forefront. There are stories to be told, and they are fresh and alive.

I’m not brave, I’m desperate. I’m desperate in wanting to share what’s happening right now with the 100 students on my rosters. But I’m terrified that I might get their stories wrong. I’m terrified that I may inadvertently amplify our small successes and diminish our big failures. I must get it right — the-truth-and-nothing-but-the-truth-so-help-me-God kind of right — or I’ll die a miserable death. Back in December, at CMC-North, Dan Meyer had invited me and two other teachers, Shira and Juana, to be part of his keynote. I’m grateful to Dan for the invitation and was excited that we teachers got to share with a wider audience.

So, if you’re in the classroom, I hope you’ll consider speaking at conferences and workshops because the number of students you currently have is the number of reasons you have to say YES. Consider co-presenting with someone if it would be your first time; it is less scary that way. My first gig was with this tall white dude. Be brave, or be desperate, I think one is just a glorified form of the other.

Posted in Shallow Thoughts, Teaching | Tagged | 6 Responses

These Twenty Things

I get to wake up without an alarm clock going off for the next 14 days. I will eat leftover grilled ribeye steak for breakfast and Cheerios for lunch. I will drink IPA for dinner and go to my mailbox in my pajamas. I will take a luxurious bath. 

As we wrap up 2016, I’d like to humbly share these 20 things that I have done — or will/want to do — and suggest that you too may want to do some of these things as a human and as a teacher.

  1. Find a reason to make caramelized onions. You can add it to your favorite pasta sauce or mashed potatoes.
  2. Call a parent to let her know how much you appreciate having her kid in your class. Maybe the kid is struggling in your class, but nonetheless, he is kind and laughs at all your jokes.
  3. Listen to country music to realize that your pain ain’t so bad after all — not the country from Garth Brooks and Tim McGraw or any of them pretty boys — I mean outlaw country music from David Allan Coe and Merle Haggard and Willie Nelson.
  4. Lie to your students that they were always on your mind during winter break, then let them hear Willie Nelson’s Always on My Mind.
  5. Watch La Maison en Petits Cubes by Kunio Katō. It won the Academy Award for Best Animated Short Film in 2008.
  6. Make hot chocolate for each kid in your favorite class. Seriously. (Point out to your non-favorite classes that they’d done a poor job in sucking up to you, hence going forward, they ought to try harder.)
  7. Stop consuming products with the label “lite” on it. Sure, it might mean 1/3 fewer calories and 1/2 less fat, but did you know it also means 1/10 of the taste?!
  8. Ask your students, “Did you know that diarrhea is genetic?” Let them ponder that for a few seconds, then say, “Yeah, it runs in your jeans.”
  9. Buy the latest book from #MTBoS: Tracy, Christopher, Mike, John and Matt, Edmund, Malke. (I’m sure I’m missing some people. Please help me out.)
  10. Treat the entire 180 days of school as flu season, spray bleach on everything in your classroom. Avoid the students’ eyeballs.
  11. Finish reading The Sound of Gravel. (For God’s sake, make time to read a non-nerdy book!)
  12. Lie — yes, again! — to your students that you’d graded all their papers over winter break. Then know that you’re fucked and must skip dinner [and life] to grade papers like a squirrel on crack that evening.
  13. Make something from scratch that you’ve never made before, like a baguette. If it comes out looking and tasting like shit, toss it immediately and buy frozen. (Ashli‘s number will be on speed dial as I attempt this.)
  14. Remind students that kindness trumps everything you do in your classroom.
  15. Be kind to yourself. Buy that item you didn’t get for Christmas from your favorite person who is now no longer your favorite. If you sleep next to this person, scream, “I hate you!” in the middle of the night like you are dreaming, except you aren’t.
  16. Connect with your students. Stand up for them. Speak up for them. Difficult decisions aren’t so difficult when we all put children first.
  17. Go to church, go to counseling, go to a friend. Reach out to someone because talking about stuff helps. Writing stuff down helps too. But it’s best to meet up with that person because a good hug is worth the drive.
  18. You are part of a team. Find the rest of your team and collaborate and share strategies and seek solutions. Leave the whiners and downers in the teachers’ lounge.
  19. Let’s not make a list of New Year’s resolutions. It’s like the goddamn pacing guide, sets us up for failure every time. Just repeat #15 above — minus the psycho screaming part, do that just once. Okay, twice. Definitely not more than three times.
  20. Critique the effectiveness of your lesson, not by what answers students give, but by what questions they ask.
Posted in General, Shallow Thoughts, Teaching | Tagged | 20 Responses

Quality Question Metrics

If I drive 60 miles per hour, my journey will take 4 hours. How long will my journey take if I drive 80 miles per hour?

Paulina volunteered, “I did sixty divided by eighty, that equals point seven five, or three-fourths. So, it would take three hours.”

When I asked Paulina why she divided 60 by 80, or what the quotient 0.75 meant, she struggled to tell me her reason. Nor could she explain how she deduced that “three-fourths… so, it would take three hours.”

We have to keep asking why-why-why all the time. Our job is to help students ask better questions. One of my question quality metrics that gets high marks is if a student can ask a question that causes the class to say Oh-shit-I-did-not-think-of-that! 

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Happy Thanksgiving!

On Sunday, I wrote a longer-than-usual email to my siblings about my intentions to begin gathering facts and etching memories for a bucket-list item of writing a book. I told them it could take anywhere between 3 to 10 years, meaning I have no clue.

I have three reasons to write this book: Nicolai, Gabriel, Sabrina.

My sister, Kimzie, replied:


I never forget how I got here, but being reminded of how I survived makes me eternally grateful for my children.

Happy Thanksgiving, everyone.

Posted in Shallow Thoughts | Tagged , , | 6 Responses

Good-Enough-for-Now Curriculum

I did my first webinar last week as a precursor to my talk at NCTM’s Innov8 Conference next month. I thought it went okay — or horribly — just tough to be the only person with the mic and not being able to actually see the attendees. It was weird.

There are a few slides from the webinar that I’d like to share here mainly because I’m still thinking about them and writing anything down helps me set the wobbly gelatin.

Two weeks ago I presented at an independent school that’s Preschool through Grade 8. Afterward, I was given a quick tour of the school — the 33-acre campus gleamed with pride in its thoughtful architecture, manicured grounds, state-of-the-art this and that, and a smorgasbord of elective offerings, including Mandarin and photography.

My school is Kindergarten through Grade 8, and the similarity between my school and this independent school pretty much ends there. I teach four classes, my smallest class has 23 8th graders, the other three, all 6th graders, have 32, 35, and 36 students.We’re a Title 1 public school.

I bring up the private school and my public school because, like apples and pomegranates, they are quite different. So, when we do PD and share whatever it is that we share about education and serving children, we need to be mindful about the space that each teacher occupies in her building and be mindful about the children who come into that space.

When someone shares something with me, one or more of these thoughts cross my mind: 1) I can see how that would work with my students, 2) I can see how I might adapt this to fit my kids, 3) This person is afraid of children or unaware that children are people, 4) Nobody cares.

Likewise, when I have the stage to share, I’m assuming you have similar thoughts of my work. But I beg you to think about the space that I share with my students.

Below is a quasi rating scale of “critical thinking demand” that I’d created to place the types of tasks that I regularly give to my students. And this scale is only possible because I’m mindful of the tasks’ contents and my own pedagogical content knowledge to facilitate these tasks.

What are these six things?

 1 & 2.  Assessment and Textbook

We’re using CPM

3.  Warm-up

Due to our new block schedule, we’ve only been doing number talks and visual patternsI’ve used and would recommend any and all of the sites below for warm-up.
open middle
fraction talks
would you rather
math mistakes
math arguments

4.  PoW

Problem of the Week, mostly from

5.  Task

My go-to resources:

Illustrative Mathematics

6.  PS (Problem Solving)

I’m secretly working on starting a math circle for young students in my county, like the ones they have at Stanford. I just need funding, time, and people. Yes, one of those secret plans that will never happen.

I get the PS from:

Math Teachers’ Circle — (I use problems that I’ve actually experienced working through at the Circles.)
Joy of Creative Problem Solving
math workshops
a lifetime love of solving puzzles

Do these 6 things align to the curriculum?

The slide below shows the 4 types of tasks that are aligned to the curriculum, or that when I pick a PoW or Task, I make sure it correlates to the concepts and skills that we’re working on in the textbook. Therefore, it’s entirely intentional that the warm-up and PS are not aligned because critical thinking and creative thinking are not objects that we can place in a box or things that I can string along some prescribed continuum.

All 6 types of tasks are of course important to me. I try to implement them consistently with equal commitment and rigor to support and foster the 8 math practices.

Which ones get graded?

I don’t grade textbook exercises, i.e., homework, because I can’t think of a bigger waste of my time. I post the answers [in Google Classroom] the day after I assign them. I don’t grade PS because that’s when I ask students to take a risk, persevere, appreciate the struggle. I don’t grade warm-up because I don’t like cats.

I’m finally comfortable with this, something I’ve been fine-tuning each year (more like each grading period) for the last 5 years. I could be a passive aggressive perfectionist — or just an asshole when it comes to getting something right — so it’s no small admission to say that I’m comfortable with anything.

It’s about finding a balance, an ongoing juggling act between building concepts and practicing skills, between problem-posing and answer-getting, between teacher talk and student talk, between group work and individual work, between shredding the evidence and preserving it. Then ice cream wins everything.

Here’s the thing. We want to build a math curriculum that makes kids look forward to coming to class everyday. I trust that that’s true for more than half of my students — this could mean anywhere between 51% and 80%. I think we’re doing something wrong when kids look forward to just Measurement MondayTetrahedron Tuesday, or Function Friday. Math should not be fun only when students get to play math “games”!

Posted in Teaching | Tagged , , , , , | 14 Responses

7 Deadly Sins of Teaching [Maths]

‘Tis the eve of FDOS (first day of school, duh), and I’m no longer nervous or anxious. I’m washing these laptop covers for my homeroom students only because that’s what some of my colleagues did with their kids’ covers. I will not let a kid taunt me on FDOS with, “How come Mrs. So and So gave her students clean covers?” Oh yeah? Well, did you ask if she’d adopt you for the next 180 days?


I committed all 7 sins at one time or another, but there’s no shame involved — says a recovering Catholic — instead, it is a reflection of sorts.

Giving extra credit. I don’t care where you teach, how old your students are, what your zodiac sign is, you’re going to have at least one kid who’ll ask for extra-credit “work” at the eleventh hour of the grading period. Don’t do it. Say no and walk away because the tears might come streaming down his/her face and you have to ration the use of Kleenex. And you should be ashamed of yourself for giving students extra-credit points for bringing in copy papers, sticky notes, dry-erase markers, tissue boxes, doughnuts. Yes, you should send me some.

Giving timed multiplication drills. Maybe there’s a well-documented success story behind this madness that I’m not aware of, but to me, it perpetuates the myth of faster-is-smarter. This practice raises self-doubt and affirms the why-should-I-even-bother mindset.

Giving out the equation. That’s like giving away life’s secrets to someone who flies to Paris to have lunch. Meaning, they don’t need it, nor did they ask for it. Your students’ conversations, their conjectures, their models — are all at the heart of a math class. To give away the equation is to passively (and aggressively!) dismiss our students’ abilities to think for themselves. It’s okay to eventually give them the equation in due time, just don’t start with the equation. Imagine if I just gave my students the equations for slope and area of a circle.

Teaching from one source. No one source is that good. The creators of that source would be fools to not concede that point. It’s like eating out at the same restaurant or boasting that you can make chicken 50 different ways. No you can’t, and nobody cares. Let one or two sources be your structural outline, your mainstay, then supplement it with your favorite lessons or other teachers’ favorite lessons. Remember, any well-crafted lesson outside of the textbook that you can bring in is your gift to your students. Tell them that. And with our prolific #MTBoS, you cannot afford not to supplement.

Talking, talking, you’re still talking. I pretty much end every workshop with this reminder: The more you talk, the less your kids learn. I plan each lesson using this as my go-to guiding principle. Math is a highly social endeavor, so for the love of Ramanujan and Lovelace, please stop talking so much so your kids may talk! Every question you pose is an opportunity for your kiddos to ponder [quietly by oneself first] and share their thoughts with peers. Every question! If you fret that your kids don’t talk in class, then I wonder about two things, 1) Do students feel safe enough to talk in your class? and 2) Is the question you’re asking interesting/worthwhile/challenging to even bother? (I must have asked hundreds of lame, boring, worthless questions, but I’m not giving up. I practice and get better.)

Keeping up with the Joneses. That colleague whose hair and complexion are always perfect is just not as funny as you are. That teacher whose students all adore her probably owns a cat that wants to kill her. And that “amazing” teacher whom everyone talks about probably sucks at everything else in life! And he might be a compulsive hoarder of all things creepy! So, don’t mind them. We’re not here to compete with one another. We’re here to make mathematics rock for our kids. There is one you and 24 hours in a day. Make time for yourself, make time for your family. We all have shitty days that rob us of our wits and sensibilities, but recognizing that and committing to having a better day tomorrow are worthy endeavors. Our students need us more than they care to admit.

Being an asshole. I already stated from the beginning that I’m guilty. No one wants to learn from someone who’s mean and angry and bossy. When we try to establish authority in the classroom, we may inadvertently end up being perceived as this person. The meaner we get, the less students want to have anything to do with us, so the angrier we get. It’s a vicious cycle, and everyone is losing. We’re the adult in the room, charged with a magnificent duty to establish a learning culture, which will not happen if we don’t behave like an adult. Children are said to be resilient, but they are also impressionable, and their impressionable minds are vulnerable — vulnerable to criticism, to shame, to false praises.

Let’s pray for more patience, more kindness, more badass. Here’s to us — and to a great school year!

Posted in Teaching | Tagged , , | 15 Responses

Growth vs. Proficiency

I usually take copious notes when attending conferences. It’s more of a self-discipline gesture to make me sit up straight and pay attention. I even try to sit in the front row. (The only time this backfired was in Psychology 101 when the professor had to talk about herpes and I was sporting a gargantuan one of the simplex 1 type on my upper lip. Not cool.)

I’m sharing the following notes from last year’s SDB Conference because we passionately voiced and argued — and confirmed and challenged — one another’s thinking and teaching practice.

Before discussing the topic of “growth vs. proficiency” within our small groups, we were asked to answer these two questions individually:

  1. What is the difference between growth and proficiency?
  2. How can we measure both accurately?

I wrote:

  1. Growth is individual progress, whereas proficiency is achieving some set of standards. One can grow but not achieve proficiency.
  2. Not sure. Can we ever measure these two things accurately?

Then the conversations began, and I jotted down some stuff:

  1. Growth is more social, proficiency is more academic.
  2. Both must be motivated.
  3. Both cannot be fostered if the classroom culture and teacher mindset are not in sync.
  4. Both must address how a kid learns best.
  5. A young child appreciates nature, but knowing the golden ratio is cool and requires knowing division.
  6. No one is born proficient, without growth, there’s no proficiency.
  7. We need to let growth and proficiency be time independent.
  8. It’s wrong that a kid can’t get a high school diploma when she’s a brilliant artist and can’t do algebra 1.
  9. We need to teach kids to reach their own goals instead of ours.
  10. To achieve proficiency, we should have standards, and we should agree on them.
  11. Types of tests should not be designed for the ease of testing.
  12. Proficiency: report card for the school, reputation of the institution, snapshot at that time. Growth: distance traveled over time, point A to point B can be a dramatic growth.
  13. Story archived on NPR: A teacher brought up a low group of kids to meet standards and was awarded, but she was penalized when she had a high group of kids and they didn’t see growth.
  14. Students think they are what their grade is.
  15. When you measure proficiency, you can see growth over time.
  16. Proficiency ≠ Excellence
  17. Setting the bar is not a teacher thing, it’s an admin thing — if you don’t make proficiency, you go to PI, and PI is hell.
  18. We teach to a bell curve, and we teachers are a bell curve. One size does not fit all.
  19. Growth is more individual, many different aspects of that kid, such as attitude, mindset, social behavior. Proficiency is more measurable, toward a whole group, show mastery, attach a number to this measure.
  20. Suggestions: students track their own progress, journaling, blogging; teacher is transparent in their expectations.

Some questions:

  1. What about teacher proficiency?
  2. Can we combine growth and proficiency? Can they be quantified?
  3. Growth is effort, how do you measure that?
  4. How do we strike a balance between growth and proficiency?
  5. How do get politicians out of this process?
  6. Who decides what proficiency means? Do students have a voice in this? (Teachers grade differently.)
  7. How can we communicate with everyone else to put equal weight between growth and proficiency? (We’re on a constant treadmill to keep up with the changes.)
  8. Portfolios are good, but how do you implement and grade them?
  9. Project-based learning has pros and cons, how do you give a grade for that?
  10. Are students learning social interactions, problem solving, communication, collaboration?

Two thoughts precipitated from this discussion for me, then and now. It might be that our schools are set up to measure proficiency, but they misinterpret those scores as measures of growth. And however we choose to define growth vs. proficiency, how do we ensure equal access to promote growth and proficiency? Professor William Tate, earlier in April at NCTM Annual Meeting, gave the Iris M. Carl Equity Address, he said, “The greatest threat to math instruction is the empty seat problem.”

Posted in Teaching | Tagged , , | 32 Responses

Catching Up

According to my online Social Security account, my earnings record shows:

  • 24 years of full-time teaching
  • 2 years of half-time teaching
  • 1 year of subbing

It also shows this not-so-fun fact:


And this, which I find morosely funny:


Had I known ahead of time how depressing these numbers would be, I would have become a farmer instead. Grow cannabis or something. 

Anyway. It’s been a busy summer. I attended my first TODOS Conference in Scottsdale, AZ. I facilitated for NCTM’s Deep Dive for three 2-hour sessions on Ratios and Proportional Reasoning. Then I presented for CUE in Brentwood, CA, on Computational Thinking.

Thankfully I managed to squeeze out some time for a week-long camping trip at Shaver Lake and a trip to Oregon for my niece’s wedding.

I have three big commitments for the remainder of the year:

  1. BCAMT in Vancouver (BC) in October
  2. NCTM’s Innov8 in St. Louis in November
  3. CMC-North in Asilomar in December

We have a new principal this year, and I have a new math colleague (she and I make up the math department). My teaching assignment this coming year remains pretty much the same — 6th and 8th grade maths — except I’ll also teach an elective period of computational thinking (problem-solving) to 6th graders. What’s new is our block schedule which I have mixed feelings about as I’m not familiar with it. Kinda nice not having to see the brats every day though. :p

In summary, I need to work until I’m 104 just to afford rent on a two-bedroom in drought-stricken southern California. But hey, where else are you going to get summer-like weather year-round?

Posted in Teaching | Tagged , | 5 Responses