My Other Math Sites
Lessons From the Classroom
Reviewing for a Test
There's not enough time or humility for me to share my teaching fails, but here's a test review routine that worked well with my students.
I pass out the review questions near the end of class and say:
Chapter 3 Test is scheduled for _____. To review for it, I need you to look through the ten problems on this paper. You're more than welcome to work on the questions, but you don't have to, not for me anyway. I need you to just examine them enough to identify two questions that you feel confident about, that you have no problem solving. Then, I need you to identify two questions that you would like to get help on most. That's it.
If you have human students as I do, they will ask questions and you'll have your answers ready.
What if I'm confident I can do more than just two?
Possible Answer (PA): That's fantastic! But, nobody cares. I just need you to tell me two.
What if I need help with more than two questions?
PA: That's why I said most... two questions that you would like to get help on most.
What if I can do the whole test? I mean I don't need help with anything.
PA: Then don't mark anything down when I ask tomorrow. Or, if you just have one question that you need help with, then identify just that one. Also, you'll get to make the test key for us.
What if I'm not confident with any of the questions?
PA: Thank you for that question! I hope you'll find time this evening to see if you can attempt two of them. Then, please send me an email letting me know if you were successful. Remember, the answers are on the back of the review questions.
You said we don't have to actually do the problems. But can we do them for extra credit?
PA: No.
How many questions will there be on the test?
PA: Ten. Or maybe thirty-five. I don't know. It'll depend on what you tell me about these review questions.
The next day, the whiteboard already has two columns drawn, and students know to give two tally marks in each.
If I were to do this again, I'd also send my students home with these two questions in Desmos instead. When you ask for a checklist in Desmos, you get an auto tally of how many students marked that choice in the teacher dashboard.I focus on the right column and say:
I want to make sure we go through all the questions on the right column. We'll start with question 4. There are at least four of you here who marked that you are confident with this question, I need one of you to please show us how to do question 4.
And we continue down the list. Notice that I ask students to work on the problems. They learn better from their peers, especially during review time. I get to check for understanding as there are occasions when the "confident" student has done the problem incorrectly. We spend class time on more targeted problems, including admitting which concept needs a serious revisit. I have them do this — looking over the questions — at home rather than in class because I need them to focus, without peer pressure and distractions. I want them to feel free to look back in their notes, in their textbook, search online, ask for help. I want honest feedback.
After we've gone over all the questions on the right column, that evening I create the test with four questions already done by choosing two from the left column that had the most tally marks, in this example, questions 1 and 2, verbatim. It's my selfish way of not having to see a test with a score of 0 which leads me to self-loathing. Then, I pick two questions from the right column that we'd gone over, also verbatim. I can actually hear the sighs of relief when students see the same questions from the review sheet.
This routine works well for regular homework too. Assign the homework without actually have them do the homework: ask for two "easy" questions and two "tough" questions. Let's not have them work on problems that they already know how to do, instead, let's spend time together in class to work on the ones they need help with. Promise me you'll only step in to show how to do the problem when no other student in the class is able to. :) But even then, I'd say, "How about someone starts out this problem, help us with that, just take it as far as you can, and then I'll take over from there."
Changing Up Popular Warm-Up Routines
As with any task, whether it's a warm-up or a curricular task, I try to think of ways to get more student engagement, tap a different thinking modality, and just to change things up.
WODB has become a common acronym in classrooms for good reasons. (Actually, does it qualify as an acronym like NATO since I've never heard it pronounced as a word? Y'all are still saying Which One Doesn't Belong, right?) Take this first one I see on the site. (I added the numbers 1-4.)
I do my homework first, in the order that the shapes come to my brain:
#2 doesn't belong because it's the only non-triangle.
#4 doesn't belong because it's the only shaded shape.
#1 doesn't belong because it's the only one with exactly one line of symmetry.
#3 doesn't belong because it's the only obtuse triangle.
With students, I ask them to give me a blank grid and get ready to draw in each box as they listen to my clues. I tell them to make quick sketches as they may need to make changes when they hear new clues. I can give them the clues in any order I choose. But, we'll stick with the order above.
#2 doesn't belong because it's the only non-triangle
A possible sketch:
#4 doesn't belong because it's the only shaded shape.
#1 doesn't belong because it's the only one with exactly one line of symmetry.
#3 doesn't belong because it's the only obtuse triangle.
Students can then share their sketches and critique each other's work. The reveal is fantastically fun.
Estimation 180 is another popular one. From the site's Day 6:
Instead of asking students for an estimation, I ask:
One of the four numbers below is the correct number of almonds in the 1/4 cup pictured. Which one is it and why did you choose it? Which number do you believe is way off?
8
15
28
40
I find their reasoning and conversations are tighter this way — more focused. I'm also one of those people who dread having to guess at something, even with a visual clue. With younger students, I'd give them 3 choices instead of 4.
Open Middle is another well-loved routine. This one is filed under Grade 4, Equivalent Fractions. (I added the digits A-G.)
Directions: Use the digits 1 to 9, at most one time each, to make three equivalent fractions.
I mark the digits 1 through 9 on red/yellow counters and put them into a baggie. I reach into the bag and randomly pull one out. Say, I pull out a 5. I call on a student [randomly] and ask, "Which space (A-G) can the number 5 not be in?"
There is a big difference between asking the above question versus, "Where do you think the number 5 goes?" The chance of answering this question correctly, if 5 is used at all, is 1 out of 7. The former question is much safer to tackle. This routine engages the whole class on one number at a time — we get deeper thinking when we can focus on one thing and while building on each other's thinking. And I very much love it when students are given opportunities to honor and build on their classmates' reasoning. After a few suggestions, a student might conclude that the number 5 can go into the discard pile. (It's also common for students to use a number more than once or use a number not allowed, so the counters alleviate this mistake.)
Also, I need to share that this problem was posed in an online workshop I attended yesterday, and I didn't even attempt it because these are my constant truths:
Someone else will come up with the answer before I do.
The answer will be revealed before I get to solve it, so no point in me ever working on it.
How many of our students also hold these truths? I understand this was a workshop for teachers and time is limited and sharing is good and all that. I'm just thinking about best practices with students though.
[Added 10/17/2020]
I meant to include a numerical WODB example because there are a lot of possible solutions, it's always fun to see what the kids come up with. Here's the first numerical one on the site. (I added the letters A-D.)
I work on it first.
D is the only non-square number.
A is the only single-digit number.
C is the only number divisible by 5.
B is the only even number.
I give the students the above clues in the same order. However, students may not erase a number as they revise, they may only cross it off so that I may see what they had originally.
MoMath
That sign is for ALL of us!!... Fucker!
I heard the angry man yell around 8:00 this morning. Our house sits at the corner of 4-way stops. Glad he was out there to give the driver a piece of our collective neighborhood mind. At our core, we sense when something is not equitable, not right. We want to speak up when someone is not abiding by the guidelines that are meant for all of us.
I saw the tweet below from Sam Shah yesterday, which I retweeted without comment after reading his post.
Howdy all, I wanted to share some concerning things I've been made aware of at the Museum of Mathematics (MoMATH). I hope the Board of Directors looks into things. https://t.co/ftLlFCBuXr#MTBoS
— samjshah2 (@samjshah2) October 11, 2020
Nothing I could add to the tweet to make it less upsetting. I had always wanted to visit MoMath when I could make it back to New York City.
Something is not equitable, not right. I spent most of my teaching at Title 1 schools: 11 years at George Middle School in Portland, Oregon, and 17 years at Mesa Union Junior High here in the same county where I live.
Good bad and indifferent, these children are mine. When I read that MoMath treated students from Title 1 schools less favorably and dismissed the voices of those who brought this fact to light, I'm made to distrust those in power, those who can do so much more to improve our children's livelihoods, instead they further marginalize them.
@MoMath1 is responsible for making itself worthy of all our students' learning and enjoyment of mathematics. That museum is for all of us.
One-Word Writing Prompts
Years ago at George Middle School (Portland, Oregon), the teachers were allowed to teach something we were passionate about. The class would be twice a week, right after lunch, for just 20 minutes. I was a science teacher at the school and asked if I may teach "writing for writing." My principal reacted with slightly more enthusiasm than my [male] colleague's "The Simpsons." I had a simple plan:
Given a prompt, we write for 5 minutes. It's imperative that I write along with my students.
We share aloud what we'd written, only if we choose to.
We also share what we'd like to do next with our 5-minute piece:
scrap it
add it to an existing piece of writing
save it for whatever whenever
get feedback on it
Typically, writing prompts come in the form of questions. Here are the first three of "34 Quick Writing Prompts for Middle School Students" from Journal Buddies:
What does the city sound like at night?
What is the coolest thing that can be found in nature?
How can you tell whether or not someone will be a good friend?
These are fine, of course. But personally, I either don't have a lot to say about the prompt or I don't care. And it's hard to think of a question that everyone cares about or have copious thoughts on. Let me try the above prompts right now, as if I were a middle schooler.
What does the city sound like at night?
The city is quiet at night. Though I'm not sure why I'm in the city at night when I should really be in bed at home. I'm a kid. Sure, there are times when I can't fall asleep or I wake up in the middle of the night. But it's still pretty quiet at night. Sometimes I can hear my Dad snore. (Or is that my Mom?) If only it would rain each night because that's the best sound to fall asleep to. But then I think about the homeless people. It sucks to be homeless, so for them, night rain is probably the worst...
What is the coolest thing that can be found in nature?
The coolest thing that can be found in nature is... I don't really know. I'm not sure if it's the "coolest" thing or even just "cool," but I like flowers and plants. I pay attention to them whenever I go on walks. My favorite house is always the one with lots of flowers, especially when they are overflowing in window boxes. They don't require a lot to grow, not like what my two cats and dog require. I like flowers that smell good.
How can you tell whether or not someone will be a good friend?
I can tell that someone will be a good friend because they are not bad. But I've been wrong lots of times before. They start out all nice and friendly, they say the right words and do the right things, then they just turn. Sometimes they turn so quickly that I have no clue what happened. It's like they have an R gear for reverse and they just shifted into that gear and ran you over. I guess there are no guarantees whether or not a person will be a good friend. Look at all the divorces and breakups! It's best to just take one day at a time.
I’m weary that a question prompt might not elicit interest or intrigue, and then I have to hear them whine pitch perfect, "But I don't know anything about that," so I give only one-word prompts.
tiny
red
outside
wax
intelligence
breakfast
sand
scream
pale
rain
My goal was writing for writing. I wanted the pen or pencil to move across the page for five minutes. I wanted the shitty first drafts. Any more than that one-word prompt might inadvertently restrict, if not constrict, their thoughts. I wanted them to feel free to write freely about the color "ecru," the noun "home," the verb "shrink," the adjective, "dull." They wrote wildly, ferociously, thoughtfully. So did I. Mainly, I didn't want any student of mine to feel the way this perennial prompt made me feel through all the school years, "What did you do last summer?" Eventually, I got tired of lying about the trips that my family couldn't afford to take and left my paper blank.
Common Denominator
I already wrote about dividing fractions here and here.
I use the explanation of "dividing by one" to explain why 5/6 divided by 2/3 is the same as 5/6 times 3/2.
But when I was asked recently about how the "common denominator" strategy worked, my muted response was, "Because it does." I didn't mean to be a jerk, rather I just hoped she'd go along with me.
I grabbed a piece of paper and wrote 5 ÷ 3 = 5/3. She was already bored with me. Then I added 1s under the numbers to show 5/1 ÷ 3/1 = 5/3. Right, right? I then changed the problem to 10/2 ÷ 6/2 = 10/6... = 5/3. Still okay, right?
Before I could give another example, she took the paper and rubbed it on my head. Rude.
***
The real common denominator is we're all in this together to #flattenthecurve. This tweet is like rainbow.
My elderly neighbor is self-isolating. So the neighbor kids are playing her a cello concert from her patio. pic.twitter.com/g6Nr2HNVho
— Jackie Borchardt (@JMBorchardt) March 16, 2020
Math Worksheets
I often create worksheets for my students, even though every district-adopted math curriculum we've had has worksheets for students. I do this for two reasons:
I sometimes want to teach differently than what the curriculum writing team was thinking.
There's a particular structure/scaffold that reflects how I see the content can unfold for learners.
Here's a sequence of practice questions for my 8th graders on rigid transformations.
Everything about this is intentional.
Item #1 is a completed sample of what's to come. This is a practice worksheet, not a problem-solving task, so I will be clear about what is expected.
I remove certain parts in item #2, while keeping it similar to item #1.
Item #3 comes before item #4 because I think it's easier to follow the stated transformations than to say what they are.
Item #6 asks for more flexibility but with an ending constraint.
Item #7 opens up the problem and allows for peer exchange.
It's esthetically easier for me to create the questions on Google Slides. I then do screenshots to toss them onto a Google Doc. Here's a screenshot of questions for 7th graders on percent change. Here's a screenshot of questions for 6th graders on ratios and rates. If you'd like copies of these:
Yes, each of these takes one unit of shit-ton of time, especially when I have to look up real products with real numbers. But it's an OCD thing too, as in If-I-can-make-it-better-I-will. Stay safe, everyone.
Six Ways to See Visual Pattern #324
I know you weren't asking, but I see n overlapping squares.
— Hunter P. (@professorpatton) November 12, 2019
What about you? How do you see the pattern in the tweet above growing? Please take a look before I completely ruin it for you.
I loved that Hunter saw overlapping squares in pattern #307 from Caden Glover. I saw the pattern like this instead — with the constant four-circle square, and groups of three circles wrapping the upper right corner, and the number of groups is one less than the step number.
Some 30 hours later, I was at my desk creating another pattern, and I started out with a diagonal of increasing cubes (highlighted in yellow), and above and below this diagonal are more cubes (marked in purple). Therefore, for any step n, I see the middle as (n + 1) and the purple ones are two equal groups of n.
It was not until I finished creating step 3 that I realized I'd made the same pattern as the one from Caden. I didn't want to scrap it because I didn't mean to copy his pattern and really had built it from scratch, and now it is pattern #324 on visualpatterns.org.
It might be my favorite one thus far because I can see it in different ways. Here are the overlapping squares that Hunter saw:
What's fun is sometimes I don't see the pattern in its "simplified" form until I've simplified the equation. (I have to do this for the answer key). The number of cubes C is related to the step number n, such that C = 3n + 1.
And I wouldn't be me if I didn't always try to see a rectangle in any pattern. (The green rectangles have dimensions of 0 by 3, 1 by 3, and 2 by 3 for steps 1, 2, and 3, respectively.)
Many of my students will try to see if the entire step can be enclosed in one rectangle, then minus the negative space. The negative space (missing cubes) of this pattern is fun to discover too!
Do you see another way?
Six years ago when I created the site I had hoped to have 180 patterns to match the number of school days. We are now at 324! I'm so grateful for all the pattern submissions and for all the ways that the site gets shared.
Jelly Beans or No Jelly Beans
What's not to love about Would You Rather; I use it with my students and always recommend it as one of the great warm-up routines. This one caught my attention last week.
Each entry always includes this statement:
Whichever option you choose, justify your reasoning with mathematics.
This statement is important, especially the word "mathematics," because you might have a student who says [truthfully] that she doesn't like jelly beans or is allergic to them and will prefer to give them all away. Then, you might have another student in the same class who [untruthfully] makes the same claim to avoid having to do any maths. There's also that thoughtful student who wants to give more to his friends and keep fewer for himself. Or a student might consider giving more away to make friends. If I were to pose this question to my students, I'd mention the above possible reasonings, but then I'd add, "This question assumes that you love jelly beans and want to keep more of it for yourself. For now, it's the mathematics we're after." I didn't ask my students this question though, I asked them to choose between this question and the one on the right.
As a student, would you rather be given the problem on the LEFT (jelly beans) or the one on the RIGHT?
Using the same image above, I asked on Twitter and of my colleagues.
As a teacher, would you give your students the problem on the LEFT (jelly beans) or the one on the RIGHT?
*****
Out of my 68 sixth graders, 71% of them chose the left problem. The words they used for their reasoning:
colorful, vivid imagination, visual, more pleasant to the eyes, interesting, engaging, not boring, not basic, better reason because you like jelly beans
My fellow educators, meanwhile, overwhelmingly chose the left problem — of the 781 people who took the survey, a whopping 90% chose the left problem.
Well, I prefer the one on the right that I’d typed up. How did I get it so wrong? I'm normally not this lame. But, truth be told, I don't love the jelly beans question. At all. Maybe the one on the right is the wrong "fix" for the left one. If I could retype the problem on the right, I'd remove the equal signs since the question is just asking which one yields a larger difference, not caring exactly what each difference is.
I want to believe that anyone who spends 5 minutes with me learns that I love mathematics. I love numbers, I love math problems and don't give two shits if they are real-world either. I'm the one who loves the problem about carrying 3,000 bananas across the desert and the one about the emperor pouring oats on every other guest's head. One of my 247 all-time favorite problems is Noah's Ark, even if I grew up in a house with a Bible in every corner and found it hard to wrap my head around this story. (However, Eddie Izzard's take on the Ark — language caution — makes me laugh.) I love problems that are simply stated, yet they beg you to savor your perseverance as you think deeply and creatively. There's great joy in solving a good problem, especially the ones that at first blush, you weren't even sure how to begin.
The jelly beans problem above is not one of these problem-solving tasks. It asks for a mathematical justification, so I'm going to assume that the mathematics is finding the difference or another arithmetic operation. I see it as a number talks problem. Students get to share their strategies. Mine might include:
364 minus 188... I'd need 12 more to go from 188 to 200, then 164 more to get to 364, so the difference is 176. Similarly, to do 281 minus 137, I'd need 63 more and 81 more, or 144. Problem A has a bigger difference of 176. Comparing the two totals, 364 and 281, the difference is 19 plus 64, or 83. While the difference between the give-away quantities 188 and 137 is 51. I start out with 83 more in problem A, but I only have to give away 51 more, so problem A leaves me with more to keep.
That's why I prefer the one on the right. Numbers are beautiful. I want students to focus on the numbers and play with them, learn to regroup, try massaging them and making them flexible, be comfortable with numbers. Math is badass, so let's do maths for maths' sake. I feel protective of numbers and don't see why they need to be dressed up in colors or dunked in forced contexts. (I suddenly think of little dogs in ridiculous outfits that I doubt if anyone asked for the dogs' permission.)
Nine out of ten of you disagreed with me. That's okay because I can make phở better anybody. But guess what though? My own 23 and 25-year-old kids chose the one on the right. This fact was comforting! Sabrina (23):
If you were one of the survey respondents, I thank you thank you thank you.
The Name is Munch, Math Munch
I’ve mentioned Math Munch before and tweeted its scrumptious bits every now and then. I wanted teachers to see the site and share its contents with their students. But truth be told, we teachers often get side-tracked, we’d be all gung-ho about something in September, only to see it vanish by the first snowfall. (I’m a nostalgic liar. Haven’t seen snow in last 11 years living in southern California. But you get my drift.)
Then as I was watching this TEDx video — thanks to Shecky Riemann at Math-Frolic! for featuring it in his post — it dawned on me to put this engaging and thoughtful resource that is Math Munch directly into my students’ hands. I figure I’m much more reliable when I involve other people in my commitments to do things. Because it’ll then be everyone else’s fault if things go awry.
Before I shared the video with the kids, I told them what I knew about Math Munch:
That three passionate teachers named Paul Salomon, Anna Weltman, and Justin Lanier created MM to help students learn and love math beyond the walls of the classroom.
That Justin is a good dancer, and what a privilege it was for me to meet and talk with him this past summer.
That Paul created this lovely art piece that hangs prominently in my home.
That last year I did a few things with my students that I learned from MM, like the Weight Puzzles and The Numbers Project.
That I’m sorry I don’t know Anna, but from her work and this video, I find her inspiring and creative.
I showed the first 8 minutes of the video. We then talked about why the team spoke of Marjorie Rice first. What is the significance of sharing her story, her discovery, her pure enjoyment of doing something mathematical. What is wrong with how math is perceived among the general public. How awful it is to confine math within the classroom and inside a textbook. How wrong, just wrong, that math is about the best and the brightest.
It was this slide in the video that gave me an idea for how I may incorporate Math Munch into my curriculum.
The five words — DO, MAKE, WATCH, READ, PLAY — efficiently categorize what my students can do with each post.
I came up with this MM_Activities handout.
My plan
Each student gets this handout, just 1 per quarter. No other handouts — saves tons of paper. All work is submitted on students’ own paper.
Initially I was going to assign MM every other week. I’m now assigning it as a quarterly project. They are to complete 15 tasks (3 in each of the 5 categories). I think this is reasonable because each MM post has 3 features, so I’m really asking kids to read and interact with essentially 5 posts in a 10-week period. The open time frame is less stressful for the kids — and for me.
At first I also thought of assigning everyone to the most recent post, say “Sept. 03, 2013” post. But MM is already 2 years old, and this would be such a waste to miss all the archived goodness. So, kids get to choose! Not to mention I’d get bored very quickly reading 142 reflections of the same post! But the real reason of having them choose is I (me me me!!!) will get to consume a lot more MM this way. Teaching is just inefficient if we don’t learn from it ourselves, right?
I’m crazy excited about this. My kids will totally rock this project because they’re awesome.
Scoring an Ordered List
My 7th graders have a question on their exam that asks them to put eight numbers (integers and fractions) in order of their distance from 0 on the number line, starting with the smallest distance.
These types of questions are tricky for me to grade, and because there are eight numbers in this sequence, the task of grading it fairly suddenly becomes thorny and irksome.
Let's change the question to this:
Put these numbers in order from least to greatest: 5, 7, 2, 3, 1, 4, 6, 8
The correct order is 1, 2, 3, 4, 5, 6, 7, 8 :) — for a possible score of 8 points. How many points would this response earn?
1, 2, 4, 5, 3, 7, 8, 6
So, only the first two numbers — 1 and 2 — are placed correctly. Is the score just 2 out of 8 then? But I want to give some credit to 4 and 5 being next to each other, likewise with 7 and 8.
I've tried to come up with some metric to score this, and then I would want to apply the same metric to different sequences to see if any would break my invisible "fairness" barometer. For example, whatever score I came up with for the above sequence, I think the below sequence should get a lower score because the 7 and 8 are farther upstream than they should be.
Anyway, I have some ideas. The above two sets are Sets A and B below.
I wonder if there's a way to score an ordered list that half of us math teachers can agree upon. I'd like for my students to think about this too. Meanwhile, here is a spreadsheet with my scores if you'd like to take a look and play along. Just enter your name in row 1 (and link your name to your Twitter, if you want) and the scores you'd assign to these sets.
[02/01/18: @MrHonner had a similar question over 4 years ago: Order These Things From Least to Greatest.)