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.
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.
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.)
Green Olives
My 7th graders are working on "percentages of" problems currently, and late last night, I saw this problem on one of Don Steward's handouts.
There are 75 olives, 40% of which are green. I eat some of the green olives until 10% of the olives that remain are green. How many green olives did I eat?
How would you solve this? I solved it using algebra. Then, immediately, I thought, Fawnzie, since when do you use algebra to solve stuff like this. C'mon, do your rectangles.
I think of 40% as 2 of 5 boxes.
So, 75 olives must split into 5 groups of 15, so there are 30 green olives.
Then, I ate some olives to end up with only 10% of the remaining olives are green.
Well, since I didn't eat any of the 45 black olives, so these 45 must make up 90% of the olives remaining [in the 9 boxes], so 45 must split into 9 groups of 5.
Oh, look! I began with 30 green olives, I now only have 5 green ones left, so I must have eaten 25 of them.Okay, your turn.
There are 80 olives, 75% of which are green. I eat some of the green olives until 20% of the remaining olives are green. How many green olives did I eat?
I’d rather use the drawings than show them my work below.
Euclid's Algorithm
I show my 6th graders this image, pointing out that this picture represents the two numbers 1 and 1 that I'd entered at the top.
I then ask them to give me two new numbers — any two positive integers [that are 10 or less, for now] — and the computer will draw a new picture. As each set of new numbers is entered and the corresponding picture is generated on the screen, I ask students to jot down their "I notice, I wonder" in Google Form and to draw a rough sketch of it in their journal. After a few sets of numbers, I ask students to imagine and/or draw a rough sketch of what they think the picture will look like before I hit the update button.
These are the pairs of numbers they'd asked for and their corresponding pictures, listed in the order that was asked.
I love that the kids are asking for...
6, 6 after the initial 1, 1
3, 8 after 3, 1
8, 3 after 3, 8
But, when sets 9, 9, and 7, 1 are asked at the end there, I say to the class, “Hey, what if figuring out this puzzle — which is how the computer draws the picture given two numbers — gets you a million dollars. And you get to ask for sample sketches like you've been asking, except that each sample costs you some money! So, make each request worth it. Let it prove or disprove your conjecture. Ask carefully."
I love the OHHHs and AHHHs after each picture is revealed. But no one is claiming that he/she had drawn the same diagram. I pause longer for them to write down their noticing and wondering.
I now say, "You may only ask for four more sets of numbers. Remember, make a request that would test your conjecture."
I ask a normally quiet student. She says, "10, 3."
Another student wants to know what "100, 5" looks like.
"What about 8, 5?" I reply, "Sure, but draw it in your journal first." They are fully engaged. Then I say, "Now, share your drawing with a neighbor."
I ask, "Did anyone sketch the same thing as their neighbor?" They're shaking their heads, and I say, "That's pretty crazy! Do you think yours is more 'correct' than your neighbor's?"
I reveal 8, 5.
The last request is 23, 75.
What some of them have written [with minor edits from me]:
When we did the same two numbers the shape didn't change but when we did different numbers it changed. Why does it divide into little parts within a square when we put 3,8? When we did 8,3 the number switched around. I wonder if the two numbers are dividing to make the shape. How can you figure out the number when it can't divide easily. My drawing for 8,5 was one whole and 5 little squares. The 23,75 was a little confusing to me.
They're different, they are the length and width, and when the two numbers are the same it's just one cube. I notice that if it can be simplified, it is. Example: 6,2 = 3,1. I don't understand 8,3, 5,9, 23,75 or 8,5. But I did notice that the smaller the parts of the shape are, the lighter shade of blue they are.
I observe that when the same numbers are entered it equals to a blue square. If the first number is bigger than 2 then it will add one more square. I wonder if you double the number for each number will it be the same shape. I wonder why for 3,8 it has one square with three parts. I observed that if you divided the first number by the second it will equal to the number of squares. For 8,5 I didn't get the right sketch. The sketch was one square with half of a square cut in half, then in one half is has a strip that is cut in half. I wonder why it has half of a square. I think that my answer for how it figures it out is right, but I don't know how it comes up with that picture for 8,5.
When you do 1 and 1 it doesn't change because we tried 6 and 6 it didn't change and if we put 3 and 1 it did change. I saw that when we did any number like 3 and 1 is 3 ones. So I think that all you have to do is divide something by something = the first number that you put in but if you can't divide by 2 then I'm wrong. I'm not sure that I got this right but this is what I think.
For the first one 1 and 1 I thought it would be a small one by one cube. What threw my off was the 6 by 6 because the size did not change. For the 8 and 5 I drew a big block and and 5 little ones, but my image was wrong. I also wondered if the first number was the amount of shapes that would appear, but I was wrong again. I don't understand yet. I tried looking for a pattern, but couldn't find one.
When I tried 8,5, my answer was almost right. I had the one big square right, the half square right, but then I got the little squares wrong. I think that the way the computer does it is dividing the first number by the second number. I am confident that if you put the numbers 10 and 5 in, it will show 2 squares. When using the diagram, the second number will represent the vertical side.
What I've been noticing was that if you put the bigger # in the front and the small # last then it would be like a rectangle. I've also been noticing that if you put the same #'s it would like keep on drawing a square. So someone said what could (8,5) look like and Ms.Nguyen showed us the drawing and the I notice that nobody got it right. I was expecting something like smaller because the #'s were small they weren't as big, but at least I tried to get it correct but I drew something a little bit smaller than that. I also wondered why when we put the same #'s together why do they all become a square that's what I wonder.
For 5, 12 I notice that it is two big squares, two smaller squares, and two tiny squares, I thought it was going to show 1 big block and another big block but that one would be cut off at the bottom or not a whole block. I also notice that the pattern is the first number multiplied by what equals the second number and the number that is missing is the amount of blocks that is created. I thought I knew it but I don't really get the ones with a bigger number first and the smaller one last. I thought I knew what 9, 23 was going to be but it the result was surprising. It didn't look at all how I thought it was going to. The website is pretty cool, but one thing I didn't understand was the placement of the small blocks and what they stand for. Like some were really tiny and some were small but I don't know what they stand for. But I bet if someone explains it to me I probably will understand perfectly and feel dumb.
The only thing that I am sure that I know is that when the first number is larger than the second, the shape is wide and when the second number is larger, the shape is tall. Other than that I am very confused.
Then, together as a whole class, they agree on the following;
When both numbers are the same, then the picture is one square.
The computer simplifies the two numbers, such that a picture for 6, 3 is the same for 2, 1.
When the second number is a 1, then the picture shows the first number of squares. For example, 7 and 1 would form a picture of 7 squares, and 100 and 1 would form 100 squares.
The first number is the horizontal dimension, while the second is the vertical.
They are all squares.
The dismissal bell is about to ring, and I want to teach forever.
Tomorrow, we'll spend some time with one set of numbers, like 10, 3 or 8, 5. We'll dissect the diagram. Play around with a few more. Practice sketching a few. We'll write out the equations that go with each diagram. I'll guide them into noticing the size of the smallest square in relation to the two numbers.
I found this investigation at underground mathematics. The site describes itself as having "rich resources for teaching A level mathematics." From what I understand, "A level" means advanced level mathematics consisting of core modules ranging from quadratic, logarithms, geometric/arithmetic series, differentiation/integration, etc.
Perfect for my 6th graders who continue to torment me with their arithmetic atrocities, such as, 3² = 6 and 5 ÷ 10 = 2.
While the original task is scripted for older and more advanced students, I found in it what I needed to make it rich and appropriately complex for my 6th graders.
Hail, Euclid's algorithm!
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:
If we know that nine halves of x is 27, then what is one half of x?
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.
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!