Intro to Lines the Visual Way

I wrote previously (apparently, a while ago) about how I introduce lines visually via patterns. After the end-of-year grades closed for Grade 7, I decided to pull together some materials and to have a go at this in the remaining weeks of the year, even though lines is really part of our Grade 8 curriculum. (My Grade 7s did very well on their end-of-year exam, that as a class there isn't much we need to go through and review again.) This intro to lines has been quite successful!! I incorporated the feedback from previous years, plus my own intuition of what works well as follow up, and came up with an introductory packet for the kids to work through.

Step 1: Writing equations of positive rates, from visual dot patterns.


Step 2: Drawing patterns from equations, in order to visualize the symbols within an existing equation.


Step 3: Writing equations with negative rate patterns, and then transitioning over to tables of values.


Step 4: Working with increasingly complex patterns with fractional rates (both positive and negative).

The result? They were hungry to get through it. They were looking for more. They were totally ready for the idea of the linear rate being "what happens, over how long it takes" by the end of this packet, and were able to write simple linear equations with no problem.

Subsequently, when I put up a table of values that had (6, 10), (9, 5), (12, 0) in it, many of the kids immediately were able to come up with the equation y = x(-5/3) + 20. Nice...! (Not all of them had really finished the packet at that point, since many of them were absent previously and were therefore a bit behind on the packet. So, this is really not bad. Even the kids who were a bit behind didn't have issues understanding why the equation would be true, when we discussed it as a class.)

So, here you go -- one of my last share-worthy materials of the year.

PS. This year, I didn't have them try to circle the missing/taken away dots in the negative rate cases. I just had them think about what is the value in stage 0. For example, if the pattern is
(1, 18), (2, 14), (3, 10), then they can first tell me that in stage 0, there are 22 dots. So, the equation is going to look like y = 22 + x(...). But, since we're not adding groups of 4, we don't want to write y = 22 + x(4). I just asked them, "How do we show that we're taking away 4 dots each time?" And the kids said, "Use negative 4!!"  So, there, we write y = 22 + x(-4) together on the board, and they can re-arrange it into y = -4x + 22 if they'd like. Easy breezy. Then, in future work with tables of values, I just had to ask them, "Are we adding dots or taking away dots?" and they'd know to fix their signs on the rate.

PPS. And, I always read equations such as y = x(-5) + 22 out loud as, "We're starting with 22 in stage 0, and taking away x groups of 5." I think if we can approach it this way, kids will not likely confuse rate or slope with y-intercept, because they'd be thinking constantly about the meaning of the multiplication. More and more, I think that it is the language that we use to describe math that has a great impact on student understanding.

Teaching Tricks I Have Learned This Year

I'm always experimenting with different ways to explain things, and each year, I am happy if I can just find one or two "biggies" that really make a big difference for my students. Here are the gems I gathered this year. There are a lot of them this year, since I taught all the same classes as I did the year before. It has been a good year to refine my teaching techniques a bit...


1. Visualizing solving equations as "unwrapping the onion" to get back to x. The kids who practiced this were awesome at re-arranging equations in terms of variables, even if the equation looks like this and they're asked to solve for Q:

R = S + 3Q - 5W
           T


2. Drilling equations of the form ax + b = c on mini whiteboards until every Grade 7 kid can do it in their sleep (including equations with fractional solutions). This really helped in the long run, because afterwards, when we moved on to much more complex equations, once they got down to an equation of this form, they could essentially do the rest while sleeping, and they wouldn't make any careless mistakes as their attention started to drift from the problem. It also helped them drill integer skills in algebra context, in the early stages of equation-solving.

Incidentally, I always read an equation like 5x = 3 as "5 copies of x is worth 3 in total. How much is each x worth?" I think this has helped my kids stay away from being confused about what to divide by what. If they can sound out the equation in their heads, the same way that I do, then it's not hard...


3. Always pronouncing fraction multiplication as "of" instead of "multiplied by", when reading an equation or expression out loud. For example, if the kids are simplifying (1/3)(x + 12), I say, "what is one-third of the quantity 'x plus 12'? Let's do it in steps. What is one-third of x? What is one-third of 12?" This helped to reinforce both the meaning of fractions and its connection to the multiplication operation.


4. Similarly, never let the kids get away with telling you that they don't know how to find a non-unit fraction, such as four-fifths of something. Always rephrase the question as, "What is one-fifth of ________? Then, how would you find four-fifths?" Push them, push them to do more in their heads. Don't let them think that fractions are harder than they are, or let them think that not knowing the meaning of fractions is OK.


5. Highlighting matching descriptions or units inside a proportion. Worked superbly to help kids set up proportions correct, consistently!


6. Teaching log by slowing down and focusing on its definition. This is a tried-and-true method for me this year; my students never had any weird issues at any point with logs this year, and I was able to repeat the same success with different students (some new transfers, some from other grades) at a later point. They were calm, independent, and their work all made sense from the start to finish during the entire unit. This had never happened to me before while teaching logs in any other way!!


7. Teaching sequences by making kids make a table of values (index vs. actual value) every single time, prior to setting up any equations. For some reason, this really slowed them down to thinking about what the word problem is giving them to work with, and they were consistently successful at tackling a variety of problems without getting confused.


8. Teaching Calculus by making kids sketch f'(x), f''(x), or f(x) graphs, given related graphs. They must do this consistently at the start of each class before moving on to work on anything else. The graphical understanding will underpin their entire algebraic understanding of Calculus, and help to bring everything together.


9. As soon as the kids differentiate a function via algebra, they must write down next to f(x) and f'(x) some word descriptions, such as "Height" for f(x) and "Gradient" for f'(x). This will build their independence in choosing the correct function to plug x-values into, and free them from having to ask you what to do at the next step of their analysis. Nag them while supervising/going over every problem, to write down these descriptions. Eventually, they won't need this anymore and they can visualize the descriptions in their heads. But, this builds their independence -- fast.


10. Repetitive quiz practice, on a complex topic, until you feel that it is quiz-worthy. This builds their confidence, while focusing their attention on a key skill, integrated with other skills they've seen before or that are nice to have.


I'm not done with the school year yet (still doing things that I'm pretty excited about, for the last few weeks of school), but I think that these are the little things that have made the most impact on my students' achievements/understanding this year. I hope that they will help you as well as they have helped me!! 

Why Exploratory Learning Makes a Difference

We were randomly reviewing word problems on the last day before the end-of-year exam, and I wrote up a word problem on the board: "The height of a certain triangle is five more than its base. The area of this triangle is 42. Find the base and height of this triangle." Nothing fancy, standard quadratic problem (not based in real-world context, I know), except we haven't worked much with triangles this year.

I walked around and noticed that all the kids had drawn and labeled a diagram, then proceeded to write down x(x + 5) =  42. I asked the class how we would find the area of a triangle, and they said, "Base times height!" And I said, "No, that's for a rectangle, not for a triangle." Within seconds, all the kids who were in my class the previous year shouted out excitedly, "--Divide by 2!!!" Meanwhile, the other kids who didn't learn the triangle area formula via its connection to the rectangle had puzzled looks on their faces; my hint that "base times height" is the procedure for a rectangular area held no apparent meaning to them with regards to generating a triangle's area formula. Rote memorization of "the simple things" is really a problem, because if you start memorizing intuitive things like this, you'll never be able to memorize everything.

The issue here for teachers (not a new discovery, obviously) is that once a teacher just gives the students the formula, the students are no longer motivated to really understand why it works. Once a kid has a teacher who has taught them a certain skill by rote, their motivation to learn the intuition behind that particular skill completely disappears as the kids simply file it lazily into the "already learned, already can do" drawer in their mind. Please don't do this, as it's really hard to un-do down the road. For every concept that you rush through in order get to the "procedural" practice, those topics will never be properly understood, explored, and developed by the child. This not only robs them of the richness of mathematics, but also creates retention issues.

I know, it's not a new discovery by any means. But, I was reminded by this small incident, of the significance of exploratory learning.

Mythical Form

My 7th-graders have been doing some lovely exploration and estimation activities on circles. It took a few days, but I think it was well worth our while, as it helped the abstract formulas make sense to them.

My students today were boggled by the fact that if pi has different digits that go on forever, that means that either the diameter or the circumference is a quantity with also digits that go on forever. That means that we have a "measurable" (ie. finite) quantity that is, in fact, not truly measurable. Trippy, eh? For a moment there, I felt the beauty of abstract math peek its head into our Grade 7 class. The kids now think the circle is a mythical, awe-inspiring form.

Survey Project - I'm Back!

I recently gave a survey project to my 7th-graders, that involved them creating/administering a survey, creating circle graphs with a protractor, drawing conclusions from graphs, and making educated predictions for a larger population.

Here are some clear photos I managed to snap of a few of their posters. I am impressed by how informative they are, considering that the kids didn't turn in any rough drafts. (Note to self: Threatening to turn it into a full-blown writing assignment really helps to bring up the quality of submitted posters.)

See full-sized yellow poster here

See the complete blue poster here.

See the full-sized cream poster here.

PS. Yes, I got married!! Here are a few pics.

Belize looks like this:

Snorkeling with sea turtles!

 

Our beach ceremony (people sat in a spiral form). It was awesome to have 50 friends and family join us from as close as El Salvador and as far as Sydney and Shanghai!


Newly weds! (My dress worked out fine in the end. It was short, with a detachable train.)

A couple of days after the wedding, we had a sunset cruise with the guests who were still around. It was lovely!!!

Visualizing Concepts

Here is an MS update. I feel pretty productive lately, as I always do during the second semester. I also feel quite productive with my Grade 11s, and as a result I'm taking on three new kids potentially, at least for a while. Grade 12's are doing OK, but the pressure is sure ramping up for their IB exams, so there's not a whole lot of "cool" instructional things that I can be doing with them...

Grade 8:

Referring to the system {5u + v = 21, 10u + 3v = 48}, one middle-of-the-road kid explained excitedly to her friend, "Oh! I got it! If 5u + v = 21, then 10u + 2v must equal 42. If you put that here into the second equation, then you will still have 1v on the left side. Which means that 42 + v = 48, and v must equal 6!"

Not bad for being day 1 of systems algebra. (Of course, they've been doing the same reasoning using shapes and visual substitution for several days now, so the transition to symbols was seamless. I didn't bother with even any examples on the board this year, and it worked out just fine without one.)

I went up to a boy who was one worksheet ahead, and pointed at the equation {y + 6x = 10,
y + 2x = 2} to ask him what immediate conclusion he could draw based on inspection. He immediately said, "X is 10 minus 2, divided by 4." I had to backtrack to ask him, "How do you know? Because 8 is equal to....?"

He finished my sentence, "4x." Great, now we're talking about the process.

I showed him the traditional notation for showing that work / reasoning via elimination, even though I know that in his head he's doing substitution from one (smaller) equation into the other.

  y + 6x = 10
-(y + 2x = 2)
        4x = 8

He was not impressed, nor surprised. When you introduce systems via pictures, the symbols become just part of the bigger concept. I explained to him that on the previous worksheet, I had let them just do much of it in their heads, but now we're going to work on the written communication piece, to carefully show our work on paper. I also gave him the hint that sometimes, instead of subtracting the smaller equation, he'll notice that he wants to add the equations instead. I said that he'll know when addition is necessary, because those equations will just "feel different." I left him on just that hint, and sure enough, 20 or so minutes later, when I came back, he had already identified the equations that needed to be added, and he was ruminating over the explicit mathematical reasons why that works. At that point, I felt that he was ready to discuss that if you have additive inverses, then you can just add them to cancel them, so we had a 1-minute discussion about that and I left him again.

I am very pleased with how this unit is going. I think I've gotten it down pretty pat; if I remember correctly, last year I didn't have to change any worksheets at all, and so far I haven't had to alter any worksheet either. This is one unit where I can just sit back and watch the kids' thinking unfold, more or less, and I can be very hands-off in their own building of the concepts of algebraic substitution and elimination. At some point, I looked around the room when I noticed the noise level rise, and saw that it was because literally every pair of kids is engaged in some sort of intense mathematical discussion with their neighbor. Awesome-o!

Grade 9:

In Grade 9, we've been working on learning / understanding / memorizing geometry area formulas via cutting and pasting. We each re-arranged, glued down a parallelogram to form a rectangle, to help them understand why A=bh for a parallelogram. Then, we each re-arranged a trapezoid into a thin parallelogram, to help them understand why A=(b₁+b₂)(h/2) for a trapezoid.  This year, I went a step further and highlighted the base 1 and base 2 sides using a green marker, so that the kids can see that they line up within the parallelogram when we do a "move and flip" of the top half of the trapezoid. This simple highlighting technique is superbly visual for my visual learners to see why the new parallelogram base MUST be (b₁+b₂).

Then, I proceeded to give them two practice trapezoid problems. In both cases, I had them draw out the parallelogram that it becomes, labeling the side lengths to emphasize the numerical connection between the trapezoid and its resulting parallelogram. It worked great! My concrete thinkers really latched on after the first example. One of the weakest kids in my class went up to the board and bravely (and correctly) drew out the trapezoid, its new parallelogram dimensions, and the calculated area, without any previous verification that he was on track. I was proud!!

I just love geometry lessons like this, because they involve both tactile and visual learners and help the concept "stick" in their heads. Slowly, I'm getting their geometry concepts up and running in order to do the 3-D project this year.

Addendum March 4, 2013: Here is a visual that Lara has made on Pinterest. Thanks, Lara!

Grade 7:

In Grade 7, we've also been doing Geometry... I'll have to report on that at another time, but I wanted to say that recently (prior to the Geometry unit), I discovered the best technique for teaching setting up of proportions. For ages it used to bother me that kids would not check that two ratios have matching units across the numerators (and across the denominators), and so half the time they would set up incorrect proportions. I figured out a trick this year. The first day we learned to set up proportions, I had them highlight the matching units across the equal sign. For that whole first day, they had to always highlight the units inside the proportions, and check that the matching colors lined up horizontally. After that first day, the highlighters went away and I never referred to them again. In the end, I didn't have a single kid mix up the positions of the numbers inside the proportions on the test. I think that mentally, the colors stuck with them and they're always visualizing that check, even when the highlighters aren't around anymore. 

Again, slowing the kids down in the beginning definitely pays off, I think. The highlighters are a trick that I'm trying to use more and more, to incorporate hidden visualization techniques that some of us "math people" tend to internalize in our minds but that teenagers who are used to rushing through things, could need more explicit instruction on. So far, so good.

Good Things and Bad Things

Good things:

• I've been thinking about little changes that have big impacts. For example, recently my colleague asked me for some articles on teaching with technology. When I was reading up on various research done about teaching via graphing calculators, I learned that how the teacher teaches with the calculator actually has a great impact on student learning and flexible problem-solving. If a teacher always emphasizes the connection between algebra and graphical analysis using the calculator, then even when you take away the graphing calculator, more of the students are able to think flexibly of multiple modes of solving problems. So, I have been pushing my Grade 8 students to be more and more reliant on the calculator as a daily tool, rather than just irregularly incorporating it.
• This change has allowed me to take on an even more passive role in my Grade 8 class (which is good, because that means they have to be even more independent). Now when I go over answers to worksheets, we only go over a subset of the answers, during which I call on a student, they provide their answer, and then I turn to the class and say, "Does everyone agree?" If they agree, we go on, and I never have to say true or false. If they disagree, then I pick a person to say step-by-step how they did the problem, and after each step I ask the class, "Do you agree with everything on the board?" Eventually, the class helps them to find their mistake, or we all agree on their answer and other kids try to figure out their own mistakes. After reviewing about half of the worksheet answers, I give the class another 10 or so minutes to verify the rest using a graphing calculator. My 8th-graders have become really good at graphing a function on the TI, adjusting window range, and then using the numerical-entry feature of Trace to quickly verify (x, y) pairs on the graph. They also know that they need to graphically check 2 points on a line in order to verify its equation, and they know how to verify their predictions along the line such as checking the value of k in (1000, k), or checking the value of n in (n, 849). On the test, I built in extra time for them to just check everything on the graphing calculator, and in the end, the kids said that the test really wasn't so bad. (Even though it had at least one quite tricky PSAT problem and other parallel, perpendicular, collinear testing problems that are fairly complex for Grade 8.)
• My 7th-graders are getting very communicative about math. Today, we played a modified Bingo Game to review for our test on Thursday. I had them write in integer values of -5 to 9 in a 4-by-4 grid, with 1 "freebie" space anywhere. Then I started writing questions on the board, one at a time. Nothing special, except we weren't going over the answers like we normally would. Once they determined the solution to a problem, they can cross that solution off of their grid, but they had to put the problem's letter (A, B, C, .... etc) next to the crossed out number, so that if they got Bingo, we could verify that they actually had all the correct answers associated with the correct problems. Sometimes I noticed while walking around that the kids were getting stuck on a problem, so I would ask, "Who can give a hint for how to start this problem?" and kids would eagerly raise their hands to offer hints. Along the way, they offered many hints like, "Cross multiply!" "Reduce before you divide!" "Find the common denominator!" "Check by putting the values into the equation!" and they also helped each other set up the percent increase/decrease problems as proportions, multiplying decimals, and finding "weird percents" like 0.1% of 3000 or 400% of 0.5. These 7th-graders are not just getting really good at algebra, but they're getting all the descriptive terminology down, too! Sometimes, they noticed that they had marked the same number as being called twice during the same game, and they had to go back to figure out which problem was solved incorrectly, and that was another way of having them self-monitor instead of me monitoring them. Eventually, when someone called out, "Bingo!" they would give me the problems and the solutions associated with those problems, and instead of me saying whether each answer was correct, I would ask the class. If the class agreed, we'd let the kid go on to the next number. Else, we stopped to go over the problem on the board. Again, I keep thinking about how I can hand over more and more of the "correctness" control to the kids, and today was a good day in Grade 7 for that.
• I recently started my weekly lunch review session with my 12th-graders. I told them right off the bat that these sessions are totally voluntary, but the kids who come tend to do a lot better on the IB exam. It's not the one-hour studying during lunch that makes the difference. In fact, when they come, they just sit and do independent mixed practice using old exams without my help really. I am helping to model what it should look like to study at home, and my physical presence builds their courage to try unfamiliar problems, I think, knowing that I can be there to help if they do get terribly stuck. The first session went very well last week. I plan to alternate between non-calculator paper and calculator paper each week, in order to build up their ability to switch gears and to think in a different mode during a different setting. So, this week we'll be doing a calculator paper. Whatever they don't finish, they'll just take home as additional homework, since I expect that they're now putting in at least a couple of hours each week to do mixed practice on their own. I have seen them grow a lot during the last year and a half, and I know that they will do well if they put their minds to it.

• In the end, I received some very positive feedback from those of my 9th-graders who had put in a lot of work into their videos project. They said that even though in the beginning, they weren't totally comfortable with the topics that they had chosen and the problems that they needed to explain, by the time that I had made them re-do and re-do it, they thought the concept was very easy in the end. The question that remains is only how I can manage this in the future for all kids, even those who put in minimal effort, and how to extend this level of articulation to all topics, and not just the one that they chose at the semester mark.

Bad things:

• I am sick and still allergic, and I feel like I am walking around in a fog. I really hope that I get well by Saturday, since I'll be seeing Geoff for the first time in over a month! (He has been working away from Germany, and finally I'll be visiting him during my February break.)
• I also lost weight recently, probably due to stress and all that jazz. It's definitely not intentional, but now my wedding dress is too big and I will probably have to take it back to the store again. I am feeling quite anxious about this, because now the clock is ticking and I don't want to risk another alteration. blah.

Intensive Feedback for Every Kid

I have said this before and I will say it again: I find that mini whiteboards are wonderful in giving immediate feedback to students and receiving immediate feedback from them. This year, after I started using mini-whiteboards on a semi-regular basis in Grade 7, I have seen my students growing leaps and bounds in their accuracy. They absolutely love those lessons because they love to be recognized for being correct. (I usually say after a complicated problem, "Pat yourself on the back if you got that one completely correct." They love patting themselves on the back. heehee. And come on, who doesn't?) By now they're used to the idea that when I say something during the lesson, it is going to help them during the mini-whiteboard practice, so their ears actually perk up to listen. That is night-and-day compared to their attention span on other days.

In one 80-minute lesson last week, we reviewed: multiplying a 2-digit number by a 1-digit number in our heads; finding the common denominator of two fractions like (3x + 5)/24 = (6 - 2x)/7 as an application of this arithmetic skill; cross-multiplication and why it works (this was following DAYS of fractional equations practice, so I felt that at this point they were ready to bypass the denominator part and ready to see why the numerators would change as such); solving various proportional equations using cross-multiplication; solving percent word problems using proportions. In fact, they were so great with this exercise that they were able to figure out that something like (3x - 5)/6 = (2x + 7)/4 would have no solution, which is a topic from a while ago that I just threw in to the mix.

In one lesson, basically all the kids practiced and understood all of these skills. Of course we'll go back to individual/paired practice this week as we build up towards a formal assessment, but having their intensely focused attention for 80 consecutive minutes and receiving/giving constant individual feedback from/to every kid is simply priceless. It does wonders for their progress towards mastery.

In fact, the word has gotten out that I use these mini whiteboards regularly and that I love them... Other-subject teachers on my floor have started to borrow them from me to use in their classes. Great!

...Wow. We're so ready as a class to move on from basic algebra to explore basic geometry. I can actually feel the anticipation! :)

Gestures, Language, and Student Understanding

I went to a fabulous AGIS session this morning on sign language in the elementary classroom. It was led by Armin Martin and Johnnie Wilson from the Munich International School. They showed videos of kids who use their hands to touch different parts of their heads (front, back, left, right, top, bottom), in order to figure out how many "faces" a cube has. The kids are able to link the mathematical word "face" to layman's definition of "face", in order to bridge the gap between the concepts.

A very interesting point that was made during this presentation was that signs can be used as an intermediary between normal language and academic language, or between home language and school-instruction language. Intentional incorporation of appropriate signs can be a strategy that works not just with our ESL / EAL population but even with our normal kids, and it ties nicely into math because when you gesture in space, you are quickly illustrating and bringing in extra dimensions that are hard to do/experience on paper. The presenters presented research that said that even when you later take away the gestures, the kids still retain the primitive, physical understanding that they had achieved through gesturing. So fascinating, because this discussion/session got me thinking about a lot of different things that previously I had thought to be disconnected:

1. Recently my 7th-graders have been working on percent word problems such as "64 is 40% of what number?" Sure, some kids can easily navigate the proportional reasoning --> 10% of that number must be 16, so 100% of that number must be 160. But, for many kids they need a different strategy, and so we have been practicing setting this up as a proportion. Even then, for kids to read a problem like this and then to consistently set it up correctly, is not trivial. They need to be able to:

a.) Parse the verbal description.
b.) Correctly associate the value given (in this case, 64) with either the fractional percent (in this case, 40%) or the whole (100%).
c.) Set up proportion accordingly.
d.) Solve algebraically.

Of this, part B is the most difficult for 7th-graders. I found this year that when I went around to conference with kids about this process and to help them get started on the assignment, I can just point at the value within the problem (64), and then gesture to them using distance between my hands to ask, "Is this the part or the whole?" This has helped them tremendously, because they can associate the rather multi-stepped numerical operations to a simple visualization, and then they only need to focus on part A (re-reading the question to themselves) in order to make that determination and to carry out the rest of the steps by themselves. This simple gesturing was able to shrink my conference time with each kid to under 1 minute; they immediately would say, "Oh, I get it now," and then proceed with the other questions which were not always phrased in the same way or giving the same information. ("42 is what percent of 70?"  "What is 20% of 95?" etc.)

2. When I taught 8th-grade back in New York, I taught one particular 8th-grader who was very economical with his words. I would always put explanation questions on the test, and he was so concise with his explanations that he could always write down the correct answer in about half of the word that I would need to use. This is an incredible skill, because in order to do this, the kid needed to:

a.) Know/master the concept and relevant vocabulary.
b.) Prioritize information in his head.
c.) Formulate his understanding in as few words as possible using the prioritized list.

3. I have read somewhere that babies can already understand physical rules in the world. If you play an optical illusion on them that is against their normal experience, they will keep staring at the place where the ball disappeared after dropping. At this stage, their understanding is far beyond their ability to verbalize it. So, I think it is definitely true that we understand a lot more than our words are able to describe, especially at a young age or as a language-learner.


So, in short, I think gestures are a fabulous way to help kids understand concepts when their language is not yet developed enough to explain or describe a concept (old or new) fully. But, this intentional incorporation of gestures should lead in mathematics to a formalization of those concepts, and attachment of specific language. Because in doing so, we are teaching the value of specificity and prioritization, which are very important skills for the older students to have/develop in the long run.

Hope this little reflection was helpful to you in jogging your brain about how to bridge abstract concepts quickly to visual/kinesthetic understanding for kids. Please share with me if you have had similar success in other examples of utilizing intentional gestures to build intuitive understanding!

Grade 7 Project: Percents and Shopping

Speaking of projects, I recently had a very lovely experience collaborating off of my colleague's percents project. I took an old project that she had written up and added extra components to it, and then gave the feedback back to her so that in the future she might incorporate the new elements. I really loved this iterative process, and she was happy as well!

Here was her original project.
Here was my add-on.

Essentially, I took the same percentage discount elements (my kids are still calculating percents by dividing by 10 to get 10%, and then working through proportional reasoning) and then added a creative component, where percent increase (shipping fee) was incorporated into this project by asking the students to come up with their own online shopping story. I also would only check the first two steps (1A and 1B) of the project, plus the calculations for their individual creative piece. The rest of the project had to be checked some other way, so they had to brainstorm ways to check their work -- and they'll be graded on this "reflective" process in addition to their overall explanations, clarity of thought/work, and accuracy.

In the end, the kids helped each other to identify a variety of math concepts (decimals, rounding, percentage, addition vs. subtraction, sequential application of percentage change, checking your work using inverse operations) that were required in this project. I'm still waiting for their writeup, to be due next week, and then probably a revised version (depending on how well the drafts are done the first time). I'm quite happy about how it has gone thus far!

These same 7th-graders also rocked their algebra, fractions, and basic percents on the first semester test. They had come in to my class not knowing integers, order of operations, or how to calculate even simple fractions. I'm so proud that they ended up doing better on the semester exam than my previous students did a year ago!!! WAY TO GO, GRADE 7. Rocking my world, one day at a time.

Review Week

It's so nice to just slow down before the semester test and to review everything we have learned. It's like taking in a deep breath, because finally we are no longer trying to cover as much ground as possible. Finally the kids have come to a reasonable stopping point before THE TEST next week.

So, this week we will "just" do some review in Grades 7 and 8. For that, up my sleeve I have:
this awesome stations review format from Amy Gruen; my normal speed game format (which I may vary up this time to have 3 kids go up at a time to the board, working individually and not keeping team score), and of course mini whiteboards.

Looking forward to the rest of this week! :) There is nothing like fun with review time (coupled with the kids doing extra review problems at home, obviously).

11 Recommendations to Middle-ish Grades Math Teachers

Maybe some of these are "radical" and "offensive", or maybe they're not. I'm just throwing them out there. Please don't hate me if you do some of these (I know a couple of them are quite commonly done by many math teachers). These are just my personal opinions, but I feel quite strongly about them. I figured I'd say it here, because some of these things drive me nuts, and I need an outlet.

11. Do not tell a kid to "move things across the equal sign and just change the signs". Preferably you don't ever say this phrase to a kid ever, but if a kid comes to you and says that someone else somewhere (ie. a parent or another teacher) has taught that to them, you need to drill into them the reason why this shortcut works. Grill them until they can articulate why the term necessarily shows up on the other side with the opposite sign.

Reason: If you teach shortcuts like this without the correct foundation of understanding, then down the road the kid cannot approach a complex situation intuitively, because they cannot visualize the equation as a balance.

10. Do not allow your students to do simple equations only in their heads / showing little or no work at the early stages.

Reason: Even if the kid is bright and they can do the equation correctly every time, you are helping them to develop a bad habit that is hard to break. Down the road, when the equations get more complicated and the kid starts to make procedural mistakes, they'll have such a hard time finding their mistakes because they've habitually skipped so many steps.

9. Do not allow your students to "open parentheses" without knowing why they do this and where it is the most useful.

Reason: Normally, it just irks me to see kids do 2(3 - 5 + 1) = 6 - 10 + 2 = -2 instead of following PEMDAS to do 2(3 - 5 + 1) = 2(-1) = -2, but it's really feckin' scary when I see students do
2 (4·5 + 7) = 8 + 10 + 14 = 32 in their second year of algebra. When all the values are known, they don't need to be distributing!!

8. Do not introduce integer operations without explaining the meaning of addition and subtraction of signed numbers! You can use number lines or you can use the idea of counting and neutralizing terms, but in the end, the kids need a framework of understanding.

Reason: I recently tutored a kid on a one-time basis (I was doing a favor for her parent, who's my friend, even though the kid doesn't go to our school), and the kid only knows memorized rules (she rattled them off like a poem), with no conceptual understanding whatsoever when I probed further. Unfortunately, that kind of problem is not really fixable in a day. It can take a teacher weeks to instill the correct habit of thinking. If you teach a kid to memorize a rule for adding or subtracting integers without understanding, then I can almost guarantee you that after the summer they won't remember it. And, if you first teach the rules by memorization, even if you try to explain afterwards the conceptual reasons for the rules, there ain't no one listening.

7. Do not teach "rise over run".

Reason: In my experience, a majority of the concrete-thinking kids out there do not know what the slope means. When you ask them, they just enthusiastically say "rise over run", but they cannot identify it in a table, they cannot find it in a word problem, they certainly cannot say that it's change of y over change of x, and half of them cannot even see it in a graph.

You have other alternatives. I teach slope as "what happens (with the quantity we care about), over how long it takes." For example, the speed of a car is "miles, per hour"; the rate of growth of a plant is "inches, per number of days"; the rate of decrease of savings can be "dollars, per number of weeks". You have alternatives to "rise over run". Learn to teach kids to visualize the coordinate plane as a sort of timeline, where the y-axis describes the interesting values we are observing, and x-axis describes the stages at which those values occur. Once they understand this, then they can correctly pick out the slope given any representation of the function.

6. Do not let any kid in your class get away with saying "A linear function is something that is
y = mx + b."

Reason: Again, in my experience, I can habitually meet a full class of kids who had spent a good part of a previous year learning about lines and linear equations and linear operations, and in the following year when they move to my class and I ask them to tell me simply what is a linear function, the only thing they can say is: Y = mx + b. That is really, really bad. The first thing they should be able to say is that it is a straight line, or that it is a pattern with a constant rate.

Please, please, do teach lines in context. Make sure that kids know what y, m, x, and b all mean in a simple linear situation.

5. Do not introduce sine, cosine, and tangent without explaining their relationship to similar triangles.

Reason: The concept is too abstract, too much of a jump from anything else the kids have done. Give the kids some shadow problems to allow them to see what the ancient mathematicians saw, and why they recorded trigonometric ratios in a chart. Let the 3 ratios arise as an outcome of a natural discovery, a natural need. And then, when you introduce the terms sine, cosine, and tangent, the kids will at least already understand them as sweet nicknames for comparisons between sides.

From then on, whenever you say sin(x) = 0.1283... , remind kids what that means simply by immediately saying, "So the opposite side is about 12.8% as long as the hypotenuse." If you do this consistently, kids will never grow afraid of those ratios.

4. Do not teach right-triangle trigonometry from inside the classroom!

Reason: Kids need to be able to visualize geometry relationships, especially in word problems. Some kids have trouble doing this on a flat sheet of paper. Don't disadvantage those learners. Take them outside, make them build an inclinometer, make them experience angles of elevation, angles of depression, and the idea of measuring heights through triangular ratios. This really only takes a day, or at most two. In the end, when you take them back to the classroom, you will be amazed at how well they can now visualize even complicated word problems involving multiple right triangles.

3. Do distinguish between conceptual and procedural errors. Your modes of intervention should look very different for those two types of mistakes, and what you communicate to the kids as recommended "next steps" should look fairly different as well.

2. Do incorporate writing into your classroom. Writing is an extremely valuable way of learning mathematics. Every project should have a significant writing component. When kids write, they are forced to engage with the subject on a personal level, and so they learn much more.

1. Do make learning fun. From a biological perspective (of primitive survival), our brain makes us remember things permanently when our emotions are triggered. Fun learning isn't a waste of time. It's necessary to help the students build long-term memory!

Proportional Reasoning with Percents

I am a big fan of estimation, probably because I am lazy. I think that most percent problems (obviously not all) can be done with just basic numerical / proportional reasoning.

For example, my Grade 7s are pretty darned good by now at doing something like "find 15% of 6.4". They can articulate that 10% of 6.4 is 0.64, and then half of that is 5% = 0.32, so 15% is 0.64 + 0.32 = 0.96 . They can also do something like "find 12% of 88" by reasoning that 10% is 8.8, and then 1% is 0.88, so 12% is 8.8 + 0.88 + 0.88 = 8.8 + 1.76 = 10.56 . Recently I looked over the big semester test that we had given last year in December, and realized that we won't have time this year before the semester exam to cover enough of proportional reasoning to do a problem like "9 is 15% of what number?" using a setup of proportions and of solving by cross multiplication. (We've only just started proportional reasoning this week.) So, today I just decided to throw up some of those types of questions on the board as the Do Now, and the kids were surprisingly clever at figuring it out! One of them said, "It's not that hard. It's like playing Math Detective."

"3 is 10% of what number?" --> This was the first "backwards" type of question I had put on the board, and the kids thought it was quite easy. You're just multipling by 10 times to get to 100%, which is 30.

"12 is 15% of what number?" --> Some kids naturally, intuitively, broke it down to 5% being 4, which is a "nice" percentage to have/know because then you know that 10% is 8, and therefore 100% is 80.

"6 is 2% of what number?" --> To do this, they were also clever. Some did it as 1% is 3, so 100% must be 300 if you know that one out of the 100 parts is 3. One kid immediately thought that you can scale both values up by 50 times in order to reach 100%, to save us from doing two steps of work.

I really liked these discussions, because I think -- in lack of a formal proportional setup -- this is the most intuitive way of approaching percentages, in my mind. It reinforces the meaning of percents and frees kids from fear of these fairly basic everyday concepts.

We discussed briefly why this is proportional reasoning. It tied very nicely to the word problems we did today on proportions, but I am not going to touch cross multiplication until January. (It's just too much, too soon for many of them, and they have already worked very hard to fill in the gaps this semester.)

So, go seventh-graders! I feel very hopeful that, despite my concerns about many of them at the start of the year, the majority of the class is nearly caught up to where they ought to be at this time of the year.

Teaching Number Definitions Meaningfully

As a kid, I was never good with mathematical terms. I was always doodling in math class and only picked up vocabulary words osmotically (which also meant not so effectively). As a teacher, I have tried to make vocab instruction more meaningful for kids.

Recently, in Grade 8, we started talking about inequalities for the first time. I started off the discussion by asking kids to give me some examples of equalities, and we wrote them on the board. After a few minutes, we started to list examples of inequalities and I went over why in math, saying -5 is less than -2 is a bit more precise/less confusing than saying -5 is smaller than -2.

Then, I asked the kids to come up with some example numbers that can satisfy the inequality
x + 2 < 15 . The kids started to list numbers, and after a few minutes I asked them for what they wrote down, and in the context of this we discussed number types. I explained to them that I think generally, in life, when you want to brainstorm options in your head, you don't want to keep listing the same types of objects over and over again, because in doing so you are limiting your vision of what is possible. The examples that the first kid gave me were {1, 2, 3, ..... , 12} and the examples that the second kid gave me were {-1, -2, -3, -100}

I drew on the board this diagram and said that "a long time ago, when cave people started counting sticks on the ground, they came up with numbers like 1, 2, 3, 4... These were called natural numbers.* Then eventually they came up with one more number to add to that set, and they called it whole numbers. Can you guess what that new number was?"

After kids enthusiastically guessed zero, they were starting to understand this diagram representation of a subset and were beginning to appreciate the historical development of numbers. Then, I added another layer of negative numbers and asked them what that is called when we started at some point to include/consider negative numbers as well.

They figured out that it was called integers. Great!

Then, I asked them what types of numbers they still know / have learned that we haven't named yet. They gave examples of decimals and fractions, and we added them to this picture.

In the end, we went back to the original topic at hand of finding example numbers that satisfy the inequality x + 2 < 15, and this time they were much better equipped to list a variety of examples and to discuss the full range of (non-discrete) solutions, which then led to the discussion of shading the solutions on the number line, and why we need the open circle rather than the closed circle sometimes.

I find this to be a more natural way to teach number types. Another time when I have done this was in teaching 9th-graders how to think of possible counterexamples that might disprove a math statement. If in their heads they are only considering a single type of example, then they're not being effective and thorough in their consideration of possibilities. As a kid, I would have appreciated this type of instruction of categorical types, followed immediately by application of its usefulness, and it would have probably helped me remember the names better. So, for me as a teacher, I always think it is important that I don't introduce the number definitions purely in isolation just because it's part of section 1.1 in the textbook. In the end, our teaching of these numerical categories should be explicitly supporting the kids' thinking, rather than just to add to the volume of disconnected rote knowledge in their heads...

*Note: By the way, I prefer this definition of natural numbers, even though I know that mathematicians don't all agree. Some refer to zero as part of the natural number set. 

Visualizing Order of Operations

This year my Grade 7s came in to my class with some missing prerequisite skills. Half of them started the year not knowing integers or order of operations or how to calculate simple fractions and percents, which are all supposed to be prerequisite knowledge for Grade 7 at our school. So, (although I did go back and fill in those gaps more or less,) teaching them algebra skills on top of this shaky foundation has been a new challenge.

At some point, I realized that it was difficult for these kids to look at a formula (in simple algebraic evaluation, for example) and to visualize what operations need to happen first, second, third, etc. They know the rules for PEMDAS, but it is just hard for them to always do it consistently. So, I came up with a trick of teaching kids to circle the operations in the given exercises in a layered manner, so that they can train their eyes to look for the operations, rather than to look at the equation from left to right.

Something like:

My special ed helper agrees that this is really helpful for them to visualize the rules. My only frustration is that they don't do this consistently because they still think they can just see it all in their heads, and in doing so they end up missing an extra negative sign here and there and throwing off the entire answer.

Anyway, I think the same trick can be used with certain types of equations to help kids see the layering and the process of peeling away the onion.

So, this is what I'll focus on for the next couple of weeks, to see if it can help solidify their foundation with this type of layered equation. (They're pretty OK with the ones with x's on both sides, since we had started off with doing balance visualization and crossing out shapes.) Basically, anything that can help them sink their teeth into abstract representation is worth a try for me. Any other ideas on how I can help these kids?

Fractions Project

My fractions project in Grade 7 is almost done! We had taken a break from it to work on some introductory algebra stuff (and to allow kids to turn in additional rough drafts to me outside of class), but on Tuesday, I gave the kids a full 80-minute period to work on their final drafts, and the polished final drafts will be due on Thursday when they walk into class. I was very pleased with the upper-end products I have seen thus far (complete with explanatory diagrams), and I think they showed me that all kids had to learn some new skills and concepts in order to complete this rather involved project. The kids towards the bottom half of the class required a lot of hands-on support on this project, and some of them will be turning it in late because they are still struggling with basic things like responsibility... (The transition from Grade 6 to my Grade 7 class has been a pretty tough reality check for some of them in terms of my expectations for what they need to complete, by what time, and how well. For three of them, I have extended the deadline and made explicit to them that they need to come conference with me later this week during lunch, and then bring me an awesome final draft following the week-long October break.)

So, I think the specs are a good starting point for a decent fractions project. They are poorly formatted though, in hindsight. For itty-bitty beginning-Grade 7 students, there was too much information all on one page, and as a result it was difficult for the kids to find on the page the culminating big-idea questions (at the bottom of the page) that they needed to answer; if I am to repeat this project in the future, I'd have to re-think how I would do this, maybe provide them with a layout or outline to help them structure their write-up, since it is the first time they have had to do a math writing of this length and complexity.

Here it is on Dropbox. If you do use it as a basis for a fractions project, please share with me your modifications so I can make mine better for next year. Thanks! If you check back on this post, I'll also update it next week to include some fragments of student explanations (and/or maybe photos of finished dartboard designs).

Fractions Project

Do you have a killer fractions project that you just love?? I made one involving designing a dartboard with specs written in fractions, but until it's done, I cannot decide whether I like the project or not. It has been quite tricky to get my Grade 7's through this project, much more difficult than I had imagined. I spent an entire lunch time with some of the weakest kids in the group, going through it step by step to show them why one-fourth of something means you are dividing by 4. And then, how to divide big numbers by single-digit numbers in order to calculate how many squared units would be in each fractional part.

After this individual calculation/planning phase (which, thank goodness, we're mostly all done with), they will design the dart board in groups, and then after they "build" the dart board, they will go back to working on individual written explanations of all the calculations. This is their first writing assignment -- possibly ever -- involving math!! Hurray! Big step for these itty-bitty incoming 7th-graders. But, it's all making me a bit stressed out that we still haven't managed to start the "regular Grade 7 curriculum." I hope that happens very soon, in a week or two. These kids are going to need to HAUL ASS to get through the "normal" Grade 7 algebra topics by December. sigh.

5-Minute Drills

I am doing daily 5-minute drills this year in grades 7, 8, 11, and 12. It started off with my frustration that kids cannot remember unit circles, even though we had worked on them, explained them, practiced applying them. I was just fed up with them not memorizing the circles. So, instead of feeling frustrated, I decided that during Grade 12 we'd do daily drills of the unit circle at the start of the year. On Day 1, I asked them to take out a piece of scrap paper and to fill out the coordinates of Quadrant I of the unit circle. They failed miserably, so we went over again the hand trick for remembering the coordinates quickly, and I said that at the end of class I'd ask them to do it again. By the end of class, there was much more success (maybe half of the kids were able to get the coordinates correct). I think the immediate feedback helped to motivate them. And then I told them that we'd do the exercise again the next class. And we did, at the beginning and at the end of the next lesson, tagging on to the unit circle basic equations to solve within the range of 0 to 360 degrees. I said that the next time I see them, this'd be a quiz collected for a grade.

In Grade 11 we're doing something similar, but primarily to review older prerequisite skills (such as writing line equations) that I think the kids should already know, and that I only wish to brush up on. We would do the same skill at the start of class, end of class, and next start of class. And then soon we'd have a mini quiz on it also at the start of a class, collected for a grade.

So, a long time ago when I taught middle school for the first time, our school implemented daily quizzes. I kind of hated them, because it was so much grading, even though it was a good practice for the kids. I really like my new 5-minute drills, because I think they are the best of both worlds. The kids still feel the time pressure and the need to be correct, but they're not graded that often and it's less work for me. For my Grade 7 and Grade 8 students, I let them do two problems a day on mini whiteboards. (I got lucky and was gifted a class set, along with markers and erasers, when I sent out a request asking to borrow them.) This is important because in Grades 7 and 8 we are working on basic skills like fractions, percents, equations, etc. The boards are a nice way to quickly check in with all kids on a daily basis, and I can see who is sure of themselves and who is not. Now that I have taught with mini whiteboards, I really don't think I can go back! I love that  kids also write their normally snarky comments on the whiteboards instead of calling them out, so that only I can "hear" those comments. Cuteness. I saw one kid write "DUH!" on his board when another kid made an obvious observation. One day I kept them over the class accidentally (since our school does not have bells), and a kid raised his board that said, "Class ended!" So, they're great for classroom management as well as daily assessment.

I'm still trying all kinds of things this year, but working close to 60 hours this week is taking its toll. Grade 9 is my baby, because this again is a very low class, and this year a bigger group. I will be doing all kinds of experimental things with them, and if it works, I'll share the strategies with y'all. So far, so good. The kids are able to graph linear functions by making tables, and they're able to write linear equations from a table. Not a bad first week for kids who couldn't graph points on Day 1!!! They also go around and check off each other's answers, which I think is so awesome because they need to be building confidence alongside content knowledge.

So, 60 hours-ish this week. A bit rough. But I am loving it!!! I also really love sharing classrooms this year. I teach in about 5 different rooms, and I just love it. Even though I have to carry my supplies everywhere and it's a pain, I am all in other people's spaces and talking to them regularly as a result. It's really nice, because it forces me out of being in the workaholic zone. Anyway, I am hoping that things will calm down soon on the department chair side, so that I can get back to a normal work schedule and re-gain work-life balance.

I hope your years are off to a wonderful start! :) Hi web, goodbye web. See you soon, hopefully.

Math Pop-Up Book Samples!

Here are some results from my Grade 7 and Grade 9 math pop-up books! They are really fantastic. Up side: The kids were really excited about the project and doing good math all the way through the last week of the year, even though it was not graded. One kid even said she wishes she had been able to finish the book before her family packed all their stuff for their move. (No worries though; I'll make sure I mail it to her.) The down side: tomorrow's the last full school day and I'm still not done giving last-minute feedback on all these Grade 7 books. I plan on getting to school early tomorrow to finish mistake-searching on all of them, and then give the kids 40 minutes in class to help each other fix them before they take them home for good -- but, fixing on our last day!! What a rush job. :( But, they've already done a terrific job and most of them have only very small fixes still to make tomorrow. I think it's doable. Crazy ambitious of me though! I hope next year, if I repeat the same project, I'll have to be smarter about pacing it as to avoid all this end-of-year stress.

http://www.flickr.com/photos/averyseriousmimi/sets/72157630208077148/

PS. This post is a follow up to this post where I had conceived the idea and written a bit to include the project descriptions. If I were to do this again though, I'd re-work the project description to include the phases of the project:
Phase 1. Gather at least 4 problems per topic, of appropriate difficulty. Need to sufficiently address all listed sub-topics with these problems. (2 to 3 days)
Phase 2. Write rough-draft explanations and get them reviewed by the teacher. (3 to 4 days)
Phase 3. Start to build the book. Get the first pages checked to make sure format of "pop-up" is actually useful and interactive. (3 days)
Phase 4. Peer review, followed by teacher review, for mistakes in arithmetic or algebra. (1 day)
Phase 5. Fix the parts of the book as necessary. (1 day)

Estimation of Circular Areas

It's my first time teaching circle formulas. Teaching circumference was easy: First, we went around the room and measured a bunch of circular objects, recording both their diameters and circumferences. Then, we put the values on the board in a table and calculated the ratios between the measurements ("how many times bigger?") and observed that it was always just over 3. Then, we practiced estimating circumferences given diameters. Then the kids learned that the exact ratio is pi, and they practiced finding circumferences, both exact (in terms of pi) and calculator approximations, when a diagram shows them only the radius.

Then... I gave them this. It's probably not special, but it worked well for us!

Questions I asked (on a worksheet, so each kid had thinking space and doodling room):

1. What is the area of the shaded square?
2. What is the approximate area of the 1/4 slice of the circle? How did you get this?
3. What is the approximate area of the whole circle? How did you get this?

After individual attempts and a brief whole-class discussion, they then practiced estimating areas of various other circles using the same technique. (On their paper, again, so they can each think at their own paces.)

We have not yet discussed the circle area formula. I am insistent on this. Area formula has to come after estimation, so that they can remember why it makes sense that (3/4 * r^2)(4 slices) leaves you with a circular area of approximately 3*r^2, or pi*r^2 to be exact.

Yay 7th-graders for being good sports and being open to a world where we develop formulas together!