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.

Quadratic Function Project Brainstorm

I'm brainstorming / laying out my end-of-year plans for my 8th-graders. After their end-of-year exam in late May, we will close grades, but we will still have about 3 or so weeks of instruction, which is enough time to do something very rich and not have to coordinate with other classes (since we use the May test to do placement for Grade 9). Last year, I used this extra time to let the 8th-graders define their own math projects, which were plenty of fun, but I wasn't entirely happy with the rigor of their mathematical results. This year, I'm toying with the idea of doing an exploratory quadratic functions unit. (Technically, quadratic FUNCTIONS are a Grade 9 topic for us, but previewing it in Grade 8 is always beneficial.)

I'm thinking of making it largely exploratory, since by then pacing won't be much of an issue and I can let them really take the time to develop their conceptual understanding of quadratic functions, which is the essential access point to a lot of higher-level algebra analysis down the road.... The timing is tight (as it was last year with my other end-of-year projects), but I think it's still doable and has a lot of potential!!!

Let me know what you think. Is it an OK approach for intro to quadratic functions / basic function transformations?? This is based on my rumination about a different way to think about flexible factorization of quadratic functions.

Day 1: Developing the understanding of how to graph y = x² + bx.

Plan - In pairs, kids will be given y = x² + 2x, y = x² + 5x, y = x²- 3x, y = x² - 7x. to graph on the calculator. They will sketch results in their notes, recording all intercepts in the form (x, y), and writing a one-sentence hypothesis about what the graph of y = x² + bx will look like.

Then, they will be reminded that in a function, we can solve for the x-intercept(s) by setting the height of the point, y, equal to zero. They will algebraically show that their hypothesis works for all b values.

Day 2:  Developing the understanding of how to graph y = ax² + bx, which is a more general version of the quadratic function.

Plan - In pairs, kids will be given y = x² + 6x,  y = 2x² + 6x, y = 3x² + 6x, y = 12x² + 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the graph. They will show how to solve for the x-intercepts using algebra only.

Day 3: Developing the understanding of how the graph is affected by the sign of its leading coefficient.

Plan - In pairs, kids will be given y = -x² + 6x,  y = -2x² + 6x, y = -3x² - 6x, y = -12x² - 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the shape of the graph. They will show how to solve for the x-intercepts using algebra.

As part of Day 3, they will do some matching between equations and pictures of graphs and to justify their choices orally.

By the end of Day 3, they should also be able to explain in writing how to graph y = ax² + bx.

Day 4: Developing the understanding of the effect of the constant term c.

Plan - In pairs, the kids will put in a function like y = x², and look at its table in the calculator. They will be asked to generate a second function that would increase all y-values by 1. They will prove their new equation works, by showing the values of both functions side by side in the calculator (y₁ and y₂), and copying down the table values. Then, they will write down the formula for the new function.

They will then look at the graphs of the two functions to determine "what happened" visually to the original graph when the equation got changed that way.

They will keep playing around with this idea, translating upwards and downwards and checking both the table and the graph to observe/verify the effect of c. 

By the end of Day 4, they will be given a graph of two functions. One of the functions will have an accompanying formula, and they will be able to see visually what happened to the points on the graph. They will then need to "guess" at the equation of the other, vertically shifted function, and verify it in the calculator.

Day 5: Putting the algebra pieces altogether

Plan - One partner of the pair will have a set of sequenced instructions, to be given to their partner one step at a time. The first step will sound like, "Sketch a graph of y = x² - 9x, labeling x-intercepts with values." Then, after that has been successfully completed, the next instruction will be given: "Now sketch the result of shifting that graph vertically up 4 units, labeling the resulting images of those original points you knew." After that has been done, the partner gives the third instruction: "Now, write the formula for this new graph." Once the partner is finished, they verify their results using the graphing calculator's graphing and table features and write a brief explanation of how they checked their results. Then, they switch, and the new partner has instructions that has to do with a downwards facing function like y =  -x² - 6x, and repeat a similar sequence of instructions to generate a new graph, a new / related equation, and to verify all results against the calculator.

Both partners will then work together to complete problems starting with functions of the form
y = ax² + bx and translating those graphs vertically to get new graphs.

By the end of Day 5, they should be able to explain the connection between y = ax² + bx and
y = ax² + bx + c, and explain how to use this connection to graph any standard-form quadratic function quickly in under 1 minute.

Day 6: Practicing/drilling the connection between quadratic function equation and graphs

Plan - In pairs, they will start with a function y = x² - 9x + 1, highlight the first two terms, sketch that function using dashed lines, and then sketch in the "real" final function using solid line. They will repeat this a few times with different functions, until they can fluidly graph any y = ax² + bx + c function. On this day, they'll also learn to visualize the axis of symmetry and to write its equation by inspection of graph.

Day 7: Going backwards from a graph to an equation

Plan - In pairs, they will be given one quadratic graph with two "nice", symmetric integer points being emphasized on the graph, one of the points being on the y-axis. They will be asked to sketch using dashed lines what this function would look like if you shifted those two points down to the x-axis, and be asked to write the function equation of both graphs. They will practice this a few times.

At the end of Day 7, they will be given a quadratic graph whose two "nice", symmetric integer points are both not on the y-axis. This tests them to see if they can figure out that the translated graph would have an equation that looks like y = (x - m)(x - n) + p instead of y = x(x - n) + p

Day 8: Playing around with the idea of adjusting "a".

Plan - In pairs, they will import Dan Meyer's basketball photo into GeoGebra. We will discuss as a class the need to find a modeling equation in order to fully predict whether the ball will make it into the hoop. From there, they will choose two nice integer points, write the equation, and graph. If they notice that the curve goes through those two points but doesn't have the correct steepness desired in order to fit the photo, then they will create a slider value in GeoGebra and toggle the value of "a" until they get a good "fit" around the graph, and record their results.

As a class, we will then go over the idea of solving for "a" using an unused point (x, y) and link it to solving for the y-intercept in linear functions. They will solve for "a" this way to compare analytical results against the technology results.

Day 9: Modeling Individually

Plan - Following a discussion of examples of parabolic applications, each pair will find and import their own photos of "real-life" parabolic shapes from the web. They will then model the function in Geogebra both using technology and using algebraic analysis.

Day 10: Creating posters

Plan - Each pair will create two posters, one with the modeled functions overlaying the photos, and one poster explaining the general process of graphing y = ax² + bx + c and the general process of fitting an equation to a parabolic graph.

Day 11: Practice presentations

Day 12: Math fair for other classes / parents?!

Totally Silly but Works

I made up a totally silly call-and-response thing this year for practicing exponent rules (after we did the initial exploration, obviously, so that they could understand why the rules work). It's mad cheesy, but the kids totally remember the rules now!! The hardest part is keeping the clapping going, but I'm not sure if it's because of my students being totally off-rhythm in general or what (they're super suburban kids).

So we clap, step from side to side, and I say, "8B, are you ready?" and they chant, "Yeah, oh yeah!"

And then I call on a random kid, "Nora, are you ready?" and she chants, "Yeah, oh yeah!"

and then I call out one of the following: "Power times power", "Power to a power", or "Power over power" while holding up fingers in each hand (up to 5, obviously, in each hand) to represent the original exponents we're working with.

Depending on which one I call out, the kids need to reply with, "You gotta add them up!" "You gotta mul-ti-ply!"  or "You gotta can-cel out!" in a sing-song voice, and that kid I named would then have to say the answer (resulting exponent) immediately after. (For example, if I am holding up 3 fingers and 5 fingers, and it's "power times power, you gotta add them up!" then the kid would shout out "8!")

And then we'd resume with me calling on the next kid randomly. It's mad cheesy, but it works! Afterwards, they were all loose and happy when practicing exponent rules. Every practice problem I would put on the board, I'd ask them which rule can be applied first or next, and they'd say it back in that sing-song voice, "you gotta mul-ti-ply!"

Go kids for being good sports!! It helps to make a boring topic a little less tedious! Next year, I'll necessarily add dance moves to help our kinesthetic learners. (I already have them. I came up with them after we did the exercise.)

Yup... I've got little shame left. :)

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.

Perpendicular Slopes and Geoboards

I'm delving back into the nitty-gritty linear concepts with my 8th-graders. I came up with an idea of teaching perpendicular slopes using Geoboards. I haven't tried it out yet, but I think it's going to work. They can first try and build a right-side-up square on the Geoboard and draw it in their notebooks on graph paper. Introduce the term perpendicular. When you ask them to draw it on their graph paper, remind them that when we look at shapes and their side lengths, we're counting space units along the side, and not the number of pegs. Then, the plan is to switch it up:

Challenge them now to build a diagonal square in the geoboard. Have them discuss when they think they have found one. Ask them to draw it on their graph paper. And then, once they've found/drawn in their notebooks a bunch of different diagonal square examples, have them look for commonalities. If I now put just one rubberband or segment down to represent one side of the square, can you find out exactly where its perpendicular side needs to extend from/to? What are their slope values? Is there a pattern? Use this to lead in to the idea of reciprocal slopes with opposite signs.

I like this activity idea, because potentially, if it works, you don't have to teach the kids the "negative reciprocal" idea like they are robots. I'll try it this week and then report back. Maybe we'll even have some photos! (I need to keep reminding myself that I need to start accumulating photos of my homeroom kids, for the yearbook.)

And, anyhow, I am excited about my first Geoboard lesson of the year!!! There's nothing quite like handing rubber bands over to 8th graders and expecting them to put them to serious use. :)

Kid Writing Samples: Round 1

I talk regularly about how I think that kids must, and I mean MUST, write in the math class. It's a lot of work for you as a teacher because you need to give time to do something lengthy, and then give some time for them to start the write-ups during class, then you need to give them copious written feedback on the first draft, and then maybe have a discussion with the whole class about how to organize their ideas logically, and then you need to still read every kid's final draft and to assign them grades, and then to post a task-specific rubric. In the end, this process is so SO worth it, even if some of the kids' products are not stellar. The important thing is that when they write, they are really learning about what it takes to put their mathematical thoughts down in words in a way that makes sense to another person outside of our class, and they're forced to re-evaluate how well they themselves understand each step. Moreover, if they are asked to make a draft and then a heavily revised draft, they begin to invest in the quality of their work and to learn to take pride in their work -- no matter how much help they might have needed along the way.

Here and here are two lab write-ups from earlier this semester (maybe November), when the kids wrote about the process of stacking cups to match a classmate's height. The two kids I posted are just two random kid athletes who happened to send me their files before they went away on a sports trip, so they definitely weren't the most polished pieces from the class, but they ARE a realistic look at what "regular" 8th-graders can do with a bit of support. (These have some errors in them, of course, but I decided to just post them as is. The kids who had really top-notch, exemplary projects didn't submit digitally, and I haven't bothered to scan them.)

I also have some interesting samples of the percent/shopping projects and the three-variable projects that my 7th- and 8th-graders worked on at the end of Semester 1, leading into Semester 2. I'll post those in a few days if you check back then. They've turned in their "first drafts", which they think are polished (last graded assignment of Semester 1), but look fairly hairy to me still, and now they'll get a chance to polish those drafts based on my written feedback, for the next round of submissions (counted as part of Semester 2).

Hope this makes you excited about writing in the math class! I love projects and I love making kids write about math. For me, I think all worthwhile math projects are inherently interdisciplinary, because even if you're teaching strictly math, as soon as the kids are asked to communicate, reflect, present, create... then you're already bringing in essential skills that they develop from other classes and helping them see that other subject skills do not stop at the classroom door. After all, we are full-time educators as much as we are math teachers. :)

PS. I have to admit that sometimes, grading the first math writing assignments of the year makes me very aggravated, because most of the kids have never done such math tasks before my class. They don't know how or when to use diagrams to visualize; they jump straight into math discussions without any introduction; they either write a lot of symbols without attaching any description OR they just give numerical results without showing the work on how they got those values; they keep repeating themselves because they cannot organize their thoughts in an efficient manner; and they seem to have developed some secret language that doesn't make sense to anyone outside of the class but me, in describing their mathematical process. But, THAT IS EVEN MORE REASON for them to write!!! By the time they leave my class, I would say that the majority of students can piece together a fairly coherent discussion of a relatively straight-forward math topic, without too much help on my part.

I just wanted to put this out there in case you are wondering, "Is it just my students who are so bad with mathematical writing?!" The answer is No. Usually I have to eat a lot of chocolate and to take lots of rest breaks when I grade the first writing assignment "rough drafts" of the year.

Visual pattern of 3 variables

Since I have multi-input relationships on my mind, here is a visual pattern of 3 variables for you to enjoy (row, column, result). I've submitted this to the good folks at visualpatterns.org, but because of the nature of the complexity of this problem, I'm not quite sure that it quite fits well with everything else over there. Anyhow, it's a good problem for your abstract thinkers and ties in the concept of both arithmetic and geometric sequences quite nicely, I think.

Above shows rows 1, 2, 3 starting at the top. It also shows columns 1, 2, 3, 4 starting from the left to the right. At this rate, in row 8, column 10, there will be 23040 cubes. Can you find out how many cubes there will be for row 20, column 15?

3-Variable Project Success!

This is a follow up to my three-variable project in Grade 8, which we're finally doing now that the semester test is out of the way. The kids are really enjoying it, and it helps to solidify the idea that multiple variables can cause the numerical output to change. (I tied this in our introductory discussion to familiar formulas like A=lw and P=2l + 2w). Prior to starting the project, I also quickly pulled up a real-world 3-variable table and asked them to tell me where the causes are and where the effects are, inside this format of the stockings sizing chart. This really helped them to understand and relate to the table setup on their given project sheet, in terms of visualizing why the table is set up that way.

They worked through the first table with guidance, and found the general formula pretty easily once I asked them "if n = 100, what is the formula for y? What if n = 372? What if n is just any n?" After this, they repeated the process for the second (and for some students, as well the third) table of values. The last two days we worked on verifying the general formulas they found, first manually -- learning to show proper work for multiple math test cases -- then with technology.

It took me a little bit to figure out how to set this up, but I made this http://bit.ly/excelTestTemplate to help my 8th-graders test their general formulas for their projects. I am very excited to see the outcome of this project, because as part of the testing procedure I also taught them how to program a very basic loop in the graphing calculator, in order to prompt for two variables, perform a basic formula, and then output the results. It looks like this for testing the first formula y = mn + n.

:While 1
:Prompt M
:Prompt N
:Disp M*N + N
:End

The kids took about 30 minutes today programming their graphing calcs to run their general formulas, testing them with a few entries from the table, and documenting it all using their camera phones in order to insert it into the math reports that they will be working all of next week. SWEETNESS. This'll be their second time writing a formal Grade 8 math paper for me, and since the first ones were pretty decent, I think this round will be as well.

I also made this project writeup guidelines to help them with framing their writeup. We discussed at various points of this project, how this mathematical process is very similar to the scientific process, with looking at the effect of a single variable at a time, then combining results to form a general hypothesis (ie. the general formula involving all variables), then test-test-testing your hypothesis using a variety of test cases and test methods. What a perfect age group to do this with, since their little minds are just opening up to the world of thinking logically and sequentially and doubting everything.

I hope you have a break in between heavy-duty algebra topics to do this with your kids. Highly recommended! Super duper multi-faceted project involving a variety of skills.

Addendum: My colleague recommended to me that at the end of the project, I show the kids  the 3-D graph using x, y, z as the variables. Wow, that's pretty crazy. I don't know if they're quite ready to handle this!

An "Understanding" Rubric for the Semester

I am reading parts of John Hattie's Visible Learning for Teachers, which has some real gems. Since this book reads dense like a textbook, I find that it is the most enticing when I flip through and just randomly stumble upon sections that are appealing to my wandering mind.

One part of the book talks about what successful feedback looks like.

The criteria for evaluating any learning achievements must be made transparent to students to enable them to have a clear overview of the aims of their work and what it means to complete it successfully.

Students should be encouraged to bear in mind the aims of their work and to assess their own progress to meet these aims as they proceed. They will then be able to guide their own work and so become independent learners. 

I think this is something that I can improve on. I give a lot of verbal feedback daily, and then a lot of individual written feedback when looking at a kid's work (ie. quizzes or homework or projects, depending on the class and the time of year). But, I am not so sure that the kids always have the big picture in mind. So, to that end, to help my 8th-graders frame their minds around what I will be looking for on their semester exam next week in terms of levels of mastery of the learned topics, I've pulled together this hopefully kid-friendly rubric/skills list which I'll go over with the kids prior to the exam. What do you think?? It's a skills list like many of you might have for SBG, but it's sort of hierarchical in terms of ordering what I think are more basic skills and what I think are more complex skills layered on top. Mimicking the MYP scale for "Knowledge and Understanding", the rubric goes up to 8 points in most topics, where a 5 is basically solidly at-grade-level "average" performance, and 7 and 8 are only if you are beginning to approach more advanced topics or skills that are somewhat beyond your grade level.

My goal is that I will go over this rubric with the students this week, and they will self assess which skills they still need to work on over the weekend, so that their efforts are not so randomly scattered during precious review time. In January, after the exams have been graded, they will again self-assess in order to figure out where their skills holes are from the first semester, if any. This type of self-analysis will help them gain independence as a more self-driven learner over time, instead of me always telling them what to work on next and feeling like I fall into the rather unappealing nagging mode.

Sounds like a plan?

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!

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. 

Three-Variable Investigation... in Grade 8!

Even after teaching all of grades 7 through 12, I still LOVE teaching 8th-graders the best. (This is my 5th year teaching 8th-graders.) They have just accumulated enough algebra under their belts to be able to do rich explorations, and they are still so naturally curious about the world. They are like math ninjas, always ready to pull out their math skills to apply to the world at a short notice, and never intimidated by the look of a problem.

In the past couple of years, I realized that it's a real shame that we don't do modeling with three variables in middle-school math. It's a shame because it would be such a terrific tie in to the scientific process, to show how in mathematics you can also hold one variable constant while examining the effect of another variable on the output, and vice versa, and then in the end generalize the results into one grand conclusion. (I realized this because, looking at the past IB portfolios, this is a required skill for 11th- and 12th- graders. This was news to me as a person coming from the American curriculum.)

This year, I decided that I will try to remedy this gaping hole in our curriculum by exposing my 8th-graders to a new assignment. Take a look! They will be completing parts of this at home, then bringing it to class for discussions as a group. And then they will take more of it home to do. Eventually, when all kids feel comfortable with the process and the results, they will write it up like a modeling report. (We have already written one lab report this year based on Dan's awesome activity, and they found it very challenging / a great learning experience. In that lab, they had to learn how to define variables, collect data, determine the type of regression, perform regression, interpret results, make mathematical predictions, test their prediction, and then do error analysis. We followed it up with a very rigorous write-up process that included carefully critiqued rough drafts and a day spent on discussing how to create / insert graphs using GeoGebra and how to structure their writeup in a logical sequence.) Since I am a firm believer that kids learn more through writing about their understanding, the gears in my head are already turning to think about this next modeling assignment.

Anyway, I am VERRRRY excited about this three-variable assignment. Since the topic is already abstract, I kept the patterns linear to make it more accessible to all kids. But, I am very hopeful that it could turn out to be an awesome learning experience.

Addendum: For you new readers, this analogy is what I am going to use to kick off the introduction of 3-variable relationships in the real world.

Solo Performance and Remembering a Formula

I was doing a bit of sorting through old posts today, and I remembered that a couple of summers ago Sam Shah had asked me to blog about the cute activity I use to teach Quadratic Formula to my kids, since he says he cannot remember it otherwise.

Here it is: I teach them how to sing the Quadratic Formula song (to the tune of Pop! Goes the Weasel ), and then of course we practice in class how to apply the formula, and then for homework they need to go to a non-math teacher on campus and sing their song from memory. (To hold them accountable, I give them a half-sheet that says something like, "Dear Ms. Yang, I hereby certify that your student ___________ indeed came to me and sang from memory the following song to the tune of Pop! Goes the Weasel: X equals negative B, plus or minus square root.... [blah blah you know the rest], Signed, ____________")

It is very cute, because henceforth they sing every time they practice the formula, and they really do become experts at it before singing solo to a teacher who checks them off! And plus, some cool teachers even make them dance while singing. How very silly! I have kids who come back to me years after and still credit this assignment/song for them never forgetting that formula.

So, here you go, one of my sillier teaching "moves". I give you fair warning that after a few years you get really, really sick of hearing that tune... 

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.

Creative End-of-Year Assignment

For the end of the year, I am thinking about asking my middle-schoolers (7th- and 8th- graders) to make a creative assignment for me. I am hopeful that we will have about 2 full weeks following the big semester exams to do this / include presentations as appropriate. Options I am throwing around:

* Informative math posters (ie. process posters about problem-solving tips OR posters comparing various methods of one important topic)

* Building a cool 3-D model of something of their creation (ie. line art, origami, toothpick model)

* Creating a math comic strip or animation that involves a topic we have studied

* Video-recording a math song or math rap of their own creation.

Any other ideas? Any suggestions on how I should structure this so that it is open-ended yet still structured? The way I envision it, kids can choose their own groups BUT they will have to fully justify the workload of every person to me, before the groups are "set." Individual projects are welcome as well. Grading will be based on criteria of accuracy of information, presentation, creativity, and sufficiency of effort (relative to # of group members they had).

If it works, then at the very end of the year we will have our own mini math fair. :) I am not teaching any of my 8th-graders next year and I know I will miss them dearly. I hope they will create things by which I can always remember them!

PS. Outline of tasks is here.

Drilling Exponents

After our successful intro to exponents in Grade 8, I feel that the students can mostly articulate how to simplify exponents and why, and that they were ready for some more focused drill / practice. We are about halfway through the exponents mini-unit -- haven't yet introduced negative or fractional exponents yet, but they should be pretty OK with everything else by now.

Since I have been looking for creative ways to stay within my printing quota, I thought of a move-around activity for my students. So, I put up a bunch of different questions around the room (a total of 22 problems) on index cards, and asked them to move around to try simplifying each one. The answer is on the back (written upside-down so that when they flip it over along the top edge, it becomes right-side up), so they can quickly check their own answers to know if they are on track. Their task was to do as many as they could, to mark the ones that they had gotten incorrect*, and to ask me for help if they really cannot figure out, even after looking at an answer, how that answer was obtained.

*I had asked them to mark the ones that they initially got incorrect, so that in case they come to me asking for more practice problems later, I would know which type of questions to make for them to address their individual issues.






It was great! Kids really got to move at their own paces, and most of them finished all 22 questions and started a new assignment. Also, because they were already moving around the room, kids who don't typically collaborate during class started working together in random corners of the room. I think the on-the-spot answer-checking was also good for the weaker students, because they can address their own gaps (and also move at their own pace) without feeling insecure about it, and if they did get something correct on the first try, it was a nice boost for their confidence.

Exponents, unfortunately, is such a hellishly boring topic. The move-around activity and also speed games** help to make it a bit more exciting, I guess. (Because we haven't played too many games this year in Grade 8, the idea of group games is still fantastically fun for them...)

**I think I wrote about my typical games format a while ago. Very simple: Two teams, each team sends up 2 people at a time. The teammates can collaborate at the board but when they raise their hands with an answer, only one will be chosen "at random" to explain the answer. Most of the time the teams figure out to pair a strong student with a weak student, and they figure out also that the weaker student will probably be asked to explain, so at the board the stronger student is trying to explain the concept to the weaker teammate -- perfect peer learning opportunity! It also keeps rotations faster so there is less crowd idleness. Problems that are unanswered at the board go to the crowd for 0.5 point, so that is an additional incentive for the audience to stay looped in and to try the problems at their desks.

Where is My Lesson Plan?

I am not a fan of full-fledged lesson plans but I am also not a fan of not making any plans. Normally, I carefully plan out the worksheets and I make answer keys in advance to gauge timing and to anticipate student questions, but I don't write out much else.

Today, I wrote out a detailed lesson plan the way I used to, when I needed to turn in weekly lessons to supervisors or when I shared lessons with others. I wrote it out because I was being formally observed, so I wanted to be proper. But then, I misplaced it DURING the lesson!!! OMG I'm such an idiot. I tried not to look frantic but I really couldn't find it after putting it down somewhere. So, I decided to play it cool and just to keep going, obviously. (What we always do when lessons hit an unexpected turn.)

Naturally -- as these things always go -- I found the printed lesson plan immediately after class. It was under my stack of extra handouts the whole time. I had fully accomplished the aim, and the lesson had gone pretty well (the kids enjoyed and understood the algebra, and they helped each other as usual while I walked around to answer questions and to check in on the weaker students), but I had had to make up check-in questions on the spot since I couldn't find the "real" plan with the pre-thought of questions. sigh. Here is to hoping that my supervisor doesn't care that I didn't actually follow the plan to a tee.

PS. I used the exponent intro lesson for my formal evaluation. It worked very well; it didn't need much introduction but I did briefly introduce the handout by going through a couple of concrete numerical examples with the whole class, just to introduce/review relevant vocabulary words and to reinforce how to interpret the meaning of the coefficient vs. the exponent:

5*5*5 = 5^3 <--- 3 is the "exponent" or "index" (plural: "indices")
5*5*5 + 5*5*5 = 5^3 + 5^3 = 2*5^3 <--- 2 is the "coefficient", and this expression means we have "2 copies of 5^3"
x*x*x*x + x*x*x*x + x*x*x*x = 3*x^4 <--- "3 copies of x^4"

I would highly recommend using this (see link above) as a lesson for introducing exponent rules for the first time!! Kids were awesome and able to articulate the big ideas by the end of the 80 minute lesson. They were either primed for starting division of terms or had already finished that part off as well. Yesss.

Substitution with Some Flair

Some of my students who prefer not to solve systems by substitution have come up with a new way of substitution that I actually REALLY like. I am going to share it here with you.

Say the problem indicates to solve this following system using substitution (I know, it's totally artificial to prescribe a specific method, but for shared assessment reasons, I want them to still be prepared for questions like this on the semester exam):

2x + 3y = 48
3x - 4y = 4

Basically, some of the kids who dislike substitution have invented a hybrid of the two methods of substitution and elimination. They first scale the equations to get matching terms:

(2x + 3y = 48)*3 becomes 6x + 9y = 144
(3x - 4y = 4)*2 becomes 6x - 8y = 8

If you solve the first equation for 6x, you get 6x = -9y + 144. Then, substitute this into the second equation, you get: (-9y + 144) - 8y = 8. Ingenious! I was impressed that they just invented this hybrid method. It's much easier than getting x = -3y/2 + 24 and plugging that into the second equation to get 3(-3y/2 + 24) - 4y = 4, because the hybrid method they have invented bypasses all of the fractions immediately and still satisfies the problem requirement of applying the concept / skill of substitution. Their method of substitution is so much more elegant than our traditional substitution.

LOVE! Amazing what kids can come up with all by themselves.

I am simply bubbling with excitement to finally start new topics this week! My short attention span really gets the better of me.