Khan Academy recently created a new “assignments” feature. Instead of using the old “recommendations”, you now create assignments with due dates. Students access the assignments directly from their profile, which is good — recommendations appeared in the old learning dashboard, which many of my students found complicated and intimidating.
These new assignments have some pro’s and con’s, but as always, what I like best is the ability to hold different students to different standards.
Here’s a tool I made to simplify grading multiple assignments at the same time — it totals up the points from multiple exercises into a single number. Then it prints out a little “report card” slip for each kid showing them their score breakdown. It also automatically gives full credit for work done on time, partial credit for work done late, and lets you specify students with permission to turn in late work for full credit (for example, if they have that accommodation in their IEP’s). I am still experimenting with ways to add extra credit that some kids can get credit for and other kids can choose to skip.
I’m kicking off a series of posts on a quadratics unit designed to have a simple narrative arc — one that’s clear enough to allow our weakest students to follow the conceptual plot. The previous post contained the outline of the whole unit. Now we’ll dive into the first lesson.
Dan Meyer has suggested his popular Will it Hit the Hoop as a good launch for quadratics study. And it is good, but the launcher is aimed in the wrong direction. Instead of aiming kids’ attention at vertex form, I think it should set up the transition to factored form. Here’s how the official Desmos version ends by making connections to vertex form:
But if we anchor our unit on factored form instead, we can expose the value of factoring early (and often). That’s great, since factoring is the skill I’m most worried kids will resist. In addition, factored form provides the clearest link between the two halves of a quadratics unit: graphing parabolas, and solving quadratic equations. Without that link, our weakest students might feel the two halves are basically unrelated to each other.
I revised the lesson to refocus it on factored form. Here it is on Desmos, and here’s how it works now:
The goal is to prompt this question: what kind of equation makes a parabola? Here are the last 2 screens of my Desmos lesson:
Hopefully, students are ready to care about zeros of a function, so it’s time for some instruction: we transition to the handout below (pdf version here). Students should NOT have access to calculators for this handout. If they have calculators, they won’t think the tedious problems are tedious, and they won’t start looking for patterns and shortcuts. I want them muttering to themselves. Here’s the handout:
You might have noticed the “Prep” questions at the top of the first page: 8 + (9) and 8(9), and (3) + (2). What’s up with those? They’re designed to remind kids that not all problem with parentheses are multiplication problems. That’s the kind of misconception that can throw our weakest students off track for the whole rest of the lesson. After that little reminder, the key connection between parabolas and zeros happens in #1, part c. That’s where the class substitutes the zeros into the equation and sees why they’re zeros in the first place, and it’s worth some class discussion to flesh it out.
At this point, we could probably start teaching kids how to take a parabola’s zeros at 2 and 5, and create the factors of (x-2) and (x-5). But personally, I don’t think they’re ready for that. Not until they feel in their bones that
(x – 2)(x – 5) is like a having a light saber
(x – 2) + (x – 5) is like wearing stormtrooper armor
(In case you never noticed, stormtrooper armor doesn’t seem to be too helpful to the stormtroopers). The rest of the handout asks a series of simple yes/no questions that gradually reveals the glory of factored form. Hopefully, this understanding emerges organically. Dan is fond of saying that if factoring is the aspirin, we first need students to experience the headache. The tedious addition problems are my headache here, and I actually think they’re a bit more effective than his headache because they’re more tightly connected to the actual concept of factoring.
Even if you disagree with everything I’ve written so far, I don’t recommend the new & improved version of Will it Hit the Hoop.
Folks who read Dan’s work regularly will find the irony here, but I think when Desmos revised this lesson, they added some new fancy tech features that detract from its power. The features help you gather individual student opinions on whether the basketball shots will go in or not (students vote “in” or “out” by clicking on their screens). But this way of voting sucks energy from something that should be a lively and social. In the class where I used them, students silently clicked through their opinions. In the classes where I didn’t, they agonized aloud — together — about where the shots would go. So I recommend not using them. Instead, have kids vote with their feet: move to one side of the room for “In” and another side for “Out”. My version of the Desmos lesson doesn’t have clickable voting.
Next blog post — Lesson 2: “You don’t play with Legos one block at a time”
Update: The official Desmos version does have one great feature my version doesn’t: the videos of the shots going in or out. When I teach this lesson, I just click over to the Desmos version to show the answers for each shot.
If you want to teach for meaning, quadratics are extra hard for a simple reason: the way quadratics are usually taught, students have to plow through weeks of difficult skill-building before becoming convinced that these new skills will make them more powerful. (Dan Meyer would say it’s hard to convince them of their intellectual need to learn quadratics).
I’d like to outline a quadratics unit that makes kids feel powerful from the beginning. It’s a conceptual approach, and for a lot of folks, that means higher-order thinking and “more rigor.” Sorry, but my goal’s a little different: a conceptual approach that will knit topics together so my weakest students feel a coherent narrative arc through the unit.
Here’s a brief outline of the whole thing. Notice that skill-building on multiplying binomials and factoring quadratics don’t appear till Lesson 8. (In later posts, I’ll describe each lesson and link to materials I have).Update 11/18/17: still haven’t gotten around to writing most of the rest of the posts, so I’m editing the below to include more detail and brief links to the materials.
1. Will it Hit the Hoop, adapted to focus on factored form. For reasons described in this blog post, I really think quadratics needs to start with factored form rather than vertex or standard form. (After all, factoring is the biggest obstacle to enjoying and understanding quadratics, right?) I rewrote Will it Hit the Hoop, byDesmos, to channel the awesomeness of that lesson to those ends. Check out the post linked above for details.
2. How do you make this parabola shape? Oh, it’s easy if you use the zeros of the function. Wait, does that mean you can make other cool shapes, too? What shapes could you not make with this skill? Here’s the main thrust of the lesson:
Lesson hook: I try to fool my students into believing that I don’t know how Lego’s work. I make it seem like I don’t know that they click together…as if I only play with one piece at a time. They’re usually surprised but quite patient in explaining to me how Lego’s work. (Kids are so nice).
Once we establish that Lego’s are meant to be assembled into larger creations, I say “Oh! Wait, that’s exactly what we’re learning in math class!” Huh? They’re puzzled.
“Well,” I say, “all year we’ve been studying equations like y=x-4 or y=x+1. They’re both lines, which is okay, but a little boring, like an individual Lego piece. How can you put them together to build something that’s cooler?”
We explore whether adding them together, as in y=x-4+x+1, would make something cooler or just another line. Why?
If they get stuck, I prompt them with the parabola concept. I display the graph of y=(x-4)(x+1) and ask, “What operation goes where the question mark is to make a parabola from y = (x – 4) ? (x + 1). Why?”
Then we go through a sequence of match-my-graph activities: I display a parabola, and they write the function.
But it’s not just parabolas we can make. Wouldn’t it be weird if someone just learned that you can click two Lego’s together, but then they always stopped after just two Lego’s? We play around with making cubic, quartic shapes. We approximate a sine curve using a lot of factors. Make it playful — let kids suggest shapes and challenge the class to write the equations (they may need to use decimals, like (x – 0.4) if you do this.
I usually include at least one graph that’s not a function, such as a sideways parabola. Someone always points out that, although all the other weird shapes could be written with various factors, this one can’t because it’s not a function.
3. But what if I need to stretch or shift my graph to make a cool shape? Here we’re still playing match-my-graph, essentially, but introducing graphs that have vertical transformations. We focus on first writing the un-stretched or un-shifted function in factored form, and then stretching or shifting it. I’ve always taught this with lame notes, but this year I hope to have a snazzy Desmos activity for it.
4. Real, actual catapult target practice that applies zeros and vertical shifts. I got this idea from Julie Reulbach, but whereas her activity is for Precalc, you can successfully adapt this for Algebra 1. The idea is to place a projectile launcher on the floor and notice the landing spot. If, say, it’s 70 cm away from launch, then y=ax(x-70) is a good approximation of the parabola, measuring distance from the launcher. You can eyeball the height of the vertex while it’s flying, adjust the value of a on a Desmos slider, and graph the actual parabola. Which lets you predict: if you launch it from a tabletop instead (introducing a vertical shift like we studied in Lesson #3), where will the projectile land? Here is a catapult design students can make — I tried a cheap store-bought one last year but wasn’t satisfied with the repeatability, so I’ll either buy an expensive electronic one for physics class this year or try the make-your own design linked above.
5. Practice. Gotta practice zeros of a function to mastery. This includes trickier problems like finding the zeros of (7x+9)(5x – 1) by setting each factor equal to zero.
6. Random skill interlude: finding factors of one thing that add up to something else. Motivated by the impossible challenge of solving x2+10x+26=2 by getting x by itself. My secret trick here is to include lots of contrasting cases where the problem can almost be done using either 2 negatives, or only 1 negative, but one of those methods gives the wrong sign for the product. It turns out this is the key difficulty to hammer home repeatedly, even ad naseum, until everyone understands. Getting kids over this hurdle addresses about 50% of their difficulty with factoring. Examples:
__ x __ = 30, and __ + __ = -13 (Answer: -3, -10)
___ x ___ = -30, and ___ + ___ = -13 (Answer: -15, 2)#MTBoS
7. Parabolas and y=x2 are connected because they both model things that are accelerating. The lesson is adapted from one I’ve already written about here. The Desmos activity linked within the post is the key that brings it all together — kids are always successful discovering the correct way to rearrange the dots into a square to calculate the the total number without adding them up.
8. Skill-building on factoring and multiplying binomials. (There are a million blog posts out there on factoring, but there’s one little thing I do that seems to make it better for weaker students).
9. Traditional quadratics topics, e.g., the quadratic formula, graphing parabolas using the vertex formula, etc.
For years I’ve been rearranging the pieces of my linear functions unit like a jigsaw puzzle, trying to optimize comprehension for weaker students. Weaker students see math as a giant bag of disconnected steps to memorize, right? Changing that can require a cultural shift in the classroom that I’m not usually able to pull off. It’s not that student engagement is so hard — there are lots of tasks that kids get excited about. But while those tasks might motivate kids to learn something like slope, they don’t always help kids internalize what slope really means.
And even if you can give them an aha! moment today, it may be lost by tomorrow. In fact, it probably will be. Once you introduce the slope formula, slope becomes that formula. It barely even matters if today’s lesson created a nice footpath in students’ brains between “slope” and the change in one quantity per unit of change in another. Once that formula comes out, your measly footpath is no competition for the 8-lane highway that’s opened up between “slope” and (y2–y1)/(x2-x1).
If the intuitive meaning is going to compete at all, you’re going to have to write a lot of drill and assessment questions that force students to traverse that footpath over and over until students notice what nice scenery it has.
Here’s a screenshot of the key question type I’ve developed to make that happen:
Here’s what I love about this question type:
You can visually interpret why m = y/x doesn’t work when the y-intercept isn’t zero. In the top picture, y/x would be 105/2=52.5 grams. Does each M&M weigh 52.5 g? No, because 52.5 g would represent 1 M&M plus half the fro-yo. On the other hand, if you just had a bunch of candies on the scale without any fro-yo, y/x would make perfect sense.
This question imposes a time cost to missing the conceptual point, but still allows students to get the question right if you don’t get the concept. If you really can’t tell whether something’s direct variation, you can always divide y/x for both examples and see if you get the same answer both times. (Sadly, that’s mainly what my state wants kids to learn about direct variation: that given a table of values, you should divide y/x and see if you get the same answer for all the ordered pairs). There will always be a few students who need to do this. But it’s much faster to notice visually that this example is not direct variation because there’s a non-zero y-intercept: the fro-yo. So I’m teaching what the state wants me to teach, but allowing students to use comprehension as a shortcut if they can see it.
You can even visually interpret why non-direct variation scenarios give you different answers to y/x. In the top picture, 105/2 represents 1 M&M and half the fro-yo. In the bottom picture, 108/6 represents 1 M&M and just one-sixth of the fro-yo. Kids can see that it should be a smaller answer.
Repeated practice with this question type causes students to associate m=(y2-y1)/(x2-x1) and y=mx+b with each other, and m=y/x and y=mx with each other. I want that association.
Soooooo, drumroll please, I now present my new linear functions unit outline:
Linear dot patterns.
The focus is on noticing which part of the pattern is repeating, and which part is staying the same, and what to do with that information. We don’t use the words “slope” and “y-intercept” yet.
Note: some dot patterns should be direct variation. Direct variation patterns can even be tricky by having 2 parts to the pattern that both repeat:
Linear story problems. These are your usual algebra story problems: a tree was 4 ft tall when it was planted, and it grows at a rate of 1.5 ft per year. Students have to interpret key words that indicate initial value or rate of change. Here we use y=mx+b, but not the words “slope” and “y-intercept”. Instead, students use their own words to describe what the m and b mean.
Linear graphing stories. I lead off with some of Dan Meyer’s graphing stories (Kenneth Lawler’s bench-press and Adam Poetzel’s Height of Waist off Ground), focusing on how the starting value shows up on the graph as the y-intercept. Then we do my own chubby bunny lesson (video below) which is more geared toward slope-intercept form. Now that rate is showing up as steepness on a graph, and the starting amount is showing up on the y-axis, it’s okay to start calling them m the “slope” and b the “y-intercept”.
Slope from 2 points, conceptually, which explores the concepts shown in the fro-yo question above. Here is my slope-from-2-points lesson. I do it as a Pear Deck lesson now, but at some point will probably convert it over to Desmos now that Desmos has the classroom conversation toolkit. After this I start giving questions like the fro-yo assessment question.
Identifying proportional scenarios. Given a scenario, can students identify it as a proportional or non-proportional situation?
Here’s a screenshot of this question type:
The idea here is to combine “Graphing stories” with “Slope from 2 points, conceptually.” The multiple choice graphing question scaffolds kids’ thinking, but it’s not just a crutch: it also improves learning by signaling to kids that the most important thing is to use common sense to tell whether the y-intercept would be zero or not. So in the question above, would a pizza with zero toppings cost $0.00 ? Compare this to the current Khan Academy exercise on identifying proportional situations:
Slope-intercept form formalism, including the all the goodies we need kids to know: graphing lines given in slope-intercept form, applying the formula for slope to random pairs of points, etc. My kids need lots of focus on distinguishing between y=2x and y=2+x, and also y=2 or x=2. Yours, too, right? The formalism can also include some more advanced work on slope and direct variation.
I’m very hopeful that at each phase of this unit outline, I’ll be able to ask quiz questions that check real comprehension of the meaning. And for a certain type of kid, if it ain’t on the quiz (and the quiz after that, and…), then you never really taught it.
How can I adaptively target content to students’ needs without restricting assessment items to the lame formats — like multiple choice — that computers are able to read?
Given the richness of the Math Twitterblogosphere, it’s pretty hard to share something new that makes a substantial contribution to our online community. I think I have something worth sharing here. It answers the question above. In other words, it’s a way to get (most of) the advantages of adaptive learning systems without all the drawbacks.
Dan Meyer has chronicled those drawbacks in many blog posts, for example, this one. I like the way commenter Dan Anderson summed up the limitations of letting a computer assess student work:
A big advantage with meatsacks [read: human teachers] over computers is the ability of a human to look at the work. Computers can only indirectly evaluate where the student went wrong; they can only look at the shadow on the ground to tell where the flyball is going. Meatsacks can evaluate directly where the student is going awry.
And yet computers do have an advantage: it’s very easy for them to keep track of what each student needs to work on and to deliver practice or assessment that’s targeted to those needs. Can we have the best of both worlds?
I’ve created a system that can make a unique printable mini-quiz for each student, depending on what skill they need to be assessed on. It draws on an item bank, categorized by skill, that can be as large as you want so questions won’t be repeated on successive retakes. Quizzes also print in order by students’ position in the seating chart, so you can simply walk down each row and breezily hand each student a personalized quiz. (Not every quiz should be personalized, though. At least half the time, I pick the topic and everyone gets the same quiz. Personalized quizzes are for efficient retakes.)
The system is free, of course, and fully editable by anyone who knows how to work a spreadsheet. Here’s how it works. Each video is only a few seconds.
Step 1: Students select the skill they want to be quizzed on.
Step 2: You display students’ current choices on-screen. The screen updates live, so students who change their minds can see their most recent selection.
Step 3: You just copy and paste the Google Form responses into the quiz generator.
Step 4: With a simple CTRL + P, you print the entire class set. It automatically prints in order by seating chart.
Step 5: Updating your seating chart is easy. Changes to the seating chart automatically update the printing order of the quizzes.
Step 6: On the next quiz, increase the “quiz generator key” by 2. This will change the questions given for each skill.
Step 7: Grading tool. This speeds up your grading process by more than a factor of 10. Duuuuude. A factor of 10. (Turn the volume on to listen to this screencast).
If this post gets decent page views, I’ll come back in and write some tech support pieces to explain how to use all the features: how to add assessment items with images (it’s not trivial to add images into a cell of a spreadsheet); how to link up the spreadsheets correctly; and how to toggle all the various options in the program.
is this not overkill?
I’m pretty sure it’s not. Let me just nail my 95 theses to this door here and see what you think. Here goes:
Students should not grade their own formative assessments. An expert needs to grade them.
That expert should be a human, not a computer, for reasons given above.
With my current grading and prep load, I’m already maxed out on how much grading I can do. I can’t check huge numbers of ungraded formative assessments in addition to grading the tests/quizzes I already give.
Therefore, formative assessments must replace some of my existing grading load, not add to it. They have to count in the gradebook.
But if they’re graded, they won’t really be formative unless students can do retakes and earn credit for improving.
[My conclusion] Formative assessments must be graded tests or quizzes that students can retake.
Should they be tests, or should they be quizzes? Many teachers use a formative assessment system with tests. Here’s Dan Meyer’s version. Let’s think about that. (If you think Dan’s is not the best example, let me know in the comments. I don’t want a straw-ish man here.)
Advantage: Tests can be comprehensive. Each test can assess the full range of skills covered so far.
Big disadvantage: Tests aren’t very frequent. Ideally, students would be able to relearn something and then earn credit for demonstrating proficiency within a couple of days, instead of waiting for the next test.
Of course, you could have a policy that students may always come in informally outside of class to demonstrate mastery, but many teachers find that students don’t really bother to come after school to do that. In fact, I think if all the students who should come really did, it would overwhelm my ability to informally generate assessments after school.
Here’s a bureaucratic reason that tests might be the wrong vehicle for formative assessments: in many districts, teachers don’t have control over the tests they give. There tends to be more flexibility and independence around teacher-generated quizzes.
Okay, let’s consider using quizzes as formative assessments. Advantages: they’re more frequent, and you can still use your district’s tests. But lots of disadvantages, too.
Advantage: Shorter, more frequent assessments are better for learning. Or so says Marzano.
Big disadvantage: How will retakes work? If you have 10 skills this quarter, and you quiz a different skill each day, a student might need to wait up to 10 class days for the chance to retake the skill they’re ready to re-do. That’s unacceptably long.
Logistical disadvantage: Even though frequent assessment is good for learning, how can I squeeze quizzes into the last 10 minutes of class consistently without losing too much instructional time? These quizzes need to be very quick to hand out (and to pass back, once they’re graded).
Solution: you need a way to let students pick the quiz topic they want to retake, so that on some days, different students can take different quizzes. If this happens frequently, students can relearn and reassess in a tight loop lasting no more than a few days.
Imagine laying out 10 stacks of quizzes on the counter, or on your teacher desk, and inviting each row of students come up and pick a quiz. If these quizzes are short (4 questions or so), the first students may be done by the time the last students have picked their quiz.
Entering grades in the gradebook is a challenge. Try typing 120 grades into the gradebook, in up to 10 different columns, while overwriting old grades (with a grading program that has no “undo” button), without making a single mistake. Not easy!
In addition, managing answer keys is a huge problem here. Try grading 5 class sets of quizzes in 10 minutes per set, when you need to make 10 different answer keys and then flip between those 10 keys to check students’ quizzes.
So maybe my assessment tool is not overkill after all.
Even if I want to assess a single skill, I can toggle an option to print out 2, 3, 4, or more different versions of the quiz, to reduce opportunities for cheating.
There’s a tool to help organize grade entry.
The tool to manage answer keys when grading a class set has nothas been created. See screencast 7 above.
Here’s how I handle passing back daily quizzes quickly: students turn their papers into a tray specific to their seating area (left, middle, or right). When I grade the papers, I keep them grouped like that. Then when I want to pass them back, they’re already grouped by seating area, and I’m not traversing the room 10 times to pass them all back. I can pass back a class set in 1 minute.
does this fix the real problem?
The root problem is that it’s hard to get kids to take the initiative and fill in their own skill gaps, even when you identify them. Here’s Michael Pershan, over at his blog
Second, I don’t think the feedback itself given in SBG [Standards-Based Grading] is helpful to kids. What’s the path from “You’re a beginner at solving linear equations” to actually learning to solve a linear equation? Some say that kids will go home and study linear equations more if you tell them they’re bad at them, which doesn’t fit with what I know about high school students. But maybe your kids are different than mine.
Not only do I agree with Michael here…I also designed this entire project as a response to his critique and Dan Meyer’s larger criticism of adaptive learning systems.
Here’s why, in my classroom, the system I’m presenting seems to avert the pitfall Michael’s pointing to. After letting students choose their retake skill on the Google Form, I let students go to different stations with study guides for their chosen skills. There’s something about signing up for a skill’s retake, and then immediately diving into that skill’s study guide (starting with circling the ones you got wrong last time) that seems to lead students to feel there’s a point in trying to relearn the skill. That what’s being asked of them is a manageable bite.
And I don’t mind making everyone do a retake, even those who had 100’s on everything. Short, frequent quizzes are good, thanks to the testing effect.
Cri de coeur
I’ve never taken a coding class. Millions of people out there could have done this better than I did. But even if I felt like waiting a few more years for a good formative assessment solution, I don’t even see one on the horizon. So I made my own. In the last 3 years, I’ve created this quiz generator, written all the quiz items (most of which I’m not publishing here for test security), made the Khan Academy grading tool in the previous blog post, and tried to rewrite as many lessons as possible to make them better. That’s a lot of time spent on tools and resources. As a teacher I’d prefer my extra time be spent on the kids rather than the tools.
Relatedly, it’s not really my dream that lots of other teachers start to use this program. My dream would be for assessment companies like MasteryConnect to include these features in their own programs so doofuses like me didn’t have to build their own quiz generator (and so teachers had a convenient platform for sharing quiz questions instead of writing them all from scratch). But almost every edtech company out there is pushing for everything to be done online. A paper-based assessment system with human graders just isn’t that interesting to them.
*Note about the title of this post: if you know me, you know I’ve worked hard to find a way to make Khan Academy a useful tool for my Algebra 1 students. So I only want to burn the computer when it comes to real assessments. As a practice tool, computerized exercises are fine with me.
I hope this is my last post on Khan Academy for a while. It’s not that central to my teaching (I’d rather be writing about Desmos or something). But I do think the tool I’ve designed to differentiate grading with Khan Academy may be useful to some folks out there.
I’ll link you to the screencasts for how it works, and then to a Google link for the actual spreadsheet file in Google Drive, but here’s the gist:
I don’t let Khan Academy automatically recommend exercises for my students to practice. I want to be in charge of selecting what kids work on.
Khan’s main value is its memory quizzes, called “Mastery Challenges”, that check if a student has forgotten something we’ve learned (if they’ve forgotten a skill, it gets added back onto the student’s agenda).
But different students need to be held to different standards of retention and accuracy, per IEP’s and observations.
My new spreadsheet allows me to exempt some students entirely from these memory quizzes, and allows other students to earn full credit with reduced expectations of retention & accuracy. Meanwhile, most students are still held to the full standard.
In addition, students can be exempted from the hardest exercises on an assignment.
On the opposite end, students who are really advanced can go ahead and earn extra credit by working on Khan Academy’s automatically recommended skills…but only after they have completed the assigned skills.
Here are the screencasts for my new grading tool. One thing to know: this year’s improvements have made it a very easy to system to maintain.
This spreadsheet explains how the teacher-facing grading system works. The student-facing end is different. It’s a technique for hacking around Khan Academy’s automatic recommendations and instead forcing kids to do the exercises you want them to do. You can find a description here at this blog post. It’s different than using Khan’s “Teacher recommendation” tool. That tool does not re-add a skill you’ve recommended when the student fails the retention quiz on the skill. So if your goal in using Khan Academy is to focus on retention, their “teacher recommendation” tool is useless.
Khan Academy’s content started weak, and some of you may not feel it’s ready for your use yet. Depends on the course. Apparently, AB and BC Calc were rewritten this summer, though I haven’t checked them out. Algebra 1 is currently being rewritten, but those changes have not gone live yet. In my spare time, I work as a volunteer to help them identify Algebra 1 improvements that need to be made. There are many. In August, I created a 30-page document suggested changes for about 25% of the course. We’ll see how many of my suggestions they take.
Future work: How can we add a feature that automatically pairs students up so each member of the pair has an assigned skill they’re able to teach the other? All the required data is there in the spreadsheet, but I can’t figure out an algorithm that makes it work.
I’ve always needed a way to motivate the study of quadratics. In the past, I’ve used materials from some of Dan Meyer’s 3-Acts: Super Mario to get students to realize that linear predictions are sometimes wrong, and Will It Hit the Hoop? to specifically focus students on quadratic graphs. But even to my teacher ears, the jump to actual quadratics skills sounded cheap: “Now that we all agree quadratic functions are important, let me teach you to multiply things like (x+1)(x+2), because it’s really important for understanding parabolas, and I’ll explain why later.” Groan.
I’d like to share a new lesson that I really liked because it:
Naturally focuses students on area models of quadratic expressions;
Shows that quadratics are the way to model something that’s speeding up or slowing down;
Has a really low barrier to entry.
A low barrier to entry means students can dabble their toes in this concept pretty easily at the start, without encountering hard math until they’ve played around a bit. Before we go on, let’s check that this blog post is worth your time. Here is the whole lesson I’m about to describe, fast-forwarded to be just 2 min long:
Still interested? Cool.
The most direct way to (a) introduce area models of quadratic expressions, and (b) make it seem like quadratic expressions are useful is to pose a question that’s directly related to area. Something like: Farmer Joe has 100 feet of fence and wants to make the largest sheep pen he can. What length and width should he use for the pen? [The answer is to model area as A = (L)(W) = (L)(100-2L) = 100L – 2L2, graph the quadratic function, and find its vertex].
In my experience, the Farmer Joe question doesn’t arouse much natural curiosity from students, and I think I know why: even students who naturally enjoy math puzzles have no inkling at the outset of their inquiry that their solution method will also help them understand the many faces of quadratics: projectiles, cars speeding up or slowing down, the famous handshake problem, etc. It’s not until you’re well into the problem, and you see that the graph of area vs length looks like the flight path of a projectile, that you have a chance of recognizing how significant quadratics might be. And by that point, you’ve already done enough hard math that you might be a bit tired or grumpy. Learning quadratics should be like hiking to a beautiful vista: look at all the things I can see from up here! The ahhhh experience of arriving at that vista needs to come sooner in the introduction or students end up feeling the way I felt on my last hiking trip in Montana: that’s a great view, but OMG I hate mosquitos–let’s get the #@%! out of here.
If you’re learning quadratics after learning linear functions, then the best way to notice you’re at a pretty awesome vista is to see that you’re looking at a pattern that’s accelerating. A pattern that’s accelerating is noticeably very different than all the patterns we’ve done so far. My class starts linear function by looking at dot patterns like Fawn’s–specifically, we focus on ones that visually distinguish the y-intercept from the slope. For example, looking at the pattern below, how many dots would be in Stage 10? Stage x?
Students get really used to asking, “How fast is the pattern growing?” or “How many dots does it add each stage?” We also do modified versions of Stacking Cups and Barbie Bungee to keep emphasizing that finding the rate is crucial for making a prediction.
In addition, the narrative in my room is that algebra is a way to predict the future by finding and expressing patterns. For example, when we study direct varation early in the year, students actually make short videos of a prediction experiment in their own lives.
Okay, against that backdrop, I present students with the following lesson to try a prediction that finally breaks the constraint of using constant-rate patterns and motivates area models for polynomial multiplication. Here’s the full, narrated video overview of the lesson.
Update 6/21/16: Here’s a Desmos activity to go with the visual “dot pattern” section at the end.
Room for improvement: As I was transitioning students to (x+2)(x+3) and drill problems, I felt that even though I’d gotten students to the vista, I need to do a better job of showing them everything they can see. What if they think these area patterns only work when the first difference in the pattern goes like +1, +3, +5, etc? I should show that if the first differences go +2, +6, +10, etc, then you can use 2x2…visually, just draw two of the x2 patterns. If you wanted +1, + 2, +3, you could use (1/2)x2 by drawing the x2 dot pattern and then cutting it in half. I should also make the connection to accelerating cars, psychology’s inverted U-shaped graph of stress vs performance graph, Farmer Joe, and everything else that’s quadratic. However, I think that’s best saved for the next lesson. We teach roughly 90-minute blocks, and I like each block to have some conceptual development and some practice. When you see kids every other day as it is, you need to squeeze in some practice to each lesson. So in the future, we’ll transition to (x+2)(x+3) and do drill just as shown above, but the following lesson I’ll take time point out all the landmarks you can see from this vista.
CCSS thought: I’m not sure how this lesson would play in a Common Core state. Do you do arithmetic series in Algebra 1, and if so, do you do them before quadratics? That would probably make this whole shtick might seem kind of lame. We don’t do CCSS here. Our state test doesn’t really assess comprehension much, so I’m not sure how much this lesson will even improve my students’ standardized test scores. My students have always been able to multiply binomials without experiencing an intellectual need for doing so. But this lesson just felt so satisfying.I hope it’s been worth your time to read about it.
Sharing the file: I’m happy to share a copy of the powerpoint file to anyone who’d like it. Just ask in the comments.