A more coherent flow of Algebra topics

You may have heard about the awesome work Jonathan Claydon (@rawrdimus) has done to reorganize topics in Algebra 2 to create a more coherent flow. When I found out about his project, I was ecstatic because my project for the last 14 years has been to do the same to Algebra 1 in a way that makes it more accessible to students with weak preparation, while simultaneously making it more meaningful (and in that sense, more “rigorous”). If you’ve followed my blog for the last 2 years, you’ve seen that I’ve begun to open up about my aspirations and my unit outlines.

Today I’m posting a tidbit of my curriculum outline here for feedback and for your reflection. If you skim it too quickly, it may look like any other curriculum. If you look at it more closely, I think you’ll see some differences between this and most other curricula (I’m curious if I’m wrong about that — please let me know).

By the way, Algebra 1 in Virginia is similar to Math 8 in CCSS states, but with quadratics thrown in at the end.

One feature of my curriculum is that topics include differentiated extensions. These aren’t just harder versions of the same problems the class is already studying. They’re also not the same as the Depth of Knowledge questions that Robert Kaplinsky likes so much. Instead, they’re designed to expose hidden connections between several topics. Students who do the extensions won’t just be more computationally proficient, and they won’t just understand this topic better — they’ll also see math as more of a coherent whole.

Here are the first 2 units. The column called “Connection to a big idea” lists connections that I think Algebra 1 students typically miss. I intend these big ideas to be themes that run throughout the whole year, similar to Jonathan’s theme of “how is the graph of an equation related to the solutions of the equation?” Of course, there are many more big ideas in Algebra, but these represent the hidden themes that I’ll work to surface wherever possible.

Topics Screenshot 1 REV

I don’t know any other teachers who do polynomial addition/subtraction in their beginning-of-year unit on combining like terms, but I think the reasoning at the bottom of the “timing” column is solid.

The differentiated extensions represent the 30% of the curriculum that I want all students exposed to, but don’t think everyone needs to practice extensively. They’re part of Algebra 1, so everyone will see them at least a bit. Students who are up-to-date on their power standards will get them in depth: they may watch a flipped-class video or get some small-group time with me, and then they’ll do independent practice. Meanwhile, students who are still working on the power standards will participate in some whole-class discussion and a few whiteboard practice problems if appropriate, but won’t study them in depth until they meet the power standard goals. (Those goals are assessed as part of my quiz generator project). What’s a power standard and what’s differentiated extension is determined by my state standards and my sense of what’s critical for Algebra 2, and a big part of this curriculum outline project is to clarify the difference.

(All students will eventually get basic exponents rules, but students who started them early via the extension from Combining Like Terms will get into more difficult exponents problems).

Here is the next unit:

Topics Screenshot 2 REV

[Update 5/31/18: I’ve been working on more differentiated extensions for the Grouping Symbols unit. Since the key idea is building up expressions, advanced students can work on modeling situations with expressions. That’s tough to teach, because students always have a zillion questions and I can’t spend the whole period with my extension kiddos — I need to have substantial time to sit with students who are struggling and reteach power standards like combining like terms, evaluating expressions, or building expressions. So I created this Desmos activity on modeling with expressions. It has so many Desmos bells and whistles it’s virtually a steam train — I worked hard on the video embedded in it. With all the embedded supports, students should be able to work toward modeling situations with expressions without too much assistance from me. Then they may be ready for Desmos’ Central Park the next class, probably with a bit of help from me. It may seem crazy to you to teach modeling with expressions as an extension rather than a core, whole-class activity, but the reality is that in Virginia, this skill isn’t really in Algebra 1. Instead, Virginia focuses modeling questions on slope-intercept or standard form, or on totally contrived questions like, “Write an expression that represents two more than the product of 5 and the sum of 3 and half a number.” Which is really more of an English grammar question than a math question, to be honest. Ugh.  /end]

You may be wondering what “building expressions” looks like on a quiz. Here’s a sample formative assessment:

Building Expressions 1


Building Expressions 2

I don’t know of any other teachers who even ask questions like these, though maybe I’ve just been living under my rock for too long. I’ve found that a few months of sporadic preparation along these lines means all students can solve literal equations, though. Pretty cool.

By the way, the Building Expressions questions above are part of my quiz generator, which I think will be getting some nice upgrades this summer. Someday the entire item bank will be big enough I can share it publicly without worrying about test security.

Writing up these outlines of my units were a good reflection for me, so I’ll keep it up until I have the whole year outlined.

Is this how you already teach? Do you have strong objections or ideas for improvement? Thank you, math teacher community, for helping us all grow.

Update: Rose Roberts writes in on Twitter to mention throwing radical expressions into the mix, too:


Conceptual Intro to Standard Form (and Linear Inequalities)

The Desmos overlay function turns out to be a really nice way to introduce 2-variable inequalities. Today we did an activity that centered around the screen shown below. A link to the activity is at the bottom of this post. For the prices shown, each student tried to create some data points that totaled more than $40, some points that totaled exactly $40, and some points that were less than $40.

Standard Form lesson

It’s pretty simple work, and it’s low-stress because students can basically try anything. We’re making use of all answers whether they total to more than $40, less than $40, or exactly $40, so they can just play with numbers without worrying that they’re doing it wrong. When you overlay their graphs, you get something like this:


This is a really nice intro to linear inequalities. The green points should all lie on a line, the blue points below the line, and the red points above. Someone else in the MTBoS blogged about a similar lesson several years ago. I don’t remember who it was, but if anyone knows, I’d be happy to credit her. (I think it was an 8th-grade teacher in Massachusetts).

Nitty-gritty teaching moves

Here are discussion notes to wring the most out of the screen shown above. Students may become confused if they (correctly) find some green points that are actually higher than some of the red points, or blue points that are the same height as red points. I reassure them that if they know their calculations are correct, the whole-class discussion will solve the mystery of the graph. Once enough data has been gathered, I show a few students’ responses, and highlight what students found interesting or confusing. For example, “Is it weird to you that the point (2, 10) is blue, but the point (8, 10) is red, and yet they’re at the same height?”

If students don’t express puzzlement, then I just ask if students see patterns for why the different colored dots ended up where they did. Prompts to use:

  • Do you see any  green points in the wrong place? How do you know?
    • If you have a SMARTboard, you can circle where the green points should be, to highlight that they’re on a line. Draw the line in green marker. Cross out any green points that are in the wrong place for the sake of clarity.
  • Why is the slope negative? 
    • Students will have a legit hard time with this. That’s because they’re still thinking in the logic of Slope-Intercept Form, not of Standard Form. That is, as you move to the right on the graph, you’re increasing the number of pizza slices. If y represented the total cost, it would increase as you move right. But here, y represents the corresponding number of candies you can buy to get the same $40 total cost. Buying more pizza means you can afford fewer candies. It’s important not to skip this discussion, because it’s the key conceptual transition from Slope-Intercept Form (where y is the output) to Standard Form (where y is another input that adjusts to keep the total amount constant, and is only an output implicitly).
  • Do you see any red or blue points that ended up in the wrong place? How do you know?
    • If any of red/blue points did end up in the wrong place, asking this question is a great opportunity to draw out the pattern. A student can circle a wrong point and say, “That point is incorrect because all the blue points are supposed to be below the green line, and the red points are supposed to be above.”

At the end of the discussion, I guide the class into translating their work into a Standard Form equation. To do this, I choose some green points and show how to prove on one line of the calculator that they make $40. For example, (5, 10) works because 4(5)+2(10)=40. And (20,0works because 4(20)+2(0)=40. The equation for this pattern is 4x+2y=40

Here I throw in the vocabulary of “Standard Form”.

Finally, I ask students to type the equation 4x+2y=40 into their expression list. Some will be impressed when they notice that the graphed line goes through all their green points.

The real kicker is when you ask them change the = to a >, and the graph shades above the line. For some reason, that gets audible gasps, and a few comments like, “Okay, that’s actually kind of cool.” 

*Admission of guilt: yes, most of this blog post is cribbed from the teacher notes I wrote for this screen of the Desmos activity. Hey, recycling, right?

Am I crazy, or do we lump Algebra topics together all wrong?

The longer I teach, the more confused I am by the topics we choose to lump together into units. This lesson is a great example. Learning to think in Standard Form, when you’ve only ever thought in Slope-Intercept Form, is a legit conceptual shift, as I described above. It’s essentially the shift from thinking of a function defined explicitly to one defined implicitly. It’s conceptually huge! And yet we lump those topics together because they both graph as lines.  From our expert perspective, they belong in the same topic. But it will take novices a week to start seeing them as they same topic!

Granted, I do teach Standard Form equations a bit in my Slope-Intercept unit, but all we do is transform them into Slope-Intercept and then graph them. They’re just puzzles… hmm, someone wrote this equation in a weird form, and we have to fix it. That’s all. We don’t discuss the logic of Standard Form at that time, because it’s almost the polar opposite of the logic we’re studying, and student brains so often mush opposite ideas together in a confusing muddle.

Instead, and as far as I know I’m the only person who does this, I teach Standard Form as my intro to 2-Variable Inequalities. Because before students really see that y (the number of candies) is really determined by x (the number of pizza slices) in a clear way that can be expressed in an explicit form, their natural thinking really is guess-and-check. Some guesses are too high, some are too low. Their natural thinking is really in terms of inequalities. So now, we’ll study graphing inequalities, both in Standard Form and in Slope-Intercept Form. This will include drill practice and application scenarios.

After that, and only after that, will we start graphing Standard Form by finding the x-intercept and y-intercept. The goal is for students to see the line generated by Standard Form as a solution set — a bunch of points, all of which satisfy Ax+By=C. If I introduce Standard Form and show graphing-by-intercepts too early, students will see the line as a shape, not a solution set. They’ll see it as the line that connects the intercepts, but they won’t see that the points between and beyond the intercepts are also solutions.

Someday, I’ll write a post explaining/ranting at other topics that I think are paired weirdly in Algebra 1. Here’s a taste of that…I’m throwing it out there in the hopes that someone in the comments or on Twitter will either agree with me or convince me that I’m wrong: Isn’t solving quadratics by taking square roots much more closely connected to the order of operations than to solving by factoring? Because the work involved is basically unwinding the equation by reversing GEMDAS. Shouldn’t it be taught early in the year, when you’ve just finished order of operations and grouping symbols, and you’re moving into solving equations?

Link to materials

Here is the full Desmos activity.

Teaching m and b for meaning

It’s time to learn how to see m and b on a graph. To dip our toes into the topic, we had a nice debate in class about the rate shown in this quick video.

Is the rate 1, or 5? They couldn’t decide, because the number of marshmallows is increasing by 1, but the time keeps increasing by 5. It’s such an incredibly simple question, but their inability to decide hooked them for an entire lesson about graphing slope and y-intercept. Part of the hook was, I think, that many knew in their guts that both answers were incomplete.

I’d bet these students would have gotten the correct rate back in 7th grade. But as you know, students sometimes over-apply recent learning in a way that erodes old knowledge. So we’re here in Algebra 1, having done an entire sequence on identifying and b, and that has skewed their thinking. We’ve looked at visual patterns, like this one with m=4 and b=8:


And we’ve looked at story problems, like this:

story problem

So far, m has always been an integer. That makes the marshmallow rate of 1/5 difficult: this lesson is the moment when we transition to m as a rate or ratio.

In this blog post, I’d like to share a Desmos lesson I made to capitalize on this moment of confusion. It turned out to be pretty engaging. You could use the lesson as-is or edit it to insert videos of yourself stuffing your face or doing something similarly silly. Here are the phases of the lesson:

1.  activate knowledge of m and b

First I shared the purpose of the lesson: how do you see m and b in a graph? We began by listing what we know about m andon the board:

  • b:
    • Beginning amount
    • stage 0 of a visual pattern
    • constant amount
    • By itself in the equation y=mx+b
  • m:
    • amount a situation grows by
    • Multiplies in the equation y=mx+b

(I’m normally a nix-the-tricks kind of guy, but m and b are arbitrary choices of letters for slope and y-intercept, so I’m okay with mnemonics here.)

2.  practice m and b

Students paired up, sharing a computer, and worked at their own pace through a bunch of card-sorts identifying m and b in this Desmos activity.

Card sorts 1

The focus is on identifying b at first. I’ve learned that when I ask students to focus on m and b simultaneously, the weakest students always mix them up and never learn the difference. If I ask only for b, they can actually learn both m and b better. Students absolutely must know the difference between m and b in story problems and visual patterns, or else they’ll never be able to comprehend representing those quantities in a graph. So keep in mind, this activity comes after 3-4 days of practicing and a little each day.

I displayed the “summary” Desmos report on the screen while student pairs did the card sorts, so they could see if they were getting correct answers by looking for the check-mark on the teacher dashboard. At this point, the Desmos pacing was set up to stop them at the end of the card-sorts.

3.  puzzle over the marshmallow video

Continuing the Desmos activity, we watched the marshmallow video as a whole class (nice laughs there) and graphed the situation, as you can see here:

video and graph

(The points can only be dragged vertically, because I didn’t want to deal with arguments about whether the marshmallow went in at 5 sec or 5.5 sec.)

The following screen asks students to identify m and b in the marshmallow-eating video. This is where we had our big debate: does m equal 1 or 5? The disagreement here drove the momentum forward — we agreed to start learning to see b on a graph, and to keep puzzling over the mystery of m. That was the end of the warmup/hook phase of the lesson.

4. practice finding b on graphs

We moved over to the next Desmos activity, which is the meat of the lesson. Its starts right off focusing students on seeing b in a graph.

Part 1 purple graph

We confirmed students’ theories by watching the video of the purple graph, and discussing m and b in the video. We were still confused about m but agreed that b=2.

Then it was time for matching graphs with their equations and values of b, starting with easy problems that require almost no knowledge transfer:

Matching with b v1

And building to more formal questions that require transfer:

Matching with b v2

5. okay, but what about m?

It was time to revisit m, but they were emotionally done being stuck on it. I needed to get them unstuck. I needed to nudge them without giving it all away. So we watched the video below with the prompt, “which half of the video shows a faster rate?”

Within a minute or so, everyone agreed that the rates were the same. So you can’t say m=1 in the bottom, because then m=2 in the top, and those are not the same. You also can’t say m=5 and m=10, because those are not the same. You need a way to say the rates that gives the same answer for both videos.

Students discussed in their pairs, and if needed, I gave another nudge: “How do you say how fast anything is happening? Like, how fast a car is driving?” Students arrived at 1/5 = 2/10. I did a live demo of the decimal rate, 0.2 marshmallows/sec, by ripping a marshmallow into 5 pieces and eating 1 piece per second. Yum. We ended by drawing the rates 1/5 and 2/10 on the graph.

Label m two ways

This led into student-paced work applying their understanding of m and b. You can see an example here:

Finally, we transitioned to drill practice on slope.  For this lesson, we did only positive slopes.

6. links to materials

  • Here are a bunch of worksheets on seeing m and b in visual patterns. You should use these before teaching the lesson in this post — they’re prerequisites.
  • Here again is the 1st Desmos activity, which prepares students by focusing on m and b in visual patterns and story problems.
  • And here is the 2nd Desmos activity, which focuses on m and b in graphs.

Maybe in the future, I’ll look back on this lesson and shake my head sadly. But the 2018 version of me thinks this lesson hits the target pretty well.

Grading Khan Academy’s “Assignments”

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.

Here is a link to the spreadsheet tool:


And here are 2 videos to show you how to use it.




Improving “Will it Hit the Hoop?”

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:

Will it Hit the Hoop 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:

Hoop conclusion1

Hoop conclusion2_v2

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:

Zeros introduction_combined

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

(– 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.





Teaching Quadratics for Meaning

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. Instead, I am front-loading the big idea of factored form of a quadratic. After all, factoring trinomials feels useless and painfully tedious unless you’re already a believer in the power of factored form. And only after you see what factored form can do will you see expressions like (x+1)(x-2) as important enough to want to multiply them out.   Update 11/18/17: Originally, I’d intended this to be a series of posts about each lesson in the unit, but instead I’ve decided to describe each lesson below and link to the materials. Update 5/1/18: I keep improving this post as I think about it. I guess it’s kind of a living document.

  • 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, by Desmos, to channel the awesomeness of that lesson to those ends. Check out the post linked above for details.
  • 2. What equations make parabolas (and other nonlinear graphs)?  Now we apply factored form to an awesome Desmos marbleslide. The little stars are all zeros of the function, which focuses student attention on the critical features in a really fun way. At the end of the lesson, students even learn to stretch or reflect their graphs to avoid the purple obstacles. Here are a couple thumbnails of the marbleslide:


  • 3. Practice and refinement. Consolidation lesson, with a slight extension:
    • Part 3.1 Drill practice on the factored form of a quadratic equation. This includes parabolas where the zeros are not integers, so you have to set the factors equal to zero. For example, in (+ 9)(5x – 1), you have to solve 5x-1=0 to find the second zero.
      • Khan Academy’s quadratics unit begins with factored form, so those exercises are a great resource if you’re into Khan Academy.
    • Part 3.2 We also need to learn to apply vertical shifts to parabolas. (This may seem out of place, but it’s necessary for the catapult-shooting that we’ll do in the next lesson).  One resource for this bit is the last 5 screens of my Desmos marbleslide. They explicitly introduce vertical shifts, and compare them to vertical stretches.
  • 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 think I’ll try to obtain the expensive model featured in tweet below:

*Note: I think you can do Julie’s projectile lab without the LabQuest interface, which saves serious $, but then you need to buy the $11 power supply for the projectile launcher.

  • 6. Random skill interlude: finding factors of one thing that add up to something else. This is a boring topic, but I have a couple of honestly excellent lessons on this. For students who don’t know their multiplication facts, this skill can be quite difficult without careful lesson design.
    • Here is my first lesson, using all positive numbers. And here is the student handout.
    • Here is my second lesson, introducing negative numbers, which are really the crux of students’ difficulty with finding factor pairs. Here is the student handout for this lesson. My secret trick here is to include lots of contrasting cases where the problem can almost be done in two different ways, using either two negatives or one negative and one positive, 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.  Here’s a screenshot of the lesson showing what I mean. Notice that the problems are almost the same: one product is -6 while the other is 6, but both problems add up to -5.

Factor Pairs


  • 7. 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. I teach factoring when a>1 first, with the box method, and then teach a=1 as a special case to students after they show they can handle the a>1 case. Making the transition in this direction seems to be much easier than in the reverse direction.
  • 8. Introduction to Standard Form: When you multiply binomials, the x2  represents the acceleration. 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.
    • Note: This used to be my lesson 7. But as I continue tinkering with this quadratics unit, I’ve realized the key content of the lesson is the different role that the quadratic and linear terms play in standard form. So I think this is best as the intro to Standard Form.
  • 9. Traditional quadratics topics, e.g., the quadratic formula, graphing parabolas using the vertex formula, etc.

Teaching linear functions for meaning

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)/(x-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)/(x-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. dott-pattern-v2
    • 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:dott-pattern-v4
  • 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 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.
A slide from my slope-from-2-points lesson
  • 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:

Screenshots of some of the Khan Academy problems
  • 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.

**Yes, that quiz question is part of my new adaptive paper-based quiz generator.