Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

## Wednesday, November 19, 2014

### Graphles to Graphles

New game! My Algebra 2 students struggle with stating the domain and range for reasons including: trouble understanding and writing inequalities, and a lack of comfort with the coordinate plane. We spent a day on looking at graphs and identifying their domain and range. We learned to deploy our wonderful domain meters and range meters that I learned about from Sam. But for maybe 25% of the students, the cluebird was stubbornly refusing to land.

So, I thought asking the question backward might be a good way to attack it. Instead of here's the graph, what's the D and R? ask, here's a D and/or R, draw a graph. I mean, I know this is pretty standard fare. The thing is, I didn't want to do examples and a worksheet, or hold-up-your-whiteboard so I could somehow assess 22 graphs in a split second. It seemed like there should be a better way.

So I did what I do, which is ask on Twitter. And I got lots of helpful ideas, but this was the one that I latched onto and ran with:
The end result is, I'd argue, more like Apples to Apples than Charades (hence the title).

To prep: Make game cards. I printed each page (docx pdf) on a different color card stock. Student play in groups of 4-ish, so plan accordingly. I printed 6 sets. (John suggested having students submit constraints, but, for this crew, I decided to unload that part and create cards with the constraints.) You'll also need a mini-whiteboard, marker, and eraser for each student. Check your dry erase markers, because nothing kills a math game buzz like a weaksauce marker. (I'll admit to a minor teacher temper tantrum where I uttered (okay, yelled) the words "I'M NOT THE MARKER FAIRY! I DON'T POOP MARKERS!" Teacher of the year, right here, folks.) Also, you'll need some kind of token that players can collect when they win a turn. I use these plastic counting chips that I use for everything, but anything would work, candy, whatever.

Doing a demo round with a few kids playing and everyone watching will pay off, in the more-kids-will-know-what-is-up sense.

Here's how the game plays:
• Someone is the referee.
• To begin the turn, the referee turns over two (or one, or three) different-colored cards, and reads them out loud. (I feel the reading aloud is important practice for interpreting inequalities.) You could do, like, first round is one card, second round is two cards, third round is three cards. Whatever suits your needs.
• The other players have one minute to sketch a graph meeting the constraints on the cards. The referee is responsible for timing one minute.
• The players hold up their mini-whiteboards so the referee can see.
• The referee disqualifies any graphs that don't match the cards, and explains why. Other players should police this, too.
• Of the remaining graphs, the referee picks his favorite. This player wins a token.
• The turn is over, and the player to the referee's right becomes the new referee.
• Repeat.
It was great! Here are things I liked about it:
• 100% participation 100% of the time. At no point should anyone be kicking back.
• Nowhere to hide. There were a couple kids who had to come to me and say, "Miss Nowak, I really don't know what's going on." which I don't think they'd be compelled to do if we were just doing some practice problems.
• Good conversations. Especially reasons for why graphs were disqualified. "You need an arrow there! The domain goes to infinity!" That sort of thing.
• Students were necessarily creating and evaluating. Take that, Bloom
• Built-in review of what makes a graph a function vs not a function.
• My chronic doodlers had a venue to express themselves. Especially if the graph didn't have to be a function.
• Authentic game play. You could use your knowledge of what a referee liked to curry favor.
Here are some action shots. Let me know if you try it, and how it goes!

## Thursday, November 13, 2014

### We Got a Problem

We spent practically the whole period (35-ish minutes) on one problem today. This one, that Justin wrote about recently, that he found on Five Triangles:

Since we just spent a few days naming pairs of angles made by parallel lines and proving what's congruent and what's supplementary, I was provisionally hoping we'd get, at the end, 10 or so minutes for students to present various solutions to the class. That did not happen in any of my three class periods. Because I didn't have the heart to interrupt them. At the 10-minutes-left mark, too many were still making passionate arguments to their small groups about why they thought their solution worked.

Here's what we did: The big whiteboards were on the tables. Groups of 3 or 4. I stated the problem while showing this diagram. Made a big deal of starting with a regular old piece of copier paper and making a single fold. 3-5 minutes silent, individual think time. No class made it to 5 minutes without a buzz starting. I wrote a time on the board and said, by this time, everyone in your group needs to be prepared to present a solution to the class. Also, the answer is not as important as the reasoning that got you there. If you are saying something like "this angle has this many degrees," you have to explain the reason why that must be true.

Then I started listening and circulating. There was almost 100% engagement, and I have to think it's due, to a high degree, to the problem itself. This problem just felt do-able, but not obvious, to every learner in the room -- the sweet spot.

When a group would start crowing, in their 9th grade way, that "MISS NOWAK. WE GOT IT," I refused to confirm or deny that their answer was right, and asked a randomly selected group member (everyone was supposed to be able to explain the solution) to walk me through the reasoning. I played the role of highly annoying and dense skeptic. "Wait, how did you know that angle was 90?" "Because it's a RIGHT ANGLE." "Wait, how do you know it's a right angle?" "... ... ...BECAUSE A SHEET OF PAPER IS A RECTANGLE." "Oohhh, right." And then, when they got to a part that was not justified (an assumed bisector, an assumed isosceles triangle, trying to use two sets of parallel lines to leap to a conclusion about congruent angles), I wasn't shy about saying I wasn't convinced. In their groups, they had already harvested the low hanging reasoning fruit. I figured my experienced eye was valuable for training a spotlight on flaws in their arguments. And they responded well, in a back-to-the-drawing-board kind of way.

Props
I'm in the habit of slagging myself on here, but I'm going to take a moment and describe a few times I witnessed and celebrated some great, inspired ideas with at least one learner today:
• You extended that line to make it intersect another line!
• You marked those lines as parallel with some arrows! And those other ones too!
• You grabbed a piece of scrap copier paper and made a physical model to look at and play with!
• You suggested your group start over and draw a clearer diagram, so more people would know what you were talking about!
The Value
This seemed evident: the importance of being able to articulate how you know things are true. I think that was one of the purposes of proof that Pershan hit on last summer... knowing why a right answer is right. In a world (of school Geometry) where if I'm careless, I'm too often asking kids to "prove" things that they think are already obvious, I want to make as much room as possible for problems like this where something is not obvious and needs justification.

Questions
I'm not entirely comfortable with leaving 72 kids hanging about what the correct solution was, and why. What I want is for this to keep bugging them, and for them to make little doodles and sketches of it in their spare time, and for them to not be able to leave it alone. I did not want to ruin anyone's fun. At the same time, I think many kids could have benefited and learned from seeing a few different correct arguments for why the angle had to measure 140 degrees. This is still an open question for me -- what's the best way to handle kids/groups that don't arrive at the correct answer? Do you let the question hang, or do you interrupt everybody so groups who made it down a valid path can have time to show what they did? I'd love to pick it up tomorrow, but I'm going to be out (whaddup, #NCTMRichmond!) so we'll have to see if anyone has any memory of what happened today on Monday.

Regrets
They should have sent a poet. I should have done this on a block day.

## Monday, November 10, 2014

### DDT, Y'all

Today in Geometry we tried Dance, Dance Transversal as popularized by Jessica and Julie. The kids dug it, and nailed an exit ticket identifying names of pairs of angles. I followed Julie's plan pretty closely. I loved that the kids were up and moving around for a good 20 minutes of class. (Was anyone else traumatized by that Grant Wiggins article? I'm very on the lookout for ways to make kids move.)

I just want to add one more resource to the arsenal: a powerpoint I made to show while playing. The slides auto-play the different moves. There are some initial slides that demonstrate where to put your feet for each cue. Then, the first two game slides are timed with a 1.5 second delay and worked well with Problem, and the second two game slides are timed with a 1 second delay and worked well with Dynamite.

Here's video. In case you're wondering, I did also play along, every period. Because their dancing did not have enough FLAVOR, and I had to demonstrate. Try to ignore the one kid doing some kind of demented hopscotch:

## Saturday, November 8, 2014

### Gallery Walk for Noticing Features of Inverse Functions

I put a call out on Twitter last week for good things for inverse functions. I got a few helpful responses but nothing that was really the thing. So here's what I made.

The day before, we had worked with inverse functions as doing and undoing equations. I started with ciphers. Students walked in and the board said, IQQF OQTPKPI, DGCWVKHWN DTCKPU! with no explanation from me. I just greeted them and took attendance and acted nonchalant. One kid sidles up and goes all sotto voce, "Miss Nowak, does the first part say Good Morning?" Since the good morning part was a pretty easy crack using context, after a minute or two someone notices that all the letters are shifted over by two, and can't keep from blurting it out, and we're off.

I had one of them explain how the encoding was done with this example. Then, they wrote secret messages using their own shift n cipher, traded, decoded. I babbled a little bit about Caesar and Enigma (I really want to show them this Numberphile video, thanks for the tip Mike Lawler). The encoding and trading and decoding only took about ten minutes, we went through one from beginning to end: what was your message, how did you encode it, how did you decode it. The alphabet was written on the board along with a counting number under each letter, the idea being that if your encoding added 5, the decoding would be subtract 5.

We spent the rest of that day couching inverses in terms of equation rules. x + 5 and x - 5 is fine and pretty obvious, but what about more complicated rules. Kids had mini whiteboards, I'd throw a function on the board and they'd try to write the inverse. Each time, they wrote down operations done in the original function, inverse operations in reverse order, then do that to an x. So for example if the given function is 3x2 - 5, they write down "square, multiply by 3, subtract 5" and then write down "add five, divide by 3, square root." Plop down an x and do those things to it. The biggest hurdles were order of operations (so they might write down "multiply by 3, square, subtract 5"). Also, always undoing the whole of what came before. So in this example, they'd be likely to write x + 5/3 instead of (+ 5)/3. But we just kept going and honestly, they didn't want to stop until they were getting them right. (I just got an idea about how to make this part better. Compute like f(3) (or something) and run the result through their inverse to see if a 3 comes out. (Instead of just you're right or wrong because I say so.) Have to figure out how to make that manageable.)

The next day, I wanted them to notice all the nice things that are true for functions and their inverses: the symmetry over y = x, that the inputs and outputs trade places, that f-1(f(x)) = x. So, each student got one of these cards. They figured out the inverse of that function using the technique from the day before. There was another student in the room with the inverse of their function, so they had to get up, talk to people, and then sit with their partner.

Each pair of students got one of these (the first page). They tacked their cards to the paper, completed the tables, graphed each function in a different color, and computed f-1(f(0)) and f-1(f(1)). They needed various levels of support interpreting instructions, but it helped to have them working in pairs on the same piece of paper - there was a natural reason for them to talk to each other to figure it out. My colleague Lois is teaching the same course, and got a coach to come in for one of her sections, which was a great move.

As the mini-posters were completed, they were hung up around the room. I said, hey, you all had different functions and now they're up there with their inverses. There are some neat things that are always true about a function and its inverse. Walk around and look at them all, and write down at least two things you notice. If you look at page 2 of this same document, the first question has space for them to write down observations.

They sat back down, and they shared their noticings with the class. I had Desmos up on the projector with some pre-loaded functions, so we had a concrete thing to point to as they were sharing.

Then they got to work on the rest of that page 2, which is lifted directly from lesson 6 of this eMathInstruction textbook (thanks Sam for pointing me to this resource). Some of them were able to just do those problems, some needed help restating the given information and what the problems were asking.

So there you have it. I especially liked this lesson for the social, discussion, get-up-and-move-around aspects. These Algebra 2 classes have not responded positively to problem-posing when they haven't been "shown how" to do a problem first, but, we have been successful with lessons like this where we break the questions into clear chunks while still requiring that they do some thinking and figuring out. It's a bit of a tightrope walk but that's how you get down a tightrope, right? One tiny step at a time?

## Friday, November 7, 2014

### Fire Up Blogging Machine

Yo. This year is hard. New building, blah blah. I'll pause a minute for no one to be surprised.

I feel, very often, like I'm not that good at this. I know that everyone does sometimes. I know, I know. I think it has very much to do with attending to formative assessment every day. (Every. Damn. Day.) Measurement: making it hard to lie to yourself since... measuring was invented.

The children are charming and testy and pathetic and confident and devious and brave, all in the same day, all in the same 45 minutes. There are 70 ninth graders that move through my room, and the thing is that a ninth grader is like the weather in Buffalo in April -- if you don't like it (or if you do), just wait five minutes.

I have lessons that I want to write up for this blog, the problem being my artifacts (documents, pictures, student work, etc) are all over the place. The file system on the school network is unreliable, so teachers all use either Dropbox or Drive or flash drives to store and/or keep a backup of everything (even though Dropbox isn't installed at school -- it's web interface only, 2005-style). Colleagues have been very generous sharing (bewildering Virginia-standards-based) materials with me. So all the stuff I've modified or created and used is on... the school file system and Dropbox and Drive and a flash drive. That puts just enough of an annoying-barrier in the way of assembling blog posts. I have got to get my computer file organizing act together.

So, those are some lame excuses for the radio silence. More coming. I'll figure this out.

## Friday, September 12, 2014

### Not One Mention of Karl Gauss

There are things I'm supposed to teach and they're strung along into a curriculum and I often get to some of these things and I'm like, "wtf? why?" So when I get to those things I think, okay, why might this ever be useful or interesting.

Why might someone want to sum an arithmetic sequence? That's a list of numbers that keep increasing or decreasing by the same amount. So for example, 3, 5, 7, 9, ... is an arithmetic sequence and so is 8, 3, -2, -7, -12, -17, ...

The only times this has ever been useful in my life is when I needed a shortcut for adding a bunch of things really fast. Which doesn't come up that much, but it does come up once in a blue moon.

So I decided to impress the children with my lightning addition capability. "Impress" is maybe a tad ambitious as words go because they are 16 but their interest can be piqued.

When they walked in I gave them a slip of paper with 20 boxes and asked them to come up with an arithmetic sequence and write down the first 20 terms. They could use their favorite numbers, or their least favorite numbers, or numbers they were indifferent toward. I made a little form special for this purpose because if one wants humans to take one seriously, one must make it obvious that one has prepared for their arrival. (And plus I'm already best friends with the guy in my department with the good paper cutter.)

The little form also had a spot for the sum of the 20 terms. They were to add them up on a calculator and write it down.

Then I asked them to trade slips with a partner and check each other's math before handing it in, because if my answers didn't match what was on their paper, there was going to be hell to pay.

They wrote their names on the list of 20 things. They cut off the sum and handed me the list of twenty things. They kept the sum. The anticipation was building. And by that I mean I said, "Children. Prepare to be amazed," and the children made me try again because I was too monotone.

So then I shuffled up the little slips of sequences and started saying, B, your sum is 210. C, your sum is 384. D, your sum is 2440. E, your sum is -24.

They were astonished! I only made two mental arithmetic errors in two class periods, which was convincing enough that they wanted to know my secret. We wrote some sequences on the board and we stared at them for a while. No one figured it out. Then I was the worst magician in the world and spilled the trick. For educational purposes.

Did they learn anything from this little stunt? No. But they were ready to learn something, which is saying something.

## Saturday, August 23, 2014

Spirited discussion kicked off my Geometry courses this year. I used a task that is in the IM task bank but not published yet. (Authored by Victoria Peacock and Yenche Tioanda with some revisions by me.)

Here it is:

For each set of shapes, write down whether you think they are the same or not the same, and explain why you think so.

Each group was given a printout of the six set of shapes. (I printed one set per half-sheet, and chopped them into 6 half-sheets before class.) Tools provided for their use (in small bins casually opened on each table without explanation of how they were supposed to use them) were patty paper and scissors.

They had five minutes of individual thinking time, and then they were asked to come to a consensus, for each set, in their groups of three or four. (I should mention that at the beginning of the class, we did ten minutes of Talking Points, described by Elizabeth here. So the expectation was set up that everyone participates, and they already had an opportunity to negotiate the awkwardness of talking to each other.)

The mathematical purpose of the task was to begin to develop a definition of "congruent" based on transformations. That a figure is congruent to another if every point on it can be matched up with every point on the other through a series of reflections, translations, and rotations.

The pedagogical purpose was to begin to illustrate the importance of justification and mutual agreement on definitions in their mathematical conversations. I intentionally used the ambiguous phrases "the same" and "not the same." The idea was that part of the work was for the group to come to a consensus about what "the same" meant in their group. In practice, this went like this:

One group: "We said set C is not the same because you have to flip it."
Me: "Great."
Other group: "Wait a minute, we said set C is the same because we thought flipping was okay."
Me: "Also great."
Yet another group: "So which is it? We said they are the same."
Me: "...   ...   ...  because... ?"
(Set C is the ribbon-y looking figures.)

Other news: my poker face is amazing.

To bring it all together, I wrote on the board "Our definition of the same" and wrote down all the things they hashed out that were okay and not okay. There was disagreement. Some particularly spirited disagreement over sets B (the lightning bolts) and D (the four equilateral triangles... or is it really only two figures?), in particular. If you saw my tweet about the kid thinking like a topologist, he had a passionate defense of his assertion that set B was the same. This was a 100-minute block period, so we had the luxury of letting the discussion happen.

Then I said, okay, so here's a little secret: what we think of as mathematics is just the result of what everyone has agreed on. We could take our definition of "the same" and run with it. In geometry there's a special word "congruent" where specific things, that everyone agrees to like a secret pact, are okay and not okay. Then, I erased "the same" and replaced it with "congruent," and made any adjustments to the definition to make it correct. They had heard the word congruent before, and had the perfectly reasonable middle school understanding that congruent means "same size and shape." I said that that was great in middle school, but in high school geometry we're going to be more precise and formal in our language.

Next up: how to describe transformations precisely.

## Wednesday, August 20, 2014

### Day 1: Sooooo.... school.

Day 1 was not so bad. I like to minimally wah wah wah about the syllabus, because they won't remember anything until the information matters, so we all did some math today.

Algebra 2
Unit 1 is series and sequences so we went in hot with Eating Grapes.
On Monday Angela ate some grapes. On Tuesday she
was hungrier and ate six more grapes than she ate on
Monday. Each day that week she ate six more grapes
than the day before. After she had eaten her grapes on
Friday she had eaten 100 grapes in all.
I read the problem as a story out loud, and asked them to tell me a few things they heard. Then I displayed the text and asked them to read silently, looking for anything that was different from what they remembered. I only showed the scenario, not the question, so next they independently wrote down anything they wondered. We got some fun wonderings like, "Does Angela have an official diagnosis of OCD, or... I mean, Miss Nowak, who counts grapes?" but focusing on questions we have the power to explore mathematically, quickly settled on "How many grapes did she eat on Monday?" They had five minutes of silent individual think time, though some couldn't help themselves from discussing with their groups and I didn't really enforce silence. Then their groups (of 3 or 4) were charged with reaching consensus on a solution and writing it on chart paper so everyone could see. (Not just the answer! Make your thinking visible! I want to see how you arrived at your answer! What was the thought process? No more than half your solution should be numbers! etc etc).

The two approaches I saw were guess and check, and writing algebraic expressions to make sense of the pattern, and then undoing. Making lists or tables were common strategies. Nobody drew diagrams. Here are two samples:

Where I struggle is, books like 5 Practices, and the anticipated answers given by Math Forum, kind of assume every group is going to do it correctly, just in a different way, and the teacher's job is to sequence the different solutions appropriately. I have seen little guidance on how to provide minimally-invasive help to students who have misunderstood something about the problem, but don't realize that their answer is wrong.

Here is the work of a group I failed spectacularly today:

They were sure they were right because 16 works in their equation, but they weren't checking if 16+22+28+34+40 added up to 100. What I did was, encourage them to use common sense and see if starting with 16 would get her to 100 grapes by the end of the week. What I should have done, I think, was interrogate them about where the 4 and the 6 came from. As a result, they fell back to guess and check, but for some reason only added up four days, because "we don't know how many she ate on Monday." They said the answer was 10 grapes on Monday, because 16+22+28+34 = 100. Interesting, right?

Here was another group I couldn't make see the light:

If you can read it, their answer was 76. They may have had two misunderstandings about the problem: that Angela ate 100 grapes on Friday instead of 100 grapes in all, or that Angela only ate 6 grapes every day Tuesday through Friday, instead of six more than the previous day. I think it was the second one. We went around and around. I tried saying "Well look. If she ate 76 grapes Monday, and 82 grapes Tuesday, she's already eaten 158 grapes. But we know that she only ate a total of 100 in the whole week" but it was like we were not speaking the same language.

Two things I have to work on:
What to do about kids who are trying to do nothing, and hope that if they are quiet, I don't notice? Group work enables this behavior, because their group can still produce something without them. I don't think "roles" is the answer, because you can be "resource manager" or whatever and still not do any math. I don't think "everyone turns in their own work" is the answer, because then there's no compelling reason to talk to each other.

How to present work so that everyone learns something about why the correct solution is correct, and ideally, learns some math I am trying to teach them? Again, 5 Practices acts as if the four anticipated solutions will show up in your classroom and it's just a matter of choosing what order to talk about them in. What about groups that do not reach a correct solution? How do you discuss their work without embarrassing them? (I think the answer is lots of deliberate growth mindset interventions, but man, it's the first day of school and the last thing I want to do is make a kid feel bad for trying.) What if (like today) no one makes a diagram? Do you generate your own teacher diagram on the fly, for illustrative purposes? What about students who are super-reluctant to speak to the whole group? Is it okay if the teacher explains their work, and maybe asks them some specific clarifying questions along the way?

Help me, people who know what you are doing. I need you.

I also taught a Geometry class! I think I will blog about that tomorrow. (Two days of block scheduling, so same lessons tomorrow.)

## Sunday, August 17, 2014

### Sound the Bell School's In Sucka

Hey! If you haven't heard, I'm back at it this year, teaching Geometry and Algebra 2 at local public high school. Mathalicious has picked up and headed west to continue doing their great work. Writing real-world lessons and helping operate a business (or trying to help, anyway) was educational and stimulating. I didn't predict how much I'd miss the day to day work of teaching class, though.

I get to work with the relentlessly charming Lois Burke! Along with many new awesome colleagues. The Geometry team, in particular is hungry to do some really good things. It's me along with three dudes I just call the Matts, because although they have many wonderful individual qualities, they are all named Matt. We have agreed to plan together and teach one new lesson per unit, observe each other, and meet weekly, which in my opinion is a prodigious start.

I don't have much to share at this point. I'm trying my best to resist the whirlwind of time-consuming stuff that I often let clutter up my thoughts about school. I don't need to read any more books. I don't need a 37 point checklist. I don't need to hang up 95 posters. I need to organize some simple tools and plan some good lessons and take many, many deep breaths.

My plan for f(t) is to record some really tedious minutiae about what I planned, how students responded, whether they learned anything, and what I did about it. I want to focus energy on the colleagues in my physical presence who are willing to negotiate ongoing collaborative work with me, but I'll also be grateful for new and continuing conversations in my online faculty lounge.

Happy new school year, everybody.

## Saturday, August 16, 2014

### Is it linear?

Hi! This is a task I helped develop that uses concrete examples to help students notice the traits of a linear relationship in different representations in service of CCSS standard 8.EE.B. I'm publishing it here to make it available. (If you know of something better to achieve the same purpose, and is easily-accessible, please let me know!)

Below are several situations which describe a relationship between two quantities which are each noted in parentheses. Your goal is to determine which of these relationships is linear. To help you decide, consider creating a table of values to compare the quantities specified in parentheses, graphing the relationship on a coordinate plane, or writing an equation to represent the relationship. Make sure to define any variables clearly, with appropriate units.
1. The price of bananas at the Italian Market is \$2 for every 3 pounds (cost and pounds).
2. Tamika plants a tree that is 3 inches tall, and it grows 6 inches per month (height and months).
3. Rhys folded a strip of paper that is 120 cm long into halves over and over so that its length shrinks by half each time (length and number of folds).
4. The perimeter of an equilateral triangle is three times its side length (perimeter and side length).
5. Malcolm bikes for half an hour at 30 miles per hour, followed by an hour at 15 miles per hour. He rests for half an hour and then bikes at 20 miles an hour for 1.5 hours (distance and hours).
6. The area of a circle is Ï€ times its radius squared (area and radius).
7. A teacher plans to grade exams for a whole day, and estimates that she grades 8 exams every 30 minutes (number of tests that have been graded and minutes).
8. (Number of squares and step)
9. (Number of pebbles vs. step)
What are some properties you used to determine whether a relationship is linear or non-linear?

### Commentary

This task uses concrete examples to help students notice the traits of linear relationships in different representations, and prepares them to see the connections between proportional relations, lines, and their equations for 8.EE.B. Since the given scenarios cover a wide range of contexts, what makes a relationship linear may not be easily discernable by reading the descriptions.  Translating them into tables, graphs, and equations enables students to spot and articulate regularity and structure in situations that produce line graphs (MP7, MP8) and to generalize these regularities as clues of linear relations.
Two of the scenarios (b and h) involve non-proportional, linear relationships. A teacher may opt to leave them out if she wants to focus on constant rate of change without the added wrinkle of accounting for a starting value.
As an instructional activity, there are many ways to implement this task. For example, a teacher could create cards showing an equation, table, and graph for each scenario, and turn it into a matching activity. This approach would decrease the level of difficulty substantially, but in skilled hands, could still provide fodder for a rich conversation about the features of a situation that imply a linear relationship.
Another approach might be to "jigsaw" the activity. One team of students might be responsible for creating tables, another graphs, and another equations. The teams could reconfigure to contain one student using each representation, and work together to see if their representations are consistent. They could write down some characteristics that tell you whether a situation is linear or non-linear, leading into a whole-class discussion.

## Solution

An equation is provided, when possible, to provide the teacher with succinct information about the relationship. Students may frame their arguments by creating tables or graphs and using them to support their reasoning, and some samples are provided. The important characteristic of a linear relation is for the quantity to exhibit a constant difference over consistent intervals.

a. Linear. Let c = cost in dollars and p = weight in pounds.
c=2p3, where p0

 p 0 3 6 9 c 0 2 4 6

b. Linear. Let m = number of months and h = height in inches. h=6m+3 Note: since the tree will not keep growing at the same rate forever, this equation only works for reasonable values of m. Students may discuss how they might limit what values were "allowed" for m.

c. Not linear. Let n = number of folds and L = length in cm.
L=120(12)n where n can only be natural numbers (you can't fold a piece of paper 4/5 of a time, for example).

 n 0 1 2 3 L 120 60 30 15

d. Linear. Let P = perimeter in any units of length and s = side length in the same units.
P=3s or s=P3

 s 0 1 2 3 P 0 3 6 9

e. Not linear. At certain time intervals, his distance traveled is linear, but he does not have a constant rate of change for his whole trip.

f. Not linear. Let r = radius in units and A = area in square units.
A=Ï€r2 or r=AÏ€
g. Linear. Let m = time in minutes and T = number of tests that have been graded.
T=8m30

 m 0 30 60 90 T 0 8 16 24

h. Linear. Let s = step and N = number of squares. N = 3s + 4, where s is a natural number.

 s 1 2 3 4 N 7 10 13 16

i. Not linear. Let s = step and P = number of pebbles.  P=(s+1)21 where s is a natural number.