Category Archives: #reasoning

Explaining “Explain”

Here is a released question from Smarter Balanced (I even answered it!!!):


Ok I lied. ¬† That was an edited version of a Smarter Balanced question – here’s the original:

ExplainingExplain1Now all of a sudden my answer doesn’t seem sufficient anymore ūüė¶ ¬† Here’s my best guess at a popular student answer:


This word “explain” is keeping me up at night lately. ¬†In this problem I’m not sure adding the word explain to the end gains us enough to warrant it. ¬†To achieve Common Core we can’t just throw the word “explain” after every problem we did last year and call it a day. ¬†By the way I’m not saying that’s what the Smarter Balanced Consortium did on this particular problem. ¬†But this use of the word “explain” does bring two things to mind:

1. ¬†It’s hard to explain your mathematical reasoning without access to drawing diagrams.

2.  If we ask students to explain something Рit should be something worth explaining.

With respect to #1 Рmy focus this year has been on explanations through multiple representations.  Basically I have students make connections between diagrams, tables, graphs, mathematical symbols, and written descriptions.  I feel underwhelmed asking students to explain with just a typed explanation.  I want explanations to look like this:


In the student work above Рimage if it was only the conclusion.  Look at how much would be lost.

There are certainly better answers to the rectangle problem from Smarter Balanced than I offered up here.  I actually really like the problem itself, I just do not think having them explain it gains us much versus just solving it.

It’s hard to explain the word explain. ¬†It’s a word that only makes sense to me until I try to explain it.


Missing Assignment Buyout Program

The Overview:

This year I wanted to do Kyle Pearce’s¬†Detention Buyout Program that Dan had highlighted in his Great Classroom Action series. ¬†The problem was that in my new school we don’t have detentions, so I didn’t think I would get much buy-in from the students. ¬†But there is something that all schools definitely do have: ¬†Missing assignments! ¬†So I created three “deals” that would allow students to pay me money in exchange for getting credit for an assignment they missed.

I used this assignment as an introduction to inequalities, but I also wanted to link the Missing Assignment Buyout Program to the linear equations we just finished covering.  That is the why as you look at this assignment, you will see a focus on connecting the information in the graphs to the information contained in the inequalities.

I sequenced this by first giving the assignment.  Then two days later I did another version of it as an opener / warmup.  And then lastly I put another version of it on their test.  Each new version offered slight modifications from the previous.

The Description:

I first offer students three possible deals for buying off their missing assignments. ¬†I poker face the whole thing and enjoy all the “Is this legal” expressions on their faces. ¬†I tell them to make sure they go home and talk to their parents about how much money they have budgeted for such as program. ¬†The first question on the worksheet asks them which deal is better for them, so as an added bonus I printed out each students missing assignments and handed it to them. ¬†This is that first worksheet:



There are a lot of interesting questions and explanations that came out of this first assignment.  For instance, having students see that x less than 5 was the same as saying x less than or equal to 4 since x could only take integer values.  Also having students see the connection between the intersection points of their graphs and the inequalities they wrote was time well spent.

A couple days later I came back to the Missing Assignment Buyout Program in the form of a opener or warmup question.  I handed the students this graph when they came into the room (two graphs per page to save paper):


Then I had students write a description of each deal, as well as the inequality and equation for each deal.  This was a slight inversion of the original assignment where I gave them the description and had them write the inequality, equation, and then graph.  Now I am giving them the graph and asking them to write the description, inequality, equation.  I have them in pairs and am checking homework and taking role while they work.  Then I randomly call on pair share partners and fill in the following table that I am projecting on the board:

Screen shot 2013-11-27 at 5.18.48 PM

Lastly to make sure that they really did understand the concept, I put a similar problem to the opener exercise in their inequalities chapter test. ¬†The test had a slight twist in a scenario where a student would want to buy the Flat Fee plan based on their number of missing assignments, but based on the money they had to spend, they would need to pick their second best option. ¬†Here’s that problem:


I initially thought having them graph each deal was kind of an unnatural excercise, because why would someone ever graph something like that? ¬†But I think it ended up working because of how the Opener and Test question both refer to the graph. ¬†All in all student engagement was high, even with the graphing portion so I think I’ll keep it next year.

The Extension:

(good idea courtesy of my principal)

Tell the students that you have decided to only offer one deal to the whole class, and they have to decide which deal they want for the class. ¬†This could open up a nice debate about fairness and equity – this deal is best for you since you don’t have any missing assignments, but what about these other students? ¬†Connect this debate to something current, like Obamacare. ¬†Discuss how math influences decisions and that often decision makers have to make decisions based on their believe on the greater good, even when the numbers indicate that some people will be negatively affected by the decision.

The Goods:




Stacking Cups Assessment

When three of your favorite bloggers all write about the same lesson (Dan, Andrew, Fawn) it is a pretty safe bet that you should do the lesson. ¬†I used Andrew’s 3Act video because my students can be pretty green and I might not hear the end of it if I couldn’t find an additional use for all these cups I was bringing into the class.

I don’t have anything to add to what was already said by Dan, Andrew, and Fawn, so I will just share a problem I created that you can put on your midterm that is a slight twist to the presentation of the original problem:


1.  How many cups would stack in a 250 cm door?

2.  What are the dimensions of the cup?  Draw it and label it with the dimensions.

I suppose you could ask for the y-intercept and slope and all that stuff too if you wanted.

Moving on from test questions РThe actual lesson went great for me and I am definitely looking forward to doing it again next year.  When I did this problem in algebra I had the students make a Stacking Cups comic that was supposed to describe how to solve the stacking cups problem.



I like the comic concept because I think this is a very visual problem, and since I didn’t provide them with actual cups they needed to create their own visuals. ¬†I have been trying to get students to give me a visual for every word problem they do this year. ¬†My stated reasoning for that has been that visuals help you give a clearer and more convincing justification for your solution.

In order for students to learn how to construct a viable argument and critique the reasoning of others (Let’s hear it for MP.3!!!), we are going to have to have an iterative process on a couple problems where they essentially hand in drafts, and we keep having them make improvements. ¬†I think this is a great problem to do for that since it has a couple nice extensions for system of equations (different sized cups) and geometry (here).

The Centauri Challenge

I’m posting this because my students enjoy it and I can’t find it anywhere online. ¬†I got it from a colleague a few years ago. ¬†I have no idea where it originally came from. ¬†It’s is a great intro to logic and proofs.


My Attempts At CCSS Word Problems

I’m trying to prepare for CCSS, so this year I have been looking at word problems with a specific goal of improving student literacy by connecting each problem to graphs and having students explain their solutions. ¬†Then afterwards coming back to their solutions and analyzing them and improving them. ¬†It’s been successful thus far based on my last assessment so well the hell – figured I share.

I started on day 1 with my Las Vegas Problem.  Then on day 2 I played this video of myself graphing the equations we wrote on day 1 for the two airports.

Then I passed out this worksheet and have the students try to figure out why Wolfram Alpha was calling 14 the “solution”.


I like the worksheet because it has the students try to explain which graph is for which airport, and this is before we have learned to graph or talked about things like y-intercepts and slopes. ¬†It is just them connecting the story to the picture of the story. ¬†The last problem on the worksheet was meant to highlight that context drives the graph, and that this particular graph should not have any negatives because you can’t have negative days. ¬†But we had a good discussion there.

Since this is day 2 and these students aren’t used to having to “explain” their reasoning, I got a lot of papers that gave an answer without any explanation. ¬†So I went back and did a Math Hospital and had the class analyze how to explain their solution.


During the Math Hospital I introduced them to one of the English languages most powerful and poetic words: “because”. ¬†I showed that all they have to do is put that word after their solution and it will literally force you to state the reason for the answer. ¬†CCSS literacy for me isn’t about having the students explain their thought process, rather it is about having the students explain why they made the decision they did.

I did a couple worksheets that were styled like the Vegas one and then I put one of them on Ch1 test (The Internets was on the test). ¬†Every single student explained their answer on the exam. ¬†After the test the class and I did the “Math Gym” where we take their healthy answers and make them healthier! ¬†Basically teaching students that it’s great to choose internet company A because it is cheaper, but we can’t just say it’s cheaper, we need to also explain why it’s cheaper.

Attached are several other worksheets similar to Vegas.  Each of the problems was taken directly from our textbook.  I just provided the graphs and asked the questions in a similar manner to the Vegas trip.  One of the things I really like about these worksheets is that they provide students with a graph of the situation and ask them to make some connection between the graphs and the situation they represent.



The Goods: