Following up on Changing Questions

I haven’t heard back from anyone (and I’m not sure I really expected to) so I’ll just relate what happened in my classroom when changed the way I asked a question.  The first thing I’ll say is that changing the way you question definitely changes the way your students think- and in most cases it is for the best.  Take for example this convo Christopher and I had yesterday…

I was writing an Open Middle problem, and as I often do I hit publish too quickly.  This is what the first draft was…


Now, I want you to know that this was up maybe 2 minutes until I realized that I hadn’t put any real constraints on the problem.  Typically with Open Middle problems you are allowed to only use a digit once, and I was modifying this for a smaller number set.  OF COURSE Christopher would be online and checking my tweets because this is what happened next…


I had also realized that it wasn’t possible if you weren’t able to repeat numbers- which would be fine for the classroom environment, but not for the Open Middle format.  This is what I was originally thinking of the problem so I published it instead (which caused Christopher’s reaction)


I was trying to reduce the numbers usable because I was wanting to target 3rd grade and was trying to put a smaller constraint on possible values for the boxes.  Unfortunately that wasn’t working, and I had to go back to the original format.  But like Christopher stated, it changed the problem and some of the thinking students needed to do.

Back to the original set of questions.  The first:

Create an addition problem where 2, 3 and 4 appear somewhere in the problem or answer.

Let me remind you that I work at a juvenile center where I have mixed classrooms.  There is a wide range of student ages and ability levels in one class.  Most of them were VERY uncomfortable with this problem.  They weren’t asked to make a direct computation, and there wasn’t any “answer” for them to work with.  After their initial shock, and Mr. A’s many many times of explaining 2,3 and 4 need to appear somewhere within the problem, they produced some nice examples.


The top left response was overwhelming, with over 92% of the class producing it.  It was the other responses that I really found interesting.  Students used combinations of grouping 2 and 3 to create a new number (like the bottom left), and others introduced additional digits with 2, 3 and 4 to create a different sum than 9 (like the top right).  I was also interested in students who created a sum equaling 234, asking students how they determined what numbers to use as addends.  Many students replied that the number they used was their favorite number, which made me a bit concerned about the student on the bottom right- students always try to sneak in inappropriate things to check and see if you are paying attention.  The bottom response was the one I found most interesting.  This is a student who HATES math class and is working at a proficiency level that is 3 grade levels lower than her enrolled grade level.  She has an IEP for Emotional/Behavior Disorders as well as Learning Disability in Mathematics.  Yet she is the one who provided me with the most elegant and interesting solution.  She loves patterns, and told me she wanted her numbers to be sequential, so she needed to use 5.  She sat and worked for over 10 minutes trying to figure out how to create a sum of only 5.  She asked me many times if she could multiply, subtract or divide- and was referred back to the wording of the problem many times.  She did not give up or throw a fit however, and I believe that is because she set the parameters she was trying to achieve.  Then, the light bulb went off, her face lit up and she asked me, Mr’ A,- can we use negative numbers?  I referred her back to the problem and asked if it indicated that she couldn’t.  She reread it and said no.  So I told her she had her answer and she was immensely pleased with herself.  She was in an incredible mood all day and even came into class the next day asking if anyone else had used negative numbers (the answer to which was no).

That second day, I once again asked them the “same” question, and highlighted how I made it different.

Create an addition problem where 2, 3 and 4 appear somewhere in the problem or  and  answer.

Even though many students found the question easy yesterday, once again the sky was falling and the math gods were against my poor innocent students.  Once again I calmly took it all in stride (while laughing to myself inside because I expected this response) and verbally explained that 2, 3 and 4 needed to appear on both sides of the equal sign.  They then went to work.


By changing one word, it created such a variety of thinking and solutions that I could have never elicited with the first question without very guided questions.  Some students used the same thinking as the first question, but I also loved how some natural properties of mathematics was produced: place value, additive identity, reflexive property.  Students introduced decimals!  When does that EVER willingly happen?  Once again, I was surprised by a student who took an extra step and made their sum by using only 2, 3 or 4 in all of their numbers (except for the 100 Mr. A, my brain started hurting).  This has left me with many a rich discussion to take with my students as we reflect on the difference of these questions, their thinking, and their solutions.

I hope you find the same in your classroom.

Changing the way we ask questions…

Reading through a book today I saw this questions posed:

Make up an addition problem where 2, 3 and 4 are used somewhere in the problem or answer.

Ask that question of your students and see what types of responses you get.

Then twist the question slightly and ask it again (this is what I was thinking when I originally read it, so I wondered what type of varied response it would get and why).

Make up an addition problem where 2, 3 and 4 are used both in the problem and the answer.

What difference did it make?  Was the change significant for teaching?

I would like to know your experiences with this, thanks for sharing.

Gemini Puzzles- Create a Headache for Order of Operations!

OK, the MTBoS has really influenced the way I approach teaching and assigning problems.  Coming back from Christmas Break, I needed a different way to get students thinking about math again, and found that thanks to Math = Love in the form of Gemini Puzzles.

What are Gemini Puzzles?  In short, they are equality statements that are missing any mathematical symbols. Here is an example:


The thing I love about this is- I put these two on the whiteboard for students to see when they first walk in and instead of asking what we are doing today or talking about what they did last night, they started thinking about what was going on.  The other great thing, I didn’t have to tell them what to do, they started playing around with the numbers to get it done.

When I asked my students what they were doing, there was a common theme:


Which is great until I ask students about the name of the problem, Gemini.  I then get answers about twins and I tell them that’s relevant to the problem.  They are given two equivalence statements, and they have to be twins.  In other words, anything they add to one statement has to be the exact same as what they add to the second.

Then students start working in a stream of consciousness and they have this as an answer:


It doesn’t take too long for a student to say that there’s problem with this.  Students are thinking “one plus one is two, times two is four”, but they are not properly showing that.  Then the talk of Order of Operations hits, and students realize the first statement is only 3.  By throwing in another set of symbols they find the correct solution:


By providing students with two equivalence statements that need the exact same symbols to make them true, you are giving students a reason to think about the order of operations and how they interact.  I have loved listening to student talk when working on these, and I even had students proudly come up to me with different solutions to a problem.  They are amazed that there can be different ways to implement mathematical operations and get the same result (see if you can find which one they discovered!)

Gemini Puzzles are a great activity for the classroom, I know I’ll be using them from now on, I hope you enjoy them as well.

I currently have a range of students from 6th to 12th grade in my classes, with various mathematical skills ranging from 4th to 12th+, and these puzzles have engaged and challenged them all.  It’s a great thing to see.

#MTBoS30- Day 9

Brute force…


This has really gotten me thinking lately.  I gave my students the following picture:


Yesterday my students and I did a problem adding and subtracting a string of numbers.  We looked at patterns that could be found in it, and even looked at why PEMDAS is misleading for order of operations.  They were finding patterns, multiple strategies and stretching their thinking.  I had hoped that this would translate to this problem- I was wrong.

#1 answer (98% of my students): I counted all of the squares.

I had tick marks, numbers, dots, colors, everything on their sheets showing how they made a 1-1 relationship with each square to find the total.  I was very disheartened, but I didn’t show it.  I asked if anyone had an alternate method, and only received a small handful (of which I will also share).  So I asked those who counted all of the squares: “How many of you still find yourself using your fingers when computing?”  All of them raised their hand.  This is a strong indication of where my students are developmentally with math, and how I need to provide them with opportunities to explore and stretching their thoughts about problems.

So, I then reminded them about yesterday and asked what we looked at with that problem.  They talked about the patterns and multiple representations- they remembered the whole segment!  So I asked, “How can that apply here?”


4+3+4+3+4+3+4 or 4×4 + 3×3


1+3+5+7+5+3+1 or 2×1 + 2×3 + 2×5 +7


15+4+4+1+1 or 15+ 2×1 + 2×4

And then the bomb dropped


5X5 square


A student came up with this and blew the minds of every one of his peers.  I let it sit there for 3 minutes before I even uttered a word.  I could see each of them processing what just happened.  Finally one student’s face lit up and he said “Oh, I totally get it!”

It took a while for them to realize that yesterday was not a “one and done” day for math, that I will expect them to do this every day they are with me.  Tomorrow we will have another pattern and see how they do with that one.

Go out and drop the bomb on your class.

#MTBoS30- Day 8

Putting it all together- that’s what my mind is trying to do.  There are a TON (really, there are) of great resources out there on the web via the #MTBoS.  There are times where it is hard to decide what to use, when to use it and where.  I am lucky enough to be slightly flexible in my curriculum (and also a curse because of the nature of the placement).

So, because I teach in MN and there are no “perfectly pre-packaged curriculum” (really who even wants that truly?), my mind is trying to wrap itself around what would be a good meld of components in the classroom.  Like I mentioned in my previous post, Science Practices in Math, I really believe that restructuring the lesson layout will help not only my students, but others as well.

Currently I have been using parts of David Wees a2i, an online curriculum for Algebra 1 and 2 as well as Geometry.  It has opening activities and I like the exploration of topics.  I also need to work with resources I find great (and my students do too) such as Estimation 180, Open Middle, WOBD, Would you Rather, Visual Patterns.

This summer will be interesting for me…