# 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.

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.

# Triangleman’s Open Middle(?) Problem.

Just for Christopher…

Directions: Using the numbers 1-6, fill in the boxes to make the following product true.  You can repeat numbers as  many times as you wish.

How many possibilities are there?

# Multiplying a Double by a Single Digit Number- Open Middle Problem

Directions: Using the numbers 1-9, each only once, fill in the boxes to make the following product true.

How many possibilities are there?

# 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.

# Reflecting Powers- Open Middle Problem (maybe?)

When working with a student today, she noticed that:

and wondered if that pattern works for any other numbers.

So, for any integer a,b when does

Would changing the constraint on a,b change the problem?  What number types would produce more values for a and b?