Tag Archives: open middle

Number Puzzles: Addition and Subtraction

After playing the Illustrative Math center Can You Build It? (Link) for a few days with my daughter, I decided to switch gears and introduce the Number Puzzles: Addition and Subtraction center (Link). I intentionally chose this center for two reasons:

  1. I like Open Middle problems (Link), and that’s what the puzzles in this center remind me of.
  2. I wanted to revisit two-digit addition before re-introducing three-digit addition.

If you’re unfamiliar with the Number Puzzles center, here’s what it looks like:

A screenshot of Puzzle 1 from the Number Puzzles center. There are four addition equations. Each equation starts with the number 75. Digits, represented by blank boxes, are missing from one addend in each equation.
Number Puzzles: Addition and Subtraction, Stage 3, Puzzle 1

Each stage includes several puzzles. In the example puzzle above, students have to make the equations true by filling in the blanks using the digits 0, 1, 2, 3, 4, and 5. They can only use each digit one time each. I like how all four equations show different ways of decomposing the number 75. I also like how each equation has the sum on the left side of the equal sign to combat the pervasive idea that the equal sign means “and the answer is…”

This center is really flexible because it has stages that span 1st grade through 4th grade math standards.

  • Stage 1: Within 10 (1st Grade)
  • Stage 2: Within 20 (1st and 2nd Grade)
  • Stage 3: Within 100 without Composing (1st Grade)
  • Stage 4: Within 100 with Composing (1st and 2nd Grade)
  • Stage 5: Within 1,000 (3rd Grade)
  • Stage 6: Beyond 1,000 (4th Grade)

For our first stage, I opted for Stage 3. I try to have my daughter practice mental math as often as possible, so I opted to start without composing so that she would feel initial success before moving on to two-digit addition with composing.

My daughter thought these were so fun! She was a little overwhelmed by the page at first so I asked her what she noticed. She said, “There are boxes. All of them have 75.” To encourage trial and error, I made her digit cards that she could move around on top of the empty boxes.

Girl thinking as she solves a Number Puzzle

After she found the missing addend in the first equation, I asked, “How did you know it was 4?”

She replied, “Because of the equal sign, this side has to equal 75 like this side. 71…72, 73, 74, 75. It’s 4.”

I love how she talked about the meaning of the equal sign without me having to ask about it at all!

I’ll admit she was a little thrown off at first by the double boxes together in the last two equations, so I did share with her that two boxes together make a two-digit number. Then she was good to go.

Here’s a picture showing her strategy for figuring out the missing addend in the last equation.

Girl with completed Number Puzzle page in front of her. Next to her is a white board showing a drawing of 75 using tens and ones. 4 tens and 3 ones are crossed out.

First, she drew a representation of 75 using base ten blocks. Then she said, “I have to take away 43.” She crossed off 43 and then counted the remaining blocks in the picture. In the future I might encourage her to try a mental strategy such as counting on from 43, but this let me know where she is comfortable working right now.

Stage 3 includes five puzzles. The first three use the digits 0-5, like you see in the example above. Puzzles 4 and 5 up the challenge a bit by adding more equations and requiring you to use all of the digits 0-9 one time each.

Girl thinking as she solves a Number Puzzle that includes even more equation

All in all, this is a pretty fun center for students to do in pairs or independently. As a teacher, I would be sure to circulate and chat with students to see how they’re grappling with the puzzles and look for places where I can nudge their thinking about addition, subtraction, and/or place value. I would also lead a few whole class conversations around strategies so students could learn from one another. While the activity is fun and gets kids thinking about addition and place value, talking and reflecting on the puzzles is going to help students get even more out of them.

My only gripe is that there are too few puzzles per stage. Usually with centers, you want students to be able to come back to them multiple times. Unfortunately some kids may finish all five puzzles the first day and they may not be interested in doing the same puzzles more than once. Thankfully, making new puzzles isn’t too much of a challenge. Here are some pointers:

  • Enlist others to help! If you work on a team of teachers, task each person with making 1-2 puzzles. The more you can share the work, the better.
  • You’ll need to think of a starting number that will be the same for each equation in the puzzle. (You could decompose a different number in each equation, but there’s power in the repeated reasoning of decomposing the same number in different ways.)
  • Consider the constraints of the stage you’re creating a puzzle for. For example, if you make another puzzle for Stage 3, you have to make sure you’re working within 100 and that none of your equations involve composing a ten.
  • Create a mix of equations. For example, have some include a two-digit addend plus a one-digit addend, while others include a two-digit addend plus a two-digit addend. You could even include three addends!
  • Think about which digits will be left blank. Be sure there’s some variety. Make sure the blanks aren’t all in the ones place in every equation, for example.
  • Try out your puzzle before putting it in front of students! Make sure that every digit gets used once. While playing with my daughter using the materials linked on the Kendall Hunt IM curriculum site, I found that Stage 3, Puzzle 2 has an error. The digit 0 is used twice and the digit 1 isn’t used at all. To fix the error, change the 88 in the second equation to 87 and all is good.
Screenshot of a Number Puzzle showing that there is an error in the puzzle

So far all we’ve tried is Stage 3, but I look forward to letting my daughter play with Number Puzzles again!

Better Questions: Math Rocks Meets Open Middle

betterquestions

This year I have been leading a cohort of elementary math educators in my district. We met for two full days in July – you can read about that here and here – and throughout this school year we’ve met every other Thursday after school.

In December, our meeting focused on the work of Robert Kaplinsky, specifically his IGNITE talk about productive struggle and his website openmiddle.com.

At the start of the session, everyone reflected on what “productive struggle” means to them. This is important because as certain phrases become popular in education, they quickly become jargon. I wanted to ensure everyone had a chance to think about how they interpret the phrase and share that with the group. Then we watched Robert’s IGNITE talk.

The image that stood out most to me from his talk was the one of the mom riding the bike for her child. It seems so silly, and yet there are many instances as teachers where we can find ourselves doing the thinking for our students instead of letting them try either on their own or with our support.

At the end of the video, Robert puts out a call to action for teachers to create opportunities for students to productively struggle. And why not start by having the Math Rocks participants do some productive struggling of their own? Regina and I posted 10 problems around the room. We let everyone loose to do some math for 15 minutes. They dove right in!

All 10 problems came from openmiddle.com. If you aren’t familiar with the open middle problem type, here’s a brief summary: (You can learn more here.)

  • they have a “closed beginning” meaning they all start with the same initial problem
  • they have a “closed end” meaning that they all end with the same answer
  • they have an “open middle” meaning there are multiple ways to approach and ultimately solve the problem

After debriefing as a group and sharing information about open middle problems, we came back around to the idea of productive struggle with this video from Michael Pershan. The whole thing is interesting, but for the purposes of our discussion, we watched the first 30 seconds of the video, and then we watched from 1:45 to 5:45.

By this point, we had made our case and it was time for the participants to take a stab at designing their own open middle problems. They had a choice of writing one from scratch or taking an existing problem from our curriculum and redesigning it as an open middle problem. A nice surprise is that our adopted textbook, Stepping Stones, already uses open middle problems in many lessons and activities! They don’t name them as such, but that’s essentially what they are.

We shared out the open middle problems they wrote. Afterward we gathered them together in this document if you’d like to see our first attempts. We closed the session with their homework assignment – giving their students an open middle problem and reflecting on it in a blog post. If you’re interested in learning more about open middle problems – especially learning from teachers trying them out for the first time! – check out our open middle blog post collection.

The consensus from the group seems to be that they can initially throw kids off if they’re not used to being asked questions like this, especially for those kids who want to neatly and easily come to the correct answer, but the questions provide opportunities for the type of thinking and struggling we want our students to engage in and we need to be using them more often.