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:

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.

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.

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.

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.

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

Play With Me

On Wednesday I had the chance to visit my first classroom this school year. Sadly, in my role as curriculum coordinator, I don’t get to do this nearly enough. So I relish opportunities like this. Even better than visiting, the teacher allowed me to play a math game with her class.

I had so much fun!

I wanted something simple and quick to get the kids engaged before moving on to another activity. I also wanted it to involve adding 3-digit numbers because her class is in the middle of a unit on that very topic. I also wanted to bring in some place value understanding and reasoning, which are very much related to adding multi-digit numbers.

Basically I brought two decks of cards – one had Care Bears on the back and the other had Spider-Man on the back. I wanted different backs to the cards so it would be easier to tell which cards were mine and which were my opponent’s in case we needed to reference them during or after the game. I also pulled out all of the 10s and face cards, with the exception of the aces. I kept those and we decided to use them as zeroes. I tell you this because if you ever want to play a game that involves digit cards, here is a great way to get some without having to painstakingly cut out cards to make your own sets. Decks of cards are cheap enough. Just use those.

The game was me vs. the class. The goal is to make two 3-digit numbers. Whoever has the greater sum wins. On my turn, I drew a card, and I had a choice of putting it blank spots that I used to create two 3-digit numbers. Once a digit was placed it couldn’t be moved. On the class’ turn, I drew the card for them, but I let them tell me where to place the digit.

My favorite part of the game was at the end when the kids started shouting out that they’d won without even finding the sum. Take a look and see why they got excited: (Just pretend I hadn’t written the sums yet. I took the picture after the game was over.)

“You have a 9 and a 4 in the hundreds place. We have a 5 and a 9.”

“Interesting, and how does that tell you you’ve won?”

“Because the 9s are the same. And we have a 5 which is greater than 4. You should have put your 5 in the hundreds place.”

“I was hedging my bets and I lost.”

Such wonderful thinking from a 3rd grader! How often do students rush to calculate and find an answer to a problem? How amazing that these students were paying attention to the place value that matters most in these numbers – the hundreds – and then comparing the digits to determine who had a greater sum?

Since I was just the lead-in to the day’s activities we only got to play once, but I would have loved to play again. I would have liked to change it up a bit. I would still construct my number on the board, but then I would have allowed everyone to create their own number at their desk using the cards that I drew on their turn. At the end we would discuss who thinks they have the greatest sum and talk about their placement of digits.

Even though I didn’t get to play again, I’ll take the time I did have. It was the highlight of my week!