Tag Archives: centers

Five in a Row: Addition and Subtraction

I’ve been blogging recently about doing math with my daughter and trying out different centers from the newly released Illustrative Mathematics K-5 curriculum. You can read about our experiences with the two centers we’ve already tried: Can You Build It? and Number Puzzles: Addition and Subtraction.

In this post, I’m going to share our experiences with a new center, Five in a Row: Addition and Subtraction. Similar to Number Puzzles, I like the flexibility of this center because it has multiple stages giving me lots of choice about what kinds of numbers we’re going to work with:

  • Stage 1: Add 1 or 2 (1st Grade)
  • Stage 2: Subtract 1 or 2 (1st Grade)
  • Stage 3: Add 7, 8, or 9 (1st Grade)
  • Stage 4: Add or Subtract 10 (1st Grade)
  • Stage 5: Add within 100 without Composing (1st Grade)
  • Stage 6: Add within 100 with Composing (1st and 2nd Grade)
  • Stage 7: Add within 1,000 without Composing (2nd Grade)
  • Stage 8: Add within 1,000 with Composing (2nd Grade)

I decided to start with Stage 6 because my daughter already reviewed adding within 100 without composing when we played the Number Puzzles center. Now I wanted to give her a chance to review adding within 100 with composing.

Here’s what the game board for Stage 6 looks like:

Five in a Row Gameboard. A grid of 25 sums students try to cover. There are 2 rows of numbers along the bottom of the page that are the addends used to create the possible sums.
Five in a Row: Addition and Subtraction, Stage 6

I’ll be honest I have some issues with this game immediately. There are 45 possible sums that can be created with all of these addends. There are 10 duplicate sums, which leaves 35 unique sums that can be created (3 of which are over 100). However, the game board only includes 25 of these possible sums. That means 10 of the possible sums are missing completely from the game board! Students could spend a lot of time adding numbers together only to get sums they can never cover.

I didn’t want to play with this game board, so I came up with two options for modifying it:

  1. Increase the size of the game board to include all 35 possible sums, or
  2. Decrease the number of addends so that the number of possible sums is 25, or as close to 25 as possible

I went with option 2. I decreased the number of addends from 10 to 8. I basically left off 26 and 48. This brings the number of possible sums down from 45 to 28. There are 2 duplicate sums, which leaves 26 unique sums that can be created. I opted to leave off 122 (65 + 57) because it’s not within 100 anyway.

Here’s my redesigned game board:

Modified Five in a Row game board

If you look closely, you’ll see another modification I made: the game is now Four in a Row, not Five in a Row. The game board feels too cramped to attempt five in a row, especially when you have an opponent actively trying to block you.

As the pictures below show, this game really got my daughter thinking!

The further we got into the game, the longer each turn took. On one hand this is great practice because it means she’s doing lots and lots of adding in her head. At the same time, it got pretty exhausting. I even found myself not wanting to be too strategic on some turns because I just didn’t care to try out all the different combinations available to me to see if I could work on making my row or block her from making her row.

Dan Finkel has some great advice about what makes a good math game which he shares in this 3-minute video:

Here are his three ideas:

  • Choice needs to be a part of the game
  • Math should be the engine of the game
  • Simple and quick to play

Choice is definitely part of this game. Students get to choose which of the two clips to move on their turn. They also get to choose to which number they move the clip. Students get to choose whether they’re going to try to place a counter such that it helps them make a row or blocks their opponent from completing a row.

Math is the engine of the game. Students aren’t just solving random addition problems. They have to look at the possible sums they might want to cover and then look at the addends to see how they can achieve their goal.

The game is simple to learn. However, where it falters is that it is not quick to play. This game took us a while, and we were only aiming for four in a row instead of five. We’ve played Stage 6 on two different days and we only played one round each day. Each round took so long, we were ready to move on after that one round.

The game does a lot right, but because of the length of time it can take to play, I don’t know that this is going to be a “go-to” game for lots of kids. Skill level probably affects playing speed quite a bit. Students who are quick and efficient at adding will have a much easier time than students who are inefficient or inaccurate. There’s also the issue of working memory. If students want to be strategic, they have to hold a lot of possible sums in their head as they plan their next move. One strategy you could try is to give students a white board so they can jot down each equation they’re considering to avoid overloading their working memories. This will also help prevent frustration that can make the game feel a lot less fun.

I asked my daughter the other day if she wanted to play Four in a Row again or go back to Number Puzzles. Considering she made comments like, “I don’t like this game,” while we were playing, I was totally surprised that she said she wanted to play Four in a Row again!

I decided to move on to Stage 7 which is about adding within 1,000 without composing. Similar to what I did with Stage 6, I redesigned the game board so that exactly 25 sums are possible.

Modified game board for Five in a Row Stage 7

This time there are two rows along the bottom. The way it works is that you always add one of the numbers from the top row with one of the numbers from the bottom row. On your turn, you’re still only allowed to move one of the clips, either the one on the top row or the one on the bottom row.

Since Stage 7 didn’t require composing, it was much less mentally exhausting to play, and my daughter seemed to enjoy it. She was particularly happy that she beat me. 🙂

I don’t want to dissuade anyone from trying out the Five in a Row center the way that it’s presented in the IM curriculum. However, if you’re interested in trying out the modified version I created, here’s a PDF of the modified game boards for Stages 6 and 7.

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!

Can You Build It?

This week I’m starting to do a little math with my daughter everyday to dust off the cobwebs before 4th grade starts in September. One of the resources I’m using is the centers from the Illustrative Mathematics K-5 curriculum (Link to Kendall Hunt’s version of IM K-5 Math).

We kicked things off on Monday with a center called Can You Build It? (Link) One thing I like about the IM centers is that they often contain multiple stages within the same center, so you can choose just the right starting point within a given concept. Since my goal was to revisit arrays and the meaning of multiplication, we started with Stage 1. In the original IM version, one person builds an array secretly and then describes it to their partner and the partner tries to recreate it.

I changed this stage into a cooperative game that turned out to be really fun for my daughter. Here’s how it works:

  1. Draw a target area card. (I created a deck of cards that have the numbers 10 – 27 on them. This means there are 18 possible target areas, which feels like a good range. The numbers are also small enough that you won’t spend all your time counting out the tiles you need before making your array.)
  2. Each player secretly makes an array with that target area.
  3. Share your arrays. If you made the same array, you collectively earn 1 point. If you each made a different array, you collectively earn 2 points. (To clarify, a 2 by 6 array is the same as a 6 by 2 array.)
  4. Earn 5 points in as few rounds as possible.

If you don’t have square tiles handy, you could use a free app like Number Frames from the Math Learning Center (Link) which can be used in a browser or downloaded onto a tablet.

Screenshot of Number Frames app. The workspace of the app shows 3 arrays: a 1 by 12, a 2 by 6, and a 3 by 4.

Or if you still want something hands-on, you could always use some crackers!

Source

After a couple of days playing Stage 1 and revisiting how to build and describe arrays, we moved on to Stage 2. There are a couple of key differences here:

  1. Instead of secretly making only one array, the goal now is to make as many different arrays as possible with the target area.
  2. The game is competitive now. The player who makes more arrays earns 2 points and the other player earns 0. If both players make the same number of arrays, they both earn 1 point. The winner is the first to 5 points. (The original IM center used a slightly different scoring scheme. I opted for something similar to the game we played for Stage 1.)

My daughter immediately started bumping into ideas related to prime numbers. Here are some highlights from our conversation as we played for the first time:

1/ Daddy: Today our game is slightly different. This time when we draw a target area, our goal is to make as many different arrays as possible. If we get the same number of arrays, we each earn 1 point. If one of us makes more than the other, that person earns 2 points.

2/ Daddy: (draws card) Our first target area is 20.
(both make arrays in secret)
Daddy: I made a 2 by 10 and a 4 by 5. How about you?
Me: I made those, and I made a 1 by 20.
Daddy: Oh! I forgot that one!
Me: You have to remember you can *always* make a 1 by array!

3/ Daddy: (draws card) Okay, this time our target area is 13.
(both make arrays in secret)
Me: Ugh! I can only make one.
Daddy: Me, too. What did you make?
Me: 1 by 13.
Daddy: Hmm, I wonder why we could only make one array.
Me: Maybe because it’s an odd number.

4/ Daddy: (draws card) Now our target area is 11.
(both make arrays in secret)
Me: No! You can only make one again.
Daddy: Huh, is this an odd number, too?
Me: Yeah.
Daddy: That’s interesting.

5/ Daddy: (draws card) Ok, our target area is 10.
Me: I’m just going to write down the 1 by array. I don’t even need to make it.
(both make arrays in secret)
Daddy: What did you make?
Me: A 1 by 10 and a 2 by 5.
Daddy: Same here. Is 10 odd?
Me: No, it’s even.
Daddy: Hmm…

6/ Daddy: You made two really interesting observations today. Do you remember what they were?
Me: …if a number is odd you can probably only make one array?
Daddy: What else?
Me: …and you can always make a 1 by array for every number!

Originally tweeted by Splash (@SplashSpeaks) on August 18, 2021.

I love how this game has a simple premise – make arrays – but it creates opportunities for students to notice deeper ideas about numbers and multiplication. If you woudl like to try this game out with your own child or students, here’s a link to the center. (Link)

If you work in a grade level that introduces prime and composite numbers, I also recommend checking out 4th Grade Unit 1 of the IM curriculum for well-designed, ready-to-go lessons. (Link)

[UPDATE] Alyson Eaglen shared a great idea on Twitter. She said that instead of using cards with pre-printed target areas, she suggests rolling three 9-sided die and the sum is the target area. What a great way to bring in some bonus addition practice! If you don’t have 9-sided dice, you could always use five 6-sided dice or whatever combination of dice yields the range of target areas you’re interested in for the game. If you don’t have physical dice handy, Polypad’s free virtual manipulatives (Link) include a variety of dice under the Probability and Statistics menu.