# Quick Shows With Ten-Frames

I was asked to come in and work with small groups (4-5 students) in Kindergarten today using ten-frames. The teacher wanted students to unitize by 5, 10, and 15, counting on the rest by ones. For example, if I asked a student, “How many do you see? How do you see them?”, she wanted the students to understand that you could find the value in a variety of ways. Here are a few of the anticipated answers she wants them to give by the end of the year:

• I counted them all. One, two, …twelve, thirteen, fourteen (Level 1)
• I saw two- 5’s, so 5, 10, 11, 12, 13, 14 (Level 2)
• I saw a ten, so 10, 11, 12, 13, 14 (Level 2)
• I saw 5’s and 1 missing. 5, 10, 15, (counting backwards) 14 (Level 3)

I had a deck of ten- and double ten-frame cards, so I decided to do some “quick shows”.  I would show a card to the kids for about 5 seconds, and they had to ‘think’ about their value (versus just shouting out the number). We rotated who gave the value first, but every child had to give the value they thought was on the card. I chose a different student to explain how they got the value, then gave every other student a chance to share their thinking. We did this for about 15 minutes per group of 4-5 students.

Here are our ah-ha’s:

• Out of the 4 groups, only one group stayed within the single 10-frame. I was getting answers from this group that were bigger than 10 every time. For example, when I showed them a card with 8 dots, one said 8, one said 12, and the other two were still counting by ones. I quickly drew a double (or triple) 10-frame and grabbed some counters (plastic circle thingies) and would show them their answer, then the original card. That worked for all but one student. For him, I kept on the table the card with the ten-frame filled in, then did the quick show. That clicked for today, but I need to do some hands-on work with this group. I also need to go back to a 5-frame and really focus on 5+ values before moving beyond 10.
• Students needed to be convinced that the two cards below each showed a value of 5. Great for starting the discussion about the commutative property! We rotated the card over and over until someone said, “It is just the same thing! You didn’t put more on or take any off. Geez!”

• The sequencing of the quick show was instrumental in students building strategies beyond counting one-by-one. The order that seemed to work the best today was as follows: 3, 5, 5 (again, upside down), 4 (to see it was 1 less than 5), 6, 8, 10, 9, 11. Notice we kept them seeing 1 more/less so they could use that strategy as well.
• For the groups that could “just see” the ten-frame, I worked up to 20. Here is the order we used with those groups: 3, 5, 5 (upside down), 8, 10, 12, 15, 14, 20, 18. 18 was tough (see the number of dots), as students really needed to push to 5’s versus  counting 10 then by ones.
• One group finished about 5 minutes early, so we played war. That way, they each had a different card and had to tell me their value before determining who had the most dots. This was interesting, as they had the cards to touch and many reverted back to one-to-one counting. We will need to think about that for next time.

What I loved about this activity was that I had 15 solid minutes to informally assess each child. I heard what they understood and where they struggled. I was able to note for the teacher which cards each child got quickly, and which he/she reverted back to counting by ones (or guessed). Every child was engaged and had to listen to their friends as each shared out their strategy. And most important to me, every child left my group smiling, asking when I was coming back to do more “quick thinking”.

For large ten-frame cards: https://lrt.ednet.ns.ca/PD/BLM/pdf_files/five_and_ten_frames/ten_frames_large_with_dots.pdf

For double ten-frame cards: We made them by cutting/pasting two ten-frames together. I am sure you can buy the cards, but this was cheapest for us.

# Cross-Out: Sums to 12

Chris asked for a new game yesterday, and I didn’t have one ready (Gasp!) So we made one up together called “cross-out”. This was quick, easy to organize, and he had fun playing it and ‘cheating’.

Materials: white board, dry-erase marker, two dice (we used dot dice, but you can use number cubes to up the level of thinking)

Objective: We played as a team. The goal is to cross-out every sum when rolling two dice (2-12).

How to Play

1. Have your child write the numbers 2-12 on the white board. This is great fine-motor practice!
2. Player 1 rolls the dice and adds up the values. Player 2 crosses out the sum on the board. I rolled a 9, so Chris had to find the 9 and cross it out (see below).
3. Player 2 rolls the dice and adds up the values. Player 1 crosses out the sum on the board. If a sum is already crossed out, continue rolling (and therefore practicing addition and counting on) until you get a sum that you can cross out. No losing turns here!
4. Once your team has crossed-out every sum, you won! Do a silly dance to celebrate your success!

Fun Note:
When we only had the 3 to cross-out, Chris asked if we could change dice to be 0-5 instead of 1-6. “Why?” I asked. “So that I have a better chance of rolling a 3! The only way I can get it is with a 1 and a 2 and that’s tough!” If I had the 0-5 dice at my fingertips, I would have totally given in. This is a great statistics insight for such a tiny human!

He rolled a few more times, got sick of rolling and decided to just roll one die. BAM! First roll he got a 3. He was very proud of his ‘cheating’ scheme!

Differentiation Ideas:

• Use a number cube and a dot die to work on counting on (Level 2).
• Use two number cubes to work on addition rather than one-to-one counting with dots.
• Use cubes that have larger values and work on the teens/twenties. I buy square wooden cubes at a hobby/craft shop and use a Sharpie to make whatever dice I want to use. Easy and cheap!
• Play against each other. Each person could write 2-12 and see who can cross-out their board first.

# Race to the Top: Kinder Observations

Last week I posted a game called “Race to the Top”. I was able to play the game with all of the students in my son’s kindergarten class. I was one of the stations during center time, and had between 4-6 students each rotation. We kept them 15 min per station, which seemed enough time for their engagement. (One group ran at 20 min and it was too long.) Each person at my station had their game card (see links below for options) and each pair had one number cube (labeled either 0-5, 1-6, or 5-10) and one dot die (1-6 dots).

Here are a few observations and patterns of misunderstandings.

1. Counting different representations is tough! The students were used to having to ‘add’ (i.e. count one-by-one) two dot dice, but hadn’t yet used a number cube WITH a dot die. When we started to play, at least one in each group said their sum was 2, no matter what was on the dice. This is because there were 2 dice. It only took a few rolls for all of them to get the hang of it.
2. For this time of year, the 0-5 number cube was too easy (with the exception of 1 pair of students). I would use the 6-10, as most of our misconceptions surfaced once they passed the number 10. Here are the misconceptions after 10.

## Example: 9 + 5 dots

• Misconception 1 (counted backwards from their number):  “9…8, 7, 6, 5, 4.”
• Misconception 2 (started counting by 10’s):  “9…10, 20, 30, 40.”
• Misconception 3 (counted them separately and got stuck): “9 and 5…pause…”
• Misconception 4 (wanted to put 14 squares on the game board, not see it as one-14)

After about 2 minutes of playing with me direct instructing, the students were successfully independent, taking turns and excited to see what their value was. In 15 minutes, I was able to watch, assess, and intervene every child in my group several times. They were engaged and excited to be playing. We averaged 18 sums per player in the 15 min, which is a LOT of addition!

## Favorite Quotes of the Day

• You cheated! You can’t keep rolling until you get your number! (This would have been me as a child!)
• Wait. We can’t play anymore? Awe.
• I only need two more 12’s to win! You need two more 7’s! It’s a tie! (I get giddy when students do more math than asked!)
• Thanks for playing games with us, Chris’ Mom!

Game Board for 0-10 (Change as needed depending on the number choices for your dice)

Race to the Top Game Board

# Race To The Top! Counting On

So my tiny human, Chris, loves to add, but always has to start from one (no matter what the values are). I finally figured out a game to play to push into Level 2: Counting On (see previous blog for more info on the levels of single digit addition/subtraction).

Objective: To get to the top of one of the columns (doesn’t matter which one) first!

Materials: 2- Race to the Top Game Boards (click on link) Race to the Top Game Board 2-dice:  Since Chris is working on making sums to 10, I created my dice. You do not have to, but it is super easy and cheap! Go to a hobby store or home improvement store and buy wooden cubes. On the first one, label with the numbers 0-5. On the second one, draw dots on it (like a normal die), but use 0-5. Notice my mad dot-drawing skills!

How to Play: First player rolls the dice. He ‘traps’ the number (in the case above, the 3) and counts on one-by-one if need-be the dots. He puts an object above the sum’s number on the game board. Please view the example. Note the time it takes for Chris to figure it out, but he still gets it. This is tough when children are used to adding with objects and we replace one set of objects with a number!

Notice how he is also doing some mental work with figuring out “how many more” he needs to win the game.

Player 2: Roll dice, count on, and place the object on her game board. Continue until someone reaches the top of a column.

Look at how many addition problems he had to complete to finish the game! He never even noticed all the fantastic work he did! (Insert evil laugh.)

### For Differentiation

Level 1 (Counting All): Use dot dice for both instead of one dot and one number cube. That way he is working on his one-to-one counting and adding all together! You can also make a 0-6 page and use 0, 1, 1, 2, 2, 3 dots for each cube.

Level 3: Use both number cubes instead of dots. You could also use number cubes 1-6 and create a game board from 2-12.

# Levels For Single-Digit Addition: Where Is Your Child?

‘Tis the season in Kindergarten for learning addition and subtraction. You may wonder where your child is with respect to these foundational operations. Most educators will think about their students’ learning of single digit addition/subtraction as a 3-level progression. I will focus on addition for sake of space.

#### Level 1: Counting All

This is where most kinders should be. They can even end the year in this level and be awesome! At Counting All, students typically use concrete objects to count one-by-one in order to find the sum. Below is an example, using 2 + 3.

#### Level 2: Counting On

This is the “trap and keep” idea. The first addend is “trapped” in your mind, and you count on from that value. This is a more sophisticated idea, because you have to understand that the first addend is a quantity of its own, and you are moving forwards from that value (versus starting at 1 every time). Often your child will “trap and keep” the first addend by taking their hand to their head to “trap” it in their mind, then use their fingers to count on. So for 5 + 3, they would “trap the 5” and count on, “siiix….sevennnnn…eighhhht”. This is a level many children live in for quite some time.

#### Level 3: “Messing with Math!”

That is actually NOT what it’s called, but I like this title much more! This is when children start realizing that there are certain “cheats” that they can use to do more of the math in their head. (Mathematically, they are called properties, but that is for another blog!) I will actually devote an entire set of blogs for this Level, as it is that important. But for now, here is an example using ten-frames with the expression, 9 + 5.

Students can see that, if they take one of the five (reds) and move it up with the nine (blues), they can make a 10. 9+5 can be renamed as 10+4=14. This is HUGE for students in terms of flexibility with numbers and algebraic thinking!

Children will ebb and flow between these three levels. The important thing is to play, explore, and play some more! The next few blogs will encourage this through games that I am trying out with my son’s Kinder class!!!

For the ten-frames (I love these because they are soft and quiet!): https://www.schoolspecialty.com/magnetic-board-set-1400695

# Basic Addition Facts For Fluency: Beyond Flash Cards!!!

This is the second of a series of blogs regarding fluency of basic facts.

Memorizing any operational facts can be easy for some (who have photographic memories) and difficult for others. I tried and tried every type of “flash card” with my oldest and all it did was provide frustration, tears, and dislike for the math. We tried flash cards, memory games, on-line shoot the sum games. And none of them worked. I freaked out, knowing that if he didn’t get these facts down, it would be an uphill battle. But is it really? Do we need to focus on memorization or understanding???

This blog is about understanding and a teeny tiny bit of memorization. Please feel free to disagree, as this is a blog after all and participation is encouraged! However, at least read on before fighting the fight for memorizing only.

Depending on whether you live in a Common Core state or not, typically we want children to know fluently all of their facts to five by the end of Kindergarten and all of their single digit facts (10 + 10 is cool too) by the end of grade 1. Note I wrote “fluently” versus “to automaticity”. So what’s the difference? It means I get three whole seconds to figure it out. I may not be able to see the answer in 0.5 seconds, but using some strategic thinking I can get it. That’s the goal. Anything beyond is cool, but this is for all children.

Fun fact: If all you are going to do is memorize the sums for two single digit numbers, the amount of facts you need to memorize is 81. 81! That is a LOT! Have fun making 81 flashcards! No thank you!

So let’s dwindle the amount to memorize. Below are two strategies to teach your kiddos to make that 81 significantly less. (There are others, but if you focus on these two you are waaay ahead of the game!)

1. Commutativity. This is a property (a rule that ALWAYS works in mathematics) of equality. Think about the distance you drive to work. And the distance you drive home. Now if you take the same route both times, you go the same distance. No matter if you start from home, or you start from work, your commute distance is the same. Commutativity works the same for numbers. I can add 3 + 5 or 5+3 and my result (the sum) is the same, 8. So if I use my commutative property, I now only have to memorize one of the two facts! I am now down to 45 facts! (You may think, “Hey, why isn’t it half of 81?” Well, the doubles such as 2 + 2 and 3 + 3 do not numerically have a commutative fact, so you are stuck with them.)
2. Making 10. This is a big stinking deal. Your kiddos need to get under their belt the ways to make ten. (1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5 and the reversals based on strategy #1.) Why? For starters, we live in a base ten world. Once you hit 9, the next number literally moves over one spot to the left and now you have a double digit value. Once you hit 99, you move over one spot to the left and you now have a triple digit. And so on. Really though, let’s be practical. Is it easier to add 9 + 8 or 10 + 7? 10 + 7, because you can add the 0 + 7 easily. So if I can manipulate my digits to make a 10 +, I am good to go. Example: 9 + 6. I know that I only need one more to make 9 a 10. So I am going to take it from the 6. 9 + 6 = 10 + 5 = 15. If I know how to make a 10 + expression from the addition fact I am working with, I now go from 45 memorization facts to 25!

What are those 25 facts? 1 + 1, 2 + 1, 3 + 1, 4 + 1, 5 + 1, 6 + 1, 7 + 1, 8 + 1, 9 + 1, 2 +2, 3 + 2, 4 + 2, 5 + 2, 6 + 2, 7 + 2, 8 + 2, 3 + 3, 4 + 3, 5 + 3, 6 + 3, 7 + 3, 4 + 4, 5 + 4, 6 + 45 + 5 (Notice the largest sums are 10, because all others can be made using a 10 + strategy.) So could your child learn these facts? ABSOLUTELY! Just remember to include the others for them to practice their two cool strategies with!

Next Blog: Fun games to play to work on making ten!

HUGE NOTE: Please do not time your children. For many, it stresses them out and they lose focus. Also, if you time children before they master the strategies, it will encourage them to count on their fingers. Though that may work for these small values, it will not help them in the long run when adding much larger numbers.