# 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

# Rolling to 100

I had the pleasure of volunteering in my son’s Kinder class for 100 day. She had a great ‘filler’ game that I wanted to adapt and share. This is a great one to take to a restaurant where they hand out crayons!

Materials: 100 chart, one per player (see below for link), die (dot for one-to-one correspondence, numbered for numeral recognition with counting), crayons (at least 2)

Objective: To be the first person to color in all 100 numbers!

How to Play

• Player 1 rolls the die and colors in that many spaces, starting at 1. Player 2 does the same on his/her gameboard. (Example: rolling a 5)
• Player 1 rolls again and, in a different color, colors in that amount. So if I rolled a 2, I would color in the next two squares in a different color. Continue playing until someone reaches 100.

How I would change it

1. Give your child a blank 100 grid and have him/her fill it in for writing practice. Then use their board to play.
2. Change out the dice as your child grows in his/her number sense. Using larger numbers will create patterns and encourage counting by larger groups instead of by ones.
3.  We are going to play to 20 and write the addition sentences on a whiteboard.         So 5 + 2=7 for the last play.  Also you could relate to counting on. Start at ______ move forward ______. I am now at _______. Starting at numbers other than 1 or 0 to count is a BIG DEAL!
4. We are going to start at 100 (or 20) and go backwards to roll to 0! Counting backwards is just as important as counting forwards!
5. Find the values. After Chris finishes up his very crumply 100 chart, he is going to guess my number. Example: My number is blue and is bigger than 23 but less than 26. Guess my number!

# War!(HUH!) What is it good for? (Absolutely lots in Math!)

With my family on the go so much during the non-lazy days of summer, we need easy games to entertain the tiny humans that don’t require mass amounts of attention from the adults (who are often in conversation). Enter the game of War. This versatile game can be used for all age groups and can really keep your child’s skills in arithmetic in check during the “summer slump”.

How to Play (Basic Version)

1. Grab a deck of cards (I keep one in my purse and in the car at all times). You don’t have to, but I prefer to take out the face cards and jokers. Shuffle the rest and divvy out to all who are playing.
2. All players shove all of their cards into a “deck” and keeps the deck face down.
3. All players (at the same time to avoid cheating) flip the first card. The player with the largest value is the winner and takes all of the cards in the round.
4. If there is a tie (that is the largest value), those players place 3 cards on their original face down and flip the fourth card. Whichever player NOW has the largest value gets all of the cards from the round.
5. Continue playing until either a) one player has all of the cards; or b) you get sick of playing. The player with the most cards is the winner.

• For younger players: Use only 2-5 from the decks and play with those. The game goes faster and they are working only with 2, 3, 4, and 5. You can use the aces as 1. Even better, use number cards or dot cards (see below for links). Print on cardstock (4 cards per number) or go online and buy a set.
• For any age: You can also play and whoever gets the smallest value wins. This is great for preK-1st graders!
• For students who need review with addition: Play two cards at a time and add them. The player with the largest sum is the winner of the round.
• For students who need review with multiplication: Play two cards at a time and multiply them. The player with the largest product is the winner of the round.
• For grades 5-7: red cards are negative values; black cards are positive values. Flip over one card. If I have a red 6 and you have a black 2, you are the winner since positive values are always greater than negatives.
• For grades 6-8 (or 7-8 if using Common Core): Play two cards and add them, using reds as negatives and blacks as positives. The player with the largest sum is the winner of the round.
• For grades 6-8 (or 7-8 if using Common Core): Play two cards and multiply them, using reds as negatives and blacks as positives. The player with the largest product is the winner of the round.
• For grades 6-8, use only values ace (for 1) through 5. Flip the first card; that is your base. Flip the second card; that is your exponent. The player with the highest value wins  the round.

Different Sets of Cards:

• You could probably look on Amazon for different card types, but I love the sets at 52 Pickup. They are of high quality and there are many different types ranging from dot cards to ten frames to cards that go through the thousands (so you can work on place value!)

https://sumboxes.com/collections/types?q=52%20Pickup%20Card%20Decks

# How Many Are Hiding?

This is a great game for tiny humans in need of some entertainment while waiting at a restaurant. Just make sure to ask for an extra children’s cup to play.

Age Level: 3-6 year olds

Materials: children’s plastic cup (any cup will do, just not transparent), a set of objects (I used goldfish for the example, as that was what was in my purse! Other options are pennies, beans, tiny annoying toys, etc.)

How to Play: Place out a certain number of ‘stuff’. Normally for me, it is however many I have in my purse! I would recommend starting with 5 or less, see how they do, and adjust as needed. The first time I played with my 5 year old, we started with 10 and it was quite frustrating for him. They catch on and you can up the number as they grow!

1. Have your child count how many there are.
2. Have your child close his/her eyes. Hide some of the objects under the cup.
1. How many do you see?
2. If there were _________ to start with, how many are hiding?
4. Allow your child to check their answer by lifting the cup.
1. How many were hiding?
2. How many were out?
3. How many in all? (Woah! It is the same as what we started with! Weird!)
6. Switch who hides and who plays.

Why play? Aside from counting one-by-one and ‘holding’ that number in their heads, students need lots of practice understanding that a number can represent an amount. That amount can be broken into parts (decomposed), but when we put them together (add them) they make the original number we had. This is part of the idea of conservation, which is critical for young mathematicians to understand (not the word but the idea) in order to add and subtract numbers.

Special thanks to my tiny human for playing this morning! Love ya, bud!

# Playing With Math: Circles and Stars

Ahhh…summer. For many of us, that means more time with the kids…waiting. Waiting at a restaurant, doctor’s office, airport, etc. For many kids, it may also mean waiting to use their brain. Research suggests that students can lose as much as 2 months of learning skills during the summer months (Oxford, 2017).  So how can we use the waiting times (or times at home when they are claiming boredom) to retain and advance their learning in mathematics? Play games/activities.

While there are many great apps for kids, I would request less screen time and more interaction with your children.  For the next three months, I will suggest a game/activity that you can use with your child. I will suggest different levels, so that you can play it often and in different ways. I use these same games with my own children, and find the time waiting goes much quicker, with less outbursts and meltdowns. Further, I am modelling playing with math, which is truly the way I feel our children learn and understand math best.

Circles and Stars (Marilyn Burns)

Grade Levels: Though used in grade 3, if all you are doing is counting the number of stars I would recommend grades 1-5. My preK has played it and just counts one by one. He cannot make the stars, so he draws x’s.

Materials: die (number cube or dots; doesn’t matter), paper or napkin, pen or pencil (I prefer a travel size Magna Doodle or whiteboard with dry erase marker)

LEVEL 1

1. Roll the die. Draw that many circles.
2. Roll a second time. Draw that many stars in EACH circle.
3. Total the stars. Whoever has the most stars wins the round. (Play as many rounds as you want. The winner could be the one with the most stars total. Woo hoo! More math!)
4. Alternative: The winner is the player with the least amount of stars.

LEVEL 2

1. Roll 2 dice (or the die twice in a row). Player chooses which die represents the number of circles and the number of stars in each circle.
2. Total the stars. Whoever has the most (or least) stars wins the round.

• If I was to switch which die represented the number of circles and stars, what would happen? (The picture would look different, but the total stars would stay the same. This is the beginning of understanding the commutative property for multiplication.)
• How could we represent what we did in words? (Example: 4 groups of 3 stars is 12 total stars.)
• How could we represent what we did as an expression?                                   (Example: 4 x 3 = 12)

# Cool Tools for Kids in Math

These past three days I got to geek it up at the NCTM annual conference in San Francisco. I have gone to several annual conferences, but this was the first time I worked in an exhibitor booth rather than attending as a participant. I was excited to be on the other side of the conference scene, but sad that I wasn’t sitting on the carpet (like so many) scouring the magazine o’ options for the perfect sessions.

As a teacher, I would ditch the exposition hall (except to get the free Legos and swag for my boys!) and attend every session I could. I would take copious notes, trying hard not to miss anything that was said in case THAT was my take away for the trip. Those fabulous notebooks that I poured my 72 hours of the conference into gather dust in a box in the garage. Don’t get me wrong; I would typically use 3-5 ideas/worksheets/tasks/quotes per conference. But was that worth the hours I sat in the back of a crowded room? Was there more to the conference than the sessions?

YES! This year I attended a single 60 min session and got several great ideas for a district I work with. The rest of my time was spent in the exhibition hall talking to reps (and long-time friends!), discussing mathematics, and truly learning from one another in a more intimate setting. I learned so much in these conversations, AND spent time at many vendor booths playing with the technology that I believe can truly make a difference in how students view mathematics. Though I am still grappling with the lack of notes in my handy-dandy notebook, I feel I am leaving with far more applicable ideas and tools than ever before!

The links below are (free!) sites you and your child can explore to really learn mathematics. They allow students to truly see what is going on and why the math “is what it is”. I hope this summer you are able to spend some time on these sites and give your students an opportunity to open up mathematics in amazing ways.

Note to teachers: These are open source and free to use on your devices at school. You are welcome!

Currently their Mastering Addition Facts app is free. Get it now before they change this! Students work on their math facts in a developmental way, understanding as they gain mastery. (They do have a multiplication app as well, but it is not free.)

DESMOS: https://www.desmos.com/

This is a site that allows you to graph functions, plot tables of data, evaluate equations, explore transformations, and much more!

Geogebra: http://www.geogebra.org/

Geogebra makes a link between geometry and algebra using visual representations students can manipulate and finally see what is going on mathematically.

Math Learning Center: http://mathlearningcenter.org/apps

These apps are amazing! So many to choose from to help students conceptually understand mathematics. There are number lines, geoboards, money pieces with a number rack, rekenreks, ten frames, pattern blocks and more! Just allowing your child to play with these apps will enhance their understanding of number! Here are just a few from the site:

Finally, this is a book that came highly recommended. Though not free, it can be a support for parents in navigating Common Core mathematics. (It does come with videos as well!)

http://www.amazon.com/Common-Parents-Dummies-Videos-Online/dp/1119013933

# Response to Confusion 43-13

So today a friend tagged me in a FB post regarding the “frightening” method that students MUST solve subtraction problems. I have posted the link below, and I believe the link is at the bottom of this post as well! Take a look.

So let’s summarize the tutor’s concerns. 1. That we are writing problems horizontally rather than vertically. 2. That students are using a strategy of “adding up” rather than “stack and subtract”. 3. They MUST use this strategy and no other.

I would like to address each of these and provide some comments.

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. So students have lots and lots of time to process what it means to subtract and its relationship to addition.
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Notice that students can use any method, with or without drawings or concrete models!!! Super awesome! Hooray for creativity! I no longer have to use one method (that I don’t really get and just memorized because my teacher told me to); I can use any method so long as I keep in mind place value and my rules (properties).
“Our classrooms are filled with students and adults who think of mathematics as rules and procedures to memorize without understanding the numerical relationships that provide the foundation for these rules. The teaching of mathematics has been viewed as a discrete set of rules and procedures to be implemented with speed and accuracy but without necessarily understanding mathematical logic. For the majority of our nation, knowledge of mathematical rules has not allowed them to use math confidently in their daily lives. With almost two-thirds of the nation’s adult population fearful of mathematics, they have simply said “NO” to math and closed the doors to careers that require higher math” (Burns, 1998; Parrish, 2010).
Let’s allow students to make sense of number relationships, what the operations MEAN, and figure out what makes sense and how to approach the mathematics based on the numbers given.

# 3 Common Subtraction Strategies

A dear friend had this posted last night:

Why?! Why would I even consider doing simple subtraction like this?! It’s about 10 steps too many, not to mention super confusing!!!

And to be honest, I had to look at the first problem (represented as a “tape diagram”) to figure out what they were asking.  So my next blog will be about the difference between CCSS standards and how they are being implemented (often poorly). Today is to help out my friend…

So, back in the day we were given a set of procedures to follow step-by-step. We never really knew why (or at least I didn’t): just follow the magical steps in order and it will produce a number that your teacher will smile at and say you are smart. If you did not follow the steps correctly, then have fun with extra practice and staying in during lunch.

CCSS is reversing this. They are allowing students (through research-based strategies kids naturally use) to explore a variety of ways to solve problems. From there, in grade 4 they will generate the traditional algorithm. It hasn’t gone away; it is just the end of the journey.

Here are 3 of the most used strategies for kiddos in subtraction.

1. Count Up. Makes sense. We spend so much time teaching how to add. Why not use it? A subtraction problem is considered a “missing addend” problem.

One of the ways to represent counting up is on an open number line. This is the one the above pic shows on a tape diagram. So you are finding how far (the distance between) it is from one value to the next.

Typically, kids will add up to the nearest ten, then hundred, and jump until they get to the number they need. Add up all of the “hops” you did and that is the distance.

28 – 9

Think about it as, “What plus 9 makes 28?

Or  “How far must I go to get from 9 to 28?

400-165

Think about it as “What plus 165 makes 400?

Or “How far do I have to go to get from 165 to 400?

This is my favorite strategy, because it takes the “borrowing” out of the math. (Why is it called “borrowing”? You will never give it back.)

2. Decompose the subtrahend. Name it what you want. You will break apart the second number in the subtraction problem to make it easier to subtract. This is typically shown by a number bond.

28 – 9

Break apart the 9 into 8 and 1. (9 = 8+1)

This allows the student to subtract the same amount of ones from ones first (8 – 8). The leftover ones (in this case, 1) can be taken away second. Again, a great strategy when the subtrahend (the second number in the subtraction problem) has a digit larger than the minuhend (the first number).

400-165

Break apart the 165 by place value (165 = 100+60+5).

This is typically a strategy that works well in your head, versus the complexity of seeing it written out mathematically. Trust me when I say that kids can do this quickly in their head; writing what they did is much harder (and much harder for us to figure out what the heck they did). You do have to know the ways to make 10’s and 100’s. So I have to know that 60 + 40 = 100 (or 6 tens + 4 tens = 10 tens = 100) to know that 300 – 60 = 240. This is the second grade standard; subtracting multiples of tens. (BTW: This is the one that gets blasted on Facebook. I have seen students use this strategy in their head time and time again. The minute we record their thinking mathematically, people get all out of whack. It is not more steps than the standard algorithm. The moves you make are written as equations using place value rather than little meaningless tick marks at the top of the problem.

3. Solve using place value. Similar to the second strategy, but you can break both numbers up as you choose.

28 – 9

Again, looks a lot scarier when I record it this way. Using a number bond is a great visual, and typically the student starts this process by just saying what they want to subtract verbally, rather than writing all of the notation down.

400 – 165

I am just showing one way to break apart the values. When I asked my son how he would do it, he said he would break up 400 into 100 + 100 + 200. That made sense to him, and in the end that is what we want. For students to use a strategy and make sense of it.

You will find that one strategy works with the numbers given better than another. For example, with the 400 – 165, I would use a counting up. With 28 – 9, I like the break up the subtrahend. The point of it all is for students to really understand what is happening when they subtract. Where is the “borrowing” or “regrouping” happening, so when they learn the traditional algorithm, it is an extension of their learning rather than some random tick mark step-by-step formula.

# Beyond Counting: Ideas and Activites For Your Little Ones

While waiting for his big brother at the orthodontist, my little boy, C,  had the following conversation…

Dr. T: How old are you, cutie?

C: I’m three!

Dr. T: How old is your brother (pointing towards my thirteen year old)

C: Four!

This was such a proud mama moment for me!

Now you may ask yourself, “Why is she getting all excited over this? Clearly, he is not four. Why is she so proud of her little boy?”

There are a number of reasons why this is a critical step towards numeracy. I truly believe that if you start children purposefully thinking about numbers early on, their chance for success in mathematics increases dramatically. So let’s highlight a few of the big ideas C is working towards.

1. Cardinality– This is the idea that the number being used is measuring some amount. It answers the question, “How many?” For example, I can ask my son, “How many bears do you see?” He would count them one by one until he got to the number six. That last number, 6, tells you the number of bears in the set. This is a big deal! The child is no longer counting from memorization; he is recognizing that the number relates to a certain amount of “things”. The more things you have, the further you have to count. C recognized that his brother was older (or “bigger”). Therefore, his brother was tagged to a number after the one he identified with, three. He did not know how many more to go, just that he had to choose a number beyond his own. Cool.

2. Inclusion– This is the idea that the number labeling “how many objects” in a group includes all of the preceding numbers. So even though we have six bears, we can also think of it as “one and some more”, “two and some more”, “three and some more”, and so on. This is critical for addition and subtraction. If I have the number 14, I can think about it as “ten and four more”, which helps me when I want to add or subtract and regroup to make the problem easier. C knew that his brother was older, and therefore had to include his age (three) and some more. Again, he isn’t at the point of knowing how much more, but is on his way. Awesome.

3. Magnitude– The size of the object. In this case, a number (or value) given to a quantity (age) for the purpose of comparing with another quantity. This idea is instrumental for estimation, particularly with very large and very small numbers. In fact, one of the posts requested of me to write is helping students compare fractions. If a child does not know the relative size of the number they are considering then it is very difficult to compare, operate or manipulate it with any real fluency or number sense. How do I know my answer is reasonable if I haven’t a clue what the numbers I am working with represent??? For C, he was able to recognize that his brother had to be a larger quantity than three, because he is older. Super rad!

These three ideas are certainly related, but each has a different feel. You can work with them simultaneously, so long as there is purpose to the questions and tasks you present to your kids. Below are some simple, but powerful, activities you can play with your little ones to build these concepts. I choose the games that you can take on the road, to the doctor’s office, to a restaurant, etc. Instead of sitting around being squirrelly, play a game while you wait. Even five minutes will have a significant impact!

1. Count and Check: Grab a handful of ANYTHING (balls, pennies, beans, cheerios, etc) and ask your child to count how many. Make sure the amount of objects is appropriate. (For example, C is working on objects through 5.) When he finishes counting the last object, ask, “How many _____ are there?” If he cannot answer, that is okay! He is working towards cardinality. He is able to say the objects one by one (which is called one to one correspondence), but hasn’t figured out that the last number he says represents the entire amount. Have him count again, and ask again. If he cannot answer again, say, “There are (say the amount) ______ here.”  You can play this at the grocery store (count the apples, bananas, etc), setting the table (How many forks?), etc.

2. Match Me! Grab a die (one dice) and a handful of ANYTHING in a baggie. (I typically do this with pennies at restaurant.) Have your child roll the die. Let’s say she rolls a five. She takes out that many pennies and lays them out for you to see. Ask her to count them one-by-one to make sure she has five. At the end ask, “How many pennies do you have?” If she doesn’t know, that is okay! Have her recount, then ask again. If she still isn’t able to tell you, say, “I see you have five pennies.” Make sure you roll next and model for your child. Take turns until you get bored or dinner comes!

To bring the difficulty up, after playing each of these, ask, “If I gave you one more item, how many would you have?” This brings in the concepts of magnitude and inclusion! If your child has to recount with one more added in, that is fine! You know he’s got it when he can answer quickly without physically adding in another item and recounting.

Need another level of difficulty? Ask, “If I took away one of the items, how many will be left?” Same idea, but working backwards, and just as important!