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. IMG_5269
  2. Have your child close his/her eyes. Hide some of the objects under the cup.IMG_5276      IMG_5275
  3. Ask your child to open his/her eyes. Ask the following:
    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.
  5. Ask the following:IMG_5274
    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!



How Many Are Hiding?

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.

  1. Most of us are used to seeing math problems vertically. Why? Well, for one, it makes the problem ready to go for  the algorithm  of “stack and subtract” (which is not the ONLY algorithm in the world, mind you). I would contend it also saves space for publishing companies in their workbooks. If the problem is already written for the algorithm, publishers do not have to provide additional space and therefore can fit more problems on a page, and save money. (Yeah, I went there. Bring on the comments!) However, when you write a subtraction problem vertically, you lose the essence of the numbers.   IMG_7211Here I only see single digit subtracted by single digit. I do not see 40 – 20, but 4 – 2 and 5 – 1. This is okay for problems where we do not have regrouping. However, some students get so stuck in the process of regrouping that they no longer see the value of the places and just write a very ‘random’ value as the difference. When I write the problem horizontally (45-21), it allows me to view it from a place value perspective. My eyes look first at the tens and then the ones, versus the horizontal example where I start with the ones and then look at the tens. Also, writing it horizontally does not constrict me to a “stack and subtract” method. (See prior blog for more info on great subtraction strategies to help kiddos.) Really, both are fine; no need to get all uppity about it. And if a teacher says they HAVE  to write it that way, it is not true, but what is true is that they learn to think about relationships and different strategies using place value and properties of numbers BEFORE learning the standard algorithm. In fact, the standard algorithm for multi-digit subtraction should not be mastered until grade 3: CCSS.Math.Content.3.NBT.A.2
    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.
  2. (and 3.) Adding up is one strategy students can use. Makes sense to me! If I already know how to add, I can simply use that to help me figure out the missing addend. This is all a subtraction problem really is: a missing addend problem! Consider the following: 43 – 16. This is really finding out 16 + ____ = 43. Now typically, students do not need to go to the next five as the tutor suggests. And really a number line is FABULOUS for modeling adding up. Here is one way to get the value. IMG_7213Notice this is a great strategy for students who struggle with regrouping, because there IS NO REGROUPING!!!  I went up 4 to the nearest ten (20), added 20 more (40), and ‘hopped’ 3 more to get to my end point (43). 4 + 20 + 3 is 27. Therefore, 43 -16 = 27.                                                               Using the tutor’s problem (43-13), I think adding up is efficient, if you move up by tens. I can simply add by tensIMG_7214 until I reach 43. 3-tens is 30. Not sure why it is so convoluted in her explanation and NO! Students do not have to add up the same way the tutor did. In fact, that is the wonderful thing about Common Core. CCSS.Math.Content.2.NBT.B.7
    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.


Response to Confusion 43-13

3 Common Subtraction Strategies

A dear friend had this posted last night:

1509183_10208482214599412_3676246263346474226_nWhy?! 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 number line

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 number line


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 number bond28 – 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 number bond


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 decompose28 – 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 decompose400 – 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.

Continue reading “3 Common Subtraction Strategies”

3 Common Subtraction Strategies