Do your students struggle with subtraction across zeroes? Here’s an activity that might help them make sense of regrouping.

Let’s start by thinking about a real life situation that might apply to the problem 301 – 164: Your dad owns a bakery. He baked 301 cupcakes today. He has an order for a wedding for 164 cupcakes. How many cupcakes are left to sell in the store today?

Suppose your dad asked you to count the cupcakes as they came out of the oven in the morning. How would you count? 1, 2, 3, 4, 5….would you count all the way to 301? Would that be efficient? Think you’d get the right answer? What would happen if your sister interrupted you and you lost count?

Counting is a lot easier if we put things in groups. This is the idea behind our place value system. Every time we get a group of 10, we call that a new unit, a ten. Then we can count by tens and the whole process of counting is much more efficient.

To help you count and keep track, suppose you have small boxes that each hold 10 cupcakes. So you count 10 at a t time. 10 (one box), 20 (two boxes), 30 (three boxes), and so on. What happens when you get to ten boxes? What is ten groups of 10? Well remember in our place value system that whenever we have 10 of something, we make another larger unit. 10 tens is the same as 100, or a new unit that we call 1 hundred. Suppose we have a large carton that can hold 10 small boxes. So we put 10 boxes (10 tens, or 1 hundred) into our large carton. We keep boxing and counting the cupcakes in this way.

Now suppose your sister interrupts you when you have 2 large cartons, 9 small boxes, and 11 cupcakes left – you are so close to finishing! Will you have to start counting all over again? No! You know that each large carton has 100 cupcakes, each small box has 10, and you have 11 left over that haven’t been boxed. So you have 2 hundreds + 9 tens + 11 ones = 200 + 90 + 11 = 301 cupcakes.

Now all you have to do is put 10 more cupcakes in a box, and you will have 1 left over. But wait, if you make one more box, you will have 10 small boxes. So you can put these together in a large carton. In the end you have filled up 3 large cartons and have 1 cupcake that is not in a box.

Now, your dad has asked you to figure out how many cupcakes will be left to sell if he has to deliver 164 cupcakes to the wedding. How do you subtract 164 from 301? And how will you give your dad 164 cupcakes if you have 3 large cartons and 1 loose cupcake? How will you get 60? How will you get 4?

This is where you are going to be thankful for your sister! When she interrupted you, you had 2 hundreds + 9 tens + 11 ones. It is easy to subtract 1 hundred + 6 tens + 4 ones from that – we just subtract hundreds from hundreds, tens from tens, and ones from ones.

2 hundreds + 9 tens + 11 ones – (1 hundred + 6 tens + 4 ones)

= (2 hundreds – 1 hundred) + (9 tens – 6 tens) + (11 ones – 4 ones)

= 1 hundred + 3 tens + 7 ones

= 100 + 30 + 7

= 137 cupcakes

Now you know how many will be left, but how will you give your dad 164 cupcakes? Well, you will have to open one of the cartons and break it back up into 10 small boxes, right? Then your dad could take 6 small boxes. Then you would have to take one of the small boxes, open it and take out the 10 cupcakes, and have 11 loose cupcakes. So now he could take 4.

This is the same reasoning you use when you write 301 – 164 vertically and regroup. You need to break up the hundred into 10 tens, then break up one of those tens into 10 ones, which means you will have 2 hundreds, 9 tens, and 11 ones.

As long as you keep the 301 cupcakes, you can group them in any way you want that makes it easier to subtract. So you could have done something like this:

301 = 100 + 100 + 100 + 1

Then 301 – 164 = 100 + 100 + 100 + 1 – (100 + 60 +4)

= (100 – 100) + (100 – 60) + (100 – 4) + 1

= 0 + 40 + 96 + 1

= 137

This would be the same as taking one whole large carton (100 – 100), opening one carton to take out 6 small boxes (100 – 60), and opening one carton, taking out one box, and taking four cupcakes from it (100 – 4). You would be left with no full cartons (0), one carton with 4 small boxes (40), and one carton with 9 small boxes and 6 loose cupcakes (96), plus the one loose cupcake you had in the beginning (1) .