Regrouping: Making Math Make Sense Part 1
This is the beginning of an extended conversation on regrouping. I will be posting student work in the next few days.
Regrouping is one of the hardest concepts for students to understand. They struggle with it when it is introduced and often times even when they gain procedural fluency, they never get the conceptual understanding. We have to give students many opportunities to see it in context. Here are a few ideas. I like to start out by acting out scenarios where the students get to make meaning of what it means to regroup. Here is one I have been using lately.
Mrs. Martinez has some cookies. There are ten in a baggie. There are 3 baggies. She wants to give 5 cookies away. What can she do? [Here the students begin to give suggestions. We decided that she has to zip open one of the baggies so those cookies are loose]. We then discuss how she can take 5 away from that group of ten that is now ten ones. We actually act this story out.
We do several problems like these with different scenarios before I ever show them the regrouping algorithm. The next step is that they work in partners with manipulatives that are in groups of tens and ones and they act out the stories as I tell them and then volunteers take turns telling the class stories to act out with their partners.
After several days, we move on to base ten blocks. After using the base ten blocks I show the traditional paper and pencil algorithm with pictures drawn beside it. Eventually, the students do just the paper and pencil algorithm as one strategy.
We then go on to talk about other algorithms (ways of doing something). We talk about how different strategies work with different numbers and how people have favorite ways of doing things. We discuss “elegant solutions” and “quick” or “efficient” ways of doing something. We work in groups to make strategy posters and display them around the room.
Some strategies we discuss: Compensation
So, to do compensation we look at the number and try to find a friendly number (one with a ten). Here we see that 29 can easily be rounded to 30 so we decide to add 1 to both numbers and we get 76-30 (which we inevitably agree is much easier to think about) and we get 46. We look for lucky 9’s and 8’s so we can use this strategy whenever possible.
Let’s look at a lucky 8 example. 62-48. We add 2 to 48 to get to 50 and we add 2 to 62 which makes 64. Now 64 -50 is much easier on the brain (we talk about it like this) and we can quickly get the answer of 14.
Compensation makes thinking easier. Another way to think about this is “Find a Friendly Number”:
I could change 74 to 75 by adding 1. That would give me 75-25 which is pretty friendly. Now, I can’t just keep that 1 I added, I have to take it away once I get my answer. So 75-25 =50. Now, 50-1 =49. That was pretty easy.
Another strategy that we talk about is Jumping Tens. So we look for Jumper Numbers and then we begin to hop back on the number line in our minds. For example, 65-27. We say that 27 is the Jumper Number. There are two jumps of ten in this number. We start at 65 and we hop back to 55 and then to 45. Now we have 7 more to go. We decide to break the seven into 2+5 and then we hop back 5—which gets us to 40. Now we only have 2 more hops and that gets us to 38.
Counting up is a 3rd Strategy.
We look at a problem like 51-27. We say “Well, mmmm, 3 more gets me to 30 and then 21 more to 51 so 24 is the answer.” We just counted up to count back ….we counted up to see how far back 27 really was….
Another Strategy is Splitting Numbers:
Here it helps if the students write it in expanded form.
70 + 0
-30 + 3
So here if subtract 3 from 0 we will get a negative number, which isn’t exactly friendly, so we’ll do something else.
Let’s break a ten into ones. We’ll rename 70 as 60+10 Now, we can continue.
– 30 +3
30 + 7
http://www.youtube.com/watch?v=CqOsj2JWbyE&feature=relatedhttp://www.eduplace.com/math/mw/models/overview/2_12_3.html (don’t use peanuts…use something else…cheerios, unifix cubes, cotton balls)