Gattegno’s Textbook 2: Halving and Doubling Numbers
In my last blog post we briefly mentioned halving and doubling numbers, and I introduced the idea of towers. This blog post will focus on halving, doubling, and tripling; why we do it; how Gattegno uses it in his second textbook; and how we can use halving, doubling, and tripling to make math problems easier.
Halving, doubling and tripling are closely tied to the factoring and towers. So we are going to sort of pick up where we left off in the last post. I hope what you begin to see is that you can't just take these ideas and separate them. Gattegno was trying to get across one main idea in the elementary years and that is that all of this is basically same stuff. We have just a few structures on which we build our math understanding - the rest is language.
Halving, Doubling, Tripling: Milestone Numbers
In textbook 2, you'll come across what Gattegno calls "milestone numbers". These are numbers built by repeated doublings and tripling prime numbers and are a springboard to richer understanding and study of multiplication, division and geometric progressions.
If you have the Gattegno Table of Products Chart and Cards, you'll find exercises for them in book 2. These exercises are meant to develop a feel for these "milestone" numbers. The products are doubles, multiple doubles and triples of the following set of numbers: 4, 5, 6, 7. 9, 10, 14, 15, 21, 25, 27, 28, 35, 45, 49, 63, and 81.
I don't use this chart, but I do use the products set as the basis for studying numbers 20-1000. If you want to use the chart, you can get it here and NRICH put out this pdf to help teachers use the products chart and cards. You can certainly make one for your own use yourself, as they still under copyright.
Gattegno felt that the products set is the most important numbers to study when studying numbers 1-1000. These are the numbers, in addition to the number 11 and it's multiples, that we concentrate on.
Beginning Halving And Doubling
If, perhaps, you skipped over the doubling work in book 1, never fear. It is not too late. Just start with 2 and practice doubling. How much is two 2's? Now that we have built that, how much is two 4's? How about two 8's?
Now start at 3, double it? How much is two 6's? What about two 12's? How about two 24's?
Are there patterns that you can find when you double? How can you learn to do this easily in your head?
Now that you've doubled 6 to get 12, what is 1/2 of 12? What is 1/4 of 12? Let's make a chart and notice and wonder.
Noticing and Wondering
What could we notice on this image?
1. When I double and double again, I am multiplying times 4.
2. When I halve I am dividing by two.
3. Half of 8 and a quarter of 16 are the same. Half of 4 and a quarter of 8 are the same.
4. Half of 12 and a quarter of 24 are the same. Half of 6 and a quarter of 12 are the same.
5. Twenty-four divided by 2 is the same as 1/2 of 12. Twenty four divided by four is 6.
There is more in the above image, but you should make the same image and notice and wonder with your kids. What happens if we double 24? What will be half of double 24? What will be 1/4 of double 16? What is 1/8 of double 16? If we double once we multiply by two. How much if we double 3 times? How much do we divide by if we halve 3 times?
Halving And Doubling: Factors
Last post, I I used a tower to show factors. We looked played with the factors and how we group them using the Associative Property. There is a lot more to explore with towers than just grouping factors.
In the above examples, we doubled both 2 and 3 three times each. That's a double, double, double. 3 x 2 x 2 x 2 allows us, to find factors of 24 depending on how we group the prime factors. Just by moving the parenthesis, we change factors. We could 3 x (2 x 2 x 2)= 24 that also equal to 3 x 8 = 24. If we move the parenthesis, we can get (3 x 2) x (2 x 2) = 24 which is also the same as 6 x 4 = 24. We can do this using towers as well.
Halving, doubling and tripling can make short work of the multiplicaiton table as well. There is only one multiplication fact that must be memorized: 7 x 7. . The tens are easy, there is a pattern. Nines shouldn't be an issue as you can follow the nine pattern. All the rest can be found by halving, doubling and tripling. Does that mean student's should memorize the multiplication table? No, that is not what I'm saying. But I am saying that drilling them until they hate math and maybe even you is a bad idea. It is better to study the table, look or patterns and practicing halving and doubling.
Number Splitting: Halving, Doubling, Tripling Large Numbers
Most of us, with practice, can double numbers in our heads. Becuase I taught my son to add and subtract mentally left to right, we count the largest place value first. The self-talk for doubling 365 would look like this: 3 + 3 is 6 hundreds plus 6 + 6 = 12 tens that gives me one hundred plus 2 tens, 7 hundred twenty plus 5 + 5 that is one more ten. Seven hundred and thirty.
This practice is called number-splitting. It allows us to deal with each place value, or a combination of place values, as separate numbers. Let's say we want to double 815 - that's 815 x 2. Mentally we split 815 into 8/15. We double the 8 first and then the 15 which gives us 16/30 or 1630. In the above example, my 8 year old splits it 3/6/5. I would split it 36/5. This allows each of us to work within our own comfort level and get to the same place.
Gattegno has students breaking apart numbers in this fashion from the very beginning so that they can manipulate them according to the rules of the operation they are working with in a way that makes sense to them. This is what I mean when I ask, "How can we take what we know about this and use to it make solving the problem in front of us easier?" Lacy, from Play, Discover, Learn 24/7, calls it, "Teaching my child to use his innate laziness to become more efficient."