I'm starting with part 2 because, I'm teaching this to my 7yr old right now. She is not a math-natural and it seems to be working like a charm. This part is about numbers whose sum is greater than 10. Part 1 is about numbers whose sum is less than or equal to 10.
In part 1, which isn't yet posted--sorry!, we emphasized "magic friends add to ten." The short story here is that every number has a friend that will sum to ten.
1 and 9
2 and 8
3 and 7
4 and 6
5 and 5
The kids should know this very well before proceeding.
Step 1 to this process is teaching kids what 10 + another number is. They should know 10+1 is 11. They don't necessarily need to understand ones place and tens place to get through this though it is helpful. If they can count to twenty, they can recognize that 10 + 2 is twelve.
In order to teach this, I start at the top.
10 + 9 is 9 teen. Emphasize the teen here.
10 + 8 is 8 teen.
Do this down to 16. They'll hear the pattern.
Next, tell them that 13 and 15 are very similar but we just pronounce them funny.
5 teen is fifteen
3 teen is thirteen
Don't forget to mention 14.
When you get to 12 and 11, mention that these numbers are got named differently since 12 is such an important number, i.e. 12 months in a year, 12 hours in a day/night, 12 in a dozen.
2 teen is twelve
1 teen is eleven
Practice adding 10 + another number until this is easy for them to do. No paper is necessary for this work.
(If the child doesn't, yet, understand that 20 + another number is just twenty and the other number, now is a good time to alert them to that fact as well. We teach that
twenty = twoty
thirty = threety
fifty = fivety
etc.
Notice, the strange naming practices end at the same spots. Twenty and Twelve have odd spelling when compared to sixty and ninety. Also, look up twelve and eleven in wiki and you'll see some interesting stuff to make math come alive for the linguists among us.)
The second step has two sub-steps and that is the end, I promise.
2a)the smaller of the two numbers in need of addition is decomposed into two smaller yet numbers (one of which is the magic friend of the larger number)
2b)the resulting 3 numbers are added pointing out that two of them are magic friends.
And example will clarify what I mean.
7 + 8
Take 7 and decompose it into 2 and 5. Then since the 2 is the magic friend of 8, the 2 and 8 make a 10. Now just add the 5 to 10 yielding five teen--15.
The key to this is knowing the magic friend of 8. And then being able to decompose 7 into that magic number and another number.
That is the concept, but now to how to teach it to a child.
First, start with the problem, I'll use a different example. 6 + 9
Write it out with lots of space below. Ask the child to recognize which of the numbers is bigger (9) and draw a line straight down from it and rewrite 9.
Now, from the smaller number, draw two lines that angle away like you would use for a family tree. Construct these lines such than when you write a number at the end of the lines, they will be level with the 9. We will be creating a new equation with the numbers beneath the lines.
Now, ask the child to give you the "magic friend" of 9. Write that number in the position that is nearest the 9. In our problem, we'll have a 1 in that position.
Ask the child what number is needed with the 1 to sum to 6 and of course that is a 5. Write that beneath the line that is empty. Put + signs between all the numbers.
So now in our new equation we have 5 + 1 + 9. Circle the 1 and the 9. Point out that this sums to 10 and then ask them to sum 5 and 10 which is of course five teen or 15.
(Note: I always find the magic friend of the larger number because 1) decomposing smaller numbers is easier than larger numbers and 2) the magic friend is smaller as well. You don't have to do it this way, but I find it works more easily.)
After they have the idea and can do this on paper easily, have them practice in their heads. Any math that can be done in the head, should be done in the head.
Wednesday, March 11, 2009
Factoring tests
There are ways to easily check to see if a large number is divisible by a smaller number.
1: All numbers are divisible by 1, even prime numbers.
2: Even numbers are divisible by 2. An even number is any number that ends in 0, 2, 4, 6, or 8.
3: When you sum the digits of your large number, if that sum is divisible by 3, then the original number is divisible by 3. Here is an example.
417
4+1+7 = 12
1+2 = 3
3 is divisible by 3 so 417 is divisible by 3.
5: If the number ends in 0 or 5 (excluding 0 itself) then the number is divisible by 5.
9: This is similar to 3. Sum the digits of the larger number, if they sum to a multiple of 9, then the original number is divisible by 9. Here is an example:
459
4+5+9 = 18
1+8 = 9
9 is divisible by 9 therefore 459 is divisible by 9.
10: If the number ends in 0 (excluding 0 itself) then the original number is divisible by 10.
Advanced hint:
When summing digits to check for 3 and 9 divisibility, you can skip adding in the 9's. One day I'll post why that is but until then, give it a try and see for yourself that you can just skip over the 9's.
1: All numbers are divisible by 1, even prime numbers.
2: Even numbers are divisible by 2. An even number is any number that ends in 0, 2, 4, 6, or 8.
3: When you sum the digits of your large number, if that sum is divisible by 3, then the original number is divisible by 3. Here is an example.
417
4+1+7 = 12
1+2 = 3
3 is divisible by 3 so 417 is divisible by 3.
5: If the number ends in 0 or 5 (excluding 0 itself) then the number is divisible by 5.
9: This is similar to 3. Sum the digits of the larger number, if they sum to a multiple of 9, then the original number is divisible by 9. Here is an example:
459
4+5+9 = 18
1+8 = 9
9 is divisible by 9 therefore 459 is divisible by 9.
10: If the number ends in 0 (excluding 0 itself) then the original number is divisible by 10.
Advanced hint:
When summing digits to check for 3 and 9 divisibility, you can skip adding in the 9's. One day I'll post why that is but until then, give it a try and see for yourself that you can just skip over the 9's.
Tuesday, March 10, 2009
Order of operations
Why do we have order of operations? Well it is pretty simple--so we can all read the same expression in the same way. It is all about communicating to one another effectively in the language of mathematics.
if What wrote a in sentence words the I of order any?
It would be really hard to understand what I meant--but not impossible. I bet with some effort you could figure out my sentence. And that is because English has some rules that we "just know" for the most part. The rest we call grammar.
We learned the basics of grammar long, long ago when we were just little kids. And mathematics is the same way. Unfortunately, we don't "speak" much math to little kids so there isn't much in the way of the basics of the mathematical language in our heads. We have to learn it the "formal way."
But that is ok. The rules are very simple--way more simple than the rules of the English language. And PEMDAS sums it all up. I have a post explaining what that means and how to use it here.
But the long and the short of it is that PEMDAS is a set of simple rules that keep us all speaking the same language.
Here is an example.
in the box
That is a nice prepositional phrase. We never see it written "box the in." There is an order that we MUST follow here. But after we do, we can put that phrase in many locations relative the thing that is "in the box."
Put the sweater in the box.
Put the sweater that I love in the box.
In the box goes the sweater.
Going in the box is the sweater.
Some sentences are better than others but you can understand the meaning of all of them.
Now, in mathematics.
(3 + 4)
This is the same as the phrase "in the box." I can move it around in an expression.
8*(3+4)
(3+4)*8
(3+4)/8
(3+4)^5
(3+4) + (5+6)
When we follow know the order of operations, we can construct very complex and terribly organized equations, yet all will be able to read it. Of course we strive to make our equations simple and beautiful--but that is were experience comes into play.
The order of operations is simply part of the grammar of mathematics making a jumble of symbols and numbers make coherent concepts.
if What wrote a in sentence words the I of order any?
It would be really hard to understand what I meant--but not impossible. I bet with some effort you could figure out my sentence. And that is because English has some rules that we "just know" for the most part. The rest we call grammar.
We learned the basics of grammar long, long ago when we were just little kids. And mathematics is the same way. Unfortunately, we don't "speak" much math to little kids so there isn't much in the way of the basics of the mathematical language in our heads. We have to learn it the "formal way."
But that is ok. The rules are very simple--way more simple than the rules of the English language. And PEMDAS sums it all up. I have a post explaining what that means and how to use it here.
But the long and the short of it is that PEMDAS is a set of simple rules that keep us all speaking the same language.
Here is an example.
in the box
That is a nice prepositional phrase. We never see it written "box the in." There is an order that we MUST follow here. But after we do, we can put that phrase in many locations relative the thing that is "in the box."
Put the sweater in the box.
Put the sweater that I love in the box.
In the box goes the sweater.
Going in the box is the sweater.
Some sentences are better than others but you can understand the meaning of all of them.
Now, in mathematics.
(3 + 4)
This is the same as the phrase "in the box." I can move it around in an expression.
8*(3+4)
(3+4)*8
(3+4)/8
(3+4)^5
(3+4) + (5+6)
When we follow know the order of operations, we can construct very complex and terribly organized equations, yet all will be able to read it. Of course we strive to make our equations simple and beautiful--but that is were experience comes into play.
The order of operations is simply part of the grammar of mathematics making a jumble of symbols and numbers make coherent concepts.
Monday, March 9, 2009
Book suggestion
13 1/2 Lives of Captain Bluebear
Science fiction hidden inside an action adventure story. Plus who can resist a title with a fraction?
Reading level: 7th grade (all ages would love it though!)
Science fiction hidden inside an action adventure story. Plus who can resist a title with a fraction?
Reading level: 7th grade (all ages would love it though!)
PEMDAS--order of operations
P= Parenthesis
E= Exponent
MD = Multiply and Divide
AS = Add and subtract
This is the order of operations.
Look at your mathematical expression and if you see any parenthesis (since it is the first item) work with that first. If there are no parentheses, look for exponents next. Continue down the list until you find the operation you need.
After you do anything to your expression, start at the top again. Check for P, then E, then MD, finally AS.
Here is an example:
(5 + 3 * 4 - 8)^(16/4)
Do we have parentheses? Yes. Let's pick one and look within it.
5 + 3 * 4 - 8 I'll look here first.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
5 + 12 - 8
Starting at the top again, but staying with this term in the first parenthesis.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? No.
Is there addition or subtraction? Yes. Let's do that.
5 + 12 - 8 = 9
Ok, we are down to a single number let's put that in the place of the parenthesis we chose to start with.
(5 + 3 * 4 - 8)^(16/4) = 9^(16/4)
Now we start again.
Is there a parenthesis? Yes. Let's work on it now.
16/4
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
16/4 = 4
Ok, we are down to a single number. Let's put that back into the problem.
9^(16/4) = 9^4
Starting again.
Is there a parenthesis? No.
Is there an exponent? Yes. Ok, let's do that.
9^4 = 9*9*9*9 = 6561
We are down to a single number so we are done.
(5 + 3 * 4 - 8)^(16/4) = 6561
E= Exponent
MD = Multiply and Divide
AS = Add and subtract
This is the order of operations.
Look at your mathematical expression and if you see any parenthesis (since it is the first item) work with that first. If there are no parentheses, look for exponents next. Continue down the list until you find the operation you need.
After you do anything to your expression, start at the top again. Check for P, then E, then MD, finally AS.
Here is an example:
(5 + 3 * 4 - 8)^(16/4)
Do we have parentheses? Yes. Let's pick one and look within it.
5 + 3 * 4 - 8 I'll look here first.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
5 + 12 - 8
Starting at the top again, but staying with this term in the first parenthesis.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? No.
Is there addition or subtraction? Yes. Let's do that.
5 + 12 - 8 = 9
Ok, we are down to a single number let's put that in the place of the parenthesis we chose to start with.
(5 + 3 * 4 - 8)^(16/4) = 9^(16/4)
Now we start again.
Is there a parenthesis? Yes. Let's work on it now.
16/4
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
16/4 = 4
Ok, we are down to a single number. Let's put that back into the problem.
9^(16/4) = 9^4
Starting again.
Is there a parenthesis? No.
Is there an exponent? Yes. Ok, let's do that.
9^4 = 9*9*9*9 = 6561
We are down to a single number so we are done.
(5 + 3 * 4 - 8)^(16/4) = 6561
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