Mis-Math Addition and subtraction Addition and subtraction can be taught using simple re-grouping.
Of course, this is also written out using numbers as 3 + 2 = 5 Introduce the concept of groups so that students understand that the above example is to combine a group of 3 with one group of 2 to make one group of 5. Suppose you have “nothing” in a group, then add “something”:
We use a '0' (zero) to represent a place that has no value in it. But don't forget, there is a place, it's just empty! It's like a chair in the room. No one is sitting there. But you know a person can certainly walk over to the chair and sit down. Then a person would be in that place. Subtraction works the same way, only you take some smaller group from a larger group.
When you do subtraction, you can match up chips from the left side of the subtraction sign with ones on the right. The ones that match up, we take away and do not count. But we need to keep track of the ones left over. Also, we can take all the chips away, leaving an empty place:
Finally, what happens if we try to take more away than we already have?
?
After we match up the chips, we still have to take care of the left-over chips on the right side. We can't just forget about them! For right now, we'll just say we still need to subtract them, but we'll turn them red so we don't forget. Until we do something else with these chips, we need to remember they need to be subtracted. We can call them “negative” chips.
Associative Property of Addition Go back to the first problem we did: 3 + 2 = 5. Don't we have a total of 5 chips on each side of the Equal sign? Does it make any difference how they are grouped, as long as we stay with 5? Think about this: Suppose you were having lunch with your friends. There are five of you all together, but three sat together, and two had something else to
talk about so they sat together but a bit apart from the first group. But one of the three moved over to the group of two, and left only two at the first table. Then another friend moved over, leaving just one at the first table, and four at the second one. In all this moving around, don't you still have five friends together? Even when one friend is alone and the other table has four, you still have a total of five friends. This is a mathematical property called the Associative property of Addition: When you add numbers, you can have all the chips in one group or in several smaller groups, as long as the total number of chips does not change.
Commutative Property of Addition Use the 3 + 2 = 5 again:
Can you swap the 3 + 3 to make 2 + 3, and still get 5? Yes. It's a lot like the associative property, but this property says you can swap the parts of an addition problem without worrying. This is called the Commutative Property of Addition. Warning! The Associative and Commutative properties do not work in Subtraction! Try it with 5 – 2 = 3:
Commutative Rule:
Associative Rule:
The ≠ means “NOT Equal to”, like 3 + 3 ≠ 5.
Multiplication We multiply numbers by adding the same amount over and over. Suppose we have three chips. Let's add in these three chips four times. (Remember, the row of chips already there counts as one, so we get three more rows to make four rows total.)
Count them, that's 12. So, 3 ∙ 4 = 12. The adding and multiplying portion is rather easy, and it can build confidence and familiarize students with working with the chips. Here we have 3 ∙ 4 = 12. The numbers on the left side of the = sign are both called factors, and the answer on the right is called the product. We'll use these terms a lot. Two (or more) factors multiplied together make a product. Take four chips, then lay them out in a row. We started out talking bout making several rows of the same number of chips. How many rows are here? How many columns? We can call this multiplying. The two factors are always the number of columns multiplied by the number of rows: 4 ∙ 1 = 4. Remember, the idea of run and jump means you run across all the columns (4 here) then jump the number of rows (1). Now add another row of 4: There are now eight. We can write this as 4 ∙ 2 = 8. We multiplied two factors (4 and 2) to get the product 8. add another row of 4:
This is 4 ∙ 3 = 12. This number 12 has six pairs of factors:
1 ∙ 12 = 12
2 ∙ 6 = 12
3 ∙ 4 = 12
4 ∙ 3 = 12
6 ∙ 2 = 12
12 ∙ 1 = 12
Commutative Property of Multiplication If you set out chips in 4 columns and 3 rows, you have 4 ∙ 3. But if you make 3 columns of 4 rows you get 3 ∙ 4, which has the same product, 12. So you could say that 3 ∙ 4 = 4 ∙ 3 = 12.
4
3 4
3
This is called the Commutative Property of Multiplication. Since mathematicians understand this, we can skip the “second” set of factors. (We'll just look at 3 ∙ 4, and know that 4 ∙ 3 is pretty much the same thing.) So, for 12's factors: 1 ∙ 12, 2 ∙ 6, 3 ∙ 4, 4 ∙ 3, 6 ∙ 2, and 12 ∙ 1 we can skip half of them! (Since 3 ∙ 4 =4 ∙ 3 and so on, we don't need to repeat.) 1 ∙ 12, 2 ∙ 6, and 3 ∙ 4 is all we need to write down. Is there an Associative Property of Multiplication? In Addition 3 + 4 = 2 + 5 = 7
But in multiplication, does 3 ∙ 4 =2 ∙ 5 (take 1 from 3 and add it to the 4)?
? Well, 3 ∙ 4 = 12 and 5 ∙ 2 = 10. And 12 ≠ 10! Try it with the chips to see,
?
4 3
5 2
So, there is a way to swap factors when you multiply using the Commutative Principal of Multiplication, but you can't move parts of one factor into the other.
Divisibility Rules
What can you do to find its factors? One way is to get a stack of 21 chips, and arrange them in different ways to find the different rectangles that 21 chips makes (that number is two rectangles for 21). Is there a way to simply look at a number, think about it a bit, and figure out some factors? Here's an easy one to get started with: Count by 2's: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 We call these even numbers, and they are all multiples of 2 (or you can say they are divisible by 2). but can you tell if a really large number, say 2,635,591 is even or not? There is a way. Look carefully at the list of multiples of 2. Look at everything about the numbers and see if you can find a pattern. Hint No. 1.:
Look at the units part of each number.
Hint No. 2.:
The units numbers all have the same property (“even”)
Hint No. 3.:
The units numbers repeat in the pattern 0, 2, 4, 6, 8, ...
If you see any number that ends in a 0, 2, 4, 6, or 8 in the units position, the whole number is divisible by 2. That long number 2,635,591? It ends in a 1. It is not divisible by 2. On a sheet of paper, for each number from 2 to 10, make a list of all the multiples between of itself and 30. Then look for patterns for each. Hint: multiples of 10 are too easy! Another hint: multiples of 7 have a rule, but it's complicated and we will skip that. Don't worry about 7. Hint: if you know the rule for 2, can you combine that with something to make rules for higher numbers (Think about other even numbers like 4 and 6 – what do their rules look like?) Be sure to think about everything about a given number. What about 27 can you figure out that will tell you it is divisible by 3? Hint: add the 2 and the 7 together. Try that with other multiples of 3. Then look at other multiples of 3, like 6 and 9 – how can this rule help you? As you discover rules, add them to a chart called Divisibility Rules. You should have a rule for every factor from 2 to 10, except for 7. For fun, take a look at multiples of 11 (work past 11 ∙ x = 150). Multiples of 12 are interesting, too. Once you have these rules, you can figure out factors for nearly any whole number (ones without fractions or decimals)
Division Up to now we have looked at addition (and subtraction), multiplication (repeated addition of the same number) and a bit of division to work out factors of numbers. Now we will look a division to divide quantities up into smaller (and equal) pieces.
Fractions as Division Recap the addition but add 1 chip under a line, showing each part is a group.
All whole numbers can be written as a single group this way. In fact addition and subtraction can 3 2 5 + = . So symbolize this problem by drawing a also be demonstrated with fractions of 1: 1 1 1 fraction line, making the green circles the numerator and a single blue circle the denominator. Here the idea of adding and subtracting is not new, but we are remembering that each part of the equation represents one group of n chips. The idea that the “bottom” number, the denominator, counts the groups above it is essential.
Division and Factors Re-write the below section for using FACTORS in DIVISION Let's make a single group with a 2 value on the denominator: This shows that there are a total of 6 green chips. Written out, this is
6 . 2
Arrange the chips so there are as many columns as there are chips in the denominator. Here there are two groups in the numerator. We know that we can make 6 chips into two groups, with 3 in each group. How can you separate the six green chips into two equal groups? How can this be represented? (Goal: we should be able to make as many equal numerator groups as there are denominator chips.)
6 3 = . The hard part in this case might be the perception that there are still six 2 1 chips on the numerator and two on the denominator. So ask the question “How many green chips are in each group?” There are 3. So the answer is expressed as “There are two groups. Each group has 3 over 1, so we can say 3 for an answer!” Here is a way:
Clarify:
In addition, the blue chip shows how many groups you need in the denominator. The number of denominator chips does not change from the left side of the equal sign to the right:
One group of six is the sum of two groups of three. For division, the operation is to make the number of groups 'requested” by the denominator. Note there is no operator between the parts on the left of the equal sign. We just indicate where all the parts go. The correct answer is one of the divided parts:
Divide one group of six into two parts. You get two single groups of three. The underlying concept is that we take a large group (dividend) and divide it into several smaller groups (divisor). The answer is always how many chips are in each (equal sized) group, or the quotient. In that case, what to do with a denominator of 3? (With a numerator also divisible by 3) How about three equal groups?
We have six green chips (the numerator) and three blue denominators. Spread the denominators out, one each, and break the six into three equal groups (2 each). So this answer is that the final groups 6 2 = are all (2 green chips). This problem is written as 3 1 The “extreme” of six chips in the numerator and six chips in the denominator:
The divided up “answer” is six sets of 1 green over 1 blue chip. But the answer to the problem is that 6 1 = . the divided up parts are 1 green chip, or 6 1 The idea behind division is to separate the dividend into as many equal groups as the divisor. The answer is the size of all the divided up parts – and they are all equal.
Remainders Starting with a simple case, an odd number divided by 2:
Separated We can divide out four into two groups.
Not Done Yet
But what about the extra one – the remainder? That needs to be divided, too. Since there's one chip to divide in half, what to do? What do you do with one cookie that you need to share with a friend? You break it in half! So the chips might look like this:
1 5 2 This is written using numbers as: . It's easier to see as = 2 1 the weird fractions shows what the chip set looks like. 2
Try this method for, say,
5 1 =2 but the version with 2 2
17 : 5
? On our first look, we can divide out 15 of the total, but we still have a math expression left over:
We can divide out 15 of the 17 into 5 groups of 3. But we still have to work on the extra 2, which does not divide by 5 easily. So you see the groups of 3 green chips over a single blue chip – we did get the 15 divided up easily enough. But the problem isn't over, so we are still left with the 2 extra green chips over 5 blue chips. We can call this 2 a remainder, like 3remainder 2 or 3r2. You would more often write 2 this answer as 3 . Usually we stop there, with a mixed number (Whole number plus a fraction). 5 2 See? The 3 comes out, and the fraction says there's some left over that won't divide easily. 5 This is one way to show division. Using fractions, the student will see there is a difference in meaning and use for the number in the numerator and the number in the denominator. Here, the denominator is the divisor (We try to make it into a 1) and the numerator is the dividend (The number we want to divide into parts).
The Basic Operations of Mathematics This completes the introduction to the four basic operations of Mathematics. Every other operation is a combination of these four operations.
Copyright Š Errol Van Stralen