Modeling Multiplication of Integers
Did you ever hear of "skip counting"? Skip counting is skipping over equal groups of numbers again and again.
Skip count by 2? That is 2, 4, 6, ... which is really the even numbers.
Skip count by 4? That is 4, 8, 12, ...which is really the multiples of 4.
So multiplication is really skip counting. One factor tells how many times to skip count by the second factor. With integers, the positive or negative of the first factor tells if we need to reverse the direction. We can model many multiplication of integer problems using a number line.
Our first model is to simplify - 4 • 2. Notice that the skip always starts at 0. The first factor tells us that we need 4 groups and the groups are going in the reverse direction of the positive 2. Our second factor tells us that we are skip counting by 2's.
What happens if we reverse the problem? Now we model 2 • - 4. What do you notice about this number line and it's answer? You see two groups of - 4. The two is positive so we stay in the direction of the negative 4.
It seems that the two answers are the same. It did not matter the order of the numbers. This in fact is the proof of the Commutative Property in Multiplication. We can switch the order and the answer comes out the same. So if you are multiplying any two numbers, switch the order if it makes the problem easier to solve!
Our next model has both factors as negatives. Simplify -2 • -3. Let's break down the problem to see what it is really asking us to do.
There are - 2 groups of - 3. The negative in front of the two (number of groups) tells us that we need to reverse the direction of our move. So instead of making two groups of negative three, we make 2 groups of positive 3. The number line shows this thinking.
Another model for multiplying with integers can be done with colored chips. View the lesson and try some problems at the site http://www.learner.org/courses/learningmath/number/session4/part_c/multiplication.html
Patterns in Multiplication and the Rules for Multiplying Integers
Another way of understanding the signs in the problem and the sign of the answer is by looking at patterns in a table. Chart 1 shows how answers become negative when going from two positives to one positive and one negative numbers multiplied together. Chart 2 shows the pattern when we start with one positive and one negative to two negative numbers multiplied together.
Looking at these two charts can you come up with some general rules for multiplying with negatives?
Try using these rules for multiplying 2, 3, even four integers at http://teachertech.rice.edu/Participants/mredi/lessons/integermultiplication.html
Modeling Multiplication of Fractions
We can model multiplying fractions just by dividing squares into equal segments and overlapping the squares.
Our model is to simplify .
We can read the above problem as one-third of two-fifths. First draw two congruent squares. Divide one vertically into thirds and shade one section in. Divide the other into fiftths horizontally and shade two sections in.
Now overlap the two squares. Notice the number of blocks has increased and there are two blocks that are purple. The purple blocks represent what one-third of two-fifths is.
The intersection of the two shaded represnts the answer. The whole has been divided into five pieces width-wise and three pieces height-wise. We get two pieces that overlap. That is the numerator. The denominator is the total number of pieces when overlapping the two grids. The denominator is 15.
Now we work our second model mathematically. We will multiply three fractions.
To start, let’s only look at the first two fractions.
We start by simplifying. We can simplify these two fractions in two different ways. We can either cross simplify the two and the four with the GCF of 2, or we can simplify two-sixths to one-third.
Let’s simplify two-sixths to one-third. Now rewrite the problem with all three fractions.
Next, we can multiply and then simplify, or we can look and see if there is anything else to simplify. One-fourth and one-third are in simplest form, four-fifths is in simplest form. Our final check is to check the diagonals.
The two fours can be simplified with the greatest common factor of 4. Each one simplifies to one.
Our final answer is .
This tutorial on multiplying fractions and mixed numbers can help you master this concept. http://www.sophia.org/multiplying-fractions-and-mixed-numbers/multiplying-fractions-and-mixed-numbers-tutorial?topic=multiplying-and-dividing-fractions--2 View it and then try the problems that are attached. See how much better you have become.
Properties Used in Multiplying Rational Numbers
Rational numbers are any numbers that can be expressed as a fraction. Examples of rational numbers are:
Rational numbers can be negative or positive.
There are 4 properties that are used in multiplying rational numbers. They are:
Commutative Property: The product of two numbers is the same whichever order the items to be multiplied are written.
Associative Property: When three or more numbers are multiplied, the product is the same regardless of how they are grouped.
Multiplicative Identity Property: The product of any number and one is the original number.
Distributive property: The multiplication of a number and the sum of two numbers is equal to the first number times the second number plus the first number times the third number.
http://www.youtube.com/watch?v=8i-QQvroJdo has another explanation of the rules of multiplication.
Here are some models to help understand how these properties are used in actual math problems.
Prove the rule for integer multiplication that a positive times a negative is a negative using the distributive property.
We set up our model using the distributive property. We simplify everything BUT the 5(- 7) because that is what we are trying to prove.
Look at the problem left. 35 plus what number equals zero? It has to be -35 because it is the additive inverse! So a positive number times a negative number is a negative number. 5( - 7) = -35.
We can multiply any two factors together first. It seems easier to multiply (1.2)(- 4) first.
Now we change either to both decimals or both fractions to multiply. If 2/3 is changed to a decimal it becomes .6666... It repeats, so it is not a good choice to change to decimals. We change - 4.8 into a fraction. Then we multiply the fractions.
What is ¾ of the product of -12 and -3.5?
First we rewrite the problem using math symbols. Product means to multiply. So we multiply -12 and -3.5 first. Multiply that answer (remember that "of" means multiply) by ¾.
Sometimes multiplication of a decimal times a whole number can even be done using mental math. How? By actually using the Distributive property. The Distributive property breaks down the math problem into simple small pieces.
Let's try to show mental math for the problem 7 x 4.3. Using the Distributive property we can break this down into 7(4 + .3). 7 x 4 = 28. 7 x .3 = 2.1. Then just add the two answers back together. 28 + 2.1 = 30.1. Math properties can really make things easy.
Show how the Distributive property can be used to multiply 6 x 5.3
Break the problem down using the distributive property.
Apply the distributive property by multiplying six by each number. This is called expanding the problem.
http://mathvids.com/lesson/mathhelp/85-multiplying-rational-numbers has a tutorial with several examples that are solved step by step. A great review that is easy to understand.
Modeling Division of Integers
Division is the inverse operation of multiplication. That means it is the opposite math operation. Addition and subtraction are also inverse operations. When solving for a variable, we always use inverse operations.
When modeling division of integers on the number line, it is always important to start at 0. Our first model has both numbers as positive to understand the grouping.
Dividing does mean to split up evenly. So to split up 8 so there are two in a group, we have 4 groups. The inverse operation to check this would be 4 x 2 = 8.
Negatives put a twist on this because the negative will mean direction.
If you were standing on the numberline and walked backward, you would be facing the negatives. Your answer is a negative. Mathematically speaking we can look at the inverse. 10 ÷ - 2 is really asking what number times -2 equals +10. Eariler in this concept we proved that a negative times a negative is a positive. Either way you look at it, the answer is - 5.
http://www.youtube.com/watch?v=Lh0tBKOTq8I is a live demonstration by students of how to use a number line in division of negative numbers.
Model the following equation on a number line and check using the inverse operation.
Our number line will have to be made going by 2's because the problem has bigger numbers in it. Set up the number line with a point on zero and a point on -16.
The negative next to the 4 tells us to "walk backward". Starting at the zero, make 4 equal groups or "hops" back to the -16.
There are 4 in each group (remember that we are counting by 2's). We also had to walk backward. When we walk backward, we are actually facing the positive numbers. The answer is positive. Therefore -16 ÷ - 4 = + 4.
To check using the inverse operation is to use multiplication. Does 4 x - 4 = -16? Yes, because 4 x 4 is 16 and we know that a positive number times a negative number is a negative number.
Patterns in Division and the Rules for Dividing Integers
Just like multiplication, we can use a table to understand division of integers. Look at the patterns when the signs change both in the problem and the answer. Chart 3 shows how answers become negative when going from two positives to one positive number divided by one negative numbers . Chart 4 shows the pattern when we start with one positive and one negative to a negative number divided by another negative number.
Look back at the two charts for patterns in multiplication. In charts 1 and 3, the pattern shows that if one number is negative and the other is positive, the answer is negative. In charts 2 and 4, the pattern shows that if both numbers are negative, the answer is positive. The rules for multiplication and division are generally the same.
We have to use two rules with one negative and one positive because division is not commutative. We also have to use the words rational number instead of integer because 4÷ -8 = -½. One half is not an integer.
http://www.youtube.com/watch?v=sEU5uaf-Tu4&feature=related is a quick review of the rules for multiplying and dividing with negative numbers.
A Triangle Trick to Multiplying and Dividing Rational Numbers
Here is a neat trick if you ever get stuck trying to figure out what the sign of your answer is when multiplying or dividing. All you need is to draw a triangle like the one above. Put two negative signs up top and one positive down below. Then cover ups the two signs in the problem. The one uncovered sign is the sign of the answer!
Here are some models:
-32 ÷ 4 = ?8 This is a negative divided by a positive. Cover up the negative and positive. There is a negative sign not covered. This is the sign of the answer. So -32 ÷ 4 = - 8.
- 17 x -1.2 = ?20.4 Cover up two negative signs. The sign of the answer is positive.
8 ÷ -2 can be written in two other ways and still mean the same (get the same answer). The first is to write the division vertically.
. This may look like a fraction, but it is actually a division bar. Yes, this does mean that every fraction is actually an unfinished division problem! The second way is to change it to a multiplication problem.
. Dividing by -2 is the same thing as multiplying by -½. Think about it - if you eat half of a pizza that has 8 slices, you eat 4 slices. ½ of (and of means x in math) 8 is (is means = in math) 4.
One more thing to remember: when a division problem is expressed using the division bar, the negative sign can appear in three different ways, yet mean the same thing.
Express the following division problem in two ways and give the final quotient.
Since the negative sign is before the entire division problem, we use parentheses when re-writing the problem.
For the first way, we re-write the problem using a ÷ sign.
For the second way, we use multiplication of the fraction .
Finally, we select any of these three expressions and solve.
Division with Zero
Using 0 in a division problem happens in two ways. One gives the answer of 0, and one has no answer. Can you tell the difference between the two math expressions and give the answers to each?
The difference between the two is the zero being before the division sign or being after the division sign.
To give the answers to each, we look at these from the inverse operation, multiplication.
is the same as asking what number times 4 equals zero. Of course the only number to multiply 4 by is zero. This is called the multiplication property of zero. So .
is the same as asking what number times 0 equals 4. Wait! Any number times zero is always zero (multiplication property of zero again). How can we ever get 4? We can't. So there is no answer to this problem. In math, when there is no answer it is called undefined.
http://www.youtube.com/watch?v=fcVnwjBgDmk is a mini lesson that reviews both multiplication and division by zero.
This is called a complex fraction. It is a" fraction in a fraction". Remembering that a fraction bar is really a division bar, the expression can be written in two different ways. Can you tell what was done?
The first way is using fractions. The second way is using decimals. Either way is fine.
The solution is 32.
http://www.sophia.org/dividing-rational-numbers--2/dividing-rational-numbers--6-tutorial?pathway=nc-skill-318 is a tutorial with concrete models and examples and solutions.
Dividing Rational Numbers using Long Division
Dividing rational numbers has only two things you need to do before it becomes the same thing as long division with whole numbers. Set up the division, move the decimal, and you are ready to divide.
If you have difficulty in completing problems in long division, check out an another way to divide using the step by step guide to using double division. http://www.doubledivision.org/
Using long division, convert into its equivalent decimal format.
Equivalent decimal format means to divide and make the fraction into a decimal.
Following the steps in the above model problem we set up the long division.
Now we divide. Two ways of division are shown, traditional division and partial quotient using only 10's, 5's, and 2's times tables. Both are good ways of dividing.
We know the decimal .875 is a rational number because it ended with 0 when we divided. A decimal that repeats is also a rational number. All fractions with a denominator of 3, 7, and 9 repeat. All but one that has 6 as a denominator (3/6 = 1/2) repeats as a decimal. And if you know what 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9 are as decimals, all you do is multiply to find out any of their fraction family decimals. Here is just one example of this "trick". First we change 1/5 into a decimal by dividing. Then we look for the pattern in the multiplication to get the fraction → decimal family.
So 3/5 would be 3 times .2, and so on. Just remember that ALL repeating decimals are rational numbers. That means that all repeating decimals can be made into a fraction.
Mixed Multiplication and Division of Rational Numbers
Multiplication and division of rational numbers follows the rules of order of operations. All multiplication and division in a math problem is solved from left to right, paying attention to all parentheses.
First multiply what is in the parentheses.
is not in lowest terms. They are both even numbers. We can find the greatest common number that divides into both, or just keep reducing by simple numbers like 2, 3, etc.
Following order of operations, parentheses is simplified first. We divide 6 by .3.
Now we can change into a decimal or change 20 into a fraction. Both numbers must be in common form. Changing 20 into a fraction, , is much easier. Now multiply the two fractions. Remember that a negative rational number times a positive rational number is a negative rational number (rules of integer multiplication).
http://www.youtube.com/watch?v=CMQrctqdz6g has a live example of a mixed multiplication that you can watch as well.