You can think of multiplication as multiple adding. When we say 5x4=_. We are saying to add 4, 5 times like in the addition problem... 4+4+4+4+4=_. You can use the old rock method with Homemade Math's Scratchpad. Drag 5 number 4 cards into a blank part of the screen so that they are lined up. Count all the dots on the number 4 cards. You should find that there are 20 dots. 5x4=20. You can reverse the order of multiplication and the answer will be the same. Try 4x5=_ instead of 5x4=_ in the problem above. Drag 4 number 5 cards into a blank area of the screen and count the total number of dots on them to find the product of 4x5, which is still 20.
Although you can reverse the order of a multiplication problem and the answer or product still comes out the same. The first number is the multiplier and the second number is the multiplicand. In the problem 6x2=, 6 is the multiplier and tells how many times 2 is to be added. 2 Is the multiplicand because it is being acted on by 6 and being multiplied. You can reverse the order and the answer will still come out the same, but one of the numbers has to be the multiplier and the other number has to be the multiplicand. Unlike addition we don't always have the same type of units for the numbers we are multiplying. One example would be that we have a piece of wood that is 3 centimeters long and we want to know how many millimeters long that would be. Since there are 10 millimeters in 1 centimeter, we can find the answer by the equation 3cm x 10mm = 30mm. cm Stands for centimeters and mm stands for millimeter.
The image above shows a tooth pick measuring 3cm, each line or graduation between the numbers representing the centimeters is 1mm. Count these graduations and you'll find there are 10 per centimeter and 30 in the length of the tooth pick. When you think of how you are adding multiple times remember that the multiplicand is the number being added and the multiplier is just telling how many times to add it. In the toothpick example we are adding millimeters to millimeters. We are still adding like units to like units as we learned in Simple Addition. We would not add millimeters to centimeters.
Zero is a special number and the product of 0 x any number is always 0. If we say 3x0=, we are saying to add 0, 3 times. If you are having trouble visualizing this try it with the number cards in the scratchpad area. The zero card doesn't have any dots on it to add so if you drag 3 zero cards onto the screen and count the dots you will have no dots and that is what zero represents. If we reverse the order of the problem to 0x3=, we are saying to add 3, 0 times, which means that we don't add 3 at all and our answer is 0.
What happens when you see a problem like, 2x3x5=? We multiply 2x3 and get 6. Then we rewrite our equation to be 6x5=. Next we multiply 6x5 and get 30, so 2x3x5=30. The order we multiply in make no difference, the answer will come out the same. Lets try 5x2x3=. 5x2=10, rewrite to 10x3=, solve 10x3=30. Note that when you run into problems like 2+3x4=, you multiply 3x4 first and then add the 2 or your answer will be incorrect. The same is true for 2x6-5=. We multiply 2x6, rewrite 12-5=, subtract getting 7. Mathematicians of long ago came up with the rule of following descending powers as we solve problems so that we would all do them the same and come up with the same answer for any given problem. We have not covered exponents or roots yet but they are the highest power and you do them first, then you do the multiplication and division and then the addition and subtraction. If for some reason we want the order which the problem gets solved in to be changed, we can use parenthesis. If we write 2x(6-5)=, we are saying to take 5 away from 6 first and then multiply by 2. Parenthesis can have a set or sets of parenthesis in them. In that case solve the problem or problems contained within the inner most set or sets of parenthesis first. Rewrite the problem with the most inner set or sets of parenthesis removed and then work the next problem inside the parenthesis. Then rewrite the equation with that set removed and after you have all the problems within parenthesis within the lager problem worked, just follow the order of descending powers in solving for the rest of the problem. Note instead of saying solve the problem within the parenthesis first, we can say, solve for the expressions within parenthesis first and this is the way you'll find it stated in math problems. The definition of expression when used in math is a set of values and operators that are to be combined. An operator is a symbol like x to show that we multiply or + to show that we add. Values are numbers like 1, 4 and 5. Values can also be symbols like a, b, x and y which represent unknown values that we solve for. If an expression contains these unknown symbols we solve for, we call it an algebraic expression. Square brackets [] and squiggly {} braces are used in place of parenthesis when writing equations on paper to help show inner and outer placement of the parenthesis. You would use the square brackets to contain a set or sets of parenthesis and then use squiggly brackets to contain a set or sets of square brackets. In a pretty complex equation you might need to go back and use regular shaped but larger parenthesis to enclose squiggly brackets enclosing square brackets enclosing regular parenthesis. In the problem 6x2=12, 6 and 2 are factors of 12. 3x4=12 so 3 and 4 are also factors of 12. Any whole number that can be multiplied by another whole number to get a product is a factor of that product. A whole number is one that doesn't contain any fraction part like 1.23 or 2 1/2. 1.23 and 2 1/2 are not whole numbers because they have fractional parts. Integer means the same thing as whole number and you will find it used in place of the words whole number in many math problems.
Copyright 2008 Robert Lee Thomas
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