In the Simple subtraction exercises you won't run into negative numbers for an answer(difference), but in regular subtraction you will. One way this happens is when you take a larger number away from a smaller number. Lets say you have 4 extra sheep and wish to sell them. A customer wants to buy 7 sheep from you. You tell him that you only have 4 sheep you can sell at the moment. Your customer says "that’s ok, you can owe me the rest of the sheep until your sheep have more kids". So you sell the 7 sheep but only give the customer 4 at this time. The number of extra sheep you have to sell becomes: 4-7=_. You can use your rocks method to find the answer. Put 4 rocks in a pile. Take a rock out and count 1. Take another rock out and count 2 and so on until you reach 4. When you have taken 4 rocks out of the pile representing the sheep you sold, you will have no rocks left, you have reached 0 rocks. You need to reach 7 as you count and take away rocks. To do this we can put some rocks in a pile we will call sheep owed. Continue counting 5 put a rock in the sheep owed pile. 6 put a rock in the sheep owed pile 7 put a rock in the sheep owed pile. You should have 3 rocks in the sheep owed pile. We write this as -3 sheep. 4-7=-3. We can show this using 1 and -1 cards from the Scratchpad. Drag 4 number 1 cards into a blank area of the screen; place them in a group close together.
Drag a 1 card away from the group of 1 cards and count 1. Drag another 1 card away, count 2. Drag another 1 card away and count 3. Drag the last 1 card away and count 4. Drag a -1 card down into the area where the 1 cards were and count 5. Drag another -1 card down into the area where the 1 cards were and count 6. Drag another -1 card down into the area where the 1 cards were and count 7. Count the number of -1 cards. You will have 3, -1 cards so 4-7=-3.
A number line is also useful in subtraction problems. 10-3 Can be found by starting at the 10 marked on the right side of the 0 or origin and counting back toward the origin 3 graduations. You will land on 7. Try 3-5=. Start at the 3 marked on the right side of the 0, count 5 graduations to the left, you land on -2. You can also take 3 from 5 and put a minus sign in front of the difference. This seems to work the quickest for me. What about 6-3-5=? You could add -3 and -5 first then rewrite the problem as 6-8=. Take 6 from 8 and put a minus sign in front of the difference. 6-8=-2. Try it on the number line. Start at 6 on the right side of the origin, count 3 graduations to the left, then count 5 more to the left, landing on -2. If we ask someone what 10-5=_, we would say to them "What is the difference between 10 and 5?". Looking at the number line, count the graduations between 10 and 5, you will count 5 and that is the difference between 10 and 5. The number you take another number away from is called the Minuend. The number that you take away from another number is called the Subtrahend. The answer to a subtraction problem is called the difference. The Minuend is the top number and the bottom number is the Subtrahend in these exercises. In a problem like 10-2=_, 10 is the Minuend and 2 is the subtrahend. We can look at the problem 4-7=-3 like 4+(-7)=-3. The 4 in the problem could also be written +4+(-7)=-3. If a number doesn’t have a plus or minus sign in front of it we assume it is positive and it saves us the time of writing the + sign. Using the main rule of algebra: What ever you do to one side of an equation do to the other side of the equation and both sides of the equation remain equal. This means in the equation +4+(-7)=-3, we can take 4 away from the left side of the equal sign and take 4 away from the right side of the equal sign and both side will still be equal.
+4-4+(-7)=-3-4
On the left side of the equation or equal sign we take 4 away from 4 and are left with -7=. On the right side we have -3-4. Since they are both negative number, we add them together and put a minus sign in front of them. -7=-7 Which is true. knowing how to use algebra is good in helping to understand what happens when we have a problem like. 4-(-7)=_. the 2 - signs cancel and become a plus sign. so that 4-(-7)=11. This has always confused lots of people. If we ask some one to solve this problem, we will ask what is the difference between 4 and -7? Look at the number line, count the spaces between 4 and -7 marked on the number line. You will find 11 spaces and that is the difference between 4 and -7. Take a look at 6-3=? We are asking; what is the difference between 6 and 3 and not asking what the difference is between 6 and -3, which would be written 6--3=?. Ponder this for a while as it is a source of confusion to lots of people. The information above is enough to solve the problems in these exercises. Below is more information that will help you out later on. Lets use algebra to show that 4-(-7)=11. First a quick lesson in how algebra works. Say you have a pile of 5 rocks and another pile of 3 rocks next to a pile of 2 rocks. You really have two piles of 5 rocks each but are expressing their values differently like in the equation 5=3+2. Lets suppose we want these two piles of rocks to remain equal in value at all times. If we put a rock in the 5 pile we need to put a rock in the 3+2 pile like 5+1=3+2+1. If we take 3 rocks from the 5 pile, we need to take 3 rocks from the 3+2 pile like 5-3=3+2-3. Most of the time you will see problems with variable like X that represent unknown values that you solve for like the problem 8=3+X. To solve for X we get X on one side of the equals sign by itself by taking 3 away from both sides of the equation. 8-3=3-3+X. 8-3=5 so we rewrite the equation as 5=3-3+X. 3-3=0 so now we rewrite it as 5=X. We don't need the plus sign because we assume X is positive unless it has a minus sign in front of it. We want to show that -(-7) or --7 is really +7. So we will take 4 away from both sides of the equation 4-(-7)=11. 4-4-(-7)=11-4 The 4s on the left side of the equation or equals sign cancel out while 11-4 on the right side equals 7 so -(-7)=7 or --7=7. You may also find it helpful to scroll back up to the number line and look at the difference between the two points +4 and -7, counting the graduations between them. What happens if we add 7 to each side of the equation. 4-(-7)+7=11+7 On the right hand side of the equation we have 11+7 which equals 18. If a --7 was negative, the 7s on the left side of the equation would cancel and we would end up with 4=18 which isn’t true. If the --7 is positive the left side of the equation becomes 4+7+7 which does equal 18 so --7 is positive. Also note that if we had combined the (-7) and the +7 on the left side of the equation, we would have 4-=18 which is incorrect. When ever you run into a double minus sign like, --, make sure and cancel your double minus signs and change them to a plus before doing any other operations or you will end up with incorrect results. In the exercises you are in at the moment you won't run into this double negative but it seems like a good place to introduce the concept.Copyright 2008 Robert Lee Thomas
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