Berikut ini contoh percakapan matematika dalam bahasa Inggris antara guru dan siswa. Apabila menginginkan materi lengkapnya silakan ontak saya via e-mail (mushadi_iksan@yahoo.co.uk) atau kunjungi:http://groups.yahoo.com/group/sbi_kaltim/
Math's Conversation in English
By Mushadi Iksan, M.Ed.
1. Integer Discussion
Pupil : What exactly is an integer?
Teacher : Well, you know what the natural numbers are, right?
Pupil : Yes, the natural numbers are the counting numbers, 1, 2, 3, 4, 5, and so on.
Teacher : That's right. The integers are a set of numbers including the natural numbers as well as zero and the negative numbers.
Pupil : The negative numbers?
Teacher : Yes, the negative numbers are the opposite of the natural numbers, also called positive numbers, and are usually expressed as -x (if x is a natural number). Think about it like this: if the number 5 is greater than zero by five units, then – 5 is less than zero by five units.
Pupil : So that means that a positive number is always greater than its negative.
Teacher : Exactly, and furthermore it means that any positive number is greater than any negative number. It is also interesting to note that as a positive number gets larger its negative counterpart gets smaller.
Pupil : Then that means – 8 is less than – 3 since 8 is greater than 3.
Teacher : It seems like you understand. Now, can you think of a situation where you would use negative numbers?
Pupil : Wouldn't you use negative numbers if, for instance, you owed more money than you had in your account. If you were in debt Rp 300,000.00 then the amount of money you have could be represented as – Rp 300,000.00.
Teacher : That's good, but can you think up one more example just to be certain you understand.
Pupil : Yes, in golf negative numbers are used to describe how many strokes a player is below par. For instance, if you were 5 strokes below par, your score could be expressed as – 5, or "five under par".
1.1 Integer Addition and Subtraction Discussion
Teacher : Now that we have established a definition for the Integers, we can perform operations on them.
Pupil : What kind of "operations"?
Teacher : Well, the easiest one would be addition. Consider this example: a friend gives you 3 pieces of candy. If you already had 2 pieces of candy, then how many pieces would you have?
Pupil : If I had three pieces and was given two more, then I would have 5 pieces of candy.
Teacher : Yes, that is right. You just added two integers. Now, to express this operation we would write, 3 + 2 = 5, which represents 3 pieces of candy plus (or in addition to) 2 pieces of candy equals 5 pieces of candy. Now what if you had three pieces of candy and someone gave you two more?
Pupil : Hmm, if I had 3 pieces and someone gave me 2 more, then I would end up with 5 pieces of candy, again. So if 2 + 3 = 5 and 3 + 2 = 5, then does 2 + 3 = 3 + 2?
Teacher : Yes, in fact, for any two integers b and c, b + c = c + b. You can add them in either order, we call this the Commutative Law of Addition. Also, if you are adding more than two numbers, you may add them in any order you like, i.e. if you were adding b + c + d, then you could first add b to c, (b + c) + d, or you could first add c to d, b + (c + d). We call this the Associative Law of Addition (parentheses are used to emphasize the order of the operations performed).
Pupil : The Integers also include the Negative Numbers, right? What happens when you add a positive number to a negative number, or two negative numbers together?
Teacher : Well, let's say we had two positive numbers b and c. Then b + (-c) would be the same as writing b - c, or b minus c. We call this subtraction. Let's return to our candy example for a moment. What if you started off with 5 pieces of candy and then someone took away 2 pieces?
Pupil : If I had 5 pieces and someone took away 2 pieces I would have 3 pieces left. So 5 - 2 = 3, right?
Teacher : Right, however, it is important to know that 5 - 2 is not the same as 2 - 5. To subtract a larger number from a smaller number simply start with the smaller number and count down. Once you hit zero, the next number will be -1, then -2, and so on.
Pupil : So to figure out what 2 - 5 equals I would count down five numbers from 2: 1, 0, -1, -2, -3. So 2 - 5 = -3, which is definitely not equal to 5 - 2 = 3.
Teacher : You certainly seem to understand addition and subtraction. Notice that if you subtracted b units from b you would end up back at zero. For any number b, b - b = 0. Now I have one last question. What do you think would happen if you added or subtracted 0 from a number?
Pupil : Well if you did not add anything to a number, it wouldn't change, so b + 0 = b if b is any integer. The same would be true if you did not subtract anything from a number, so b - 0 = b.
Teacher : Yes, and since adding or subtracting 0 from any number changes nothing, we can drop it out of the computation completely. This leaves us with b = b, for any b, which we call the identity equation. It tells us that any number b is always equal to itself.
1.2 Integer Multiplication Discussion
Pupil : So now that I understand addition and subtraction, are there more operations?
Teacher : Yes there are. The next operation is called multiplication. We write multiplication problems in the form a x b, or a times b. For an example we will consider 5 x 3. What this means is that 5 is being added to itself 3 times (5 + 5 + 5), but a better way to think about is that it means 5 groups of 3 units each. Therefore, to solve 5 x 3 you would count the number of units in each group.
Pupil : I feel a little confused.
Teacher : Alright, we will use an example you can visualize. Picture 5 separate plates, each with 3 quarters on them. How many quarters are there all together?
Pupil : Well, if there are 3 quarters for each of the 5 plates, then there are 15 quarters all together. So 5 x 3 = 15.
Teacher : Exactly. Now, what if we had 3 plates, each with 5 quarters?
Pupil : Then there would be 5 quarters for each of the 3 plates, meaning 15 quarters altogether. So 3 x 5 = 15. Does that mean that multiplication is commutative like addition?
Teacher : Yes, multiplication is commutative. For any a and b, a x b = b x a. It is also associative. If you are multiplying three numbers, then (a x b) x c = a x (b x c). Now, what do you think would happen in the case where a number was multiplied by 0 or 1? If you have trouble just try to visualize what it means.
Pupil : Hmm, if you had any number of plates p, each with 0 quarters, you would have 0 quarters altogether. So any number times 0 equals 0. Now, if you had 1 plate with q quarters, then there would be q quarters altogether. So any number times 1 is just the number itself.
Teacher : That is absolutely right. You thought through that very well.
Pupil : This isn't so hard, but what about negative numbers? I know how to add them, but how do you multiply negative numbers?
Teacher : Well, first I should explain a little more about what a negative number is. Even though the negative numbers are less than zero, this does not mean that they are less than nothing. A negative number is merely a negation of a positive number. For instance, what would happen if you traveled one mile east and then backtracked one mile west?
Pupil : You would end up where you started.
Teacher : Exactly, the one mile you traveled west negated the mile you traveled east. Another way to express your movement west would be to say you traveled -(one mile east), or negative one mile east. So we can think of the negative sign as an operator, just as the multiplication sign is an operator. Its operation would be to replace positive units with negative units.
Pupil : Where exactly is this all leading us?
Teacher : Precisely where we want to be. Remember that a positive unit plus a negative unit always equals zero, therefore, the relation between positive and negative is reciprocal. This means that the negative of a positive is a negative and that the negative of a negative is a positive. Knowing this, we can now move on to multiplication by negative numbers.
Based off the definition I gave you earlier, what do you think 2 x (-3) equals?
Pupil : Well, 2 x (-3) would be 2 groups, each with (-3) units, so that would be (-6) units.
Teacher : Good, that one was tricky, but mostly straightforward. How about this one: What is (-2) x 3?
Pupil : That would be (-2) groups, each with 3 units. How do you count negative groups, though?
Teacher : First, let me rephrase it as negative 2 groups, each with 3 units. Now, a negative group is merely a group that would negate a positive group. What kind of group would negate a group of 3 units?
Pupil : A group of 3 units would be negated by a group of (-3) units, right?
Teacher : Yes, therefore (-2) x 3 = 2 x (-3), which we already know is equal to (-6) units. Now what do you think (-2) x (-3) equals?
Pupil : Well, that would be negative 2 groups, each with 3 negative units. So I need to know what kind of group would negate a group of 3 negative units . . . a group of 3 positive units! So (-2) x (-3) = 2 x 3 = 6. I think that's right.
Teacher : That's exactly right. So what you can see from these few examples is that a negative number times a positive number will give you a negative number and a negative number times a negative number will give you a positive number.
Pupil : What exactly is an integer?
Teacher : Well, you know what the natural numbers are, right?
Pupil : Yes, the natural numbers are the counting numbers, 1, 2, 3, 4, 5, and so on.
Teacher : That's right. The integers are a set of numbers including the natural numbers as well as zero and the negative numbers.
Pupil : The negative numbers?
Teacher : Yes, the negative numbers are the opposite of the natural numbers, also called positive numbers, and are usually expressed as -x (if x is a natural number). Think about it like this: if the number 5 is greater than zero by five units, then – 5 is less than zero by five units.
Pupil : So that means that a positive number is always greater than its negative.
Teacher : Exactly, and furthermore it means that any positive number is greater than any negative number. It is also interesting to note that as a positive number gets larger its negative counterpart gets smaller.
Pupil : Then that means – 8 is less than – 3 since 8 is greater than 3.
Teacher : It seems like you understand. Now, can you think of a situation where you would use negative numbers?
Pupil : Wouldn't you use negative numbers if, for instance, you owed more money than you had in your account. If you were in debt Rp 300,000.00 then the amount of money you have could be represented as – Rp 300,000.00.
Teacher : That's good, but can you think up one more example just to be certain you understand.
Pupil : Yes, in golf negative numbers are used to describe how many strokes a player is below par. For instance, if you were 5 strokes below par, your score could be expressed as – 5, or "five under par".
1.1 Integer Addition and Subtraction Discussion
Teacher : Now that we have established a definition for the Integers, we can perform operations on them.
Pupil : What kind of "operations"?
Teacher : Well, the easiest one would be addition. Consider this example: a friend gives you 3 pieces of candy. If you already had 2 pieces of candy, then how many pieces would you have?
Pupil : If I had three pieces and was given two more, then I would have 5 pieces of candy.
Teacher : Yes, that is right. You just added two integers. Now, to express this operation we would write, 3 + 2 = 5, which represents 3 pieces of candy plus (or in addition to) 2 pieces of candy equals 5 pieces of candy. Now what if you had three pieces of candy and someone gave you two more?
Pupil : Hmm, if I had 3 pieces and someone gave me 2 more, then I would end up with 5 pieces of candy, again. So if 2 + 3 = 5 and 3 + 2 = 5, then does 2 + 3 = 3 + 2?
Teacher : Yes, in fact, for any two integers b and c, b + c = c + b. You can add them in either order, we call this the Commutative Law of Addition. Also, if you are adding more than two numbers, you may add them in any order you like, i.e. if you were adding b + c + d, then you could first add b to c, (b + c) + d, or you could first add c to d, b + (c + d). We call this the Associative Law of Addition (parentheses are used to emphasize the order of the operations performed).
Pupil : The Integers also include the Negative Numbers, right? What happens when you add a positive number to a negative number, or two negative numbers together?
Teacher : Well, let's say we had two positive numbers b and c. Then b + (-c) would be the same as writing b - c, or b minus c. We call this subtraction. Let's return to our candy example for a moment. What if you started off with 5 pieces of candy and then someone took away 2 pieces?
Pupil : If I had 5 pieces and someone took away 2 pieces I would have 3 pieces left. So 5 - 2 = 3, right?
Teacher : Right, however, it is important to know that 5 - 2 is not the same as 2 - 5. To subtract a larger number from a smaller number simply start with the smaller number and count down. Once you hit zero, the next number will be -1, then -2, and so on.
Pupil : So to figure out what 2 - 5 equals I would count down five numbers from 2: 1, 0, -1, -2, -3. So 2 - 5 = -3, which is definitely not equal to 5 - 2 = 3.
Teacher : You certainly seem to understand addition and subtraction. Notice that if you subtracted b units from b you would end up back at zero. For any number b, b - b = 0. Now I have one last question. What do you think would happen if you added or subtracted 0 from a number?
Pupil : Well if you did not add anything to a number, it wouldn't change, so b + 0 = b if b is any integer. The same would be true if you did not subtract anything from a number, so b - 0 = b.
Teacher : Yes, and since adding or subtracting 0 from any number changes nothing, we can drop it out of the computation completely. This leaves us with b = b, for any b, which we call the identity equation. It tells us that any number b is always equal to itself.
1.2 Integer Multiplication Discussion
Pupil : So now that I understand addition and subtraction, are there more operations?
Teacher : Yes there are. The next operation is called multiplication. We write multiplication problems in the form a x b, or a times b. For an example we will consider 5 x 3. What this means is that 5 is being added to itself 3 times (5 + 5 + 5), but a better way to think about is that it means 5 groups of 3 units each. Therefore, to solve 5 x 3 you would count the number of units in each group.
Pupil : I feel a little confused.
Teacher : Alright, we will use an example you can visualize. Picture 5 separate plates, each with 3 quarters on them. How many quarters are there all together?
Pupil : Well, if there are 3 quarters for each of the 5 plates, then there are 15 quarters all together. So 5 x 3 = 15.
Teacher : Exactly. Now, what if we had 3 plates, each with 5 quarters?
Pupil : Then there would be 5 quarters for each of the 3 plates, meaning 15 quarters altogether. So 3 x 5 = 15. Does that mean that multiplication is commutative like addition?
Teacher : Yes, multiplication is commutative. For any a and b, a x b = b x a. It is also associative. If you are multiplying three numbers, then (a x b) x c = a x (b x c). Now, what do you think would happen in the case where a number was multiplied by 0 or 1? If you have trouble just try to visualize what it means.
Pupil : Hmm, if you had any number of plates p, each with 0 quarters, you would have 0 quarters altogether. So any number times 0 equals 0. Now, if you had 1 plate with q quarters, then there would be q quarters altogether. So any number times 1 is just the number itself.
Teacher : That is absolutely right. You thought through that very well.
Pupil : This isn't so hard, but what about negative numbers? I know how to add them, but how do you multiply negative numbers?
Teacher : Well, first I should explain a little more about what a negative number is. Even though the negative numbers are less than zero, this does not mean that they are less than nothing. A negative number is merely a negation of a positive number. For instance, what would happen if you traveled one mile east and then backtracked one mile west?
Pupil : You would end up where you started.
Teacher : Exactly, the one mile you traveled west negated the mile you traveled east. Another way to express your movement west would be to say you traveled -(one mile east), or negative one mile east. So we can think of the negative sign as an operator, just as the multiplication sign is an operator. Its operation would be to replace positive units with negative units.
Pupil : Where exactly is this all leading us?
Teacher : Precisely where we want to be. Remember that a positive unit plus a negative unit always equals zero, therefore, the relation between positive and negative is reciprocal. This means that the negative of a positive is a negative and that the negative of a negative is a positive. Knowing this, we can now move on to multiplication by negative numbers.
Based off the definition I gave you earlier, what do you think 2 x (-3) equals?
Pupil : Well, 2 x (-3) would be 2 groups, each with (-3) units, so that would be (-6) units.
Teacher : Good, that one was tricky, but mostly straightforward. How about this one: What is (-2) x 3?
Pupil : That would be (-2) groups, each with 3 units. How do you count negative groups, though?
Teacher : First, let me rephrase it as negative 2 groups, each with 3 units. Now, a negative group is merely a group that would negate a positive group. What kind of group would negate a group of 3 units?
Pupil : A group of 3 units would be negated by a group of (-3) units, right?
Teacher : Yes, therefore (-2) x 3 = 2 x (-3), which we already know is equal to (-6) units. Now what do you think (-2) x (-3) equals?
Pupil : Well, that would be negative 2 groups, each with 3 negative units. So I need to know what kind of group would negate a group of 3 negative units . . . a group of 3 positive units! So (-2) x (-3) = 2 x 3 = 6. I think that's right.
Teacher : That's exactly right. So what you can see from these few examples is that a negative number times a positive number will give you a negative number and a negative number times a negative number will give you a positive number.
Posts
0 Comments