One matching would be to show how they pair up for that one dance: maybe A with X, B with Z, C with W and D with Y –that is Matching 11, as shown below. Each boy needs to be paired up with one girl for a dance, so there will be four couples dancing for one dance. For instance, think of a group of 4 boys–A, B, C and D and 4 girls–W, X, Y and Z. Everyone from each set gets exactly one partner from the other set to dance with, and everyone gets to dance. To show all of the 1-1 correspondences, or matchings between two sets, I think of all the ways two sets might partner up to do a single dance on the dance floor. MATCHING #1 MATCHING #2ĭraw a one-to-one correspondence between the elements of the sets below. Note I've used different formats than in exercise 5. Let's match up the subset of Circles with the subset of Squares now. That is why it is necessary to have the same number of elements in each set in order to represent a matching. If your first set had 5 girls and your second set had 4 boys, they wouldn't match because one girl would be left without a partner. Continue doing this until all elements are paired up and dancing! This is a matching. Now that they are "dancing" together, take another element from the first set and pick someone in the second set for it to dance with. Think of showing a matching between two sets up by taking one element in the first set and pairing it up to dance with one element in the second set. It might be easier to think of the word pairing instead. NOTE: Two sets that match DO NOT have to match in a natural or pleasing way the way we might match the colors of our clothes together. Make sure each large triangle is "dancing" with one of the small circles. There is more than one possible way to match them up. Using pictures, show a matching between the large triangles and the small circles. We say that the set of Blue A-blocks match the set of Red A-blocks because we can pair each element of the Blue A-blocks with exactly one element of the Red A-blocks. Below is another way to show Matching #1: Pictures may be used to represent a matching. Show a different way to match them under Matching #3 below. We will match each element in the Blue A-blocks with an element in the Red A-blocks. Take the subset of Red A-Blocks and the subset of Blue A-blocks. Take out your A-blocks and divide them into subsets according to color. When people recognized that a collection of four pebbles and a collection of four bananas had a common property, they were learning the concept of matching. Although it took seven symbols to write the numeral two million, note that there are only two different symbols used to write that numeral –a two (2) and a zero (0). Notice, it took one symbol to write the number for three and seven symbols to write the number for two million (we don't count the commas as symbols because they aren't essential to writing the numeral). For instance, in the Hindu-Arabic system, the number three is represented by writing the numeral three which we write like this: 3 and the number two million is represented by writing the numeral two million which we write like this: 2,000,000. In the Hindu-Arabic system we use, the numerals are made up of one or more of these ten symbols –0,1,2,3,4,5,6,7,8 and 9. A numeral is made up of one or more symbols. The symbols that we see and touch that are used to represent numbers are actually numerals. Well, a number is an abstract idea that represents a quantity. The number four, for instance, is used in many ways –four years old, room 4, four feet high, 4 children, four hours, four minutes after eight, mail station 4, 4 fingers, a grade of 4 on a quiz, count to 4, four times bigger than something, 4th place, four wishes, fourth in line, call 444-4444, code 44, etc. One difficulty with the concept of numbers as we use them today is that they are concepts and not necessarily used only for counting objects. Some tribes even have different number names for different types of objects. Although there are still some cultures that have not assigned names for numbers larger than three, most cultures have developed this concept. This was a considerable mathematical advancement. Eventually they used words, which we will call number names, to describe sets of objects (remember sets?!) in terms of their size. Imagine what a breakthrough it was when people first thought to give names to objects. In fact, we take language for granted –but that topic is too involved to go into right now. Most of us in the modern world pretty much take the concept of numbers and counting for granted. What do you remember about learning about numbers or how to count? Have you ever helped to teach a young child how to count? If so, what was it like? How long do you think it takes for someone to learn how to count for the first time?
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