Which principle of counting is that the number of a set is the number given to the last number counted?
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Adapted from Young Children's Mathematics: Cognitively Guided Instruction in Early Childhood Education By: Thomas P. Carpenter, Megan L. Franke, Nicholas C. Johnson, Angela Chan Turrou, and Anita A. Wager Capturing a child’s understanding of the cardinal principle while they are counting can be challenging, as children don't necessarily end the process of counting by explicitly stating the total amount that they have in their collection. A child may know that counting objects involves reciting a sequence of numbers, but not that the outcome of this process is a number that represents the total quantity. A child may say “1,2,3,4” as they count a collection of four, but this does not necessarily mean that the child understands that there is a quantity of four objects. Applying the cardinal principle requires that children name the set according to the last number used in their count. In this case, that last number used was four, so there are four objects in the collection. Because the process of counting and what the count tells you are not necessarily the same thing, figuring out what a child knows about the cardinal principle often requires waiting for a child to complete their count and then asking a question like, “So, how many do you have in your collection?” Other ways to get at the cardinal principle could include saying to the child: “Here are some blocks. How many are there?” Or “Do you have enough to give me 4?” Asking children to make a group of counters of a given size rather than counting a given collection also can focus them on the cardinal principle. Watch this video of Gracie as she counts 31 pennies. What do you notice about her counting? message
Supporting children to make sense of the cardinal principle occurs as teachers follow up on children’s counting by asking how many they have in their collection. You can provide additional support by checking the quantity with the student. When you ask how many, and the student is not sure, you can say: “Let’s see, are there 4? Let’s check together.” You can also support cardinality when you gesture over the entire collection while restating the final number, indicating that the final number used tells the amount in the collection, or when you work with small collections, where students can easily count and see the quantity. To Learn More about Young Children's Mathematics: Cognitively Guided Instruction in Early Childhood Education and to download a sample chapter, click here. Topics: Video, Education, Megan Franke, Thomas Carpenter, Young Children's Mathematics, Angela Chan Turrou, Anita Wager, Children’s Mathematics, Elementary, Kindergarten, Math, Megan Loef Franke, Nicholas Johnson Fundamental TheoremsArithmeticEvery integer greater than one is either prime or can be expressed as an unique product of prime numbers AlgebraEvery polynomial function on one variable of degree n > 0 has at least one real or complex zero. Linear ProgrammingIf there is a solution to a linear programming problem, then it will occur at a corner point or on a boundary between two or more corner points Fundamental Counting PrincipleIn a sequence of events, the total possible number of ways all events can performed is the product of the possible number of ways each individual event can be performed. The Bluman text calls this multiplication principle 2. FactorialsIf n is a positive integer, then n! = n (n-1) (n-2) ... (3)(2)(1) n! = n (n-1)!A special case is 0! 0! = 1PermutationsA permutation is an arrangement of objects without repetition where order is important. Permutations using all the objectsA permutation of n objects, arranged into one group of size n, without repetition, and order being important is: nPn = P(n,n) = n!Example: Find all permutations of the letters "ABC" ABC ACB BAC BCA CAB CBAPermutations of some of the objectsA permutation of n objects, arranged in groups of size r, without repetition, and order being important is: nPr = P(n,r) = n! / (n-r)!Example: Find all two-letter permutations of the letters "ABC" Shortcut formula for finding a permutationAssuming that you start a n and count down to 1 in your factorials ... P(n,r) = first r factors of n factorial Distinguishable PermutationsSometimes letters are repeated and all of the permutations aren't distinguishable from each other. Example: Find all permutations of the letters "BOB" To help you distinguish, I'll write the second "B" as "b" BOb BbO OBb ObB bBO bOBIf you just write "B" as "B", however ... BOB BBO OBB OBB BBO BBOThere are really only three distinguishable permutations here. BOB BBO OBBIf a word has N letters, k of which are unique, and you let n (n1, n2, n3, ..., nk) be the frequency of each of the k letters, then the total number of distinguishable permutations is given by: Consider the word "STATISTICS": Here are the frequency of each letter: S=3, T=3, A=1, I=2, C=1, there are 10 letters total 10! 10*9*8*7*6*5*4*3*2*1 Permutations = -------------- = -------------------- = 50400 3! 3! 1! 2! 1! 6 * 6 * 1 * 2 * 1You can find distinguishable permutations using the TI-82. CombinationsA combination is an arrangement of objects without repetition where order is not important. Note: The difference between a permutation and a combination is not whether there is repetition or not -- there must not be repetition with either, and if there is repetition, you can not use the formulas for permutations or combinations. The only difference in the definition of a permutation and a combination is whether order is important. A combination of n objects, arranged in groups of size r, without repetition, and order being important is: nCr = C(n,r) = n! / ( (n-r)! * r! )Another way to write a combination of n things, r at a time is using the binomial notation: Example: Find all two-letter combinations of the letters "ABC" AB = BA AC = CA BC = CBThere are only three two-letter combinations. Shortcut formula for finding a combinationAssuming that you start a n and count down to 1 in your factorials ... C(n,r) = first r factors of n factorial divided by the last r factors of n factorial Pascal's TriangleCombinations are used in the binomial expansion theorem from algebra to give the coefficients of the expansion (a+b)^n. They also form a pattern known as Pascal's Triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1Each element in the table is the sum of the two elements directly above it. Each element is also a combination. The n value is the number of the row (start counting at zero) and the r value is the element in the row (start counting at zero). That would make the 20 in the next to last row C(6,3) -- it's in the row #6 (7th row) and position #3 (4th element). SymmetryPascal's Triangle illustrates the symmetric nature of a combination. C(n,r) = C(n,n-r) Example: C(10,4) = C(10,6) or C(100,99) = C(100,1) Shortcut formula for finding a combinationSince combinations are symmetric, if n-r is smaller than r, then switch the combination to its alternative form and then use the shortcut given above. C(n,r) = first r factors of n factorial divided by the last r factors of n factorial TI-82You can use the TI-82 graphing calculator to find factorials, permutations, and combinations. Tree Diagrams The first event appears on the left, and then each sequential event is represented as branches off of the first event. The tree diagram to the right would show the possible ways of flipping two coins. The final outcomes are obtained by following each branch to its conclusion: They are from top to bottom: HH HT TH TTTable of Contents What are the principles of counting?The first three principles—stable order, one-to-one correspondence, and cardinality—are considered the “HOW” of counting. Research is clear that these are essential for building a strong and effective counting foundation. The remaining two principles—abstraction and order irrelevance—are the “WHAT” of counting.
Which number principle states that the last number named in a set is the number of items in the set?The cardinal principle This principle says that, on condition that the one-one and stable-order principles have been followed, the number name allocated to the final object in a collection represents the number of items in that collection.
What is the last number in counting?There is no ending. because numbers are infinite.
What is the addition principle of counting?In combinatorics, the addition principle or rule of sum is a basic counting principle. Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways of doing another thing and we can not do both at the same time, then there are A + B ways to choose one of the actions.
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