Q) I want my pupils to look at adding sets of consecutive numbers and a "trick" I learned in teacher training would be great, but I can't remember how to do it.
A) Suppose we wish to add the numbers 1 to 10, inclusive. Write the sequence normally, then back to front, and add the two sequences together: This is 10 X 11 = 110, but as we want to add only a single sequence we divide the result by two, to give 110 V 2 = 55.
We can create a general sequence as follows: 1 + 2 + 3 + I + n, where n is the number of terms, the first term is 1, the second 2 and so on. So the nth term is n, and in this case n is also the last number in the sequence.
The addition is now written as: This is written as 1 + 2 + I + n = n(n2+ 1) But what about a sequence that does not begin at 1, such as the following?
In this case, 5 x 28 = 140; 140 V 2 = 70.
To create the general rule for this sequence, let the first term be called a and the number of terms be n. The second term is therefore (a + 1), the third is (a + 2), and so on. The last term will be (a + n - 1).
This gives a more general formula for the sum of a sequence of consecutive numbers: a + (a + 1) + (a + 2) + I + (a + (n - 1)n(2a2+ n - 1) The initial formula given above can be found by substituting a = 1 into this formula, giving n(2 x 12+ n - 1n(2 +2n - 1n(n2-1) as found in the first part of this reply.
Q) Please can you explain how a Carroll diagram could be used for questions involving probability?
A) Carroll diagrams are used with categorical data, so they can be helpful in solving probability questions. I have created a school scenario for Year 11. The students are given a slip of paper and asked to write their strongest subject for entering a quiz. They are allowed only one choice.
The data are to be used to randomly select the strongest Year 11 quiz team to represent the school. There is to be a mixed gender team of five (three girlsboys and two boysgirls). The data are collected and put in a chart: The totals to the right show how many selected each category - eg 26 (12 boys 14 girls) chose sport. The column totals show the gender distribution: 48 boys64 girls.
To select the strongest team, we look at the probabilities that each category reflects the best topic, so the probabilities for each category by gender can be calculated. Ten of the 48 boys indicated general knowledge as their best topic. So when a boy is selected, the probability that his best topic is general knowledge is 0.21 (1048 = 0.21 (correct to two decimal places). For the girls, the probability that this is their best topic is 0.14 (964 = 0.14). Calculations are then made in this way for each of the topics: To select the team, each row is examined and the gender with the highest probability is chosen - general knowledge: boys; geography: boys; science: girls; entertainment: girls; sports: boys. The best probable team will be composed of three boys and two girls, who should be randomly selected from the year group.
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