Get serious about series
Q) I am confused. One of my students asked me the difference between a sequence and a series of numbers. I thought one was a list of numbers generated by a rule and the other a sum of a list of numbers generated by a rule. The student said that if this was the case then they didn't understand the explanation on the BBC AS Guru website (www.bbc.co.ukeducationasgurumaths13pure03sequences16sequencesindex.s html).
A) I checked this page and then asked myself how I should explain it to my students. What is the difference between a sequence of numbers and a series of numbers?
I had a look at this web address and can see how you might be confused, as sequences and series are mentioned in the same sentence. A sequence of numbers is exactly that - a collection of numbers which may be finite or infinite.
A sequence is usually written as, say, s1, s2, s3, I ,sn.
It is true that at school level there is usually a rule for generating a sequence. The even numbers are an obvious example: 2, 4, 6, 8 .... But 8, 6, 4, 2 ... is also a sequence. However, in general there does not need to be any rule between the numbers. The numbers might just be some collection of numbers, for example, the prime numbers: 2, 3, 5, 7, 11 ...
A series is a sum of terms, so 2 + 4 + 6 + 8 + ... is an arithmetic series with a difference of 2. Usually, we talk about the sum to n terms of the series. If the terms of the series are a1, a2, a3 , etc, then the sum to
terms is usually written Sn = a1 + a2 + ... + an.
Thus there is an obvious sequence, S1, S2, I of partial sums, associated with a series.
I suppose it is true to say that every series has an associated sequence; however, it is not generally the case that each sequence has a series naturally associated with it.
Q) I teach in a small primary school. There are three classes and this year I am teaching a mixture of Years 5 and 6. Many of these pupils don't have English as a first language. I am trying to collect activities that show how mathematical language is used. Having mixed year groups can often inhibit the older children who are afraid of getting it wrong and being laughed at. Have you any fun suggestions?
A) A focus on shape and space and mental imagery might be helpful for your class, as this combination offers a challenge.
Write the focus words for the activity (square, vertex, base, point, line, side, centre, circle, diameter, perpendicular and vertical) on pieces of card that can then be left in the classroom as a reminder for repeats of the activity.
Pupils need an A4 wipeable whiteboard and a ruler. Ask them to draw a square with sides of about 8cm on their board.
Then ask them to point to named parts of the shape in turn: "Point to a vertex of the square"; "Point to two parallel sides"; and so on. Discuss their replies.
You can ask them reverse questions as well. Point to the base of the square and ask them if they can remember the name. Repeat for the names of the circle.
Now for the fun. Tell them that you have in front of you a diagram that uses the words you have discussed and that you are going to give them a set of instructions to follow. Perhaps you can have a go now.
* Sketch a square with sides of 6cm in the centre of your board.
* Draw a straight vertical line twice as long as the side of the square.
This line cuts the square in half and the lowest point of the line sits on the base of the square.
* In the square, draw a 3cm straight line that is perpendicular to the right-hand side of the square and meets the centre of the square.
* In one of the new squares you have made, draw a circle with a diameter 3cm that touches the base and right-hand side of the original square.
* Mark the centre of the circle.
* Join the right-hand vertex of the square to the highest point on the line with a straight line.
* Next, join the left-hand vertex of the square to the highest point on the line with a straight line.
Show pupils what your drawing looks like and ask if anyone has anything similar. Discuss the stages on the board if only a few have it correct.
This activity can be extended to encourage children to use the language themselves.
Split them into pairs. Give one of each pair a diagram and ask them to describe it to their partner while their partner draws. Extend this so they create their own diagrams.