A lesson and a half;Beat the inspector
Decimal fractions on a cold autumn morning with a middle set of 31 Year 6 children. The new school buildings and wealth of resources belie high levels of social deprivation. The lesson objectives are derived from the national numeracy strategy teaching programme for Year 6. Earlier lessons had been concerned with vulgar fractions (12 as opposed to .5) and some simple percentage equivalence.
There are some quick-fire questions from the teacher: "Give me a mixed fraction between 1 and 2, 2 and 3. Give me a vulgar fraction between 0 and 1, and another, and anotherI" and so on. "How many halves in 1? How many quarters in 112? How many tens in 20, 30 ... 100, 200? How many tenths in one, two? How many hundredths in one, two... a tenth, two-tenths? One divided by 10? Two divided by 10?" And so on.
We are in the mental arithmetic start of the lesson (I await the "half of three quarters", which really did "beat the inspector" when the chief inspector Chris Woodhead was confronted with it on the Radio 4 programme You and Yours).
Children respond eagerly and mistakes are sensitively followed up with explanations proffered by others. The last questions centre on tenths and hundredths and their answers are recorded on the board.
The teacher uses an acetate on the overhead projector.
She points to the numbers across the rows of both tables and reads: "Two hundreds, three tens, four make two hundred and thirty four." Pupils chant with her (they have obviously been here before). A few giggle as someone chants "six hundred and onety three", but this is picked up by the teacher and she writes on the board: five tens ' fifty
six tens ' sixty
The pupils recite each line after it is written: seven tens ' seventy
two tens ' twenty
one ten ' "teen"
We return to the number columns and a few more are added and read successfully. The teacher moves on to explain the decimal point. Pupils are asked what they think it is for. "A full stop", "A punctuation mark", are some of the replies.
* Phase 3
The teacher puts another acetate on the OHP: Pointing to the first question mark, the teacher asks what they think might continue the pattern. Pupil: "Zero"; Teacher: "Why?"; Pupil: "Nine"; Teacher: "Why?". After some discussion the teacher asks: "How many ones in ten?" Pupils: "Ten". Teacher: "How many tens in a hundred?" Pupils: "Ten". Teacher (pointing to the columns): "A hundred divided by ten?" Pupils:
"Ten". Teacher: "Ten divided by ten?" Pupils: "One". Teacher: "One divided by ten?" a few reply: "A tenthI Ah!" Teacher: "A tenth divided by ten?" Pupils: "A hundredth". The headings were discussed and redrawn.
The decimal point is describedas where decimal fractions start and end. Decimal fractions are described as a particular sort of vulgar fraction.
Some more quick-fire questions:
"How many hundredths in one tenth, two tenths, five tenths...?" The teacher writes on the board:110 = 10100, 210 = 20100 saying that two tenths is equivalent to twenty hundredths. Back at the OHP the teacher writes:
"So we can write 310 as .3 and 1510 as 1.5; The top frame shows decimal fractions and the bottom frame shows the same numbers in decimal notation."
Three pupils are invited to write 1710, 39910, and 4100 on each frame. The rest of the children watch with interest and nod their approval when they agree.
* Phase 4
The teacher uses decimal place-value cards to reinforce her explanation and to show how zero is used when the place is empty. (The class is used to whole number place-value cards.) The card "point 01" is shown against the frames. The teacher says: "Point zero one: one hundredth." Pupils chant other hundredth cards when they are held up. By overlaying the cards, inthis case the .2 and .04, the teacher presents: Pupils respond, initially with prompt: "Point two four; two tenths and four hundredths." After a few other examples the teacher returns to: The pupils are then asked how many hundredths. A few respond "twenty hundredths". Their explanation helped others, viz "There are ten hundredths in a tenth so there are twenty hundredths in two tenths." Nods of understanding. "How many hundredths in .4, .5, .6I" presents no problem! And so on to .24: "How many tenths?" "Two." "How many hundredths?" "24".
* Phase 5: the plenary
The teacher and pupils discuss some questions for that night's homework. It provides an opportunity for children to bring together their new understanding andclarify what is expected from them. The homework exercise is summarised below: Write in decimal notation:
210 310 710 4100 6100 9100 310 and 4100
510 and 5100
46100 78100 42100
Write as decimal fractions:
0.2 0.6 0.5
0.03 0.02 0.09
The teacher tells the inspector after the lesson that the task set for homework was originally intended as a class-work task.
The teacher had prepared a different plenary activity, which related to putting a set of cut-out numbers, written in decimal notation, into order. However, she had adjusted the pace of the lesson in order to pick up on misconceptions and to backtrack.
Why was this lesson good?
Because the class made progress:
* They ended the lesson knowing something new.
* Children built on their previous understanding of place value of whole numbers and of fractions.
* The teacher used her own knowledge and understanding to give clear explanations of why and how.
* The resources were used to illustrate the teacher's explanations.
* Correct mathematical words were used to enhance the teacher's explanations.
* Children participated throughout the lesson, suggesting answers and explanations.
* Children were encouraged to use and extend their mathematical reasoning.
* The lesson had a brisk pace, a planned shape and a very clear sense of purpose.
* The teacher modified her original plan in the light of the time she had spent picking up on misconceptions.
* The orderliness and logic of the mathematics was revealed so that decimal fractions were made as accessible as whole numbers.
* Although there was a single focus to the lesson, the different approaches and activities interconnected. The new learning was well and truly attached to what pupils already knew and what they could do; the lesson enabled a cross bracing of the scaffold of learning.
Peter Lacey is deputy director of education for North East Lincolnshire Council and an OFSTED inspector