Aims of the Session To build understanding of mathematics and it’s development throughout KS2 To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods) To enhance subject knowledge of the pedagogical approaches to teaching mathematics

What comes first? Fractional Understanding? Equal parts Larger denominators means smaller parts Year 3 recognise that tenths arise from dividing an object into 10 equal parts and in dividing one – digit numbers or quantities by 10.

What comes first? Place Value and scaling? Dienes Place Value Counters Bead bars and beadstrings Money (problem solving) Counting Other ideas? 3 times as tall

The Decimals & Percentages Journey Visual Appreciation Formal Operations Informal Operations Rounding Decimals Equivalence between Fractions, Decimals & Percentages Percentages Try to have consistent approaches across year groups.

Count spaces or lines? What's the difference? Appreciation of Decimals Year 3 Pupils connect tenths to place value, decimal measures - Sort place value first – Make early links Use Dienes Creatively Count in tenths: Prepare for misconceptions when bridging units… U 1 10 Need to use the lines when counting in decimals – so that there are values between the points Count spaces or lines? What's the difference?

Possible misconception Appreciation of Decimals Year 4 Recognise and write decimal equivalents of any number of tenths or hundredths - Count in tenths Count up and down in hundredths and be able to compare (differentiation opportunity) Plan for misconceptions when bridging units & tenths!  Possible misconception http://www.mathsisfun.com/numbers/number-line-zoom.html

Appreciation of Decimals Year 5 And on to thousandths… providing pupils have understood the build up in year 3 and year 4, this will be a natural progression. But, if pupils have had difficulty, differentiation is needed – some might need to be more “hands on” with kinaesthetic resources, others will appreciate visually on a number line. http://www.mathsisfun.com/numbers/number-line-zoom.html

Comparing & Ordering Decimals Year 5 Read, write, order and compare numbers with up to 3 decimal places Just add zeros and make the numbers all the same length???? Good diagnostic question: “Which is bigger: 0.2 or 0.20?” “Which is bigger: 0.47 or 0.470?” U T H 1 10 1 100

Formal Operations Year 4 Multiplication and division by 10, 100 and 1000 (starting with a one or 2 digit no.) Year 6 Multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 dp How many places does the decimal point move?? Sliders - let them see and do it!! Try the following on your sliders - What misconceptions or problems could arise? With Slider: a) 2.3 x 100 b) 1.2 ÷ 10 c) 1.8 ÷ 100 d) 6060 ÷ 1000 Without Slider: e) 0.093 x 100 f) 3 ÷ 100 U T H 1 10 1 100 Top Marks Moving Digits

Addition & Subtraction Year 5 Practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places Why does the column method work?? e.g 9.8 + 3.41 U 1 10 1 100 Physical – Visual – Dual - Formal Written

Addition & Subtraction Year 5 Practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places Why does the column method work?? e.g 9.8 - 3.41 U 1 10 1 100 Physical – Visual – Dual - Formal Written

Multiplication & Division Year 6 Multiply one-digit numbers with up to 2 decimal places by whole numbers Why do the written methods work?? e.g 1.25 x 3 1 0.2 0.05 3 1 10 1 100 U 1 0.1 0.1 0.01 0.01 0.01 0.01 0.01 1 0.1 0.1 0.01 0.01 0.01 0.01 0.01 Column method – multiply the whole numbers and putting decimal point back in - do they realise that it is scaling up and down 1 0.1 0.1 0.01 0.01 0.01 0.01 0.01 Physical – Visual – Dual - Formal Written

3 7 . 8 6 Multiplication & Division Year 6 Use written division methods where the answer has up to 2 decimal places Why does the bus stop method work?? e.g 7.86 ÷ 3 1 10 1 100 U 1 0.1 0.01 3 7 . 8 6 1 0.1 0.01 1 0.1 0.01 1 0.1 0.01 1 0.1 0.01 What about 121 ÷ 4 1 0.1 0.01 1 0.1 0.1 Physical – Visual – Dual - Formal Written

Informal Operations Year 4 Solve simple measure and money problems Mentally add and subtract decimals and whole numbers and decimals. Work with complements of 1 (eg: 0.83+0.17=1). Use real money first! Can we use scales? Progression of examples with money - in what order would you do these?? £7 - £4 £7.20 + £0.60 £4.62 + £3.53 £4.80 - £0.30 £9.42 + £1.36 £8.21 - £3.43 £15.59 - £3.27 £1 + £2 £4 - £2.21 £10 - £3.27 Number bonds to 10 and 100 are key!!

Rounding Decimals This journey should have started in Year 3 and 4 as you build up to be able to round any whole number to the nearest 10, 100, 1000 Round 185 to the nearest 10, 100, 1000 NB – which ten is “closest” to 185

Rounding Decimals Year 4 Round decimals with 1 decimal place to the nearest whole number e.g. 1.7 to the nearest whole number: Year 5 Round decimals with 2 decimal places to the nearest whole number and to 1 dp e.g. 3.81 to the nearest whole number and to 1 d.p. e.g. 0.45 to the nearest whole number and to 1 d.p. (Misconception opportunity?) Extension: Round 9.99 to the nearest whole number and to 1 d.p. Year 6 Solve problems which require answers to be rounded to specified degrees of accuracy. e.g. Two-stage problems – 1st stage = calculation, 2nd stage = round appropriately.

Equivalence between FDP Year 4 Recognise and write decimal equivalents to 1/4, 1/2, 3/4 Year 5 Understand what percent means Know percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 Percent - What does it mean? What can the pupils relate it to in their prior learning? How can we use this to find simple equivalents?

Equivalence between FDP Year 5 Write percentages as fractions Year 6 Associate a fraction with division and calculate fraction equivalents Recall and use equivalences between fractions, decimals and percentages %  Fractions 17% = 78% = 48% = 3.5% = Fractions  % 27 100 = 17 50 = 2 25 = 2 3 = Decimals  % 0.15 = 0.05 = 0.015 = 1.5 = %  Decimals 70% = 7% = 0.7% = Fractions  Decimals 27 100 = 17 50 = 2 25 = 3 8 = FDP National Strategy Interactive FDP and compliments Starters: Equivalent Fractions and x & ÷ by 100 Allow pupils the chance to explore

Percentages of an Amount Year 6 Solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] 50, 75, 25

Percentages of an Amount Year 6 Solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] Based on 10 Percentage strips

Aims of the Session To build understanding of mathematics and it’s development throughout KS2 To have a stronger awareness of when and how to progress from non-formal to formal methods at the appropriate stage for your pupils (and the pitfalls of formal methods) To enhance subject knowledge of the pedagogical approaches to teaching mathematics

Where now? I am going to trial/action... * Where next? Disseminate amongst colleagues