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Mathematics Development in KS2: From Informal to Formal Methods

mathematics, KS2, pedagogical approaches, decimals, fractions, rounding, operations, place value

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Mathematics Development in KS2: From Informal to Formal Methods

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  1. 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

  2. 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.

  3. 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

  4. 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.

  5. 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 Count spaces or lines? What's the difference?

  6. Appreciation of Decimals Year 4 Recogniseand 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

  7. 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

  8. 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?” T U H

  9. 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 T U H Top Marks Moving Digits

  10. Addition & Subtraction Year 5 Practiseadding 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.g9.8 + 3.41 U Physical – Visual – Dual - Formal Written

  11. Addition & Subtraction Year 5 Practiseadding 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.g9.8 - 3.41 U Physical – Visual – Dual - Formal Written

  12. 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 0.2 0.05 1 3 U 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.1 0.1 0.1 0.1 0.1 0.1 1 1 1 Physical – Visual – Dual - Formal Written

  13. 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.g7.86 ÷ 3 U 0.01 0.01 0.01 0.01 0.01 0.01 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1 1 1 1 1 1 1 What about 121 ÷ 4 Physical – Visual – Dual - Formal Written

  14. Informal Operations Year 4 Solve simple measure and money problems Year 5 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!!

  15. 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

  16. 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.

  17. Equivalence between FDP Year 4 Recogniseand 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?

  18. 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  % = Decimals  % 0.15 = 0.05 = 0.015 = 1.5 = %  Decimals 70% = 7% = 0.7% = Fractions  Decimals = FDP National Strategy Starters: Equivalent Fractions and x & ÷ by 100 Allow pupils the chance to explore Interactive FDP and compliments

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

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

  21. 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

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

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