The Progression is structured by differentiating between Knowledge and Strategy: Strategy describes the mental processes children use to estimate and solve problems Knowledge describes what we want children to be able to recall without having to think about it Learning is a cycle, where knowledge provides the foundation for developing strategies, and use of strategies develops new knowledge. Both strategy and knowledge have been split into different areas for assessment and tracking, although these will often be connected and interlinked in learning and teaching: Strategy Knowledge Addition and subtraction Number Sequences and Order Numeral Identification Grouping and Place Value Basic Facts Written Recording Multiplication and division Fractions, decimals and percentages Addition/subtraction and multiplication/division should be taught simultaneously to reinforce understanding of inverse properties. See the page of the Highland Numeracy Blog for diagnostic assessments for each strategy and knowledge area, and for progression trackers at each level for knowledge and strategy. CfE levels link to Maths Recovery and NZ Maths stages in this way: HNP Description of Stages of Thinking Example Maths Recovery NZ Maths Early * Pupils are unable to consistently count a given number of objects because they lack the knowledge of counting sequences and/or the ability to match things one-to-one. I could not collect 7 counters. 1, 2, 4, 5, 10 Emergent Stage 0 Emergent Stage 0 Early ** Pupils can count up to ten but cannot solve simple problems that involve joining and separating sets like 4 I collected 4 counters and collected another 3 counters but N/A One-to-one
6 bundles of ten and 7ones = 67 or one ten and 26 ones or 36 ones. e.g. 330= 2 hundreds and 13 tens or 33 tens or 330 ones. Increase and decrease by tens on decade (within 100) Efficiently Increase and decrease by tens on/ off the decade. e.g. Place 3 ten strips on the table. Add one ten strip, add a 3 dot strip and two more ten strips. How many dots altogether? [ 63] Increase and decrease by tens off the decade e.g. 15 add ten = 25 add two tens = 45 Increase and decrease by 100s on the century to 1,000 e.g. 6 hundred, 7 hundred, etc… Increase and decrease by hundred off the century e.g. 100 square and 2 ten dot strips. Add multiples of 100 squares and keep count, 120, 220, 320… Increase by tens and ones, e.g. 24 add ten = 34 add 3 ones =37 Efficiently increase and decrease by tens and ones e.g. = 57 Children are told what number is under the screen but have to visualise it and then add one ten and three ones. Efficiently increase and decrease by 10s and 100s off the century and off the decade, without seeing any materials. e.g. 32, 142, 262….902 Estimate first and then use counting in 2s, 5s and 10s to work out how many in a collection of concrete materials (e.g. grouping in pairs and tens] points to consider. Estimate first and then use counting in 5s, 10s and 100s to work out how many in a collection of concrete materials. Estimate first then count a large collection by systematically counting in 10s, 20s, 25s, and 100s. Basic Facts Basic Facts Basic Facts Recall quickly 5 plus facts within 10 e.g. 5 + 1, 5 + 2, 5 + 3, 5 + 4 Instantly recall subtraction facts using 5 plus facts, e.g. 9 – 5 = 4, because 9 is 5 plus 4 6 – 5 = 1 because 6 is 5 plus 1 7 – 5 = 2 because 7 is 5 plus 2 Know by heart and recall quickly 5 plus facts within 14 and subtraction facts using 5 plus facts e.g. 5 + 6, 5 + 7, 5 + 8, 5 + 9, 14 -5, 13 – 5, 12 – 5
Record multiplication problems using the terms ‘groups of’, ‘lots of’, ‘sets of’, ‘shared equally’ and ‘remainders’. Begin to use symbols (x, ÷) to represent the operations of multiplication and division. Confidently use the symbols (x, ÷) to represent the operations of multiplication and division. Experiences and outcomes
Solve multiplication problems by using a rounding and compensation method (tidy numbers) e.g. 5 x 19 5 x 20 and then subtract 5 100 – 5 = 95 5 x 21 5 x 20 and then add 5 100 + 5 = 105 Children should be presented with daily opportunities to solve a mix of calculations in the form of equations and word problems where children choose the strategy from * and ** and discuss confidently the most efficient method chosen. Begin to solve multiplication or division problems involving a 3 - digit number by a 1-digit number by a short written algorithm Solve division problems by partitioning using place value. e.g. 186 ÷ 3 = (180 ÷ 3) + (6 ÷ 3) = 60 + 2 = 62 Solve division problems using known addition, subtraction and multiplication facts within the 2, 3, 4, 5 and 10 times table. E.g. How many threes are there in 15? 15 ÷ 3 = 5 5 threes makes 15 How many equal sets of 3 makes 15 15 – 3 – 3 – 3 – 3 – 3 = 0 3 + 3 + 3 + 3 + 3 = 15 Begin to solve division problems by using a rounding and compensation method (tidy numbers) e.g. 248 ÷ 7 How many 7s are there in 280? 7 x 40 = 280 280 – 248 = 42 42 ÷7 = 6 so 40 sevens subtract 7 sevens equals 33 sevens.
Solve division problems by using a multiplication strategy and by using the inverse rule. Children should be presented with daily opportunities to solve a mix of calculations in the form of equations and word problems where children choose the strategy from * and ** and discuss confidently the most efficient method chosen. Begin to split a number to reveal its factors to help solve a multiplication and division problem e.g. 4 x 32 = 2 x 2 x 32 = 2 x 64 = 128 8 x 57 = 2 x 2 x 2 x 57 = 2 x 2 x 114 = 2 x 228 = 456 76 ÷ 4 = 76 ÷ 2 ÷ 2 =38 ÷ 2 =19 168 ÷ 8 = 168 ÷ 2 ÷ 2 ÷ 2 = 84 ÷ 2 ÷ 2 = 42 ÷ 2 = 21 Represent multiplication/ division stories using a variety of ways, including story, pictorial, concrete and abstract. e.g. Tommy arranged his 15 football trophies evenly on three shelves. How many trophies were on each shelf? Fiona distributed 650 flyers for her new restaurant evenly amongst 13 streets. To how many houses did she deliver the pamphlets in each street? Solve a division problem by using multiplication e.g. 72 ÷ 8 = how many 8s are there in 72? Use division for sharing or grouping a collection. Remainders can be left or shared out depending on the context of the story e.g. There were 33 people at the family reunion dinner. The restaurant only had tables that would sit four people. How many tables did the family need to book? Anya has bought 57 eggs from the farmer but they are all in one basket. She wants to pack them safely into egg Children should be presented with daily opportunities to solve a mix Solve a division problem that involves remainders and can express the remainder as a whole number, fraction or decimal depending on the context.