The Number, Addition and Subtraction spine is divided into 31 segments. For each of these segments we have produced a detailed teacher guide, including text and images. The images are also presented as animated PowerPoint slides, which further enhance teacher knowledge and can be used in the classroom (for best results, please view these in ‘Slideshow’ view; for some slides, supporting notes are provided in the ‘Notes’ section). These materials are not, in any way, lesson plans, but they can be used in conjunction with a high-quality mastery textbook to support planning. For some segments, there’s also an introductory video summing up what each segment covers. Click on the ‘+’ signs below to view the materials.
Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Year 6
1.1 Comparison of quantities and measures
Explore the relationship between numbers and introduce children to the important concept of equivalence; focus on the correct use of comparative language, as well as use of mathematical symbols (<, = and >).
1.2 Introducing 'whole' and 'parts': part–part–whole
Introduce children to the concept of partitioning, which underpins many of the subsequent segments, and build towards use of the part–part–whole model.
1.3 Composition of numbers: 0–5
Apply the partitioning structure to the numbers to five, and introduce children to new concepts such as subitising, ordinality and the bar model.
1.4 Composition of numbers: 6–10
Extend the partitioning structure to the numbers six to ten, explore the five-and-a-bit structure of the numbers, and introduce children to the concept of odd and even numbers.
1.5 Additive structures: introduction to aggregation and partitioning
Progress to the use of abstract notation (+, − and =) as a way of representing the part–part–whole structure.
1.6 Additive structures: introduction to augmentation and reduction
Introduce children to addition as augmentation, and subtraction as reduction (take away), using a ‘first…, then…, now…’ story representation and abstract notation (+, − and =); explore the inverse nature of the two operations.
1.7 Addition and subtraction: strategies within 10
Equip children with a range of useful strategies for addition within ten, including adding and subtracting zero and one, commutativity, adding and subtracting two to/from odd and even numbers, and doubling and halving.
1.8 Composition of numbers: multiples of 10 up to 100
Explore multiples of ten, including counting in tens to 100; apply number facts within ten to addition and subtraction for multiples of ten.
1.9 Composition of numbers: 20–100
Build on multiples of ten, by introducing non-zero values in the ones place; apply the partitioning structure to these two-digit numbers, decomposing them into tens and ones.
1.10 Composition of numbers: 11–19
Explore the ten-and-a-bit nature of the numbers 11–19, using the partitioning structure; apply number facts within ten to addition and subtraction of single-digit numbers to/from the numbers 11–19.
Easy-download: Year 1 (All of the teacher guides and representations for segments 1.1–1.10 in one easy-to-download zip file.)
1.11 Addition and subtraction: bridging 10
Apply the aggregation and augmentation structures of addition to three single-digit numbers, exploring commutativity and associativity, to work towards strategies for adding and subtracting across ten.
1.12 Subtraction as difference
Introduce children to subtraction as difference, the third and final subtraction structure; review consecutive numbers, as well as consecutive odd/even numbers, in the context of difference.
1.13 Addition and subtraction: two-digit and single-digit numbers
Build on segments 1.8, 1.9 and 1.10 to equip children with useful strategies for addition and subtraction of a single-digit number to/from two-digit numbers.
1.14 Addition and subtraction: two-digit numbers and multiples of ten
Explore counting on, and back, in ten from any two-digit number; apply number facts within ten to the addition and subtraction of multiples of ten.
1.15 Addition: two-digit and two-digit numbers
Build on segments 1.13 and 1.14 to equip children with useful strategies for addition of two or more two-digit numbers, partitioning two-digit numbers into tens and ones before calculation.
1.16 Subtraction: two-digit and two-digit numbers
Build on segments 1.13 and 1.14 to equip children with useful strategies for subtraction of one two-digit number from another, partitioning two-digit numbers into tens and ones before calculation.
Easy-download: Year 2 (All of the teacher guides and representations for segments 1.11–1.16 in one easy-to-download zip file.)
1.17 Composition and calculation: 100 and bridging 100
Explore the additive and multiplicative composition of 100; draw on known strategies and number facts to calculate across the 100 boundary.
1.18 Composition and calculation: three-digit numbers
Explore the composition of three-digit numbers; use place-value and partitioning knowledge to support additive calculation, and extend known additive strategies to three-digit numbers.
1.19 Securing mental strategies: calculation up to 999
Build on segments 1.15 and 1.16 to equip children with useful calculation strategies for bridging hundreds boundaries, and three-digit numbers; continue to use the partitioning structure to facilitate calculation.
1.20 Algorithms: column addition
Introduce children to the column algorithm for addition calculations, applying the algorithm to a variety of aggregation and augmentation contexts for two-digit and three-digit numbers; explore regrouping (column total is ten or greater) in detail.
1.21 Algorithms: column subtraction
Introduce children to the column algorithm for subtraction calculations, applying the algorithm to a variety of partitioning, reduction and difference contexts for two-digit and three-digit numbers; explore exchange (insufficient quantity to subtract from in a column) in detail.
Easy-download: Year 3 (All of the teacher guides and representations for segments 1.17–1.21 in one easy-to-download zip file.)
1.22 Composition and calculation: 1,000 and four-digit numbers
Explore the composition of 1,000 and four-digit numbers, using the partitioning structure, and make links to measures; introduce children to calculation across thousands boundaries, and extend column algorithms and rounding to four-digit numbers.
1.23 Composition and calculation: tenths
Introduce children to tenths using both the partitioning structure and ideas of place value; apply additive facts and strategies, including column algorithms, and rounding to numbers with tenths.
1.24 Composition and calculation: hundredths and thousandths
Building on segment 1.23, introduce children to hundredths (and thousandths) using both the partitioning structure and ideas of place value; apply additive facts and strategies, including column algorithms, and rounding to numbers with hundredths (and thousandths).
1.25 Addition and subtraction: money
Building on segments 1.23 and 1.24, introduce children to conventions for expressing monetary value and explore the equivalence of 100 p and £1; encourage children to select column algorithms or equivalent calculations where most appropriate.
Easy-download: Year 4 (All of the teacher guides and representations for segments 1.22–1.25 in one easy-to-download zip file.)
1.26 Composition and calculation: multiples of 1,000 up to 1,000,000
Explore the composition of six-digit, whole-thousand numbers, using the partitioning structure; apply knowledge and strategies from segments 1.17 and 1.18 combined with unitising in 1,000s, as well as column methods and rounding.
1.27 Negative numbers: counting, comparing and calculating
Introduce children to negative numbers, making links to everyday contexts; explore addition and subtraction below zero and across zero.
1.28 Common structures and the part–part–whole relationship
Extend the part–part–part–whole structure (three or more parts) to solve missing part/whole problems in a range of contexts; draw on number composition and additive concepts from across the spine, focusing on the structural equivalence of the problems.
1.29 Using equivalence and the compensation property to calculate
Explore the effect on the sum of changing the value of one or both addends; explore the effect on the difference of changing the value of the minuend, the subtrahend or both. Apply knowledge of compensation properties and inverse operations to calculate and balance equations.
Easy-download: Year 5 (All of the teacher guides and representations for segments 1.26–1.29 in one easy-to-download zip file.)
1.30 Composition and calculation: numbers up to 10,000,000
Building on segment 1.26, explore six-digit numbers that are not whole thousands, and then extend to seven-digit numbers; apply additive facts and strategies, including column algorithms, and rounding to these numbers.
1.31 Problems with two unknowns
Equip children with strategies for solving problems with two unknowns, including using the bar model to represent relationships between known numbers, and working systematically.
Easy-download: Year 6 (All of the teacher guides and representations for segments 1.30–1.31 in one easy-to-download zip file.)
NCETM and Maths Hubs would like to thank the following for their contribution to the materials:
- Primary mathematics specialists: Martin Adsett, Adrian Cannell, Clare Christie, Suzanne Coxon, Katie Crozier, Jonathan East, Claire Gerrard, Rebeka Goh, Joanna Caisová, Rebecca Holland, Alison Hopper, Viv Lloyd, Sally O'Brien, Emma Patman, Cat Stone, Valda Varadinek-Skelton, Debbie Weible, Andrew Whitehead
- Educational consultant: Dr Alf Coles, University of Bristol
- Editorial and production: Liam Benson, Design and Define; Nicholas Bromley; Jalita Jacobsen; Emily Marchant; Rachel Houghton; Yvette Sturdy, DiGiV8 Ltd.
- Illustrations: Steve Evans; Alphablocks Ltd. (1.1 Comparison of quantities and measures, step 2:2, image 1; 1.3 Composition of numbers: 0–5, step 1:1, image 1; 1.4 Composition of numbers: 6–10, step 1:1, image 1)