# Mathematics

Piagetian Level
Schemata 2A, 2A/B: Early/Mid Concrete Operations 2B: Mature Concrete Operations 2B*: Concrete Generalisation 3: Formal Operations
Properties of Numbers
- ordination
Add and subtract small number of objects, solve whole number problems, order and count to 100. Work out sequence backwards or forwards. Mentally add/subtract two single-digit numbers, the decades. Know where 1,000 comes in number sequence. Can give
change from £1 coin. Use addition and subtraction to find differences between numbers up to 1,000; can handle units of 10s and 100s.
Understand use of negative numbers in practical context. Mental addition and subtraction of two-digit numbers, or on paper any addition or subtraction of whole numbers. Enter money in f./p on calculator. Use negative numbers in multiplication. Make estimates in calculations.
Properties of Numbers
- cardination
Read/write to 100, then to 1,000. Count accurately 100 objects. Read/write multiples of 100 up to 10,000. Adding zero does not change value. Whole number approximation of numbers with one decimal place and interpolate 10ths. Read or enter any number to 1 million. Read scale marked in 100ths, labelled in 10ths.
Multiplication and Division Can double numbers up to 10. Practically knows half a set of objects. Identify halves and quarters. Know and can use multiplication tables. Solve problems using multiplication and division. Use multiples 2, 5, 10. Practical quartering. Two inverses. wo-digit x one digit.
Multiplication commutative.
0.5 = 50% =1.
Identify fractions, percentages.
Mapping fractions onto 0-100 scale. Equivalence of fractions and decimals. Scaling as operation preserving multiplicative relations.
Expressions and Equations Devise repeating patterns. Recognise use of symbol. (Can solve 3 + G = 4 for G.) Can generate and continue a number pattern. Without enumerating intersection, can see problem of dividing pupils into those with pets and those with blue eyes. Provide many values of G & E which fit G + E = 9. Use formulae to express words. Sets leading to bracketing. Can find member in each area of intersecting Venn. Make general statement about series (e.g. 1, 2, 4, 8 ... 'This doubles each time"). Can evaluate simple expression, e.g. wage = hours x rate/hour. Solve one-step linear equations, e.g. C x 32 = 192.
Functions Sort or order a set of objects using one criterion. Sort on two criteria, two-stage ordering, use of 2 x 2 sorting table. Three or more successive stages of ordering. 'Function machines, interrogate and interpret data in computer database. Linear functions as generalised number. Express a simple
function symbolically.
Shape and Dimensions Identify 2D shapes. Estimate a metre by eye. Identify common 3D shapes. Construct rectangle given dimensions. Sort 2D shapes in different ways. Read scales in units and 10ths. Meaning of five major kinds of triangles. Order and estimate angles, know area units, understand volume. Construct triangles, angles in triangle sum 180°. Areas of more complex figures. Use of 271 relationship in circles.