## Equivalent fractions

### Teacher’s notes

- Understands numerator and denominator.
- Understands that both parts of the fraction have to be multiplied/divided by the same number.
- Is aware that there can be more than one equivalent fraction.

### Next steps

- Use knowledge of equivalence to order fractions with different denominators.
- Develop understanding of equivalence between fractions, decimals and percentages.

## Carroll diagrams

### Teacher’s notes

- Understands the vocabulary multiple, factor and square.
- Sorts numbers by comparing them and recognising properties.
- Calculates mentally to generate examples.

### Next steps

- Use sorting conventions for Carroll diagrams more consistently, for example use ‘smaller than 70’/‘not smaller than 70’.
- Search for different relationships and write more challenging labels, for example ‘multiples of 3’, ‘square numbers’, rather than ‘smaller than 70’.

## Squares and circles sequences

### Teacher’s notes

- Continues the sequence using the pattern given.
- Identifies differences between numbers by looking for patterns.
- When prompted, identifies a rule for the sequence using multiplication and addition.
- Uses the inverse operations to work backwards through a sequence, when prompted.

### Next steps

- Use the = sign correctly, in working.
- Use simple algebra to express patterns and rules.
- Evaluate expressions by substituting numbers.
- Find the rule for the nth term.

## What the teacher knows about Polly’s attainment in Ma2

Polly recognises and describes number patterns and sequences. She has a good understanding of multiples, squares and factors and recognises some prime numbers under 100. She continues sequences forwards and backwards and writes simple rules or general statements in words to explain them. Polly clearly understands the effect of multiplying and dividing whole numbers by 10, 100 and 1000.

Polly recognises proportions of a whole and uses simple fractions and percentages to describe them. She knows the place value of the digits within a decimal and how to order decimals with up to three decimal places, for example 0.706, 0.76, 0.67, 0.607. She recognises equivalences between commonly used fractions, decimals and percentages, for example $\frac{2}{5}=0.4=40\mathrm{\%}$; and calculates simple fractions and percentages of given amounts, for example $\frac{3}{8}$ of 48 or 60% of £40. Polly is beginning to understand and use ratio although she does not yet use ratio notation.

Polly understands the relationships between the four operations with whole numbers. She uses inverse operations to find missing numbers. She is less confident, however, when working to two decimal places. Polly understands the use of brackets to indicate the order of operations, and understands the role of = in showing that expressions on each side of the symbol have the same value, for example $36\times 4=150-\mathrm{\square}$.

Polly calculates mentally using all four operations. She recalls multiplication facts up to 10 × 10 quickly and derives corresponding division facts. She uses her knowledge of complements to 100 and place value to calculate complements to 1000, for example $360+\mathrm{\square}=1000$. She uses her knowledge of multiplication facts and place value to calculate with larger numbers, for example 80 × 7. She understands that halving then halving again is the same as finding $\frac{1}{4}$ or dividing by 4.

When solving problems Polly deals with two constraints: for example, she finds a number that is a multiple of two and not a multiple of three. She works through both steps of a two-step problem, for example using multiplication and subtraction to find the change from £10 when buying three comics at £1.35 each. She reads temperatures and understands negative numbers in this context. When using a calculator, Polly appreciates the value of the digits in the display. She interprets the calculator display when solving money problems and knows to record 4.6 as £4.60. Polly checks the reasonableness of her answers and explains why she believes an answer to be correct. She is beginning to use simple formulae expressed in words, although she does not yet use letters and symbols. She plots and interprets coordinates in all four quadrants accurately.

Polly uses efficient methods to add and subtract four-digit numbers and decimals with up to two decimal places. Having previously used the grid method, Polly now multiplies a three-digit by a two-digit number using a standard efficient method. She multiplies decimal numbers by a single digit: for example, to calculate 7.5 × 8, Polly multiplies 75 by 8 and uses her understanding of place value to position the decimal point. She uses short division to divide whole numbers by a single digit.