Learning overview

Children consolidate their counting on and back in steps of 2, 3, 4, 5, 6 and 10. They recognise when the numbers in a counting sequence are odd or even. For example, counting in steps of 4 from 3 will generate odd numbers only, while counting in steps of 3 from 4 the numbers alternate between odd and even. Children count in steps of two-digit numbers such as in 12s from 3, using a 10 and 2 count to generate 3, 13, 15, 25, 27, 37, 39, 49, 51... alternately whispering quietly and shouting aloud the numbers involved.

This offers an opportunity to assess children's understanding of numbers and the number system and their ability to recognise a wide range of sequences by asking questions such as how they know if any number is or is not in a particular sequence. For example:

If I start with 3 and count in 2s, will 21 be in the sequence?

Hayley makes a sequence of numbers. Her rule is: 'Find half the last number then add 10'. Write in the next two numbers in her sequence. 36, 28, 24, ,

Look for children being able to recognise multiples of 2, 5 and 10.

Children solve problems and puzzles involving all four operations. They identify relevant information and select the appropriate operations in order to solve word problems such as:

There are 12 stamps in a sheet. Each stamp costs 28p. I buy a quarter of the sheet. How many stamps is this?

I pour out 180 ml and then 270 ml from a one-litre bottle of squash. How much is left?

This offers an opportunity to assess problem solving and the ability to select the mathematics that needs to be used in a range of classroom activities by asking children what resources they could use to model the problem and what method they would use to solve the problem. It also allows the opportunity to assess the ability to solve numerical problems, including those that have two steps, by asking children how they are interpreting the language of the problem to identify the steps that they need to take.

Children use counting strategies and partitioning to add and subtract combinations of one-digit and two-digit numbers. They add two-digit numbers by partitioning one or both of the numbers. For example, they work out 58 + 74 by partitioning 58 into 50 and 8 then adding 50 and 8 onto 74. Children use a similar strategy for subtraction, for example working out 94 - 58 or 294 - 58 by partitioning 58 and subtracting 50 then 8. They use counting-up strategies where appropriate, as in 124 - 68 where they count up from 68 to 70 to 100 to 124, recording and adding the steps 2, 30 and 24. Children use a number line to note the steps and to explain how they did the calculation. Children also subtract by counting on from the smaller to the larger number in their heads when the difference is small, as in 305 - 297, making notes to support calculation.

This offers an opportunity to assess mental methods and the ability of children to add and subtract two-digit numbers mentally by asking children how they are solving calculations mentally. In particular, look for children being able to make decisions about whether to partition numbers or to count up.

Children develop their use of the empty number line to support their calculations. They begin to record vertically addition and subtraction calculations that cannot be easily done mentally. They partition one of the numbers and add or subtract the units, tens and hundreds separately:

76+47

267-149

Children recognise the relationship between the vertical presentation and the steps on the number line. They begin to use an expanded layout that underpins the standard written method. For example, for 76 + 47 and 83 - 48 children use:

This offers an opportunity to assess the ability to add and subtract three-digit numbers using a written method by asking children to explain their written method. Look for children being able to add and subtract three-digit numbers both with and without bridging.Children round two-digit and three-digit numbers to the nearest 10 and 100 and use this to give approximate answers to addition and subtraction calculations. For example, they recognise that the answer to 247 + 76 will be just less than 250 + 80 or 330, and 183 - 48 is about 180 - 50 or 130. They understand that finding an approximate answer is a useful strategy for checking a calculation.

This offers an opportunity to assess children's understanding of the number system by asking children to demonstrate using a number line how they are making decisions to round up or down, or using examples such as 'Estimate the number marked by the arrow'.

In particular, look for children being able to order numbers up to 1000 and justify their answers through an understanding of place value.Children begin to work systematically, using lists or tables to organise their solutions to problems such as:

• A farmer has cows and chickens on the farm. Altogether the animals have 24 legs. How many cows and chickens could there be on the farm?

Using problems such as these gives the opportunity to assess children's ability to communicate their mathematics, and in particular to show how they are looking to develop organised approaches to recording their work.

Children use their understanding of place value to support multiplication and division involving multiples of 10 to answer questions such as:

• Three pencils cost 90p altogether. How much does each pencil cost?
• Rani picks up seven 50 g weights. How much do these weigh altogether?
• Sam is making cards. Each card takes 20 minutes. He starts at 4:30 and makes four cards. What time does he finish?

Children begin to use partitioning to multiply and divide two-digit numbers. For example, they calculate 24 × 4 by partitioning 24 into 20 and 4 and working out 20 × 4 + 4 × 4, and 96 ÷ 3 by partitioning 96 into 90 and 6 and dividing each part by 3 to get the answer 32. They identify remainders in related calculations such as 95 ÷ 3, and begin to round the remainder up or down when the context demands it. For example, if cars can each transport up to 4 people, they work out that 12 people would need 3 cars but 13 people require 4 cars.

This offers an opportunity to assess written methods. In particular, look for children being able to multiply and divide two-digit numbers by 2, 3, 4 or 5.Children solve problems such as:

• Use three of the digits 2, 3, 4, 5 and 6 to create multiplication calculations (e.g. 34 × 6). What products can you make? What is the largest/smallest product?

They work in pairs or groups, with all children in the group contributing to decisions about the methods they use, whether they will use resources and how they will record their work.

This offers an opportunity to assess children's ability to discuss their mathematical thinking and begin to explain their thinking. In particular, look for children being able to make general statements such as saying where the largest, middle and smallest digits go when making the largest product above.

Resource links to existing published material

Mathematical challenges for able pupils Key Stages 1 and 2

Intervention programmes

Supporting children with gaps in their mathematical understanding (Wave 3)

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