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Year 6 Block E – Securing number facts, relationships and calculating

Unit 1

Learning overview

In this learning overview are suggested assessment opportunities linked to the assessment focuses within the Assessing Pupils' Progress: Assessment guidelines . As you plan your teaching for this unit, draw on these suggestions and on alternative methods to help you to gather evidence of attainment, or to identify barriers to progress, that will inform your planning to meet the needs of particular groups of children. When you make a periodic assessment of children’s learning, this accumulating evidence will help you to determine the level at which they are working.

To gather evidence related to the three Ma1 assessment focuses (problem solving, reasoning and communicating), it is important to give children space and time to develop their own approaches and strategies throughout the mathematics curriculum, as well as through the application of skills across the curriculum.

In this unit the illustrated assessment focuses are:

  • Ma1, Communicating
  • Ma2, Written and calculator methods
  • Ma2, Solving numerical problems.

Children recall multiplication and division facts and use these to derive related facts involving decimals, such as 8 × 0.9 or 3 ÷ 0.6. They count on and back, for example in steps of 0.3, relating the steps to the 3 times-table. They use their knowledge of number facts, relationships between numbers and relationships between operations to solve problems and puzzles such as:

Find two numbers with a product of 899.
Solve 3.2 ÷ y = 0.4.
Using all the digits 2, 4, 5 and 8, place one in each box in the calculation ☐ ☐ ☐ ÷ ☐ to make the smallest possible answer.
Write in the missing number: 32.45 × ☐ = 253.11

Children use efficient written methods to add, subtract, multiply and divide integers and decimal numbers. They calculate the answer to HTU ÷ U or U.t ÷ U to one or two decimal places, and solve problems such as:

Find the total length of three pieces of wood with lengths 167 cm, 2.8 m and 1008 mm.
Find 78% of 14.8 m.
A tree trunk is 6.5 metres long. Frank cuts the tree trunk into four equal lengths. How long is each length?

Children choose methods to solve these problems efficiently, and consider the accuracy of the answer in the context of the problem.

Assessment focus: Ma2, Written and calculator methods

As they solve problems, look for evidence of the calculation methods children choose to use. Look out for children who use multiplication facts up to 10 × 10 and place value within their written methods of multiplication and division. Look for children who are beginning to use written methods to multiply or divide decimals by a single-digit number. Look for the ways in which children choose to calculate with fractions. Look at the examples for which they choose to use a written method, and other examples for which they use a calculator. Look for evidence of children recording the calculations they perform with a calculator and how they check their accuracy.

Children tabulate information, working systematically, to help them to solve problems and explain their conclusions. For example, they explore a problem such as:

In a village where all the roads are straight, every time two streets intersect a street lamp is required. Investigate the number of street lamps required for 2 streets, 3 streets, 4 streets, …
What is the minimum and maximum number of lamps needed for 5 streets? n streets?

They explain their methods and reasoning, using symbols where appropriate.

Assessment focus: Ma1, Communicating

As they investigate situations, look for evidence of children recording results systematically, to help reveal patterns and gain insights into the situation. Look out for children considering how to record individual results to check more easily for repeats. For example, if children are finding all of the different solid cuboids that can be built with 72 linking cubes, look for those who list the dimensions of individual cuboids in size order, so that 2 × 4 × 9 and 4 × 2 × 9 are not listed as different results. Look for children who review results and put them into order to check for omissions. For example, with the cuboids, look for children who record their results beginning with 1 × 1 × 72, 1 × 2 × 36 and 1 × 3 × 24. Notice those children who look for ways to record systematically from the outset.

Children express a quotient as a fraction, for example 19 ÷ 8 = 2 3/8 or 3 ÷ 4 = 3/4, simplifying the fraction where appropriate. They solve problems, giving their answers as a fraction, for example:

Share 9 pizzas equally between 4 people.
Divide a 28 m length of wood into 6 equal pieces.

Children express a larger whole number as a fraction of a smaller one using practical contexts or diagrams. For example, they compare a bag containing 10 grapes and a bag containing 25 grapes, grouping the 25 grapes into groups of 10 (with a group of 5) to establish that the larger bag contains 2 1/2 times as many grapes as the smaller bag. They simplify fractions by cancelling and use equivalent fractions to compare one fraction with another. For example, they use fraction strips to show that 1/3 lies between 1/4 and 2/5.

ITP 'Fractions'

Children find fractions and simple percentages of amounts, identifying the appropriate steps towards finding the answer. They solve problems involving fractions and percentages, using calculators where appropriate, and identifying and recording the calculations needed. For example:

A class contains 12 boys and 18 girls. What fraction of the class are boys? What percentage of the class are girls?
25% of the apples in a basket are red. The rest are green. There are 21 red apples. How many green apples are there?

Children build on their understanding of direct proportion to solve, for example:

This cup holds 40 ml. How many cups can I pour from a 1/2 litre bottle?

They represent this problem as 40 ml × ☐ = 500 ml.

They scale numbers up or down by converting recipes for, say, 6 people to recipes for 2 people:

In a recipe for 6 people you need 120 g flour and 270 ml of milk. How much of each ingredient does a recipe for 2 people require?

Assessment focus: Ma2, Solving numerical problems

Look for evidence of children solving problems with and without a calculator. Look for children interpreting the problem, deciding the information that is relevant and the calculations that are needed. Look for evidence of children checking how reasonable their results are by referring to the context or the size of the numbers. Look out for those children who check calculations, for example by repeating the calculation with a calculator or by using inverses. Look for children who estimate, using approximations to check results are reasonable.

Objectives Children's learning outcomes are emphasised Assessment for learning

Tabulate systematically the information in a problem or puzzle; identify and record the steps or calculations needed to solve it, using symbols where appropriate; interpret solutions in the original context and check their accuracy

I can record the calculations needed to solve a problem and check that my working is correct

What could you draw to help you solve this?
Does your answer make sense?
How do you know you have identified the maximum number of intersections for 5 streets?
Explain how making a table could help you to solve this problem.

Parveen has the same number of 20p and 50p coins. She has £7.00. How many of each coin does she have?

Explain reasoning and conclusions, using words, symbols or diagrams as appropriate

I can talk about how I solve problems

[Give children a completed table, e.g. for the number of handshakes made between a given number of people.]
What does this table represent? How would you explain this table to other children?

Solve multi-step problems, and problems involving fractions, decimals and percentages; choose and use appropriate calculation strategies at each stage, including calculator use

I can work out problems involving fractions, decimals and percentages using a range of methods

Find another way of expressing:
175% 331/3% 11/4
Explain how you would solve these problems. Would you use a calculator? Why or why not?

185 people go to the school concert.
They pay £1.35 each.
How much ticket money is collected?
Programmes cost 15p each.
Selling programmes raises £12.30.
How many programmes are sold?

Use knowledge of place value and multiplication facts to 10 × 10 to derive related multiplication and division facts involving decimals (e.g. 0.8 × 7, 4.8 ÷ 6) (end of year objectives)

I can use place value and my tables to work out multiplication and division facts for decimals

What multiplication table does this image represent? How do you know? What other numbers will you see in the boxes outside?

ITP 'Number dials'

Use efficient written methods to add and subtract integers and decimals, to multiply and divide integers and decimals by a one-digit integer, and to multiply two-digit and three-digit integers by a two-digit integer

I can use efficient written methods to add, subtract, multiply and divide whole numbers and decimals

What do you expect the mean length to be? Why?
Make up an example of a calculation involving decimals that you would do in your head, and one that you would do on paper.
Write in the missing digit. The answer does not have a remainder.

Example mathematics

Use a calculator to solve problems involving multi-step calculations

I can, when needed, use a calculator to solve problems

Here is a set of instructions on cards for using a calculator to solve a problem. Put the cards in the correct order.
What is the answer to the problem? Is it a sensible answer?
Write in the missing number: 50 ÷ ☐ = 2.5

Express a larger whole number as a fraction of a smaller one (e.g. recognise that 8 slices of a 5-slice pizza represents 8/5 or 1 3/5 pizzas); simplify fractions by cancelling common factors; order a set of fractions by converting them to fractions with a common denominator

I can write a large whole number as a fraction of a smaller one, simplify fractions and put them in order of size

What clues did you look for to cancel these fractions to their simplest form?
How do you know when you have the simplest form of a fraction?
Karen makes a fraction using two number cards. She says,
'My fraction is equivalent to 1/2. One of the number cards is 6'
What could Karen's fraction be?
Give both possible answers.

Relate fractions to multiplication and division (e.g. 6 ÷ 2 = 1/2 of 6 = 6 × 1/2); express a quotient as a fraction or decimal (e.g. 67 ÷ 5 = 13.4 or 13 2/5); find fractions and percentages of whole-number quantities (e.g. 5/8 of 96, 65% of £260)

I can find fractions and percentages of whole numbers

Harry said: 'To calculate 10% of a quantity you divide it by 10, so to find 20% of a quantity you must divide by 20.' What is wrong with Harry's statement?
Explain how you would solve this problem:

There are 24 coloured cubes in a box. Three quarters of the cubes are red, four of the cubes are blue and the rest are green.
How many green cubes are in the box?
One more blue cube is put into the box. What fraction of the cubes in the box is blue now?

Solve simple problems involving direct proportion by scaling quantities up or down

I can scale up or down to solve problems

Two rulers cost 80 pence. How much do three rulers cost?
Here is a recipe for pasta sauce.

Pasta sauce
300 g tomatoes
120 g onions
75 g mushrooms

Josh makes the pasta sauce using 900 g of tomatoes. What weight of onions should he use? What weight of mushrooms?
A recipe for 3 portions requires 150 g flour and 120 g sugar. Desi's solution to a problem says that for 2 portions he needs 80 g flour and 100 g sugar. What might Desi have done wrong? Work out the correct answer.

Participate in a whole-class debate using the conventions and language of debate, including Standard English

I can take part in a debate

How might we set about solving this problem on percentages? What ideas do you have?
What are the advantages and disadvantages of multiplying the two numbers like this? Could you use a more efficient method?

Resource links to existing published material

Mathematical challenges for able Key Stages 1 and 2
Activities Resources
Activity 55 – Money bags Puzzles and problems for Year 5 and 6
Activity 76 – Slim Jick
Intervention programmes
Objectives for Springboard intervention unit Springboard unit
Identify and use the inverse relationship between multiplication and division Springboard 6, lessons 1–11
Order fractions by converting to a common denominator Springboard 6, lessons 1–11
Supporting children with gaps in their mathematical understanding (Wave 3)
Diagnostic focus Resources
Is not confident in making reasonable estimates for multiplication and division 4 Y6 ×/÷
Wave 3 (4 Y6 ×/÷) Teaching activities to help children make estimates when multiplying or dividing

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