Year 6 Block E:Three 3-Week Units

Redbridge Version 201426 of 26 The National Strategies  Primary
Year 6 Block E: Securing number facts, calculating, identifying relationships

Year 6 Block E:Three 3-week units

Securing number facts, calculating, identifying relationships

Objectives / Units
1 / 2 / 3
•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 / 
•Explain reasoning and conclusions, using words, symbols or diagrams as appropriate /  / 
•Solve multi-step problems, and problems involving fractions, decimals and percentages; choose and use appropriate c alculation strategies at each stage, including calculator use / 
•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) / 

Use efficient written methods to add and subtract integers and decimals

/  / 

•Use efficient written methods to multiply and divide integers and decimals; to divide numbers up to 4 digits by a two digit whole number using formal written methods of short division where appropriate, interpreting remainders according to contextUse 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 up to fourtwo-digit and three-digit integers by a two-digit integer

/  / 
•Solve multi-step problems, and problems involving fractions, decimals and percentages; choose and use appropriate calculation strategies at each stage, including calculator use.Use a calculator to solve problems involving multi-step calculations /  /  / 
•Simplify fractions by cancelling common factors; order a set of fractions by converting them to fractions with a common denominator /  / 
• 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 135 pizzas) / 
•Express one quantity as a percentage of another (e.g. express £400 as a percentage of £1000); find equivalent percentages, decimals and fractions Solve problems involving the calculation of percentages. /  /  / 
•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 132/5); find fractions and percentages of whole-number quantities (e.g. 5/8 of 96, 65% of £260); associate a fraction with division and calculate decimal fraction equivalentsRelate 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 1325); find fractions and percentages of whole-number quantities (e.g. 58 of 96, 65% of £260) /  / 
•Solve problems involving ratio and proportionSolve simple problems involving direct proportion by scaling quantities up or down /  /  / 
•Add and subtract fractions with different denominators and mixed numbers using the concept of equivalent fractions;
•mMultiply simple pairs of proper fractions writing the answer in its simplest form (¼ x ½ = 1/8); d
•Divide proper fractions by whole numbers (for example 1/3 divided by 2 = 1/6) / 
 /  / 
• Divide numbers up to 4 digits by a 2 digit whole number using the formal written method of long division and interpret remainders as whole number remainders, fractions or by rounding, as appropriate for the context, / 
• Associate a fraction with division and calculate decimal fraction equivalents (0.375) for a simple fraction (3/8) / 

Key aspects of learning: focus for the block

Enquiry / Problem solving / Reasoning / Creative thinking
Information processing / Evaluation / Self-awareness / Managing feeling
Social skills / Communication / Motivation / Empathy

Vocabulary

problem, solution, calculator, calculate, calculation, jotting, equation, operation, symbol, inverse, answer, method, strategy, explain, predict, reason, reasoning, pattern, relationship

add, subtract, multiply, divide, sum, total, difference, plus, minus, product, quotient, remainder, multiple, common multiple, factor, divisor, divisible by

decimal fraction, decimal place, decimal point, percentage, per cent (%)

fraction, proper fraction, improper fraction, mixed number, numerator, denominator, unit fraction, equivalent, cancel

proportion, ratio, in every, for every, to every

Building on previous learning

Check that children can already:

•solve one-step and two-step problems involving whole numbers and decimals

•use understanding of place value to multiply and divide whole numbers and decimals by 10, 100 or 1000

•use efficient written methods to add and subtract whole numbers and decimals with up to two decimal places, to multiply HTU×U and TU×TU, and to divide TU÷U

•find equivalent fractions

•understand percentage as the number of parts in every 100, and express tenths and hundredths as percentages

•use sequences to scale numbers up or down

•find simple fractions of percentages of quantities.

Year 6 Block E: Securing number facts, calculating, identifying relationships

Extracts from the New National Curriculum

The national curriculum for mathematics aims to ensure that all pupils:
 become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
 reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
 can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
Number – Addition and Subtraction
Pupils should be taught to:
 add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)
 add and subtract numbers mentally with increasingly large numbers
 use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy
 solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why / Notes and guidance (non-statutory)
Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics Appendix 1).
They practise mental calculations with increasingly large numbers to aid fluency (for example, 12 462 – 2300 = 10 162).
Number – Multiplication and Division
Pupils should be taught to:
 identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
 know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
 establish whether a number up to 100 is prime and recall prime numbers up to 19
 multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers
 multiply and divide numbers mentally drawing upon known facts
 divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context
 multiply and divide whole numbers and those involving decimals by 10, 100 and 1000 / Notes and guidance (non-statutory)
Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Mathematics Appendix 1). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.
They use and understand the terms factor, multiple and prime, square and cube numbers.
Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 = 98/4 = 24 r 2 = 24½ = 24.5 ≈ 25).
Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres.
Distributivity can be expressed as a(b + c) = ab + ac.
They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 92 x 10).
Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13 + 24 = 12 + 25; 33 = 5 x ).
Number – fractions, decimals and percentages
Pupils should be taught to:
 use common factors to simplify fractions; use common multiples to express fractions in the same denomination
 compare and order fractions, including fractions > 1
 add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
 multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, ½ x ½ = 1/8]
 divide proper fractions by whole numbers [for example, 1/3 ÷ 2 = 1/6]
 associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8 ]
 identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places

multiply one-digit numbers with up to two decimal places by whole numbers

use written division methods in cases where the answer has up to two decimal places
solve problems which require answers to be rounded to specified degrees of accuracy
recall and use equivalences between simple fractions, decimals and percentages, including in different contexts. / Notes and guidance (non-statutory)
Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. They should start with fractions where the denominator of one fraction is a multiple of the other (for example, ½ + 1/8 = 5/8) and progress to varied and increasingly complex problems.
Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle.
Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if ¼ of a length is 36cm, then the whole length is 36 × 4 = 144cm).
They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.
Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to two decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.
Pupils are introduced to the division of decimal numbers by one-digit whole number, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication.
Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations. This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers.
Ratio and proportion
Pupils should be taught to:
 solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts
 solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison
 solve problems involving similar shapes where the scale factor is known or can be found
 solve problems involving unequal sharing and grouping using knowledge of fractions and multiples. / Notes and guidance (non-statutory)
Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).
Pupils link percentages or 360° to calculating angles of pie charts.
Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.
Pupils solve problems involving unequal quantities, for example, ‘for every egg you need three spoonfuls of flour’, ‘3/5 of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.

Year 6 Block E: Securing number facts, calculating, identifying relationships

Unit 1Unit 1

Objectives Unit 1 / Assessment for Learning
•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
What needs to be added to 3.63 to give 3.13?
What needs to be added to 4.652 to give 4.1?
I multiply a number with three decimal places by a multiple of 10. The answer is approximately 3.21
What was my number and what did I multiply by?
When I divide a number by 1000 the resulting number has the digit 6 in the units and tenths and the other digits are 3 and 2 in the tens and hundreds columns. What could my number have been?
•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)
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?

Learning overview

Contained in this learning overview are suggested assessment opportunities linked to the assessment focuses within the Assessing Pupils’ Progress (APP) guidelines. As you plan your teaching for this unit, draw on both these suggestions and 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 167cm, 2.8m and 1008mm.
Find 78% of 14.8m.
A tree trunk is 6.5metres 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=23/8 or 3÷4=3/4, simplifying the fraction where appropriate. They solve problems, giving their answers as a fraction, for example: