Category: Maths news

Well, it is not uncommon for me to make a mistake with the articles and worksheets published here, but I don’t have a team of proof readers to help me. Far more serious was the error in this years Maths AS Paper.

Over 6000 students took the paper and the question with the highest number of marks was impossible to answer! The question carried 8 marks out of 72 being awarded for the whole paper. This meant that many students would have agonised over the question, wasting many minutes in futilely trying to reach a correct answer.

A spokesperson for the exam board said,

“We very much regret that there was a mistake… and that our quality assurance procedures failed to identify this error.

“Because we have been alerted to this so early, we are able to take this error into account when marking the paper. We will also take it into account when setting the grade boundaries. We have sent a letter to all schools and colleges explaining in more detail what we shall do.

“We do apologise again that this has happened.”

Obviously some of the students think that this is a less than satisfactory outcome as some would have spent much longer than others on the question, leaving less time for the rest of the exam.

Read more

 

We all know that smaller classes, uniforms and primary homework are ways of boosting schools’ performance. Or are they? Not according to a recent report from the Sutton Trust. In fact, reducing class sizes, setting homework during primary school, and introducing school uniforms are among the least effective ways of improving school results.

Looking at class sizes they found that benefits, “are not particularly large or clear, until class size is reduced to under 20 or even below 15”.

Another myth seems to be that hiring more classroom assistants is effective. This is at odds with what most teachers think as 44% said that hiring more teaching assistants was one of their top three priorities. The report says that hiring more teaching assistants  is associated with “very small or no effects on attainment”.

Significant gains in attainment meanwhile come from proven classroom approaches – providing effective feedback on pupil’s performance, encouraging students to think about their own learning strategies, and getting pupils to learn from each other. Implemented correctly, these approaches can increase pupils’ performance by an extra eight or nine months in a school year for a very low cost, according to the guide.

Key findings include:

On effective feedback – “One study even estimates that the impact of rapid feedback on learning is 124 times more cost effective that reducing class sizes.”

On peer tutoring – “Benefits are apparent for both tutor and tutee, though the approach should be used to supplement or enhance normal teaching, rather than replace it.”

On meta-cognitive approaches – “Studies report substantial gains equivalent to moving a class from 50th place in a league table of 100 schools to about 25th.”

On homework – “It is more valuable at secondary school level and much less effective for children of primary school age.”

On teaching assistants – “Most studies have consistently found very small or no effects on attainment.”

On school uniforms – “No robust evidence that introducing a school uniform will improve academic performance.”

On reducing class sizes – “Overall the benefits are not particularly large or clear, until class size is reduced to under 20 or even below 15.”

On one-to-one tuition – “Pupils might improve by about 4 or 5 months during the programme, but costs are high as the support is intensive.”

On ability grouping – “There may be some benefits for higher attaining pupils, but these are largely outweighed by the negative effects on attitudes for middle and lower performing learners.”

The full report can be found at:

The Sutton Trust

 

Should maths lessons be quiet is clearly a tricky one to answer. There should certainly be allowances made for children to discuss the work they are doing, explore ideas, work in small groups on a tricky investigation, but there is a big difference between quietly discussing options and shouting at each other. There are ceertainly times when everyone should be quietly concentrating.

The most recent research also suggests that the acoustics of the room can make it more difficult for children to learn. Researchers have found that 14-16 year olds working in a noisy environment performed at a similar level as 11-13 year olds working in a quiet classroom.

Dr Daniel Connolly of the Institute of Education said the study shows that poor noise levels are not good for learning.

“In a lot of tasks, higher noise levels take the older age group back to the same level as the younger age group. There will always be a certain amount of noise in a school, but if you put students in an acoustically poor room it will amplify that,” he said.

When you went to your child’s classroom was it noisy, even with nobody talking? I remember working in one classroom where the chairs made a horrendous scraping noise every time someone moved. It certainly was disturbing and onlty solved with fitting carpets. Noise from outside can also be disturbing – with some classrooms right next to the school playing fields, or schools under the Heathrow flightpath!

See Independent for more.

It might sound like their worst nightmare for many teenagers, but Michael Grove has just announced that students who do not achieve a good GCSE pass in maths will be forced into continuing to study the subject until they do, or at least until they are 18. A good GCSE pass is considered to be Grade C and last summer 45% of 16 year olds failed to reach this standard. Also the CBI have stated that more than one third of businesses are unhappy at the level of numeracy shown by college leavers.

Following a review by Alison Wolf the government intends to bring in this change from 2015, so it will directly effect those children who at present are in the Upper Primary stage. Could the thought of an extra two years studying maths prove to be an incentive to study harder now? Possibly, but children’s long term planning has generally always been poor and I am not sure I would like to be the teacher taking the maths for the extra two years!

This is the final article in our series on what maths children should know, although I will be returning to this theme again and again in the future.
Much of the work carried out in year 5 eg addition and subtraction will need to be reinforced and is not covered in these targets for year 6. Please note that this is only a summary of the key areas of maths to be covered in the year. Some of the work here will be too difficult for many year 6 children: it is very much a set of targets published by the Primary Framework rather than a realistic review of what children can actually achieve. It does give you a good idea of what teachers are hoping to achieve with their children by the end of year 6.
I believe that if a child can successfully meet these challenges then they have a very bright future in maths ahead of them.

Counting and understanding number:

By the end of year 6 children should
• be able to add or subtract using negative numbers in contexts such as temperature.
(eg The temperature is -7 degrees. It rises by 3 degrees. What is the new temperature?)

• understand decimals up to thousandths and partition numbers.
(eg 4.567 = 4 units + 5 tenths + 6 hundredths + 7 thousandths.)

• round decimals.
(eg round 4.56 to the nearest tenth.)

• order decimals up to 3 decimal places and place them on a number line.
(eg order these numbers: 0.7, 0.77, 0.707.)

• understand and use improper fractions.
(eg recognise that 8 pieces of a 5-piece chocolate cake is 8/5 or 1 and 3/5.)

• simplify fractions by cancelling.
(eg 6/10 = 3/5.)

• write one quantity as a percentage of another.
(eg write £30 as a percentage of £300.)

• find equivalence between percentages, decimals and fractions.
(eg 1/10 = 10% = 0.1.)

• solve simple problems involving ratio and proportion.
(eg Sarah has 15 sweets. She gives Tasha 1 sweet for every 4 she keeps herself. How many sweets does Tasha get?)

Knowing and using number facts:

By the end of year 6 children should
• use knowledge of tables to multiply decimals.
(eg multiply 0.6 by 8.)

• use knowledge of tables to divide decimals.
(eg divide 4.8 by 8.)

• use knowledge of tables to work out square numbers up to 12 x 12.
(eg the square of 11?)

• use knowledge of tables to work out the squares of multiples of 10.
(eg the square of 30?)

• understand that prime numbers only have two factors and identify prime numbers up to 100.
(eg what is the next prime number after 17?)

• find the prime factors of 2-digit numbers.)
(eg find the prime factors of 12.)

• use various methods to check answers to calculations, including rules of divisibility.)
(eg if the sum of the digits of a number is divisible by 3 then so is the number.)

Calculating:

By the end of year 6 children should
• mentally add decimals.
(eg 4.5 + 3.2)

• mentally subtract decimals.
(eg 5.4 – 3.8)

• mentally multiply decimals.
(eg 3.6 x 5)

• mentally divide decimals.
(eg 4.5 divided by 9.)

• use efficient (standard) methods to add numbers, including decimals.
(eg £6.78 + £9.85)

• use efficient (standard) methods to subtract numbers, including decimals.
(eg £9.78 – £5.85)

• use efficient (standard) methods to multiply numbers, including decimals.
(eg 345 x 67)

• use efficient (standard) methods to divide numbers, including decimals.
(eg 456 /8)

• relate fractions to multiplication and division.
(eg 8 divided by 2 = ½ of 8 = 8 x ½)

• know the term quotient and state as a fraction or a decimal.
(eg 54 divided by 5 = 10.8 or 10 and 4/5.)

• find fractions of whole numbers.
(eg find 4/5 of 60.)

• find percentages of whole numbers
(eg find 60% of 48.)

• make sensible use of a calculator to solve multi-step problems.

Understanding shape:

By the end of year 6 children should
• classify 2-D and 3-D shapes according to their properties, including recognising parallel and perpendicular sides/edges.
(eg know the diagonals of a square intersect at right angles.)

• become increasingly accurate when drawing or making shapes.

• be able to reflect, translate and rotate shapes through 90 degrees or 180 degrees.
(eg sketch the reflection of a 2-D shape on a grid.)

• use co-ordinates in the first quadrant to draw and complete shapes.
(eg sketch the position of a triangle after it has been translated 2 units down.)

• learn to use a protractor to measure and draw angles.
(eg draw an angle of 45 degrees.)

• calculate the angles in a triangle.
(eg if two angles of a triangle are 60 and 70 degrees, calculate the third angle.)

Measuring:

By the end of year 6 children should
• choose and use to a suitable degree of accuracy metric units of measurement.
(eg measure 450 ml of water in a jug.)

• convert between units using decimals to two places.
(eg change 3750 ml to 3.75 litres.)

• recognise that measurement is approximate and decide on the necessary degree of accuracy when measuring.
(eg measure the length of a tray to the nearest 1 cm.)

• calculate the perimeter and area of rectangles.
(eg know the formula for finding the area of a rectangle.)

• estimate the area of irregular shapes by counting squares.
(eg count any square more than half covered as a whole square and any less than half as zero.)

Handling data:

By the end of year 6 children should
• use the language of chance or likelihood to describe and predict outcomes.
(eg answer the question; what is the probability of throwing a 5 with a 1 to 6 dice?)

• collect, sort, process and present data to solve a problem and interpret the results.
(eg find the most common period of time that goals are scored in the Premier League.)

• construct frequency tables, bar charts, line graphs and pie charts to show results of investigations.
(eg show frequency table for times of goals scored.)

• identify the difference between discrete and continuous data.
(eg draw a line graph in which intermediate values have meaning.)

• use the terms mode, range, median and mean.
( eg know that mode means the most common.)

Using and applying mathematics:

By the end of year 6, when investigating, children should
• choose appropriate operations to solve multi-step problems.

• decide whether to calculate mentally, on paper, or by calculator.

• use tables to show the steps taken.

• record results and check accuracy.

• suggest plans and lines of enquiry.

• review methods used.

• begin to use simple formulae in words and then symbols.

• explain orally, in words or diagrams their reasoning and conclusions.

In a very exciting development, I have just heard that Mangahigh is going to be absolutely free from tomorrow for schools to use. In fact, going to the site, it is free now!! From tomorrow the mangahigh.com games-based resources will be completely free for UK and Irish schools. Previously Mangahigh charged for access but they feel that a free service is more attractive for teachers.

Toby Rowland, CEO and founder of mangahigh said,

‘We know that schools across the UK and Ireland are short on funding, and the situation may get worse before it gets better. That’s why we think it’s the perfect time to launch a completely free maths resource for schools.’

Incorporating more than 45000 maths questions and 11 unique maths games as well as powerful analytic tools it must be well worth a visit.

mangahigh

Much of the work carried out in year 4 eg addition and subtraction is reinforced in year 5. Efficient written methods for all addition, subtraction and multiplication are expected and moving towards this with division.
Please note that this is only a summary of the key areas of maths to be covered in the year.

Counting and understanding number:

By the end of year 5 children should
• read and write larger numbers.
(eg read 3 456 789.)

• count back beyond zero.
(eg count back 7 from 3.)

• count on and back in decimal steps.
(eg count on 6 tenths from 4.7)

• explain what each digit represents in whole numbers.
(eg say what the value of the 7 is in 347 450.)

• explain what each digit represents in decimals up to two places.
(eg say what the value of the 7 is in 34.75)

• partition numbers.
(eg 4.56 = 4 units + 5 tenths + 6 hundredths.)

• round whole numbers.
(eg round 3 501 to the nearest thousand.)

• order decimals.
(eg which is bigger 4.3 or 4.09?)

• write a smaller whole number as a fraction of a larger one.
(eg recognise that 7 out of 10 is 7/10.)

• find equivalent fractions.
(eg 6/10 = 12/20.)

• relate fractions to decimals.
(eg know that 7/10 = 0.7)

• begin to understand percentage.
(eg know that 40% means 40 out of 100 or 40/100.)

• begin to solve problems using ratio and proportion.
(eg a chicken must be cooked for 50 minutes for every kilo. How long must a 3 kilo chicken be cooked?)

Knowing and using number facts:

By the end of year 5 children should
• use their knowledge to add two decimals mentally.
(eg 7.5 + 3.6)

• use their knowledge to subtract two decimals mentally.
(eg 7.5 – 3.4)

• use their knowledge to double or halve two decimals mentally.
(eg double 5.7)

• recall rapidly all tables up to 10 x 10.
(eg 9 x 7.)

• use knowledge of tables to multiply pairs of multiples of 10 or 100.
(eg 90 x 70.)

• use knowledge of tables to derive division facts.
(eg 800 divided by 4.)

• find pairs of factors of a number.
(eg find the factors of 15.)

• find common multiples of two numbers.
(eg find the common multiples of 9 and 12.)

• use knowledge of number to estimate and check answers.
(eg know that 36 x 19 is about 35 x 20.)

Calculating:

By the end of year 5 children should
• respond rapidly to oral and written questions using addition and subtraction.
(eg increase 190 by 24.)

• use efficient (standard) methods for addition, including decimals.
(eg add 3.45 to 1.98)

• use efficient (standard) methods for subtraction, including decimals.
(eg subtract 368 from 4134.)

• extend mental calculations to multiply a 2-digit number by a single digit
(eg 16 x 5.)

• extend mental calculations to multiply a 2-digit number by a 2-digit number
(eg 16 x 25.)

• use understanding of place value to multiply numbers by 10, 100 or 1000.
(eg multiply 4.5 by 100.)

• refine written methods of multiplication to multiply HTU by U, TU by TU, U.t by U.
(eg multiply 24 by 35.)

• refine written methods of division to divide HTU by U.
(eg divide 374 by 5.)

• find fractions using division.
(eg find 1/100 of an amount by dividing by 100.)

• Find percentages of numbers and quantities.
(eg find 15% of £40.)

• Use a calculator to solve problems involving larger numbers, decimals and fractions.
(eg find ¾ of 450g.)

Understanding shape:

By the end of year 5 children should
• be able to recognise and describe the properties of rectangles, triangles, regular polygons and 3-D solids.
(eg recognise that rectangles have four right angles.)

• identify and draw nets of 3-D solids.
(eg find different nets for an open cube.)

• read and plot co-ordinate points.
(eg know the convention (3,2) for describing a point.)

• recognise parallel and perpendicular lines in shapes and use a set square and ruler to draw shapes.

• draw the position of a simple shape after reflection or translation.

• estimate, draw and measure acute angles and obtuse angles using a protractor.
(eg draw accurately an angle of 60 degrees.)

• calculate angles in a straight line.
(eg use knowledge that an angle of 180 degrees is a straight line.)

Measuring:

By the end of year 5 children should
• use standard metric units of length, weight and capacity.

• convert larger units to smaller units using decimals to one place.
(eg change 3.7 kg to 3 700g.)

• correctly take a reading that lies between two numbered divisions on a scale.
• draw and measure lines to the nearest millimetre.
(eg draw a line 3.6 cm long to within one millimetre accuracy.)

• measure and calculate the perimeter of regular and irregular polygons.
(eg measure one side of a regular pentagon and multiply by 5 to calculate the perimeter.)

• use the formula for the area of a rectangle.
(work out the area of a rectangle 6 cm long and 4 cm wide)

• read timetables and use the 24 hour clock.
(convert 7:00 pm to 19:00 h.)

Handling data:

By the end of year 5 children should
• use the language of probability to describe events, including: fair, unfair, likely, unlikely, certain, uncertain, probable, possible, impossible.
(eg “It is likely that I will watch TV tonight.”)

• answer questions by collecting, sorting and organising data and identify further questions to ask.
(eg test the statement: “You are just as likely to throw a 6 as any other number on a dice.”)

• construct frequency tables, pictograms, bar and line graphs.
(eg draw a line graph showing room temperature over time.)

• find and interpret the mode of a set of data.
(eg find the most common item in a list.)

Using and applying mathematics:

By the end of year 5 children should
• solve one-step and two-step problems involving money, measures, time, and numbers by choosing appropriate calculations, including decimals and calculators.
(eg decide what calculations to carry out to reach an answer.)

• solve puzzles and present the solution.
(eg arrange the numbers in a magic square.)

• suggest a line of enquiry and collect the necessary information to find the answer.
(eg describe a short way to work out the area of a rectangle.)

• make a statement and test it with examples.
(eg angles on a straight line add up to 180 degrees.)

Now we have come to what is expected of children aged between 8 and 9: as defined by the English Primary Framework for Mathematics (part of the National Curriculum). In year 4 there is a greater emphasis on written methods, with the standard or efficient method expected for both addition and subtraction. Children are expected to know all their tables and there is a lot more work on decimals.

Counting and understanding number:

By the end of year 4 children should
• read and write whole numbers up to thousands.
(eg write in words 4009.)

• be able to continue number sequences counting on and back in even steps.
(eg count back 60 in tens from 645.)

• partition 4-digit numbers into thousands, hundreds, tens and units.
(eg know that 2547 is 2 000 + 500 + 40 + 7.)

• order and round 4-digit numbers.
(eg round 3456 to the nearest hundred.)

• use negative numbers and place them on a number line.
(eg reading a thermometer below zero.)

• state inequalities using < and > signs.
(eg -4<0)

Continue reading “What maths should children know by the end of Year 4?”

In year 3 one of the major areas of progress is in using written methods for working out harder addition, subtraction, multiplication and division. At first these will not be the shortened ‘standard’ methods which we use, but steps towards them and may well take the form of jottings. However, the emphasis is still on mental calculations and a question should only be done on paper if it can’t be done mentally. To help with mental arithmetic more tables are to be learned by heart including the 6x table.
Remember to look at the year 2 expectations to see the progression.

Counting and understanding number:

By the end of year 3 children should
• be able to read, write and order whole numbers up to at least 1 000.
(eg write in figures: four hundred and nine.)

• be able to order numbers and position numbers on a number line.
(eg which is more 315 or 513?)

• be able to count on and back in single digits.
(eg count on six from 97.)

• be able to count on and back in multiples of 10.
(eg count back in tens from 83 to 23.)

• partition 3-digit numbers into hundreds, tens and units.
(eg know that 547 is 500 + 40 + 7.)

• round 2-digit or 3-digit numbers to the nearest 10 or 100.
(eg round 255 to the nearest 100.)

• give estimates for addition and subtraction questions.
(eg 56 + 43 is about 100.)

• read and write proper fractions.
(eg 3/5.)

• know and understand the meaning of denominator and numerator.
eg know that the denominator in 3/7 is seven.)

• recognise fractions of shapes and compare fractions and then begin to see equivalence between fractions.
(eg 2/3 = 4/6.)

Knowing and using number facts:

By the end of year 3 children should
• know by heart all addition and subtraction facts up to 20.
(eg 8 + 6 = ?)

• derive and recall sums of multiples of 10.
(eg 40 + 50.)

• derive and recall differences of multiples of 10.
(eg 60 – 40.)

• derive and recall number pairs that total 100.
(eg 45 + 55.)

• know 2, 3, 4, 5, 6 and 10 times tables and derive corresponding division facts.
(eg work out 54 divided by 9 by knowing that 9 x 6 = 54.)

• recognise multiples of 2, 5 or 10 up to 1 000.
(eg know that 65 is a multiple of 5.)

• use their knowledge to estimate and check answers.
(eg give an estimate for 96 + 57.)

Calculating:

By the end of year 3 children should
• be able to add combinations of 1-digit and 2-digit numbers mentally.
(eg add 5, 11 and 4.)

• be able to subtract 1-digit numbers from 2-digit numbers mentally.
(eg subtract 8 from 25.)

• use written methods for addition of 2-digit and 3-digit numbers: but not necessarily the standard method.
(eg 86 + 67.)

• use written methods for subtraction of 2-digit and 3-digit numbers: but not necessarily the standard method.
(eg 63 – 38.)

• multiply 1-digit or 2-digit numbers by 10 or 100.
(eg 26 x 10.)

• use informal written methods to multiply 2-digit numbers
(eg 17 x 3).

• use informal written methods to divide 2-digit numbers.
(eg 17 divided by 3.)

• know when to round remainders up or down.
(eg the number of 5 seater taxis needed for 16 people: 3 remainder 1, means 4 taxis are needed.)

• understand that division is the inverse of multiplication and vice versa.
(eg if 7 x 5 = 35 then 35 divided by 7 is 5.)

• Find unit fractions of numbers and quantities.
( eg find 1/5 of 20.)

Understanding shape:

By the end of year 3 children should
• be able to relate 2-D and 3-D shapes to drawings of them.
(eg using cubes make a solid from its picture.)

• describe, draw and make shapes.
(eg put two triangles together to make a quadrilateral.)

• draw the reflection of a shape in a mirror line.
(eg find letters of the alphabet with reflective symmetry.)

• use the 4 compass directions to describe movement on a grid.
(eg use squared paper and a counter to move 2 squares north…etc.)

• read and record the vocabulary of position, direction and movement.
(eg know and use diagonal, forward, backward etc.)

• use a set square to draw and identify right angles.

• compare angles with a right angle, saying whether they are more or less.

• recognise that a straight line is equivalent to two right angles.

Measuring:

By the end of year 3 children should
• know the relationship between kilometres, metres and centimetres.

(eg know there are 1000 metres in a kilometre.)
• know the relationship between kilograms and grams.

• know the relationship between litres and millilitres.

• choose and use appropriate units to estimate and measure.

• read scales to the nearest division and half division.

• measure and draw to a suitable degree of accuracy.

• read the time on a 12 hour digital clock.

• read the time to the nearest 5 minutes on an analogue clock.

• calculate time intervals between events.

Handling data:

By the end of year 3 children should

• answer a question by collecting, organising and interpreting data.

• use tally charts, frequency diagrams, pictograms, and bar charts.

• use Venn diagrams and Carroll diagrams to sort data and objects.

Using and applying mathematics:

By the end of year 3 children should
• solve one step problems involving numbers, money, measures, time.
(eg an egg weighs about 50g. About how much do 6 eggs weigh?)

• solve two step problems involving numbers, money, measures, time.
(There are 15 books on the top shelf and 13 books on the bottom. 6 books are removed. How many are left?)

• solve puzzles using numbers or diagrams.
(eg solve a magic square.)

• investigate by following a line of enquiry, organising information in tables, graphs etc.
(eg find how many biscuits are eaten each day for a week.)

• investigate patterns in numbers and shapes and use these to solve problems.
(eg if you multiply numbers either way round the answer is always the same.)

• describe and explain choices made in deciding how to solve a problem, orally and in writing.
(explain how to go about finding a pair of numbers which has a sum of 7 and a product of 10.)