Transcript Improving Mental Mathematics in Schools
Improving
Mental Mathematics in Schools Rob Perkes Tom Garner
Aims of Session:
To familiarise ourselves with key mental calculation strategies that should be taught throughout the school; To develop an understanding of progression in mental calculation across the school; To learn how mental recall can be developed to facilitate mental calculation; To identify opportunities for and the importance of developing visualisation; To understand how to teach and not just test times tables.
Why is mental calculation so important?
Activity 1:
Discuss on your tables and be prepared to feedback.
What does working mentally in mathematics mean?
What skills and attitudes do children need to be able to work mentally?
What opportunities do children need in order to develop these skills?
Mental calculation is all about patterns and relationships between numbers. Children need to be able to learn how to solve problems by recognising which strategies and known facts to apply.
How and where does mental calculation start?
FS Yr1 Yr2 Yr3 Yr4 Yr5 Yr6
OFSTED – Understanding the score
Too often pupils are expected to remember the methods, rules and facts without grasping the underpinning concepts, making connections with earlier learning and making sense of mathematics so they can use it independently.
Activity 2 – part 1:
On your own have a go at writing down as many different mental calculation methods as you can.
Activity 2 – part 2:
Working with a partner have a go at filling in the grid. Which of the mental calculation methods you have thought of do you think go where?
Year 1 Year 2 Year 3
Progression in mental calculation
Year 4 Year 5 Year 6
Progression in mental calculation
Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
Number bonds to 10. Extend to 20.
Halving.
Doubling.
Identify near doubles.
Counting in steps (1, 2, 5 & 10) backwards and forwards).
Addition facts (number to 10) – early partitioning.
Partitioning (tens and units).
Addition (up to 10+10).
Subtraction (up to 10).
Partition into 5 and a bit when adding 6, 7, 8 or 9.
Bridge through 10/20 when adding a single digit number.
Add 9 to a single digit number by adding 10 and subtracting 1.
Add more than 2 numbers.
Put largest number first to add.
Times tables (2, 5 & 10).
Doubling/halving – multiples of 5/10.
Counting in steps of 3 and 4 (backwards and forwards).
Counting through hundreds numbers.
Partitioning 2-digit numbers (tens and units).
Partition into 5 and a bit when adding 6, 7, 8 or 9.
Extend addition and subtraction to 100.
Bridge through a multiple of 10 when adding a single digit number.
Identify subtraction facts corresponding to addition calculations (inverse).
Add/Subtract 9 and 11 by adding/ subtracting 10 and adjusting by 1.
Add/Subtract 19 and 21 by adding/ subtracting 20 and adjusting by 1.
Multiplying by 10.
Recall pairs of multiples of 10 that total 100.
Round numbers to nearest 10.
Times tables (3, 4 & 6).
Identify near doubles using doubles already known (e.g. 80 + 81).
Counting through hundreds and thousand numbers.
Flexible Partitioning: Choose and use appropriate strategies for a mental calculation (5 and a bit, pairs, add 10 and adjust, largest number first).
Bridge through a multiple of 10 and adjust.
Use patterns of similar calculations.
Complements to 100 any pairs of 2-digit numbers.
Multiplying by 10 and 100.
Inverse operations (derive division facts from multiplication facts).
Multiplication – understanding it can be done in any order.
Division – related to multiplication.
Count up to find small differences.
Rounding (to nearest 10/100).
Use known number facts and place value to add/subtract mentally.
Times tables (7, 8 & 9).
Doubles: all whole numbers to 50, multiples of 10 to 500, multiples of 100 to 5,000 and corresponding halves.
Identify near doubles using known doubles (e.g. 150 + 160).
Counting through thousand numbers (count on and back in steps of 1, 10 and 100).
Flexible partitioning.
Multiplication.
Division.
Add or subtract to the nearest multiple of 10 and adjust.
Pairs of multiples of 50 with a total of 1,000.
Rounding (to nearest 100).
Use known number facts and place value to add/subtract mentally including any pair of 2 digit whole numbers.
Extend tables to 11 and above – use of partitioning, e.g. 13 x 8 = (10 x 8) + (3 x 8).
Use doubling and halving (partition numbers first).
Identify near doubles (e.g. 1.5 + 1.6).
Counting through tens of thousands.
HTU partitioning.
Use closely related facts (e.g. partitioning to multiply).
Add or subtract to the nearest multiple of 10 or 100 and then adjust.
Add several numbers (4 or 5 single digit numbers).
Find differences by counting up through the next multiple of 10, 100 or 1,000.
Calculations to 1 d.p.
Equivalent calculations.
Use factors.
Use known number facts and place value to multiple and divide mentally
Use closely related facts for multiplication, e.g. 49 x 51 – multiply by 50 and adjust.
Approximation (e.g. 6.1 x 7.8).
Prime numbers.
Square numbers of multiples of 10 to 100.
Doubles of 2-digit numbers, including decimals and corresponding halves.
Doubles of multiples of 10 to 1,000 and multiples of 100 to 10,000 and corresponding halves.
Counting in decimals.
Use closely related facts (e.g. partitioning to multiply).
Add or subtract to the nearest multiple of 10 ,100 or 1,000 and then adjust.
Calculation to 2 d.p.
Fractions/ Percentage/Decimal equivalence.
Use factors.
Use known number facts and place value to add, subtract, multiple and divide mentally (including with decimals).
Year 1 Year 2 Year 3 Year 4
Progression in mental calculation
Year 5
Number bonds 20. Use for addition and subtraction Halving.
Doubling.
Identify near doubles.
Counting in steps (1, 2, 5 & 10) forwards and forwards backwards to 100.
Addition/subtraction facts for numbers up to 20 – early partitioning.
Partitioning (tens and units).
Partition into 5 and a bit when adding 6, 7, 8 or 9.
Bridge through 10/20 when adding a single digit number.
Add 9 to a single digit number by adding 10 and subtracting 1.
Add more than 2 numbers.
Put largest number first to add.
Recall addition and subtraction facts to 20 fluently.
Times tables (2, 5 & 10).
Doubling/halving – multiples of 5/10.
Counting in steps of 2, 3 & 5 (forwards and backwards).
Count in steps of 10 forwards and backwards from any given number.
Counting through hundreds numbers.
Partitioning 2-digit numbers (tens and units).
Partition into 5 and a bit when adding 6, 7, 8 or 9.
Extend addition and subtraction to 100.
Derive and use facts up to 100.
Bridge through a multiple of 10 when adding a single digit number.
Identify subtraction facts corresponding to addition calculations (inverse).
Add/Subtract 9 and 11 by adding/ subtracting 10 and adjusting by 1.
Add/Subtract 19 and 21 by adding/ subtracting 20 and adjusting by 1.
Multiplying by 10.
Recall pairs of multiples of 10 that total 100.
Round numbers to nearest 10.
Use commutative law for addition and multiplication.
Times tables (3, 4 & 8 ).
Identify near doubles using doubles already known (e.g. 80 + 81).
Counting through hundreds and thousand numbers.
Addition and subtraction, including HTU & U, HTU & TU, HTU & HTU.
Count in multiples of 4, 8, 50 and 100.
Flexible Partitioning: Choose and use appropriate strategies for a mental calculation (5 and a bit, pairs, add 10 and adjust, largest number first).
Bridge through a multiple of 10 and adjust.
Use patterns of similar calculations.
Complements to 100 - any pairs of 2-digit numbers.
Multiplying by 10 and 100.
Identify and recall 10/100 less/more.
Inverse operations (derive division facts from multiplication facts).
Multiplication – understanding it can be done in any order.
Division – related to multiplication.
Count up to find small differences.
Count up and down in tenths.
Rounding (to nearest 10/100).
Use known number facts and place value to add/subtract mentally.
All times tables (up to 12 x 12).
Count in multiples of 6, 7, 9, 25 and 1,000.
Count backwards through zero including negative numbers.
Use commutative laws.
Doubles: all whole numbers to 50, multiples of 10 to 500, multiples of 100 to 5,000 and corresponding halves.
Identify near doubles using known doubles (e.g. 150 + 160).
Flexible partitioning.
Multiply TU X U numbers.
Division using multiplication facts.
Add or subtract to the nearest multiple of 10 and adjust.
Multiply and divide (including by 0 and 1 and multiplying 3 numbers).
Pairs of multiples of 50 with a total of 1,000.
Count up and down in hundredths.
Round decimals with 1 d.p. to the nearest whole number.
Rounding (to nearest 10, 100 and 1,000 ).
Use known number facts and place value to add/subtract mentally including any pair of
Extend tables beyond 12 x 12 [using partitioning, e.g. 13 x 8 = (10 x 8) + (3 x 8).] Use doubling and halving (partition numbers first).
Identify near doubles (e.g. 1.5 + 1.6).
Counting through tens of thousands.
Count forwards ad backwards in steps of powers of 10 for any number up to 1,000,000.
HTU partitioning.
Use closely related facts (e.g. partitioning to multiply).
Multiply and divide numbers and decimals by 10, 100 and 1,000.
Add or subtract to the nearest multiple of 10 or 100 and then adjust.
Add increasingly large numbers.
Find differences by counting up through the next multiple of 10, 100 or 1,000.
Calculations to 1 d.p.
Equivalent calculations.
Use factors, common factors and multiples.
Recall prime numbers to 19.
Square numbers and cube numbers.
Round decimals with 2 d.p to the nearest whole number and to 1 d.p.
Use known number facts and place value to multiple and divide mentally.
Use closely related facts for multiplication, e.g. 49 x 51 – multiply by 50 and adjust.
Perform mental calculations including mixed operations and large numbers.
Approximation (e.g. 6.1 x 7.8).
Prime numbers to 100, common factors, and common multiples.
Doubles of 2-digit numbers, including decimals and corresponding halves.
Doubles of multiples of 10 to 1,000 and multiples of 100 to 10,000 and corresponding halves.
Counting in decimals.
Use closely related facts (e.g. partitioning to multiply).
Add or subtract to the nearest multiple of 10 ,100 or 1,000 and then adjust.
Calculation to 2 d.p.
Fractions/ Percentage/Decimal equivalence.
Round any number to a required degree of accuracy.
Use factors.
Use known number facts and place value to add, subtract, multiple and divide mentally (including with decimals).
Models and Images
‘Teaching Children to Calculate Mentally’ publication.
Activity 3:
Have a look through the booklet. What is its main focus? What do you notice?
Teaching children to calculate mentally
• Addition and subtraction p4-7 • Multiplication and division p8-11 • Addition and subtraction strategies p26-50 • Multiplication and division strategies p51-71
Ensuring mental and oral opportunities are planned across a range of mathematics:
The 7 strands within the framework:
- Using and applying mathematics; - Counting and understanding number; - Knowing and using number facts; - Calculating; - Understanding shape; - Measuring; - Handling Data.
Transum
The 6 Rs of Oral and Mental Work:
Rehearse Recall Refresh Refine Read Reason
Rehearse
To practise and consolidate existing skills, usually mental calculation skills, set in a context to involve children in problem-solving through the use and application of these skills, use of vocabulary and language of number, properties of shapes or describing and reasoning.
Recall To secure knowledge of facts, usually number facts, build up speed and accuracy, recall quickly names and properties of shape, units of measure or types of charts, graphs to represent data.
Refresh
To draw on and revisit previous learning; to assess, review and strengthen children’s previously acquired knowledge and skills relevant to later learning; return to aspects of mathematics with which the children have had difficulty; draw out key points from learning.
Refine
To sharpen methods and procedures; explain strategies and solutions; extend ideas and develop and deepen the children’s knowledge; reinforce their understanding of key concepts, build on earlier learning so that strategies and techniques become more efficient and precise.
Read
To use mathematical vocabulary and interpret images, diagrams and symbols correctly; read number sentences and provide equivalents, describe and explain diagrams and features involving scales, tables or graphs; identify shapes from a list of their properties, read and interpret word problems and puzzles; create their own problems and lines of enquiry.
Reason
To use and apply acquired knowledge, skills and understanding; make informed choices and decisions, predict and hypothesise; use deductive reasoning to eliminate or conclude, provide examples that satisfy a condition always, sometimes or never and say why.
The Six Rs of Oral and Mental Work
Activity 4:
Working with a partner, think of a mental oral starter for each of the six Rs.
(The starter can be for any year group).
The Six Rs of Oral and Mental Work
Six Rs Rehearse Recall Refresh Refine Read Reason Learning Focus
To practise and consolidate existing skills, usually mental calculation skills, set in a context to involve children in problem-solving through the use and application of these skills, use of vocabulary and language of number, properties of shapes or describing and reasoning.
Possible activities
Interpret words such as
more
,
less
,
sum
,
altogether
,
difference
, subtract; find missing numbers or missing angles on a straight line; say the number of days in four weeks or the number of 5p coins that make up 35p; describe part-revealed shapes, hidden solids; describe patterns or relationships; explain decisions or why something meets criteria.
To secure knowledge of facts, usually number facts, build up speed and accuracy, recall quickly names and properties of shape, units of measure or types of charts, graphs to represent data.
Count on and back in steps of constant size; recite the 6 times table and derive associated division facts; name a shape with five sides or a solid with five flat faces; list properties of cuboids; state units of time and their relationships.
To draw on and revisit previous learning; to assess, review and strengthen children’s previously acquired knowledge and skills relevant to later learning; return to aspects of mathematics with which the children have had difficulty; draw out key points from learning.
Refresh multiplication facts or properties of shapes and associated vocabulary; find factor pairs for given multiples; return to earlier work on identifying fractional parts of given shapes; locate shapes in a grid as preparation for lesson on co-ordinates; refer to general cases and identify new cases.
To sharpen methods and procedures; explain strategies and solutions; extend ideas and develop and deepen the children’s knowledge; reinforce their understanding of key concepts, build on earlier learning so that strategies and techniques become more efficient and precise.
Find differences between two two-digit numbers, extend to three-digit numbers to develop skill; find 10% of quantities, then 5% and 20% by halving and doubling; use audible and quiet counting techniques to extend skills; give co-ordinates of shapes in different orientations to home concept; review informal calculation strategies.
To use mathematical vocabulary and interpret images, diagrams and symbols correctly; read number sentences and provide equivalents, describe and explain diagrams and features involving scales, tables or graphs; identify shapes from a list of their properties, read and interpret word problems and puzzles; create their own problems and lines of enquiry.
Tell a story using an interactive bar chart, alter the chart for children to retell the story; start with a number sentence (eg 2 + 11 = 13) children generate and read equivalent statements for 13; read values on scales with different intervals; read information about a shape and eliminate possible shapes; set number sentences in given contexts; read others’ results and offer new questions and ideas for enquiry.
To use and apply acquired knowledge, skills and understanding; make informed choices and decisions, predict and hypothesise; use deductive reasoning to eliminate or conclude, provide examples that satisfy a condition always, sometimes or never and say why.
Sort shapes into groups and given reasons for selection; discuss why alternative methods of calculation work and when to use them; decide what calculation to do in a problem and explain the choice; deduce a solid from a 2-D picture; use fractions to express proportions; draw conclusions from given statements to solve puzzles.
Always, Sometimes, Never
The three angles in a triangle are different.
A triangle can not have an internal angle that is a reflex angle.
Numbers with a unit digit of 3 are divisible by 5.
Mathematical Language
What language should be used in each year group?
Is there a progression in the use of mathematical language?
Mathematical Language
Mathematical Vocabulary booklets.
Note: This was produced for the original NNS. Due to the fact that some objectives moved year groups in the 2006 Renewed Framework some vocabulary may need to be introduced in earlier year groups.
Developing Mathematical Language
ATM - Fourbidden Cards.
F ourbidden is a mathematical card game to
promote the use of mathematical language devised by Phil Dodd and published by ATM.
There are now two packs of Fourbidden cards, the latest designed with KS3 students in mind. There are 52 cards in each set, on each card a familiar mathematical term is printed on the left, with four related words shown on the right hand side of the card. There is a good explanation on different ways of using the pack.
Break
Tea, coffee, fruit and biscuits available at the back of the hall.
How mental recall can be developed to facilitate mental calculation.
There is a heavy reliance on known facts.
Conclusion:
If children haven’t learnt the facts in the first place they can’t: a) Recall them; b) Use them to help them calculate; c) Develop those facts further (i.e. for larger numbers).
Foundation Stones
Activity 5:
Look at the progression document again. Choose two year 6 objectives and identify earlier objectives from previous year groups that would need to be embedded for each objective to be understood.
How would you solve it?
(calculation sorting) Don’t solve the calculation… …identify the most appropriate strategy.
How would you solve it?
(calculation sorting)
Activity 6:
With a partner, work through the calculations on the yellow sheet in the middle of your table.
DO NOT SOLVE THEM!
For each one identify the most appropriate strategy that should be used to solve it.
How would you solve it?
(calculation sorting) There is no right or wrong answer.
The point is… About stopping and thinking; Making things easy for yourself; Using known facts to solve the problem (by doing as little maths as possible!)
How would you solve it?
(calculation sorting) • Should be introduced from year 3 • Should be used to develop lateral thinking about strategies • Should develop and build on .
previously learned strategies.
Mental Maths Practise Tests
Mental Maths Practise Tests Use weekly:
• As a teaching opportunity to discuss strategies; • To cover and practise a whole range of Maths; • To practise rapid recall of facts/information; • For speed.
The most important part is NOT the testing or the mark achieved, but the discussion that follows the test.
Children need regular timed practise to speed up their recall.
Other Strategies
• Times Tables (and division) Clubs • Mathletics/Education City/RM Maths
Mental Oral Starters
Pace is very important.
Any recall of facts should be rapid.
Chris Moyles’ Quiz Night
http://www.sheffieldmaths.co.uk/Ch ris%20Moyles%20Starters.html
What is
Visualisation?
Activity 7:
Skim read the article ‘Thinking Through, and By, Visualising.
We rely on visualising when we solve problems. Sometimes we create an image of the situation that is being discussed in order to make sense of it; sometimes we need to visualise a model that can represent the situation mathematically before we can begin to develop it, and sometimes we visualise to see 'what will happen if ...?'. But are there other ways in which we visualise when solving mathematical problems and if so how can we encourage, value and develop visualising in our classrooms?
Children need to have had the opportunity to hold, turn, examine and work with objects before they can visualise them.
• Which column would number 12 be in?
• Find me 2 numbers that add up to 10.
• Give me a number that will appear in the middle column.
• What can you tell me about the numbers in each column?
• Find the sum of the first row. Is it a multiple of 3?
• Are there any other rows and columns with multiples of 3?
• Look at the first column. If extended, would 73 be included?
• Give me 2 numbers between 50 and 60 tat would appear in the final column.
• Will there be a multiple of 100 in the middle column?
Shape Visualisation
Begin by visualising a square:
With one fold make it into a rectangle. Describe the properties of the rectangle.
Make it into a smaller square. Make it into a triangle. What kind of triangle have you made?
How do you know? How do you know it is not an equilateral triangle?
Fold it back into a square. Make it into a pentagon. Can you visualise one fold to make a pentagon with one line of symmetry? How do you know this is a line of symmetry?
Try using a post-it note to scaffold.
Shape Visualisation
Imagine a pyramid.
Walk around the pyramid. What can you see?
Imagine you can fly and fly above it.
What can you see now?
Imagine you can lift it up with a magic spell. Spin it around, invert it and then put it back down again. What can you see now?
Now talk to the person next to you and talk about the similarities and differences between your two pyramids.
Number visualisation
• Imagine the number five hundred and thirty two drawn in the air in front of you.
• Which digit is in the middle? Which is on the left? Which is on the right?
• Replace the middle digit with a four. What number can you see now?
• Swap over the middle digit with the one on the left. What number can you seen now?
• Remove the middle digit and push the two together so that they are next to each other. What number can you see now?
Conclusion:
Visualisation is important in all areas of Maths.
It is important for working out properties of shapes, positions of shapes or objects that have been reflected, rotated or translated and nets of 3D shapes.
It is also important for measuring.
Conclusion:
It is also extremely important for calculating.
If you can imagine, partition and recombine in your mind’s eye, you are more likely to be able to flexibly use and adapt known facts to assist in mental calculation.
Conclusion:
Visualisation can be supported with practical equipment.
Use of a broad range of vocabulary is vital.
Some more ideas for visualisation activities are provided in booklets for you.
Other activities are available on the internet via a Google search (Nrich is a good website).
How to teach and not just test times tables.
Times tables are extremely important.
Here are a few reasons why: Mental calculation of areas of shapes Mental Division (use of inverse operation) Use of known facts for other calculations Mental multiplication of 2-digit by 1-digit numbers Use of the grid method (as a written calculation)
Progression in Mental Multiplication
Addition facts (known facts) Number bonds to 10, then 20….
Complements to 100 (multiples of 10 first) Multiplication as repeated addition Times tables Extend knowledge of times tables (e.g. 16 x 5)
Times Tables Progression
2, 10 then 5 3 and 4 6 7, 8 then 9 Beyond 10 x 10
Times Tables Progression
Begin in year 1 Consolidate in year 2 2, 10 then 5 3 and 4 6 Learn remaining facts from these tables in year 4 7, 8 then 9 Begin 3s in year 2 Consolidate in year 3 Learn 4s in year 3 Progress on to 6s Beyond 10 x 10 From year 5 onwards
Progression in teaching tables
Counting forwards and backwards in steps As above, but ‘skipping around’ and introducing missing numbers.
Chanting, e.g. 2, 4, 6, 8… Reciting times tables Continued
Progression in teaching tables
Year 3: Inverse operation should be introduced Counting, chanting, etc. should be ongoing through Key Stage 2.
Loop cards
Teaching ideas
Refer to models and images sheets.
Counting stick http://www.topmarks.co.uk/flash.aspx?f=countingstickv4
Teaching ideas
Counting hoop Pendulum Good to link in with time and counting around a clock face
Teaching ideas
Arrays are important for the understanding of multiplication as well as division.
Can be used from year one upwards.
5 x 3 = 15 3 x 5 = 15 15 ÷ 5 = 3 15 ÷ 3 = 5
Teaching ideas
Year 3: Inverse operation should be introduced Fact families for multiplication and division (derive division facts from known multiples) 3 x 2 = 6 2 x 3 = 6 6 ÷ 3 = 2 6 ÷ 2 = 3
Teaching ideas
Counting stick challenge Count along the top in 3s Count along the bottom in 4s
SIMULTANEOUSLY.
Let’s have a go!
Teaching ideas
Counting with a twist
A B C
Group A count in steps of 0.1
Group B count in steps of 0.3
Group C count in steps of 0.5
Teaching ideas
Stack ‘em up games Multiplication and Division 3 24 9 30 21 Three Times Table
Stack ‘em up
12 15 27 6 18 http://www.fairhaven.ik.org/p_Printable_Maths_Games.ikml
Key Messages
• Mental maths is wider than just mental calculation and is not just a times tables test.
• Use of jottings and practical resources are acceptable ways of supporting mental calculation.
• Children need daily and fun whole class, guided and/or independent opportunities to practice and apply their mental maths skills and improve their confidence and efficiency.
• Opportunities to develop mental maths should include developing reasoning and communication.
All resources used today and links to websites referred to can be accessed via our website: http://www.fairhaven-cpd events.wix.com/info
Click on the ‘Resources’ tab
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