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Concrete resources
Concrete – students should have the opportunity to use concrete objects and manipulatives to help them understand what they are doing.
Pictorial
Pictorial – students should then build on this concrete approach by using pictorial representations. These representations can then be used to reason and solve problems.
Challenge 1 • 12 + 15 =
• 12 + 11 = • 15 + 14 = • 17 + 12 = • 13 +13 = • 15 + 14 =
Challenge 2 • 16 + 15 =
• 17 + 18 = • 16 + 19 = • 19 + 13 = • 17 + 15 = • 15 + 17 =
Challenge 3 • 31 + 25 =
• 82 + 23 = • 66 + 25 = • 59 + 22 = • 177 + 146 = • 165 + 132 =
Abstract With the foundations firmly laid, students should be able to move to an abstract approach using numbers and key concepts with confidence.
Whistle stop tour of calculation policy – non formal to formal !
•Addition
•Subtraction
•Multiplication
•Division
Whistle stop tour of calculation policy – non formal to formal !
• A deep understanding of place value and times tables facts are vital for true fluency and understanding of number operation.
Models of Addition
• There are 12 girls and 3 boys. How many children altogether? • The chocolate bar was 12p last week, but today the price went up by 3p.
What is the price now?
Make up a word problem on a slip of paper that represents the calculation
12 + 3 = 15
Addition Do children definitely know what addition is? It could represent an “two items being totalled” (aggregation) or “add on more to the first” (augmentation) e.g. 6 + 9
What is the “ideal approach”? A visual “bar model”.
6 9 ?
Starting Point Before launching in to the expectations of KS2, the following are the new National Curriculum expectations for year 2:
Solve problems with addition and subtraction: - Using concrete objects, pictorial representations, including number and measure. - Apply their increasing knowledge of mental and written methods.
Recall addition & subtraction facts up to 20 fluently, and derive facts up to 100.
Add and subtract numbers using concrete, pictorial & mentally, including: - TU + U - TU + T - TU + TU - U + U + U
Understand that addition is commutative, but subtraction is not.
Use inverse relationships between addition and subtraction to check calculations & solve missing number problems
Trading Game – Addition
H T U
30
4
Excellent activity for the end of KS1 which develops conceptually the “regrouping” required for KS2.
Building the Journey Year 3 Addition and Subtraction up to 3 digits using formal methods Year 4 Addition and Subtraction up to 4 digits using formal methods (Solve simple measure and money problems involving decimals to 2 dp) Year 5 Addition and Subtraction more than 4 digits using formal methods (Solve problems involving number up to 3 dp) (They mentally add and subtract tenths, and one-digit whole numbers and tenths) (They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1)). All years groups also refer to "estimation", "inverse operations" to check, "problem solving"
Three-Digit Column Addition – Stage 1 Students will still, require concrete resources. This is likely to be Dienes (or equivalent) to model two digit addition. No re-grouping to take place. Students record concrete and abstract calculations together.
T U
2 3 4 1 +
T U
Three-Digit Column Addition – Stage 2 Students will still, require concrete resources. This is likely to be Dienes (or equivalent) to model two digit addition. No re-grouping to take place. Students record concrete and abstract calculations together.
T U
2 5 4 7 +
T U
Three-Digit Column Addition – Stage 1 remodelled Students still require concrete resources. Some students will want to move away from Dienes, and handle resources less cumbersome (as they now have a feel for “size”) – e.g. place value counters
T U
2 3 4 1 +
T U
10 1 1 1
1
10
10 10 10 10
Three-Digit Column Addition – Stage 2 remodelled Students still require concrete resources. Students are now ready to tackle problems requiring “re-grouping”. There are different way students could effectively communicate their thoughts. In time students won’t need counters.
T U 2 5 4 7 +
T U 10 1 1 1 10
10 10 10 10
1 1
1 1 1 1 1 1 1
T U 2 5 4 7 +
Three-Digit Column Addition – Stage 3 Students still require concrete resources. Students are now ready to tackle problems requiring “re-grouping”. There are different way students could effectively communicate their thoughts. In time students won’t need counters.
T U 2 5 4 7 +
T U 10 1 1 1 10
10 10 10 10
1 1
1 1 1 1 1 1 1
T U 2 5 4 7 +
10
2
2
1
1
6 0
Three-Digit Column Addition – Stage 3 Students still require concrete resources. Students are now ready to tackle problems requiring “re-grouping”. There are different way students could effectively communicate their thoughts. In time students won’t need counters.
T U 2 5 4 7 +
T U 10 1 10
10 10 10 10 1
T U 2 5 4 7 +
10
2
2
1
1
6 0 7 2
7
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10 12
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10
2 1
10 10
+
1
1 1 1
1 1
1
1 1
1
1
1
10
10
10
10
10
2 1
5 2 7 4
10 10
+
1
1 1 1
1 1
1
1 1
1
1
1
10
10
10
10
10
2 1
5 2 7 4
10 10
+
1
1 1 1
1 1
1
1 1
1
1
1
10
10
10
10 10
2 1
5 2 7 4
10 10
+
1
1 1 1
1 1
1
1 1
1
1
1
10
10
10
10
10
2 1 7
5 2 7 4
Year 3 Essentially year 3 becomes a time when more "formal methods are introduced". : e.g. 34 + 21 35 + 19
U T
1
1
1
1
1
1
1
1
1 10
10
10
10
10
10
U T
Addition with 3-digits (an end of Year Objective)
U T H
e.g. 345 + 126 283 + 142 364 + 159
Why is this a poor example?
1 10 100
U T H U T H
1 10 100
1 10 100
1 10
1
1 10 100
1 10 100
1 10 100
1 10
10
10
1 10 100
1 10 100
1 10
10
10
10
10
10
Alternatively, set the addition into an application phase: this is still largely fluency
Find the perimeter of this shape:
Show this pattern goes up by the same amount each time. Then find the next number in the pattern: 325, 462, 599, …
352
Play the role of the teacher: Reasoning – why is it incorrect – explain….. Mark the following questions. If they are right give a tick, if they are wrong, explain why you think the mistake has been made: Reflection: is column addition the most efficient way to tackle this question? 346 + 300 – 5 Where does planning allow for reflection?
9 5 4 6 +
1 1 1
2 3 9 4
9 5 4 6 +
6 1
2
2 3
9 3
U T H Th 10Th
100
10
1
More than Four-Digit Addition
1000
U T H
10000
Year 5
Year 5 They mentally add and subtract tenths, and one-digit whole numbers and tenths) They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1) Here the most important concept to introduce is 0.9 + 0.1 ≠ 0.10 Two helpful strategies:
http://www.mathsisfun.com/numbers/number-line-zoom.html
Subtraction
Make up a word problem that represents the calculation
9 – 3 = 6
Subtraction problems
• I had 9 apples but my rabbit ate 3 of them. How many did I have left? • I had 9 apples. My friend Harry had 3 apples. How many more apples did I
have?
3 ? 9
The Bar Model – How does it support understanding?
Robber maths
43 – 13
The number you need to subtract is small enough to “pick it up and
take it away”
Mind the gap
74 - 69
The gap between the two numbers is smaller so it is more efficient to find the difference (probably by counting
on)
Robber maths? – Mind the gap?
101 – 99
63 – 21
84 – 78
1006 – 999
86 – 14
Trading Game
H T U
30
4
10 10
10
10
10
10 10
1 1 7 2 7 4 -
10
1 6
1 1
1 1
1
1 1 1
1
1
10 10
10
10
10
10 7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
10 10
10
10
10
10 7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
10 10
10
10
10
10 7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
10 10
10
10
10
10 7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
10 10
10
10
10
10 7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
5
10 10
10
10
10
10 7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
5
10 10
7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
5
10
10
10
10
10 10
7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
5 10
10
10
10
10 10
7 2 7 4 -
1
1 1 1
1 1
1
1 1
1
1
1
1 6
5 2 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10 12
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10
2 1
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10
2 1
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4 10
10
10
10
10
2 1
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4
10
10
10
10 10
2 1
10 10
5 2 +
1
1 1 1
1 1
1
1 1
1
1
1
7 4
10
10
10
10
10
2 1 7
Year 3 Essentially year 3 becomes a time when more "formal methods are introduced". : e.g. 57 - 23 52 - 27
1
1
1
1
1
1
1
1
1
10
10
10
10
10
10
10
10
10
10 U T U T
Expanded method of subtraction 273- 147 =
200 70 3
100 40 7
Subtraction with 3-digits (an end of Year Objective)
e.g. 345 - 236 523 - 136 300 - 159
U T H
1 10 100
U T H U T H
1 10 100
1 10 100
1 10
1
100
100
100
1 10 100
1 10 100
1 100
100
100
Spicing up Addition and Subtraction: Problem solving through deep reasoning. Arithmagons Number Walls
Some of these will need real resilience, but the sense of achievement will be much greater once solved!
Ink Blots/ Missing Digit
Magic Squares Darts?!
Cryptarithms/ Alphametics
T U 2
2 + 3 5
T U 7
5 + 3 5 1
H
Year 4 The strategies met in year 3, extend into year 4 - with addition and subtraction now with four digits. 1. Although students might be able to naturally extend the method, revisit the
kinaesthetic examples so they link their new objective to prior learning. If some students need longer working kinaesthetically than others - fine!
2. Some very visual learners will even remember "counter colours" from the previous year, so ensure complete consistency between year groups.
Note - all the "livening up" and "problem solving" skills from year 3 should also be embraced in years 4 & 5. As too should estimation and inverse operations to check.
Models of multiplication
• I had four bags and they each contained six books. How many books do I have?
• I had six pens. Tom had four times as many? How many did Tom have?
Year 3 solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects
Models for multiplication
74
Scaling
3 times as tall
This can be generalised to include any multiplier including those less than
one – i.e. making smaller
6 6 6 6 6 6 6 6 6
6 + 6 + 6 + 6 Additive
reasoning
6 x 1 6 x 4 Multiplicative
reasoning
Arrays
2 lots of 3 make 6
3 lots of 2 make 6
There are two 3s in 6
There are three 2s in 6
2 x 3 = 6
3 x 2 = 6
6 divided by 2 = 3
6 divided by 3 = 2
An image for 7 u 8 = 56
77
Multiplication At the heart of success of this topic is clearly mastery of times tables. It is the one area where deviating from your year group and extending has value. Fluency with tables opens up so many other topics in maths (e.g. fractions and area) and conversely closes off success in other topics if they haven't been mastered at a young age. Guide: Yr 2 (2, 5, 10) Yr 3 (3, 4, 8) Year 4-6 (upto 12 x 12) Is this the wisest plan? Practice, practice, practice is the key. Use every (daily throughout the year) opportunity open to you, especially: - Lining up to assembly, getting changed for PE, etc. - Parents! They can really support the regularity of practice.
Note: DK Times Table App – many can access at home.
Arrays to Solve Multiplication
10 3
4
10 x 4 = 40 4 x 3 = 12
40 + 12 = 52 13 x 4 = 52
13 x 4 =
Year 4 (set questions involving all their tables targets - though this will need differentiation throughout the classes) Always go back to the kinaesthetic example when re-introducing (even if only for a few seconds) 43 x 6 247 x 3
U T
U T H
Year 5 Multiply upto 4 digits by one-digit or two-digit, using a formal written method, including long multiplication for two-digit numbers. re-visit: Hands-On? Short-Multiplication Long-Multiplication 4253 x 7 4253 x 7 253 x 37
Year 5 Multiply multi-digit numbers up to 4 digits by a 2-digit whole number.
18
13
18 10 8
13
3
10 100 80
24 30
Progressing towards the standard algorithm
1 0 8
1 0
3
1 0 0 8 0
3 0 2 4
10 8
10
3
100 80
30 24
1 8
u 1 3
1 8 0
5 4
2 3 4
?
When?
How?
Year 5 Multiplying (and Dividing) by 10, 100 and 1000 4 x 10 13 x 10 6 x 100 4.3 x 10 0.12 x 1000
U T H 110
1100
http://www.topmarks.co.uk/Flash.aspx?f=MovingDigitCards
Year 5 Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers. Know the vocabulary of prime numbers and non-prime numbers.
Differentiation
Year 5 Recognise square numbers and cube numbers - and notation. Square Numbers Cube Numbers
Year 6 Multiply multi-digit numbers up to 4 digits by a 2-digit whole number. Identify common factors, common multiples and prime numbers.
Regarding the "mastery learning" approach, if a skill is mastered by the end of the Autumn or Spring term, consider the following for the Summer Term: Set questions in a "worded context" Apply skills to a more "problem solving" question, e.g. "The school hall measures 12m by 26m. Miss Smith is going to carpet the hall using square carpet tiles which are 50cm long. How many tiles are needed?“ Extend further??
1
1
Using Inequalities 3 x 42 5 x 21
9 3 2
3
5 4 4 x
Links with other topics… Area Pictograms Measure
Division
At the heart of success of this topic is clearly mastery of times tables. The more fluent a student is at their tables, the easier they will find division.
Grouping and Sharing
12 ÷ 3 = 4
Grouping – we know how many are in each group but not how many groups there will be. The answer is the number of groups.
Sharing - we know how many groups there are but not how many are in each group. The answer is the number in each group.
Grouping
Building the Journey Year 3 Pupils can derive associated division facts e.g. if 6 ÷ 3 = 2, then 60 ÷ 3 = 20 Pupils develop reliable written methods for division, progressing to the formal written methods of short division. Year 4 Pupils can derive associated division facts e.g. if 28 ÷ 7 = 4, then 2800 ÷ 7 = 400 Pupils practise to become fluent in the formal written method of short division with exact answers Year 5 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. Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, e.g. in scale drawings or in converting between metric units. Year 6 Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context. Divide numbers up to 4 digits by a two-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. Solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts. Solve problems involving similar shapes where the scale factor is known or can be found.
Ella has 48 plasticine legs to make animals for a display. How many cows could she make?
How many beetles could she make?
How many spiders could she make?
An image for 56 y 7
Either:
• How many 7s can I see? (grouping)
Or:
• If I put these into 7 groups how many in each group? (sharing)
An image for 56 y 7
5 6 7
8 5 6 7 8
The array is an image for division too
363 ÷ 3 =
3 6 3 3
1 2 1
364 ÷ 3 =
3 6 4 3
364 ÷ 3 =
3 6 4 3
1 2 1 rem 1
345 ÷ 3 =
3 4 5 3
1 1 1
5
Year 4 The journey is now about fluency with short division. Although not explicit, three-digit divided by one-digit seems a sensible goal by the end of the year. There should be exact answers (no remainders). What if students "master" the process in the autumn term? Do you have "tricks" as a teacher to ensure there are no remainders?! 462 ÷ 2 725 ÷ 5 537 ÷ 3 474 ÷ 6 738 ÷ 9
1
10
100 H T U
Year 6 Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context. Divide numbers up to 4 digits by a two-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. (Note: as a mathematician, I never use long division, and do not see its value in the new National Curriculum... but you have to teach it!) 12 5 4 2 1 12 5 4 2 1 Careful then writing recurring decimals...
Year 6 Solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts. This is ratio! "mixing paint"... let your students take ownership of their learning...
(Hidden) Applications of Ratio: If 3 pencils cost 45p, how much did one pencil cost? If 2 pencils cost 60p, how much would 5 pencils cost? If 5 pencils cost 70p, how many pencils could I buy for £2.10? Ingredients to make 16 gingerbread men 180 g flour 40 g ginger 110 g butter 30 g sugar How much of each ingredient would you need to make ...... gingerbread men?
Year 6 Solve problems involving similar shapes where the scale factor is known or can be found. Language: If two shapes are identical, we say they are ........................ Similar shapes means the two shapes are .......................... ................................................ If you double the sides, does everything double?
Maths activities to support the new curriculum (other subjects too!): http://www.theschoolrun.com/ Various maths games: http://www.transum.org/Software/Game/ http://mathszone.co.uk/number-facts-xd/ http://www.primaryinteractive.co.uk/maths.htm Open-ended maths puzzles: http://nrich.maths.org/ Maths triangles: http://www.helpingwithmath.com/printables/others/fac0201fact_triangle01.htm
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