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Maths Terminology Mental calculation requires strategic thinking. If students are to develop their ability to think strategically, then it is important that a common language is taught and plenty of opportunities are provided for practice. The following terms should be introduced to students and the associated concepts explored, practised and extended so that the language forms an integral part of every number exploration. Terminology Subitise Comment Building on Knowledge Meaning to suddenly know, subitise is taken from the Latin word subito. Subitising is an important skill that lays the foundation for many later mathematical concepts. Perceptual subitising means looking at a single group of objects and knowing just by looking at them how many there are. Conceptual subitising builds on this skill: for 6 and 2 thrown on two dice students can be encouraged to subitise the 6 and count on 2 more. Count on Requires that students understand altogether as the joining of groups of objects or numbers and that we can count on from the numbers other than 1 and thus break the counting sequence. Touching and counting every object does not practise early addition skills. Students need to realise that it is more efficient to subitise than count on. How does knowing count on 1, 2 or 3 help you with 50 + 30? How does knowing count on 1, 2 or 3 help you with 70 - 20? Double Doubles lay the foundation for more sophisticated strategies such as ‘near double’, are often part of number splitting strategies and provide a direct link to the concept of halving when working with subtraction. How does knowing your doubles facts help you with 60 – 2? How does knowing your doubles facts help you with 70 – 35? How does knowing double 3 = 6 help with double 33? How can we apply doubling strategies to situations involving measurement (e.g. time, length etc)? Rainbow Facts Also known as build to 10 they identify all the pairs of numbers that add to 10. Rainbow facts lay the foundation for understanding friendly numbers and the later bridge through 10 strategy. The concept can be extended to include pairs of numbers that add up to 1, 100, 1 000 etc. What number pairs add to 1, 100, 1 000 etc.? How do rainbow facts help us with subtraction? e.g. 100 – 70 = 30 Turnarounds Refers to the commutative property of numbers in addition. Students need to understand that the order in which numbers are added makes no difference to the answer but it does make a difference to the ease of adding the numbers. For example, for 2 + 6 it is faster to turn the numbers around and then count on 2 from 6. The concept of turnarounds also applies to multiplication. How does knowing your turnarounds for addition help you with multiplication? Near double Sometimes described as ‘double + 1’ or ‘double – 1’. Near double allow students to build on a known strategy, a process that becomes very important as students learn to extend their number facts. How does knowing 6 + 7 is near to 6 + 6 help with 60 + 70? Adding zero It might seem obvious that adding zero to a number leaves it unchanged but it is necessary to ensure students have developed this understanding. What is the outcome when we multiply by 0? Friendly Numbers Numbers ending in 0 are called friendly because they are very easy to add a single digit number to. It helps lay the foundation for the bridge through 10 and compensation strategies. How does being able to work easily with numbers ending in 0 help us with calculations like 9 + 7 or 80 + 50? TERMINOLOGY cont’d Bridge through Builds on knowledge of friendly numbers and rainbow facts. For example, to use bridging to add 9 + 4, the 4 is split into 1 + 10 3 giving 9 + 1 + 3. The rainbow pair 9 + 1 makes 10, and 3 is easily added to the friendly number 10 to make 13. Strategy can easily be extended into 10s, 100s and beyond e.g. 90 + 50 becomes 90 + 10 + 40. (See Compensating) How does being able to bridge through 10 help us with 130 – 50? How does being able to bridge through 10 help us with 39 + 26? Adding 9 Relating + 9 to 10 – 1 makes it easier for students to make connections with their understandings about friendly numbers. What thinking could I use to help me with + 90 or + 100? Landmark Numbers The term given to numbers such as 25, 50 and 100. Knowing that double 25 is 50 makes computing double 26 easy. Also facilitates discussion around mathematical facts such as 75 + 25 = 100. Concept can also be linked to percentages and fractions. What are the landmark fractions and percentages? Number Splitting Numbers can be split in many ways to make computations easier e.g. 26 + 37 could become 26 + 4 + 33, allowing students to strategise with reference to rainbow facts and friendly numbers. Numbers can also be split for other operations e.g. 13 x 4 could become (10 x 4) + (3 x 4) How can we use number splitting to help us complete calculations involving the other operations? Compensating Making friendly numbers Rounding, estimating and number splitting are the foundation for the strategy of compensating when adding or subtracting. Rounding 51 to the friendly number 50 makes it easier to add 17. For an exact answer, compensation in the form of adjusting is required. The 1 taken off must now be added back. Rounding and adjusting is one of the most frequently used compensation strategies. (See Bridge through 10) Changing operations Inverse operations Halving/ doubling Students may find it easier to think addition when subtracting or think multiplication when dividing. For example, 13 – 8, students can think 8 add what makes 13? Using developed doubling and halving strategies, students can be encouraged to simplify more complex calculations. For example, students could work out 17 x 4 by thinking 34 x 2 What sequence of numbers can be halved down to 1? Partitioning Refers to the myriad of ways that numbers can be divided into parts. There are standard place value partitions such as 582 = 500 + 80 + 2, but 582 can also be viewed as 58 tens + 2 ones or even 382 + 200. The partition 382 + 200 is far more helpful when calculating 582 – 198. In what ways can partitioning be used to make calculations easier? Commutative Law The commutative law says that when you add or multiply numbers the order doesn’t matter – you can swap numbers over and still get the same answer, e.g. 2 + 3 = 3 + 2 = 5 or 2 x 3 =3x2=6 See Turnarounds Associative Law The associative law says that it doesn’t matter how you group the numbers (which you calculate first) when you add or when you multiply, e.g. (2 + 4) + 5 = 6 + 5 = 11 or 2 + (4 + 5) = 2 + 9 = 11 Distributive Law The distributive law says that when you add up some numbers then do a multiply or you multiply separately than add them, you will get the same answer e.g. 2 x (5 + 2) = (2 x 5) + (2 x 2)