Mental math
Mental math is not arithmetic done faster. It is arithmetic done differently: 7 + 8 as "double seven and one more", 63 - 29 as "take thirty, give one back". Each strategy is a small rearrangement that makes a hard sum into an easy one, and a child who has three or four of them is quicker than a child who has memorised twice as many facts.
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Near doubles
7 + 8
- 8 is one more than 7, so this is nearly a double.
- Double 7 is 14.
- One more makes 15.
- 7 + 8 is 15.
Worth knowing. This works because doubles are the easiest facts to remember. Learning ten doubles and one adjustment covers a great many more than ten sums.
Compensating
63 - 29
- 29 is awkward; 30 is not.
- Take 30 instead: 63 take 30 is 33.
- You took one too many, so give it back.
- 33 and one more is 34.
Worth knowing. The direction of the adjustment is the part that catches people out. Taking away too much means giving back, and it is worth saying that sentence out loud rather than memorising which way to go.
Where it usually goes wrong
- Adjusting the wrong way after compensating, so 63 - 29 becomes 32.
- Reaching for the column method for a sum that a strategy would answer instantly.
- Trying to hold a whole column algorithm in the head, which is what mental strategies exist to avoid.
The Professor adds 29 by adding 30 and giving one back. It feels like cheating and it is just arithmetic.
Practise it
All level pathsQuestions teachers and parents ask
Should mental strategies replace the written methods?
No — they sit beside them. Written methods work on any numbers and never run out of room; strategies are faster on the numbers they suit and useless on the rest. A child wants both and wants to know which is which.
Which strategy should be learned first?
Doubles, then near doubles, then pairs that make ten. Those three between them cover most of the addition facts within twenty, and each one is built on the one before.
Is it worth learning several strategies for the same sum?
Yes, and it is a good sign when a child argues about which is quicker. 8 + 7 can be a near double, or eight and two more to make ten and five after that. Both are right, and choosing between them is the skill.
When is this taught?
Fluency within 20 using strategies is grades 1 and 2 (1.OA.C.6, 2.OA.B.2), and the properties that justify them — commutative, associative — are named in grade 1 (1.OA.B.3). The strategies keep working on bigger numbers long after that.
Where this shows up
Standards
- 1.OA · Operations and algebraic thinking
- 2.NBT · Number and operations in base ten
- 3.OA · Operations and algebraic thinking
- CCSS.MATH.CONTENT.1.OA.B.3 — Apply the commutative and associative properties of addition as strategies.
- CCSS.MATH.CONTENT.1.OA.C.6 — Add and subtract within 20; fluently add and subtract within 10.
- CCSS.MATH.CONTENT.1.NBT.C.5 — Given a two-digit number, mentally find 10 more or 10 less without having to count.
- CCSS.MATH.CONTENT.2.OA.B.2 — Fluently add and subtract within 20; know all sums of two one-digit numbers from memory.
- CCSS.MATH.CONTENT.2.NBT.B.8 — Mentally add or subtract 10 or 100 to a number between 100 and 900.
- CCSS.MATH.CONTENT.3.OA.B.5 — Apply properties of operations as strategies to multiply and divide.