Reasoning · Grade 4-2 Adding and Subtracting Decimals
Problem
Order and grouping are free to change.
Rearranging a sum is just walking round the same pile of counters and picking them up in a different order - the pile does not change, so the total cannot either.
1.OA.B.3Organize Information In More WaysIn a long addition the order of the numbers and the way they are grouped are both free, so the terms may be rearranged at will.
Swapping two numbers in a sum cannot change what they come to, so the list may be reordered however is convenient.
Which two numbers you push together first makes no difference either, so brackets can be moved to make friendly pairs.
Pairing the ends, (1) is 18.
Moving one step in from the left adds 0.2 while moving one step in from the right takes 0.2 away, so the pair total never budges - that is why every pair is the same 3.
5.NBT.B.7Look For A PatternPairing decimals that complete wholes, (2) is 22.
Making the decimal part disappear turns each pair into a whole number, and whole numbers are the easiest thing there is to add up.
5.NBT.B.7Look For A PatternReading down the columns, (3) is 49.995.
Sorting the same digits by place value instead of by row turns nine messy additions into one easy one, 1 + 2 + ... + 9 = 45, done four times over.
6.NS.B.3Organize Information In More WaysPeeling off the 0.99, (4) is 1.48.
Each subtract-then-add pair only nudges the total up by one hundredth, so 98 terms collapse into a single multiplication by 49.
5.NBT.B.7Look For A PatternCounting the shortfalls from 2, (5) is 17.000000001.
It is far easier to add nine 2s and take away one tidy number than to line up nine ragged decimals - looking at what is missing instead of what is there is the whole saving.
6.NS.B.3Change Focus Count The ComplementThe five answers are 18, 22, 49.995, 1.48, 17.000000001.
Writing all five together makes it easy to spot a stray decimal point, since each answer should sit in a size range you can predict at a glance.
5.NBT.A.3Organize Information In More WaysBefore you add a long list of decimals, look for the rule that built it - pair the ends, pair the partners that make a whole, add down the columns, or count what each number is missing.