AMC 10 · 2014 · #9
Grade 6 algebraPick an answer.
Everything hangs on one letter, a, so Tool #4 (Introduce a Variable) lets me write all five integers as a, a+1, a+2, a+3, a+4 and turn the words into an exact expression for the average. Tool #5 (Look for a Pattern) supplies the shortcut that does the heavy lifting: for five consecutive integers the average is always the middle one, so I never have to add long lists twice.
Name the five integers
The start names all five numbers.
Consecutive integers just march up by one, so one letter plus 0,1,2,3,4 names them all.
6.EE.A.2Introduce A VariableAverage them to get b
Their average is the middle one.
The mean is the sum shared equally, and for a balanced run of numbers that share lands on the middle one.
6.SP.B.5Introduce A VariableMiddle term is the average
The same rule applies to the second run.
Five in a row always balance on their center value, so the middle term is the average every time.
Five consecutive integers always balance on their centre value, so the middle term is the average every time.
▸ Why?
Terms spaced by the same step sit symmetrically about the middle, so the outer ones cancel against each other.
▸ Why?
An average is the total shared over the count, and that shared value lands exactly on the middle term.
Substitute b = a + 2
Substituting gives a plus 4, choice (B).
Each averaging step nudges the start up by 2, and doing it twice moves a up by 4.
6.EE.A.3Introduce A VariableThe average of five numbers in a row is just the middle one, and averaging twice bumps the starting number up by 2 each time — so a climbs to a+4.
- Name the five integers
- Average them to get b
- Middle term is the average
- Substitute b = a + 2