Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #4
Grade 6 arithmeticPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The data is given in a jumbled order, so the first move is Tool #2 (Make a Systematic List): sort the eight numbers from smallest to largest. That single ordered list makes the median and mode obvious by inspection and keeps us from miscounting duplicates. Tool #7 (Identify Subproblems) is the natural frame for the whole problem because the question is really three small problems — find mean, find median, find mode — followed by one easy addition.
Sort the data
Sort the eight numbers from smallest to largest so repeats and middle values stand out.
Lining up data on a number line — exactly what a Grade 5 line plot does — makes the order easy to read.
5.MD.B.2Make A Systematic ListFind the mode
One value repeats most in the sorted list, making the mode 3.
Picking the most frequent value is the simplest Grade 6 measure-of-center reading.
6.SP.B.5Identify SubproblemsFind the median
With eight values, average the 4th and 5th sorted terms to get the median 2.5.
Averaging the two middle values when the count is even is part of the Grade 6 "summarize a data set" standard.
6.SP.B.5Identify SubproblemsFind the mean
Add every value and divide by eight to get the mean 2.
Sum-over-count is the textbook Grade 6 definition of the mean.
The mean of the eight numbers equals 2.
▸ Why?
The mean is the eight values' total shared out equally among them; that total is 16, and 16 shared among 8 values comes to 2 each.
▸ Why?
Adding the eight separate values gives one combined total of 16, because parts joined back with nothing left out make up the whole.
▸ Why?
Sharing that total of 16 equally among the 8 values is exactly what the mean is — the total divided by the count — so 16 split into 8 equal shares is 2 each.
Add the three values
Add the three centers — mean, median, and mode — to reach the final answer.
Adding whole numbers and a decimal to the tenths is a Grade 5 decimal-arithmetic skill.
5.NBT.B.7Identify SubproblemsThis AMC 8 problem only needs Grade 6 measures of center — mean, median, and mode — you already know!
- Sort the data
- Find the mode
- Find the median
- Find the mean
- Add the three values
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