Problem
Evaluate a nested defined operation
A new symbol is defined by a three-step recipe. Multiply the two numbers, add them, then divide the first by the second. The expression nests one inside brackets. Find the value of the expression.
Reasoning · Grade 6-2 Mixed Fraction and Decimal Calculations
Your answer
Watch out It looks messy, but two equal numbers give half of one straight away.
How to solve
Strategy Identify Subproblems — A nested symbol is really two ordinary questions stacked up, so I split it into the inner one and the outer one and do them in that order, never trying to hold both in my head at once. Each subproblem is just 'multiply, add, divide' with the two numbers I have been handed. Afterwards I look for a pattern in the definition to check the work: 6 club 6 and 3 club 3 are both of the form a club a, so if there is a simple rule for that case it should reproduce both of my answers independently. Rewriting the definition upside down gives a second, quite different check.
1STEP 1
Read the definition as a three-step recipe
The symbol means the product over the sum.
a ♣ b = (a × b)/(a+b)
A made-up symbol can be scary, but the definition is a filling-in-the-blanks instruction. Once you say out loud 'product on top, sum on the bottom', the strangeness disappears and it becomes ordinary Grade 5 arithmetic.
6.EE.A.2Organize Information In More Ways2STEP 2
Do the inner club first, because it is inside the brackets
The bracketed one comes to 3.
6 ♣ 6 = (6 × 6)/(6+6) = 36/12 = 3
Order of operations puts brackets first, and here the bracket collapses to a single tidy whole number, which makes the second step no harder than the first.
5.OA.A.1Identify SubproblemsThe inner bracket has to be finished first, because its answer is one of the two numbers the outer rule needs.
Why?
The outer expression is built from two finished values, so it cannot be evaluated until both of them are numbers.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The definition puts each input into one definite slot, so the inner answer has one place to go and cannot be swapped with the other.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Feed that answer back into the outer club
Feeding it in, the outer gives 1 1/2.
3 ♣ (6 ♣ 6) = 3 ♣ 3 = (3 × 3)/(3+3) = 9/6 = 3/2 = 1 1/2
A fraction bar simply means divide, so 9/6 is 9 shared into 6 equal parts — one whole and a half left over. No new skill is needed for the outer club; it is the identical recipe run a second time.
5.NF.B.3Identify Subproblems4STEP 4
Find the pattern for a club a and use it as a check
The halving rule confirms it.
a ♣ a = (a × a)/(a+a) = (a × a)/(2 × a) = a/2
Spotting that the whole problem is 'halve 6, then halve the result' turns it into 6, 3, 1 1/2 — a check a young solver can do in their head, arrived at along a completely different route from the arithmetic above.
6.EE.A.3Look For A PatternAnswer
1 1/2
9 ÷ 6 = 3/2
The club of two positive numbers must always come out smaller than either of them, because the top a x b is being divided by a + b, which is bigger than either a or b on its own. Both results obey that: 6 club 6 = 3 is less than 6, and 3 club 3 = 1 1/2 is less than 3. The size is sensible too — halving twice takes 6 down to 1.5, not to something near 36 or 12, so an answer in the ones rather than the tens is expected. The answer is a pure number with no units, as it should be, and 3/2 is in lowest terms and written as the mixed number 1 1/2 the way the book asks.
Takeaway
A new symbol is just a recipe — do the inside bracket first, and here the recipe turns out to mean nothing more than 'halve it'.
- Read the definition as a three-step recipe
- Do the inner club first, because it is inside the brackets
- Feed that answer back into the outer club
- Find the pattern for a club a and use it as a check