Problem
Evaluate nested defined operations
Two new signs are defined by a●b = a×b-1 and a▲b = a+b+1. The expression nests the two signs inside parentheses. Work out the value of 5▲(3●(6▲4)).
Reasoning · Grade 5-1 Finding the Hidden Rule
Your answer
Watch out Left-to-right is tempting, but the innermost bracket comes first.
How to solve
Strategy Identify Subproblems — The expression looks intimidating only because two unfamiliar symbols are stacked inside one another. Peeling it apart turns it into three ordinary one-line calculations, each of which is easier than anything in a normal Grade 5 mixed-operation exercise. So I identify the three subproblems from the inside out, and before starting I rewrite each strange rule in plain arithmetic words so that substituting is mechanical rather than mysterious.
1STEP 1
Say each new rule in ordinary words
Write each rule out in words.
a ● b = a × b - 1, a ▲ b = a + b + 1
Testing a rule on a pair of small numbers costs ten seconds and catches a misread definition before it can wreck three steps of work.
5.OA.A.2Solve An Easier Related Problem2STEP 2
Find the innermost bracket and do it first
The innermost bracket gives 11.
6 ▲ 4 = 6 + 4 + 1 = 11
Exactly the same habit as with ordinary mixed calculations: whatever is wrapped in the innermost parentheses is a finished number before anything outside it may be touched.
5.OA.A.1Identify Subproblems3STEP 3
Substitute and do the middle operation
Substituting, the middle gives 32.
3 ● 11 = 3 × 11 - 1 = 33 - 1 = 32
The arithmetic inside is small — 3 × 11 = 33 is a Grade 3 multiplication fact — so the only thing that can go wrong is forgetting the '-1' the rule tacks on.
3.OA.C.7Identify Subproblems4STEP 4
Do the outermost operation
The outermost gives 38.
5 ▲ 32 = 5 + 32 + 1 = 38
The last step is a two-digit addition; all the difficulty of the problem was in the order, not in the numbers.
5.OA.A.1Identify Subproblems5STEP 5
Lay the three steps out as a chain
Lay the three steps out as one chain.
5 ▲ (3 ● (6 ▲ 4)) = 5 ▲ (3 ● 11) = 5 ▲ 32 = 38
Rewriting a nested expression as a short vertical chain removes the need to hold three half-finished thoughts in your head at once.
5.OA.A.1Organize Information In More Ways6STEP 6
Confirm that the order really mattered
Left to right gives 58, so order matters.
wrong order: 5 ▲ 3 = 9, 9 ● 6 = 53, 53 ▲ 4 = 58 ≠ 38
Seeing a wrong order give a visibly different answer is the most convincing argument that reading parentheses carefully is not a formality.
5.OA.A.1Organize Information In More WaysThe order really mattered, because doing the operations the other way round lands on a different value.
Why?
One worked example where swapping the order changes the answer is enough to prove the operations are not free to be rearranged.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
Each bracket is a finished value fed into the next rule, so the chain is built from the inside out and cannot be reassembled at will.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
38
5 + 32 + 1 = 38
The answer is a plain whole number, which is right since both rules only add, multiply and adjust by 1 on whole numbers. A rough size check supports 38: the last operation adds 5, adds 1, and adds whatever the bracket gave, so the answer must be 6 more than the bracket's value of 32 — and 38 is exactly 32 + 6. Likewise the bracket 3 x 11 - 1 = 32 must be just under 33, which it is, and 6 + 4 + 1 = 11 is just over 10 as expected. Each step's result is checked by re-reading its rule: multiply-then-minus-one for the bullet, add-then-plus-one for the triangle.
Takeaway
A made-up symbol is just a recipe — read it out loud, do the innermost bracket first, and the scary-looking expression becomes three easy lines.
- Say each new rule in ordinary words
- Find the innermost bracket and do it first
- Substitute and do the middle operation
- Do the outermost operation
- Lay the three steps out as a chain
- Confirm that the order really mattered