Problem
Compute with newly defined operations
Two invented symbols come with recipes: a ▲ b = (a ÷ b) + (b × a) and a ♣ b = (a + b) ÷ a × b. The number on the left is a and the one on the right is b. Find the value of 0.6 ▲ 1 1/2 and of 2 2/5 ♣ 1.2.
Reasoning · Grade 6-1 Fraction and Decimal Calculations
Your answer
Watch out Filling a and b either way is tempting, but the left one is a and swapping changes the value.
How to solve
Strategy Identify Subproblems — Each definition is really two or three small calculations glued together with + or ×, so the safe way through is tool 7: compute one bracket at a time, write the value down, and only then combine. Before any of that, tool 15 does the single most useful piece of housekeeping — rewriting every decimal as a fraction so the whole calculation stays in one language instead of switching between decimals and fractions in the middle of a division. Tool 5 is what makes the substitution reliable: the definition is a template, and the job is to match the pattern, dropping the left number into every a slot and the right number into every b slot.
1STEP 1
Read the new symbol as a fill-in-the-blanks template
The new symbol is just a fill-in template.
Treating the definition as a form to fill in stops the commonest error, which is remembering the shape of the recipe but plugging the two numbers in the wrong way round.
5.OA.A.1Look For A PatternReading the new symbol as a fill-in-the-blanks template stops the two numbers being plugged in the wrong way round.
Why?
The definition puts each of the two inputs into one definite slot, so the first number and the second are not interchangeable.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Once the slots are filled the expression is just its ordinary pieces put together, so no new arithmetic is needed.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Change every decimal into a fraction first
Change every decimal into a fraction first.
0.6 = 3/5, 1.2 = 6/5, 1 1/2 = 3/2, 2 2/5 = 12/5
Fractions and decimals are two spellings of the same numbers, and picking one spelling for the whole problem means every step uses one familiar rule instead of two.
4.NF.C.6Organize Information In More Ways3STEP 3
(1) Fill the triangle template and split it into two pieces
The first template splits into two pieces.
0.6 ▲ 1 1/2 = (3/5 ÷ 3/2) + (3/2 × 3/5)
The plus sign in the middle guarantees the two brackets never interfere with each other, so each one can be finished and written down before the other is even looked at.
5.OA.A.1Identify Subproblems4STEP 4
(1) Compute the two pieces and add
Adding the pieces gives 1 3/10.
2/5 + 9/10 = 4/10 + 9/10 = 13/10 = 1 3/10 = 1.3
Dividing by 3/2 shrinks a number to two-thirds of itself and multiplying by 3/2 grows it by half again, so one piece is smaller than 0.6 and one is larger — and their sum landing just above 1 is exactly what those two effects predict.
6.NS.A.1Identify Subproblems5STEP 5
(2) Fill the club template and respect left-to-right order
The second must keep the left-to-right order.
2 2/5 ♣ 1.2 = (12/5 + 6/5) ÷ 12/5 × 6/5 = 18/5 ÷ 12/5 × 6/5
Multiplication and division sit on the same level, so a chain of them is read like a sentence — strictly left to right — and noticing that is the whole difference between the right answer and a wrong one here.
5.OA.A.1Look For A Pattern6STEP 6
(2) Finish the chain
Finishing the chain gives 1 4/5.
18/5 × 5/12 × 6/5 = 3/2 × 6/5 = 18/10 = 9/5 = 1 4/5 = 1.8
Because every number was already written in fifths and halves, each step is one cancel-then-multiply, and no long decimal division is ever needed.
6.NS.A.1Identify SubproblemsAnswer
1 3/10, 1 4/5
2/5 + 9/10 = 13/10
Both answers are the right size. In (1), 0.6 ÷ 1.5 = 0.4 and 1.5 × 0.6 = 0.9, and 0.4 + 0.9 = 1.3 — the decimal arithmetic agrees with the fraction arithmetic exactly, so no cancelling slipped. In (2), 2.4 + 1.2 = 3.6, then 3.6 ÷ 2.4 = 1.5, then 1.5 × 1.2 = 1.8, again matching. As a sanity check on the order of operations in (2), doing it the wrong way (3.6 ÷ (2.4 × 1.2) = 3.6 ÷ 2.88 = 1.25) gives a different number, which confirms that the left-to-right reading really is doing work and was not an accident.
Takeaway
A new symbol is just a fill-in-the-blanks recipe: put the left number in every a slot, the right number in every b slot, and then obey the ordinary order of operations.
- Read the new symbol as a fill-in-the-blanks template
- Change every decimal into a fraction first
- (1) Fill the triangle template and split it into two pieces
- (1) Compute the two pieces and add
- (2) Fill the club template and respect left-to-right order
- (2) Finish the chain