Problem
Apply a newly defined symbol rule
A new symbol is defined by a ◆ b = (b − a) × b. Both boxes hold single digits. Find every single-digit pair making □ ◆ □ = 24 and write each as a ◆ equation.
Reasoning · Grade 2-2 Rules and Defined Operations
Your answer
Watch out It looks swappable like multiplication, but the second number is used twice, so 2 ◆ 6 and 6 ◆ 2 differ.
How to solve
Strategy Make a Systematic List — Since the result 24 = (b − a) × b, the second box b must be a number that divides 24. There are only a handful of single-digit values for b, so we list each one, work out what a would have to be, and keep only the pairs where a is also a single digit.
1STEP 1
See what b must be
The rule forces the second number to divide 24: 1, 2, 3, 4, 6, 8.
(b - a) × b = 24
Listing the factors of 24 narrows a huge search down to just six candidate values for b.
4.OA.B.4Make A Systematic ListBecause b has to divide 24 exactly, the search shrinks from every number down to the six divisors of 24.
Why?
A number that 24 splits into evenly hands over its partner at the same time, so the divisors can be listed in pairs and none is missed.
🧱Divisors come in pairsEach divisor hands over its partner for free, so divisors pair off two by two unless one is its own partner.
Why?
Every value of b that leaves a remainder is thrown out immediately, so only the handful that divide cleanly are worth testing.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
2STEP 2
For each b, find the matching a
Testing each, only 6 and 8 leave the first number a single digit.
b=6: (6-2) × 6 = 24, b=8: (8-5) × 8 = 24
Treating 24 ÷ b = (b − a) turns each guess into a quick division-and-subtraction check, and only two of them land on a single-digit a.
3.OA.A.4Guess And Check3STEP 3
Write the two equations
So the true equations are 2 ◆ 6 = 24 and 5 ◆ 8 = 24.
2 ◆ 6 = (6-2) × 6 = 24, 5 ◆ 8 = (8-5) × 8 = 24
Collecting the surviving pairs from the list gives the complete answer with nothing missed and nothing extra.
3.OA.C.7Make A Systematic ListAnswer
2 ◆ 6 = 24, 5 ◆ 8 = 24
Check both: (6-2) × 6 = 4 × 6 = 24 ✓ and (8-5) × 8 = 3 × 8 = 24 ✓. Because b had to be a single-digit factor of 24, and we tested all of them, no other single-digit pair is possible.
Takeaway
Spotting that one box must divide 24 turns the puzzle into a short list you can check by hand — just Grade 3–4 factor sense!
- See what b must be
- For each b, find the matching a
- Write the two equations