Problem

Evaluate a custom-defined operation symbol

The symbol $\heartsuit$ is defined as follows. Find the number that belongs in $\square$ .

$a \heartsuit b = a \times 7 + a \times b$

$\square \heartsuit 8 = 135$

Operations

Show worked solution try it yourself first ✦

1 · Understandwhat's really being asked

A new symbol heart is defined so that a heart b means 'multiply a by 7, then add a times b.' We are told that (box) heart 8 equals 135, and we must find the number that goes in the box.

Givens

The rule: a heart b = a times 7 + a times b
box heart 8 = 135

Unknowns

The number that belongs in the box

Constraints

The box stands for one whole number
Both products use the same first number (the box)

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check#7 Identify Subproblems

Substituting into the defined rule turns a heart 8 into a single multiplication once we notice both terms share the box as a factor (the distributive pattern). Then a quick guess-and-check (or dividing) recovers the box.

3 · Execute3 carry out the plan

1Substitute into the rule

#7 Identify Subproblems 3.OA.D.8

Replace a with the box and b with 8 in the rule a heart b = a times 7 + a times b.

\square \heartsuit 8 = \square \times 7 + \square \times 8

Following a defined symbol is just plugging numbers into a recipe, a Grade 3 skill.

2Combine the two products

#5 Look for a Pattern 3.OA.B.5

Both products multiply the same box. Adding 7 of the box and 8 of the box gives 15 of the box.

\square \times 7 + \square \times 8 = \square \times 15

Seven groups plus eight groups of the same size is fifteen groups; this is the distributive property kids already use.

🧠 Why? · Feynman

$\square \times 7 + \square \times 8 = \square \times 15$

Why?

The two products both multiply the same box, so they merge into the box multiplied by seven-and-eight, and seven and eight make fifteen.

Why?

Because $\square \times 7$ and $\square \times 8$ share the box as a factor, adding them is just the box taken over 7 and over 8 combined into one product, $\square \times (7 + 8)$ .

🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.

Why?

Inside that single product the 7 and the 8 join to make 15, so $\square \times (7 + 8)$ is $\square \times 15$ .

🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

3Find the box

#6 Guess and Check 3.OA.A.3

We need box times 15 = 135. Since 9 times 15 = 135, the box is 9.

\square \times 15 = 135 \Rightarrow \square = 9

Testing the friendly factor 9 (because 9 times 15 = 135) confirms the answer at once.

Answer: 9

4 · Reviewdoes it hold up?

Check: 9 heart 8 = 9 times 7 + 9 times 8 = 63 + 72 = 135, which matches the given total exactly.

Another way: Instead of combining, compute each part as an unknown total: box times 7 plus box times 8 = 135, then divide 135 by 15 directly (work backwards, tool 11) to get 9.

Standardsmin grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Substituting values into the defined two-step operation
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Combining box times 7 and box times 8 into box times 15 using the distributive property
3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the box from box times 15 = 135

💡Takeaway. A made-up symbol is just a recipe; once you see both pieces share the same number, 7 groups plus 8 groups is simply 15 groups!

▶ Practice — 10 problems