A new symbol can define a custom operation

4.OA.A.35.OA.A.1 · take · grade 5

Archetype: Apply a Newly Defined Operation · step in a 5-type progression

For any two numbers $a$ and $b$ , the operation $\blacklozenge$ is defined by $a \blacklozenge b = a \times b + 30$ . Find the value of $(32 \blacklozenge 15) \blacklozenge 23$ .

Show solution

Understand

A new symbol is defined so that a-diamond-b means a times b, then plus 30. Using this rule, first compute 32-diamond-15, then take that result diamond 23.

Givens

The operation is defined by a ◆ b = a × b + 30
We must evaluate (32 ◆ 15) ◆ 23

Unknowns

The value of (32 ◆ 15) ◆ 23

Constraints

The inner parentheses must be evaluated first
Each ◆ means multiply the two numbers and add 30

Plan

#7 Identify Subproblems · also uses: #5 Look for a Pattern

The expression has a parentheses-first structure, so we split it into two subproblems: first evaluate the inner 32 ◆ 15, then feed that result into a second application of the same rule with 23. Each application follows the identical pattern multiply-then-add-30.

Execute

#7 Identify Subproblems 5.OA.A.1

Apply the rule to 32 and 15: multiply, then add 30. 32 x 15 = 480, and 480 + 30 = 510.

32 \blacklozenge 15 = 32 \times 15 + 30 = 480 + 30 = 510

The parentheses tell us to finish the inside first, exactly like order of operations says.

#5 Look for a Pattern 4.OA.A.3

Now apply the same rule to 510 and 23: 510 x 23 = 11730, and 11730 + 30 = 11760.

510 \blacklozenge 23 = 510 \times 23 + 30 = 11730 + 30 = 11760

The custom symbol is just a recipe; reusing the same multiply-then-add-30 pattern finishes the job.

Answer: 11760

Review

The answer is dominated by 510 x 23 = 11730; adding 30 nudges it to 11760, which is sensibly just above 11730. Both steps follow the defined rule exactly, and the inside-first ordering was respected.

Use Look for a Pattern (tool 5) by noticing every application adds the same constant 30, so you could compute both products first (480 and 11730) and add 30 once per step, confirming 11760.

Standards · min grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Carrying out the multiply-then-add steps required by the custom operation.
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Evaluating the inner parentheses (32 ◆ 15) before the outer operation.

💡 A made-up symbol is just a recipe -- do the inside first, then follow the same multiply-and-add-30 steps again!