Name: ___________________________ Date: ______________ Class: ____________
LEVEL 2 · ENRICHMENT

The Number System & Expressions

Standards: 6.NS.A.1 (Dividing Fractions) · 6.NS.B.4 (GCF/LCM) · 6.EE.A.1–4 (Exponents & Expressions)

Directions: Solve each challenge. Show all steps and use correct order of operations. For "Explain" prompts, write at least one complete sentence justifying your reasoning. Finish with the Stretch Problem.

Challenge Problems

1. Evaluate using order of operations: 2³ + 3² − 4 × 2 + 1 MULTI-STEP

2. Find the GCF and LCM of 24 and 36 using prime factorization. REASONING

Explain why GCF uses the lowest powers of shared primes and LCM uses the highest powers.

3. Simplify: 6(2x + 3) − 4(x − 1) MULTI-STEP

4. A recipe uses ⅔ cup of flour per batch. How many full batches can you make with 5 cups of flour? REAL-WORLD

5. Write an expression for "five less than three times a number n, squared," then evaluate when n = 4. OPEN-ENDED

Explain how the placement of the square changes the answer.

6. Two numbers have a GCF of 8 and an LCM of 96. One number is 32. Find the other. REASONING

Use the rule GCF × LCM = product of the two numbers.

7. Simplify: 2(3x + 4) + 3(2x − 1) − (x + 5) MULTI-STEP

8. Without a calculator, find: 2⁵ − 3³ + 4² REASONING

9. Pencils come in packs of 8 and erasers in packs of 6. What is the smallest number of each you can buy to have the same number of pencils and erasers? How many packs of each? REAL-WORLD

10. Which is larger: 2⁸ or 8²? EXPLAIN

Explain your answer using powers of 2 instead of just calculating both.

Real-World Application

11. A party for 48 guests needs plates (packs of 8), cups (packs of 12), and napkins (packs of 10). (a) How many packs of each are needed? (b) Which item has the most leftovers? (c) Find the LCM of 8, 12, and 10. INVESTIGATION

★ Stretch Problem

This is a preview of 7th-grade equations. If 3(2x − 1) = 4x + 9, what is the value of x? Show every step, then check your answer by substituting back in.

Answer Key

  1. 10 — 8 + 9 − 8 + 1 = 10
  2. GCF = 12 (2²×3), LCM = 72 (2³×3²). 24 = 2³×3, 36 = 2²×3².
  3. 8x + 22 — 12x + 18 − 4x + 4 = 8x + 22
  4. 7 full batches — 5 ÷ ⅔ = 5 × 3/2 = 15/2 = 7.5, so 7 full
  5. (3n)² − 5 = 139 when interpreted as (12)² − 5; if only n is squared, 3n² − 5 = 43
  6. 24 — 8 × 96 = 768; 768 ÷ 32 = 24
  7. 11x — 6x + 8 + 6x − 3 − x − 5 = 11x
  8. 21 — 32 − 27 + 16 = 21
  9. 24 each — LCM(8,6) = 24; 3 packs pencils, 4 packs erasers
  10. 2⁸ is larger — 8² = (2³)² = 2⁶, and 2⁸ > 2⁶
  11. (a) 6 plate packs, 4 cup packs, 5 napkin packs; (b) napkins leftover 2; (c) LCM(8,12,10) = 120
  12. Stretch: x = 6 — 6x − 3 = 4x + 9 → 2x = 12 → x = 6. Check: 3(11) = 33 = 24 + 9 ✓