Math Mastery Series

1.0 pt Answer: Fully correct quotient and remainder explicitly written.

Quick Answers (Part A): 1: 87 • 2: 122 R 3 • 3: 369 • 4: 364 R 1 • 5: 504

Long Division Single and Double Divisor • Key Page 1 of 3 • Part A Solutions (10 pts) →

Teacher Reference

Long Division Single and Double Divisor • Key

Part B: 10 Points

2 Part B Solutions: Two-Digit Divisors

2 pts each: 1 pt complete work • 1 pt correct quotient/remainder (10 pts total)

6 468 ÷ 12

Exact (2 pts)

39

12

468

-36 

108

-108

0

Steps (1 pt work):

• 46 ÷ 12 = 3 R 10

• 108 ÷ 12 = 9 R 0

Check: 39 × 12 = 468 ✓

Answer (1 pt): 39

7 895 ÷ 14

Remainder (2 pts)

63 R 13

14

895

-84 

55

-42

13

Steps (1 pt work):

• 89 ÷ 14 = 6 R 5

• 55 ÷ 14 = 3 R 13

Check: (63 × 14) + 13 = 895 ✓

Answer (1 pt): 63 R 13

8 1,512 ÷ 24

Exact (2 pts)

63

24

1512

-144 

72

-72

0

Steps (1 pt work):

• 151 ÷ 24 = 6 R 7

• 72 ÷ 24 = 3 R 0

Check: 63 × 24 = 1512 ✓

Answer (1 pt): 63

9 3,745 ÷ 15

Remainder (2 pts)

249 R 10

15

3745

-30  

745

-60 

145

-135

10

Steps (1 pt work):

• 37÷15=2, 74÷15=4

• 145÷15=9 R 10

Check: (249 × 15) + 10 = 3745 ✓

Answer (1 pt): 249 R 10

10 6,804 ÷ 27

Exact (2 pts)

252

27

6804

-54  

1404

-135 

54

-54

0

Steps (1 pt work):

• 68÷27=2, 140÷27=5

• 54÷27=2 R 0

Check: 252 × 27 = 6804 ✓

Answer (1 pt): 252

Part B Quick Reference (10 pts)

Q6: 39

Q7: 63 R 13

Q8: 63

Q9: 249 R 10

Q10: 252

Part A (10) + Part B (10) = 20 pts Part C Next →

Long Division Single and Double Divisor • Key Page 2 of 3 • Part B Solutions (10 pts) →

Teacher Reference

Long Division Single and Double Divisor • Key

Part C: 8 Points

3 Part C Solutions: Word Problems

2 pts each: 1 pt equation & work • 1 pt answer with units (8 pts total)

11 School Field Trip

2 Points

Equation (1 pt work): 258 ÷ 8

32 R 2

8

258

-24 

18

-16

2

• 25 ÷ 8 = 3 R 1

• 18 ÷ 8 = 2 R 2

Check: (32×8)+2 = 258 ✓

Remainder Interpretation (1 pt): 32 vans hold 256 people. The 2 extra people require 1 more van, so 33 vans are needed.

Final Answer (1 pt): 33 vans

12 Art Markers Distribution

2 Points

Equation (1 pt work): 1,365 ÷ 5

273

5

1365

-10  

365

-35 

15

-15

0

• 13 ÷ 5 = 2 R 3

• 36 ÷ 5 = 7 R 1

• 15 ÷ 5 = 3 R 0

Check: 273 × 5 = 1365 ✓

Interpretation: The shipment splits completely evenly across all 5 schools with 0 markers remaining.

Final Answer (1 pt): 273 markers

13 Bakery Packaging

2 Points

Equation (1 pt work): 1,520 ÷ 24

63 R 8

24

1520

-144 

80

-72

8

• 152 ÷ 24 = 6 R 8

• 80 ÷ 24 = 3 R 8

Check: (63×24)+8 = 1520 ✓

Remainder Interpretation: 63 represents full boxes (0.5 pt); 8 represents the leftover cookies (0.5 pt).

Final Answer (1 pt): 63 boxes, 8 left over

14 Arena Seating Sections

2 Points

Equation (1 pt work): 3,192 ÷ 14

228

14

3192

-28  

392

-28 

112

-112

0

• 31 ÷ 14 = 2 R 3

• 39 ÷ 14 = 2 R 11

• 112 ÷ 14 = 8 R 0

Check: 228 × 14 = 3192 ✓

Interpretation: Each of the 14 identical sections holds exactly 228 seats with zero remainder.

Final Answer (1 pt): 228 seats

Long Division Single and Double Divisor • Key Page 3 of 3 • Part C: 8 pts • Assessment Total: 28 Points

Numerical Problem Solving

Distributive Property Expressions Quiz

Name:

C Part 3: 4-Term Numerical Distribution (No Variables)

2 pts each: 1 pt for distributing across all terms • 1 pt for evaluated total.

11 (2 pts)

4-Term Sum

4(50 + 30 + 10 + 5)

Calculated Value:

12 (2 pts)

4-Term Mixed

6(40 + 20 - 5 - 3)

Calculated Value:

13 (2 pts)

Double Distribute

5(20 + 6) + 3(40 + 8)

Calculated Value:

14 (2 pts)

Double Distribute

8(30 + 5) - 4(20 + 5)

Calculated Value:

15 (2 pts)

Expanded Form

3(1000 + 400 + 60 + 8)

Calculated Value:

Distributing Across 4+ Terms

The 4-Term Rule:

a(b + c + d + e) = ab + ac + ad + ae

Key Tip: Multiply the factor outside by every single term inside before summing!

Part 3: 10 Points Turn to Page 4 →

Distributive Property Expressions Quiz • Page 3 of 4 (Part 3: 10 pts) Continue to Page 4 for 4-Term Decimals & Applications →

Numerical Problem Solving

Distributive Property Expressions Quiz

Name:

D Part 4: Decimals, Fractions & Real-World 4-Term Problems

2 pts each: Distribute across all terms, show work, and evaluate final total.

16 (2 pts)

4-Term Decimals

4(10 + 5 + 0.5 + 0.25)

Calculated Value:

17 (2 pts)

4-Term Fractions

8(10 + 4 + ½ + ¼)

Calculated Value:

18 (2 pts)

4-Term Expression

7(30 + 8) + 45 + 15

Calculated Value:

19 (2 pts)

4-Item Equipment

A coach buys 5 gear packs. Each pack has 12 basketballs, 8 soccer balls, 6 volleyballs, and 4 dodgeballs. Evaluate using: 5(12 + 8 + 6 + 4)

Total Balls:

20 (2 pts)

4-Item Catering

Caterers prepare food for 8 tables. Each table receives 25 rolls, 15 skewers, 10 mini quiches, and 5 tarts. Evaluate using: 8(25 + 15 + 10 + 5)

Total Food Items:

Checking Multi-Term Work

You can verify your distributed answer by adding inside the parentheses first and multiplying:

• 5(12 + 8 + 6 + 4) = 5(30) = 150 ✓

• 8(25 + 15 + 10 + 5) = 8(55) = 440 ✓

Part 4: 10 Points Total Quiz: 40 Points

Distributive Property Expressions Quiz • Page 4 of 4 (Part 4: 10 pts) End of Assessment • Total: 40 Points

B Part 2 Solutions: Advanced Terms & Combining Like Terms

2 pts each: 1 pt distribution/combining steps • 1 pt simplified expression (10 pts)

6 -3(-4k - 8)

Double Negative

Expanded Steps (1 pt work):

= (-3 • -4k) - (-3 • 8)

= 12k + 24

Teacher Note: Both products result in positive terms because negative × negative = positive.

Answer (1 pt): 12k + 24

7 2x(5x - 3)

Variable Factor

Expanded Steps (1 pt work):

= (2x • 5x) - (2x • 3)

= 10x² - 6x

Teacher Note: \( x \cdot x = x^2 \). Writing \( 10x - 6x \) or \( 4x \) is a major misconception.

Answer (1 pt): 10x² - 6x

8 ⅓(9p - 24)

Fractional

Expanded Steps (1 pt work):

= (⅓ • 9p) - (⅓ • 24)

= 3p - 8

Teacher Note: Dividing both terms by 3 is mathematically equivalent: \( \frac{9p}{3} - \frac{24}{3} = 3p - 8 \).

Answer (1 pt): 3p - 8

9 5(2n + 4) + 3n

Combine Like Terms

Steps (1 pt work):

1. Distribute 5: 10n + 20 + 3n

2. Combine like terms: (10n + 3n) + 20

= 13n + 20

Teacher Note: Do not add 5 and 3n first; distribution has multiplication precedence!

Answer (1 pt): 13n + 20

10 7 - 2(3x - 4) + 5x

Multi-Step

Steps (1 pt work):

1. Distribute -2: 7 - 6x + 8 + 5x

2. Combine: (-6x + 5x) + (7 + 8)

= -x + 15 (or 15 - x)

Key Pitfall: Distributing \(-2\) gives \(+8\), not \(-8\). Do not compute \(7 - 2 = 5\).

Answer (1 pt): -x + 15

Part 2 Quick Reference (10 pts)

Q6: 12k + 24

Q7: 10x² - 6x

Q8: 3p - 8

Q9: 13n + 20

Q10: -x + 15 (or 15 - x)

Part 1 (10) + Part 2 (10) = 20 pts Part 3 Next →

Distributive Property Expressions • Key Page 2 of 4 • Part 2 Solutions (10 pts) →

Teacher Reference

Distributive Property Expressions • Key

Part 3: 10 Points

C Part 3 Solutions: 4-Term Numerical Distribution (No Variables)

2 pts each: 1 pt for expanded distributive products • 1 pt for evaluated total (10 pts)

11 4(50+30+10+5)

2 Points

Distribute to all 4 terms (1 pt):

= (4×50) + (4×30) + (4×10) + (4×5)

= 200 + 120 + 40 + 20

= 380

Check: 4 × (95) = 380 ✓

Calculated Value (1 pt): 380

12 6(40+20-5-3)

2 Points

Distribute to all 4 terms (1 pt):

= (6×40) + (6×20) - (6×5) - (6×3)

= 240 + 120 - 30 - 18

= 312

Check: 6 × (52) = 312 ✓

Calculated Value (1 pt): 312

13 5(20+6)+3(40+8)

2 Points

Double Distribution (1 pt):

= (5×20) + (5×6) + (3×40) + (3×8)

= 100 + 30 + 120 + 24

= 274

Check: (5 × 26) + (3 × 48) = 130 + 144 = 274 ✓

Calculated Value (1 pt): 274

14 8(30+5)-4(20+5)

2 Points

Double Distribution (1 pt):

= (8×30) + (8×5) - (4×20) - (4×5)

= 240 + 40 - 80 - 20

= 180

Check: (8 × 35) - (4 × 25) = 280 - 100 = 180 ✓

Calculated Value (1 pt): 180

15 3(1000+400+60+8)

2 Points

Place-Value Distribution (1 pt):

= (3×1000) + (3×400) + (3×60) + (3×8)

= 3,000 + 1,200 + 180 + 24

= 4,404

Check: 3 × 1,468 = 4,404 ✓

Calculated Value (1 pt): 4,404

Part 3 Quick Reference (10 pts)

Problem 11: 380

Problem 12: 312

Problem 13: 274

Problem 14: 180

Problem 15: 4,404

Parts 1–3 Total: 30 pts Part 4 Next →

Distributive Property Expressions • Key Page 3 of 4 • Part 3 Solutions (10 pts) →

Teacher Reference

Distributive Property Expressions • Key

Part 4: 10 Points

D Part 4 Solutions: 4-Term Decimals, Fractions & Word Problems

2 pts each: 1 pt for numerical steps • 1 pt for final value with units (10 pts)

16 4(10+5+0.5+0.25)

2 Points

Distribute to 4 terms (1 pt):

= (4×10) + (4×5) + (4×0.5) + (4×0.25)

= 40 + 20 + 2 + 1

= 63

Check: 4 × 15.75 = 63 ✓

Calculated Value (1 pt): 63

17 8(10+4+½+¼)

2 Points

Distribute to 4 terms (1 pt):

= (8×10) + (8×4) + (8×½) + (8×¼)

= 80 + 32 + 4 + 2

= 118

Check: 8 × 14.75 = 118 ✓

Calculated Value (1 pt): 118

18 7(30+8)+45+15

2 Points

Distribute & Add (1 pt work):

= (7×30) + (7×8) + 45 + 15

= 210 + 56 + 45 + 15

= 326

Check: (7 × 38) + 60 = 266 + 60 = 326 ✓

Calculated Value (1 pt): 326

19 4-Item Gear Packs

2 Points

Expression: 5(12 + 8 + 6 + 4)

= (5×12) + (5×8) + (5×6) + (5×4)

= 60 + 40 + 30 + 20

= 150 balls

Total Balls (1 pt): 150 balls

20 4-Item Catering

2 Points

Expression: 8(25 + 15 + 10 + 5)

= (8×25) + (8×15) + (8×10) + (8×5)

= 200 + 120 + 80 + 40

= 440 items

Total Items (1 pt): 440 items

Part 4 Quick Matrix

Q16: 63

Q17: 118

Q18: 326

Q19: 150 balls

Q20: 440 items

Part 3 (10) + Part 4 (10) = 20 pts Total Quiz: 40 Points

Distributive Property Expressions • Key Page 4 of 4 • End of Key • 40 Points Total