Factor Pairs Slides Grade 4 Mathematics • Number Theory
Factor Pairs Hunt
Finding All Factor Pairs for Whole Numbers 1 to 100
Core Concept
A factor pair is two whole numbers that multiply together to yield a specific product.
Key Strategy
Systematic listing using a T-chart starting at 1 prevents missing hidden factors.
Target Range
Mastering factor relationships for any whole number from 1 through 100.
Standard: CCSS.MATH.CONTENT.4.OA.B.4 Lesson 1 • Direct Instruction
Concept Foundation
What Is a Factor Pair?
Mathematical Definition
A factor is a whole number that divides evenly into another whole number with no remainder.
A factor pair consists of two numbers whose product is the target number:
Factor × Factor = Product
Example: Factor Pairs of 12
1 × 12 = 12 Pair: 1 and 12
2 × 6 = 12 Pair: 2 and 6
3 × 4 = 12 Pair: 3 and 4
All individual factors of 12 in order: 1, 2, 3, 4, 6, 12
Rule: Every whole number greater than 1 has at least one factor pair (1 and itself). Slide 2
Methodology
The Systematic T-Chart Strategy
1 Start with 1
Every number is divisible by 1. Always write down 1 × [Number] as your first factor pair.
1 × 24 = 24
2 Test in Order
Test integers in ascending order: 2, 3, 4, 5, etc. Check using mental math or division. Skip numbers that leave a remainder.
2 × 12, 3 × 8, 4 × 6
3 Know When to Stop
Stop testing when the numbers meet or repeat. If you test 5 and it fails, and the next number is 6 (already listed), you are done!
No factor pairs will be missed!
Golden Rule of Factors: Never guess randomly! Systematic order guarantees 100% accuracy. Slide 3
Guided Walkthrough
Worked Example: Finding All Factor Pairs of 36
Factor T-Chart for 36
1
2
3
4
5 (no)
6
36
18
12
9
(remainder)
6
Factor Pairs of 36:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
Special Case: Square Numbers
Since 6 × 6 = 36, 6 is paired with itself. When listing all individual factors, write 6 only once:
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 total factors)
Square numbers have an odd number of total factors because one factor is repeated. Slide 4
Independent & Partner Practice
Try It: Find Factor Pairs for 48 and 75
Challenge A: Target = 48
Test integers systematically starting with 1:
• 1 × 48 = 48
• 2 × 24 = 48
• 3 × 16 = 48
• 4 × 12 = 48
• 5 × ? (Does not divide evenly)
• 6 × 8 = 48
• 7 × ? (Does not divide evenly) → Next is 8 (stop!)
Challenge B: Target = 75
Test integers systematically starting with 1:
• 1 × 75 = 75
• 2 is even, 75 is odd (Skip 2)
• 3 × 25 = 75
• 4 × ? (Does not divide evenly)
• 5 × 15 = 75
• 6, 7, 8 do not divide evenly
• 9 × 8 = 72, 9 × 9 = 81 (Over target! Stop!)
Ready for Worksheet Practice: Set up your T-charts and check each divisor carefully! Slide 5
Factor Pairs Worksheet Name: ____________________________________
Date: __________________
Score: ______ / 20
Factor Pairs Hunt Worksheet
Standard: 4.OA.B.4 • Find all factor pairs for whole numbers from 1 to 100.
Page 1 of 2
Directions: Find all factor pairs for each given number. In the workspace, write your systematic tests (start at 1, then test 2, 3, 4, etc.). Then, write all factor pairs as multiplication equations and list the factors from least to greatest.
1. Target Number: 18 (2 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
2. Target Number: 32 (2 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
3. Target Number: 45 (2 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
4. Target Number: 60 (3 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
Turn to page 2 to complete numbers 5 through 8 • Check your multiplication carefully
Name: ____________________________________
Target Numbers 1–100: Continued
Page 2 of 2
5. Target Number: 72 (3 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
6. Target Number: 91 (2 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
7. Target Number: 100 (3 pts)
Computation Workspace (T-chart / test division):
Factor Pairs: ___________________________________
All Factors: ___________________________________
8. Mathematical Reasoning and Error Analysis (3 pts)
Marcus evaluated the number 42 and claimed: "42 has only three factor pairs: 1 × 42, 2 × 21, and 6 × 7." Did Marcus find all factor pairs? Prove whether he is correct or identify any missed factor pair(s).
Proof and Computation Workspace:
Is Marcus correct? (Circle one): YES / NO
Explain your mathematical reasoning: ____________________________________________________________________
Check your list: Did you arrange all factors in order from least to greatest?
Factor Pairs Key TEACHER ANSWER KEY & DIAGNOSTIC GUIDE
Lesson 1 • 4.OA.B.4
Total Points: 20
Factor Pairs Hunt Answer Key
Solutions and exemplar systematic work for Target Numbers 18, 32, 45, and 60.
Page 1 of 2
1. Target Number: 18 (2 points) Full Solution
Exemplar Systematic Testing:
1 × 18 = 18
Factor Pairs: 1 × 18, 2 × 9, 3 × 6
All Factors: 1, 2, 3, 6, 9, 18 (6 factors)
Diagnostic note: Watch for students omitting 3 × 6 or forgetting that 18 is even so 2 must be tested.
2. Target Number: 32 (2 points) Full Solution
Exemplar Systematic Testing:
1 × 32 = 32
Factor Pairs: 1 × 32, 2 × 16, 4 × 8
All Factors: 1, 2, 4, 8, 16, 32 (6 factors)
Diagnostic note: Common error is forgetting 2 × 16 due to division beyond basic times tables.
3. Target Number: 45 (2 points) Full Solution
Exemplar Systematic Testing:
1 × 45 = 45
Factor Pairs: 1 × 45, 3 × 15, 5 × 9
All Factors: 1, 3, 5, 9, 15, 45 (6 factors)
Diagnostic note: Students frequently miss 3 × 15 because 45 ends in 5, leading them straight to 5 × 9.
4. Target Number: 60 (3 points) Full Solution
Exemplar Systematic Testing:
1 × 60; 2 × 30; 3 × 20
Factor Pairs: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10
All Factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (12 factors)
Diagnostic note: 60 is highly composite. Emphasize checking 4 (half of 2's pair: 30/2=15).
See Page 2 for solutions to problems 5 through 8 and the Error Analysis rubric.
TEACHER ANSWER KEY • CONTINUED
Target Numbers 72, 91, 100 & Error Analysis
Page 2 of 2
5. Target Number: 72 (3 points) Full Solution
1 × 72; 2 × 36; 3 × 24;
Pairs: 1×72, 2×36, 3×24, 4×18, 6×12, 8×9
Factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 (12 factors)
Scoring: 1 pt for systematic workspace, 1 pt for all 6 factor pairs, 1 pt for ordered list.
6. Target Number: 91 (2 points) Full Solution
1 × 91 = 91
Factor Pairs: 1 × 91, 7 × 13
All Factors: 1, 7, 13, 91 (4 factors)
High-risk problem: 91 looks prime to many students! Testing 7 is essential.
7. Target Number: 100 (3 points) Full Solution
1 × 100 = 100; 2 × 50 = 100
Pairs: 1×100, 2×50, 4×25, 5×20, 10×10
Factors: 1, 2, 4, 5, 10, 20, 25, 50, 100 (9 factors)
Square number reminder: 10 is listed only once in the individual factor list.
8. Mathematical Reasoning and Error Analysis (3 points) Rubric & Answer
Correct Circle: NO (1 point)
Missed Factor Pair: 3 × 14 = 42 (1 point)
Complete Factor Pairs for 42: 1 × 42, 2 × 21, 3 × 14, 6 × 7.
Exemplar Explanation: "Marcus is incorrect because he skipped testing 3. Since 3 × 14 = 42 (or 42 ÷ 3 = 14 with no remainder), 3 and 14 form a factor pair that he missed." (1 point)
Scoring Criteria & Next Steps:
18–20 pts (Mastery): Ready for multiples and divisibility tests.
14–17 pts (Approaching): Review systematic T-chart; check 3 and 4 tests.
< 14 pts (Needs Support): Re-teach arrays and basic facts with multiplication charts.
Multiple Masters Slides Grade 4 Mathematics • Number Theory
Multiple Masters
Connecting Factors and Multiples for Numbers 1 to 100
Key Relationship
A whole number is ALWAYS a multiple of every single one of its factors.
Divisibility Test
If number \(N \div d\) has a remainder of 0, then \(N\) is a multiple of \(d\).
Target Skills
Determine if numbers up to 100 are multiples of any given 1-digit number.
Standard: CCSS.MATH.CONTENT.4.OA.B.4 Lesson 2 • Direct Instruction
Concept Connection
Factors vs. Multiples: The Two Sides
4 × 7 = 28
Looking Inward: FACTORS
The numbers that build 28 are 4 and 7 .
Factors are smaller than or equal to the target number.
4 is a factor of 28 • 7 is a factor of 28
Looking Outward: MULTIPLES
The product created is 28 .
Multiples are greater than or equal to the starting number.
28 is a multiple of 4 • 28 is a multiple of 7
Key Standard Takeaway: A whole number is a multiple of EACH of its factors. Slide 2
Number Patterns
Generating Multiples by Skip Counting
To generate multiples of any number, multiply that number by counting numbers: 1, 2, 3, 4, 5, ...
Multiples of 6 (up to 60):
6 12 18 24 30 36 42 48 54 60
Multiples of 8 (up to 80):
8 16 24 32 40 48 56 64 72 80
Common Multiple Notice: 24 and 48 appear in both lists! Both 24 and 48 are multiples of both 6 and 8.
A number has infinitely many multiples, but we focus on values from 1 to 100. Slide 3
Verification Strategies
Testing: Is \(N\) a Multiple of a 1-Digit Number?
Method 1: Division Test
Divide the target number by the one-digit number.
• If Remainder = 0 → YES, Multiple!
• If Remainder > 0 → NO, Not a multiple.
Example: Is 78 a multiple of 6?YES! (Since \(6 \times 13 = 78\))
Method 2: Landmark Multiples
Use friendly multiples of 10 to break down the number.
• Target: Is 91 a multiple of 7?
• Friendly Landmark: \(7 \times 10 = 70\)
• Remainder to check: \(91 - 70 = 21\)
• Is 21 a multiple of 7? Yes! (\(7 \times 3 = 21\))
Conclusion: \(70 + 21 = 91\), so \(7 \times 13 = 91\) → YES!
Strategy Tip: Always show your division or friendly landmark math to verify your answer! Slide 4
Active Practice
Practice: Multiple or Not?
Question 1
Is 54 a multiple of 4?
Check: \(54 \div 4\)
\(4 \times 10 = 40\)
\(54 - 40 = 14\)
\(14 \div 4 = 3\) R 2
Answer: NO
Question 2
Is 84 a multiple of 7?
Check: \(84 \div 7\)
\(7 \times 10 = 70\)
\(84 - 70 = 14\)
\(14 \div 7 = 2\) R 0
Answer: YES (\(7 \times 12\))
Question 3
Is 96 a multiple of 8?
Check: \(96 \div 8\)
\(8 \times 10 = 80\)
\(96 - 80 = 16\)
\(16 \div 8 = 2\) R 0
Answer: YES (\(8 \times 12\))
Now open your Multiple Masters Worksheet and complete the computation proofs! Slide 5
Multiple Masters Worksheet Name: ____________________________________
Date: __________________
Score: ______ / 20
Multiple Masters Worksheet
Standard: 4.OA.B.4 • Factor-Multiple Relationships & One-Digit Multiples
Page 1 of 2
Directions: Complete each section below. In the calculation workspaces, show your division or multiplication reasoning clearly. A whole number is a multiple of each of its factors.
Part 1: Connecting Factors and Multiples (6 points)
1. Consider the equation: 7 × 9 = 63
Is 63 a multiple of 7? YES / NO
Is 63 a multiple of 9? YES / NO
Explain why using the term factor : ________________________________________________________________
2. Complete each statement with either FACTOR or MULTIPLE:
a. Since 5 × 8 = 40, the number 5 is a ____________________ of 40.
b. Since 5 × 8 = 40, the number 40 is a ____________________ of 8.
3. Write the first five non-zero multiples for each one-digit number:
Multiples of 6: ___, ___, ___, ___, ___
Multiples of 9: ___, ___, ___, ___, ___
Part 2: Testing One-Digit Multiples (Workspace Required)
4. Is 76 a multiple of 4? (2 pts)
Computation Workspace (Division or Landmark Multiplication):
Circle one: YES / NO
Reason / Equation: ____________________________________
5. Is 68 a multiple of 6? (2 pts)
Computation Workspace (Division or Landmark Multiplication):
Circle one: YES / NO
Reason / Equation: ____________________________________
Turn to page 2 to test additional numbers and solve the mystery number challenge
Name: ____________________________________
Multiple Verification & Applications
Page 2 of 2
6. Is 85 a multiple of 5? (2 pts)
Computation Workspace (Show your division or rule check):
Circle one: YES / NO
Reason / Equation: ____________________________________
7. Is 91 a multiple of 7? (2 pts)
Computation Workspace (Division or Landmark Multiplication):
Circle one: YES / NO
Reason / Equation: ____________________________________
8. Is 94 a multiple of 8? (2 pts)
Computation Workspace (Division or Landmark Multiplication):
Circle one: YES / NO
Reason / Equation: ____________________________________
9. Mystery Code Number Challenge (4 pts)
A secret number is between 40 and 60 . It is a multiple of 3 , a multiple of 4 , and a multiple of 6 . Find all possible numbers that fit this description.
Computation Workspace (List candidate multiples in the range):
Possible secret number(s): ________________________________________________________________
Explain how you know both 3 and 4 are factors: ____________________________________________
Review your division work • A remainder of zero confirms a number is a multiple
Multiple Masters Key TEACHER ANSWER KEY & DIAGNOSTIC GUIDE
Lesson 2 • 4.OA.B.4
Total Points: 20
Multiple Masters Answer Key
Solutions for Part 1 (Concepts) and Part 2 (Problems 4 & 5).
Page 1 of 2
Part 1: Factor-Multiple Connections (6 points)
1. Equation: 7 × 9 = 63 (2 pts)
Is 63 a multiple of 7? YES • Is 63 a multiple of 9? YES
Explanation: "Since 7 and 9 multiply to make 63, both 7 and 9 are factors of 63, which means 63 is a multiple of each factor."
2. Vocabulary Completion (2 pts)
a. The number 5 is a FACTOR of 40.
b. The number 40 is a MULTIPLE of 8.
3. First Five Multiples (2 pts)
Multiples of 6: 6, 12, 18, 24, 30
Multiples of 9: 9, 18, 27, 36, 45
Part 2: Testing One-Digit Multiples
4. Is 76 a multiple of 4? (2 pts) YES
Division Proof:
Multiplication Proof:
Diagnostic note: Students can decompose 76 into 40 + 36. Since 4 divides both 40 and 36, it divides 76.
5. Is 68 a multiple of 6? (2 pts) NO
Division Proof:
Landmark Proof:
Diagnostic note: Because 68 falls between 66 and 72, it has a remainder of 2 and is not a multiple.
Turn to page 2 for solutions to problems 6 through 9 and the mystery number analysis.
TEACHER ANSWER KEY • CONTINUED
Problems 6–8 Solutions & Mystery Challenge
Page 2 of 2
6. Is 85 a multiple of 5? (2 pts) YES
Division / Fact:
Divisibility Rule:
Full credit requires either the division showing remainder 0 or citing the ones digit ending in 5.
7. Is 91 a multiple of 7? (2 pts) YES
Division Proof:
Decomposition:
Diagnostic note: 91 is a frequent source of student errors; decomposition (70 + 21) is the clearest strategy.
8. Is 94 a multiple of 8? (2 pts) NO
Division Proof:
Surrounding Multiples:
Scoring: 1 pt for circling NO, 1 pt for computation showing remainder of 6 or listing 88 and 96.
9. Mystery Code Number Challenge (4 pts) Answer: 48
Step 1 (Candidate Multiples of 4 between 40 and 60): 44, 48, 52, 56
Step 2 (Candidate Multiples of 3 between 40 and 60): 42, 45, 48, 51, 54, 57
Step 3 (Common to 3 and 4): 48 (Note: \(3 \times 16 = 48\), \(4 \times 12 = 48\)).
Step 4 (Check multiple of 6): \(6 \times 8 = 48\). It works!
Note on endpoints: If inclusive of 60, 60 is also a multiple of 3, 4, and 6 (\(3\times 20, 4\times 15, 6\times 10\)). Accept 48 or 48 and 60 (if student interprets "between" inclusively).
Remediation Guide for Multiples:
Confusing Factor & Multiple: Remind students: "Factors divide in, Multiples multiply out." Multiple is the product.
Division Struggles: Encourage landmark multiples of 10 (\(d \times 10\)) to subtract friendly chunks.
Prime Composite Slides Grade 4 Mathematics • Number Theory
Prime and Composite
Classifying Whole Numbers 1 to 100 by Their Factor Counts
Prime Number
A whole number greater than 1 with exactly two distinct factors: 1 and itself.
Composite Number
A whole number greater than 1 that has more than two factors.
Special Cases
The number 1 is neither prime nor composite. The number 2 is the ONLY even prime.
Standard: CCSS.MATH.CONTENT.4.OA.B.4 Lesson 3 • Direct Instruction
Classification Criteria
Prime vs. Composite Definitions
PRIME NUMBERS Exactly 2 Factors
Has only 1 factor pair : 1 and the number itself. Cannot be divided into equal groups other than 1 group of \(N\).
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
• Factor pair of 13: only \(1 \times 13\)
COMPOSITE NUMBERS 3+ Factors
Has at least two different factor pairs . Can be divided evenly into multiple equal groupings.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...
• Factor pairs of 9: \(1 \times 9\) and \(3 \times 3\) (3 factors)
Crucial Rule for the Number 1: 1 has only ONE factor (1). Therefore, 1 is NEITHER prime nor composite! Slide 2
Fast Strategy
The 4-Step Prime Test for Numbers 1 to 100
Because \(10 \times 10 = 100\), you only need to test one-digit prime numbers: 2, 3, 5, and 7 !
Step 1: Test 2
Is the number even and greater than 2?
Ends in 0, 2, 4, 6, 8 → COMPOSITE
Step 2: Test 5
Does the number end in 5 (and > 5)?
Ends in 5 → Divisible by 5 → COMPOSITE
Step 3: Test 3
Do the digits sum to a multiple of 3?
Ex: 51 → \(5+1=6\) → Div by 3 → COMPOSITE
Step 4: Test 7
Divide by 7 to check remainder.
Check 49, 77, 91 → All divisible by 7 → COMPOSITE
Conclusion: If a number ≤ 100 passes all 4 tests with NO factors found, it is PRIME! Slide 3
Misconception Alert
Beware: "Odd" Does NOT Mean "Prime"!
Many 4th graders assume that every odd number is prime. This is false! Many odd numbers have factors other than 1 and themselves.
Common Misconception:
"51 is odd and ends in 1, so it must be prime."
Truth: \(3 \times 17 = 51\) → 51 is COMPOSITE!
The Tricky Odd Composites Under 100:
• 27 (\(3 \times 9\))
• 33 (\(3 \times 11\))
• 39 (\(3 \times 13\))
• 49 (\(7 \times 7\))
• 51 (\(3 \times 17\))
• 57 (\(3 \times 19\))
• 77 (\(7 \times 11\))
• 87 (\(3 \times 29\))
• 91 (\(7 \times 13\))
• 93 (\(3 \times 31\))
Always check divisibility by 3 and 7 before deciding an odd number is prime. Slide 4
Mastery Check
Classify and Prove: Prime or Composite?
Target: 47
• Even? No
• Ends in 5? No
• \(4+7=11\) (Not div 3)
• \(47 \div 7 = 6\) R 5
PRIME
Target: 63
• Even? No
• \(6+3=9\) (Div by 3!)
• \(3 \times 21 = 63\)
• Also \(7 \times 9 = 63\)
COMPOSITE
Target: 79
• Even? No
• Ends in 5? No
• \(7+9=16\) (Not div 3)
• \(79 \div 7 = 11\) R 2
PRIME
Target: 87
• Even? No
• \(8+7=15\) (Div by 3!)
• \(87 \div 3 = 29\)
• \(3 \times 29 = 87\)
COMPOSITE
Ready for the Unit Assessment: Demonstrate your complete factor and multiple mastery! Slide 5
Prime Composite Unit Assessment Name: ____________________________________
Date: __________________
Score: ______ / 30
Number Code Breakers Unit Assessment
Standard 4.OA.B.4 • Factor Pairs, Multiples, and Prime/Composite Numbers (1–100)
Page 1 of 2
Directions: Read each question carefully. Use the provided computation workspaces to show all arithmetic proofs, T-charts, and division tests. Clear mathematical work is required for full credit.
Section 1: Finding All Factor Pairs (8 points)
1. Target Number: 48 (4 pts)
Computation Workspace (Show systematic tests from 1 upward):
Factor Pairs: ___________________________________
All Factors (Ordered): _____________________________
2. Target Number: 75 (4 pts)
Computation Workspace (Show systematic tests from 1 upward):
Factor Pairs: ___________________________________
All Factors (Ordered): _____________________________
Section 2: Factor-Multiple Relationships & Divisibility (7 points)
3. Factor-Multiple Connection Reasoning (3 pts)
A classmate claims: "Because 6 is a factor of 54, that means 54 must be a multiple of 6." Is this claim true or false? Explain the mathematical relationship and write the related multiplication fact.
Written Explanation Workspace:
Related Equation: ________________________________________________________________
4. Testing One-Digit Multiples: Target Number 84 (4 pts)
a. Is 84 a multiple of 7? Workspace:
Circle: YES / NO • Proof: ____________
b. Is 84 a multiple of 8? Workspace:
Circle: YES / NO • Proof: ____________
Turn to page 2 to complete prime/composite classifications and multi-step problem solving
Name: ____________________________________
Unit Assessment: Continued
Page 2 of 2
Section 3: Prime vs. Composite Classification (8 points)
Classify each number as PRIME or COMPOSITE . In each workspace, write the factor pairs that justify your answer.
5a. Number: 29 (2 pts)
Factor Test Workspace:
Classification: PRIME / COMPOSITE
5b. Number: 51 (2 pts)
Factor Test Workspace:
Classification: PRIME / COMPOSITE
5c. Number: 61 (2 pts)
Factor Test Workspace:
Classification: PRIME / COMPOSITE
5d. Number: 91 (2 pts)
Factor Test Workspace:
Classification: PRIME / COMPOSITE
Section 4: Mathematical Reasoning & Synthesis (7 points)
6. Special Numbers: 1 and 2 (3 pts)
a. Why is the number 1 classified as neither prime nor composite?
b. Why is the number 2 the only even prime number?
7. Number Code Breaker Mystery Problem (4 pts)
Maya needs to open a combination lock with a 2-digit secret number less than 100. The number satisfies all three clues:
• Clue 1: It is an odd, 2-digit number less than 100.
• Clue 2: It is a multiple of 7.
• Clue 3: It has exactly 3 total factors.
Computation and Proof Workspace:
Secret Combination Number: _______
List the exactly 3 factors: ________________________________________________________
Review your answers and make sure every computation workspace contains clear mathematical work
Unit Assessment Scoring Guide UNIT ASSESSMENT SCORING GUIDE & RUBRIC
Grade 4 • Standard 4.OA.B.4
Total Points: 30
Number Code Breakers Assessment Key
Complete solutions and scoring criteria for Sections 1 and 2.
Page 1 of 2
Section 1: Finding All Factor Pairs (8 points total)
1. Target Number: 48 (4 points) Exemplar
Systematic Tests (1 pt):
Factor Pairs (2 pts): 1×48, 2×24, 3×16, 4×12, 6×8All Factors (1 pt): 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 (10 factors)
Deduct 1 pt if 3 × 16 is omitted (common oversight when students only test basic multiplication facts).
2. Target Number: 75 (4 points) Exemplar
Systematic Tests (1 pt):
Factor Pairs (2 pts): 1×75, 3×25, 5×15All Factors (1 pt): 1, 3, 5, 15, 25, 75 (6 factors)
Deduct 1 pt if student skips 3 × 25 because 75 is odd.
Section 2: Factor-Multiple Relationships & Divisibility (7 points total)
3. Factor-Multiple Reasoning (3 points) Answer: TRUE
Rubric Breakdown:
• 1 pt: States that the claim is true.
• 1 pt: Explains that since \(6 \times 9 = 54\), 54 is the product/multiple of multiplying 6 by a whole number.
• 1 pt: Writes related equation: 6 × 9 = 54 (or \(54 \div 6 = 9\)).
4. Testing One-Digit Multiples: 84 (4 points) (2 pts per subpart)
a. Is 84 a multiple of 7? YES (1 pt)Proof (1 pt): 84 ÷ 7 = 12 R 0 or \(7 \times 12 = 84\) (or \(70 + 14 = 84\)).
b. Is 84 a multiple of 8? NO (1 pt)Proof (1 pt): 84 ÷ 8 = 10 R 4 (or \(8 \times 10 = 80, 8 \times 11 = 88\)).
Turn to page 2 for Sections 3 and 4 solutions and the mastery rubric
ASSESSMENT SCORING GUIDE • CONTINUED
Sections 3 & 4 Solutions & Standards Rubric
Page 2 of 2
Section 3: Prime vs. Composite Classification (8 points total • 2 pts each)
5a. Target: 29 PRIME
Proof: Only factor pair is 1 × 29. Not divisible by 2, 3, 5, or 7. Exactly 2 factors.
5b. Target: 51 COMPOSITE
Proof: Divisible by 3 (5+1=6). Factor pairs: 1 × 51 and 3 × 17 (4 factors).
5c. Target: 61 PRIME
Proof: Only factor pair is 1 × 61. Passes tests for 2, 3 (\(6+1=7\)), 5, and 7 (\(61\div 7=8\) R 5).
5d. Target: 91 COMPOSITE
Proof: Divisible by 7. Factor pairs: 1 × 91 and 7 × 13 = 91 (4 factors).
Section 4: Mathematical Reasoning & Synthesis (7 points total)
6. Special Numbers: 1 and 2 (3 points)
a. Status of 1 (1.5 pts): 1 has only ONE factor (itself). A prime must have exactly two distinct factors; a composite must have more than two factors. Therefore, 1 is neither.
b. Status of 2 (1.5 pts): 2 has exactly two factors (1 and 2), making it prime. Every other even number is divisible by 2, giving it at least three factors (1, 2, and itself), making all other evens composite.
7. Secret Combination Problem (4 points) Answer: 49
Step 1 (Multiples of 7 that are odd and 2-digit): 21, 35, 49, 63, 77, 91 (1 pt)
Step 2 (Testing factor counts):
• Factors of 21: 1, 3, 7, 21 (4 factors)
Factors listed: 1, 7, 49 (1 pt)
Standard 4.OA.B.4 Proficiency Scale:
Level 4 (28–30 pts): Advanced. Perfect systematic reasoning, error-free proofs, fluent prime testing.
Level 3 (24–27 pts): Proficient. Meets grade standard; minor calculation slip with solid conceptual grasp.
Level 2 (18–23 pts): Developing. Struggles with tricky composites (51, 91) or skips systematic T-chart.
Level 1 (< 18 pts): Intervention required. Confuses factors and multiples; re-teach using grid arrays.
