Divisibility Detectives Slides
Math Sleuth Academy
Core Rules: 2, 3, 5, and 10
Divisibility Detectives
Crack the code of large numbers to find common factors and simplify fractions in seconds.
Goal: Simplify fractions using smart factor clues Case File 01
The Problem & The Clue
01 / 08
The Old Way: Guessing
Staring at \(\frac{48}{72}\) and guessing numbers to divide by wastes time and causes mistakes.
"Does 4 work? Does 6 work? Did I miss something?"
The Detective Way
Use quick visual tests on the last digit or digit sum to instantly spot shared factors!
Both even \(\rightarrow\) Divide by 2 immediately!
Golden Rule: Whatever you do to the numerator, you MUST do to the denominator.
2
Rule of 2: The Even Number Clue
02 / 08
The Rule
A number is divisible by 2 if its last digit is even.
0
2
4
6
8
Look ONLY at the very last digit in the ones place!
148
Ends in 8 (Even) \(\rightarrow\) Yes!
3,250
Ends in 0 (Even) \(\rightarrow\) Yes!
857
Ends in 7 (Odd) \(\rightarrow\) No!
If both numerator and denominator end in 0, 2, 4, 6, or 8, you can divide both by 2!
5 & 10
Rules of 5 & 10: The Fast Endings
03 / 08
Rule of 5 Ends in 0 or 5
A number is divisible by 5 if the last digit is 0 or 5.
✓ 45 (ends in 5)
✓ 120 (ends in 0)
✗ 34 (ends in 4)
Rule of 10 Ends in 0
A number is divisible by 10 if the last digit is 0.
✓ 70 (ends in 0)
✓ 530 (ends in 0)
✗ 45 (ends in 5, not 0)
Pro Detective Tip: If a fraction ends in 0 over 0, divide by 10 first to knock off the zeros! \(\frac{30}{70} = \frac{3}{7}\)
3
Rule of 3: The Secret Sum Test
04 / 08
The Secret Formula
Add up all the digits! If the sum is divisible by 3, the whole number is too!
Multiples of 3 to recognize:
3, 6, 9, 12, 15, 18, 21, 24...
Don't just look at the last digit—add every single digit together!
Test 258 Divisible by 3
\(2 + 5 + 8 = \mathbf{15}\)
15 is divisible by 3 \((3 \times 5 = 15)\) ✓
Test 145 Not Divisible
\(1 + 4 + 5 = \mathbf{10}\)
10 is NOT divisible by 3 ✗
Essential for odd numbers like 51 \((5+1=6)\) or 87 \((8+7=15)\) that look prime but aren't!
The 3-Step Simplification Protocol
05 / 08
1
Scan & Test
Run both numerator and denominator through your tests: 10, 5, 2, or 3.
Find a match for BOTH!
2
Divide Both
Divide the top and bottom by that shared common factor.
\(\frac{A \div k}{B \div k}\)
3
Repeat / Stop
Check the new fraction. Can you simplify again? Stop when common factor is only 1.
Simplest Form!
You don't have to find the greatest factor on step 1—scaffolded steps will always get you there!
Case File #1: Walkthrough \(\frac{24}{36}\)
06 / 08
Simplify Fraction
\(\frac{24}{36}\)
What clues do you see?
Clue 1: Both 24 and 36 are EVEN
Divide top & bottom by 2
\(\frac{24 \div 2}{36 \div 2} = \frac{12}{18}\)
Clue 2: Both 12 and 18 are still EVEN
Divide top & bottom by 2 again
\(\frac{12 \div 2}{18 \div 2} = \frac{6}{9}\)
Clue 3: 6 and 9 are both divisible by 3
Divide top & bottom by 3
\(\frac{6 \div 3}{9 \div 3} = \frac{2}{3}\)
Case Solved! \(\frac{24}{36}\) in simplest form is \(\frac{2}{3}\).
Case File #2: Walkthrough \(\frac{45}{60}\)
07 / 08
Simplify Fraction
\(\frac{45}{60}\)
45 ends in 5, 60 ends in 0!
Clue 1: Ends in 5 and 0
Both are divisible by 5
\(\frac{45 \div 5}{60 \div 5} = \frac{9}{12}\)
Clue 2: Check 9 and 12
Sum test: 9 and \(1+2=3\), divisible by 3!
\(\frac{9 \div 3}{12 \div 3} = \frac{3}{4}\)
Case Solved! \(\frac{45}{60}\) simplifies directly to \(\frac{3}{4}\).
Detective Challenge: Your Turn!
08 / 08
Mission A
\(\frac{18}{30}\)
Rule to try: 2, 3, or 6?
Mission B
\(\frac{35}{50}\)
Rule to try: 5 or 10?
Mission C
\(\frac{27}{36}\)
Rule to try: Sum for 3 or 9?
Ready to solve on your worksheet? Let's begin the investigation!
Divisibility Detectives Worksheet
Divisibility Detectives
Scaffolded Fraction Simplification Investigation
Name:
Date:
Period:
Detective Clue Reference
Rule of 2: Ends in 0, 2, 4, 6, 8
Rule of 3: Sum of digits is \(\div 3\)
Rule of 5: Ends in 0 or 5
Rule of 10: Ends in 0
1 Warm-Up Clue Check: Mark all divisibility rules that apply to each number (\(\checkmark\)).
| Number | Divisible by 2? | Divisible by 3? | Divisible by 5? | Divisible by 10? |
|---|
| 60 | | | | |
| 45 | | | | |
| 84 | | | | |
2 Guided Fraction Simplification: Test top and bottom before dividing.
Case A: \(\frac{20}{50}\) Ends in zeros
1. Divisibility Clue: Both end in 0 \(\rightarrow \div 10\)
2. Show Division:
3. Simplest Form:
Case B: \(\frac{18}{24}\) Test 2, 3, or both
1. Divisibility Clue:
2. Show Division:
3. Simplest Form:
Case C: \(\frac{35}{60}\) Check endings for 5 or 10
1. Divisibility Clue:
2. Show Division:
3. Simplest Form:
Case D: \(\frac{27}{45}\) Sum of digits test for 3
1. Sum Clue: \(2+7=\_\_\), \(4+5=\_\_\)
2. Show Division:
3. Simplest Form:
Divisibility Detectives • Mathematics Practice Page 1 of 2
Part 2: Independent Investigation & Case Reviews
Divisibility Detectives
3 Solve the Cases: Test divisibility, show each division step, and circle your final answer.
Case 1: \(\frac{42}{56}\) Clue: Both even
Rule(s) used: [ ] 2 [ ] 3 [ ] 5 [ ] 10
Case 2: \(\frac{54}{72}\) Clue: Try 2 or 3 sum
Rule(s) used: [ ] 2 [ ] 3 [ ] 5 [ ] 10
Case 3: \(\frac{75}{100}\) Clue: Ends in 5 / 0
Rule(s) used: [ ] 2 [ ] 3 [ ] 5 [ ] 10
Case 4: \(\frac{90}{120}\) Clue: Knock off 0s first!
Rule(s) used: [ ] 2 [ ] 3 [ ] 5 [ ] 10
4 Crime Scene Error Analysis: Spot the suspect's mathematical blunder!
Suspect Sam tried simplifying \(\frac{18}{24}\) with this work: \(\frac{18 \div 2}{24 \div 3} = \frac{9}{8}\)
Divisibility Detectives Answer Key
Teacher Guide & Answer Key
Divisibility Detectives • Complete Solutions
ANSWER KEY
1 Part 1: Warm-Up Clue Check Solutions
| Number | Divisible by 2? | Divisible by 3? | Divisible by 5? | Divisible by 10? |
|---|
| 60 | ✓ (Ends in 0) | ✓ (\(6+0=6\)) | ✓ (Ends in 0) | ✓ (Ends in 0) |
| 45 | ✗ (Ends in 5) | ✓ (\(4+5=9\)) | ✓ (Ends in 5) | ✗ (Not 0) |
| 84 | ✓ (Ends in 4) | ✓ (\(8+4=12\)) | ✗ (Not 0 or 5) | ✗ (Not 0) |
2 Part 2: Guided Fraction Simplification Solutions
Case A: \(\frac{20}{50}\) Answer: \(\frac{2}{5}\)
Clue: Both end in 0 \(\rightarrow \div 10\)
\(\frac{20 \div 10}{50 \div 10}\)
\(\frac{2}{5}\) (Prime numbers, simplest form)
Case B: \(\frac{18}{24}\) Answer: \(\frac{3}{4}\)
Clue: Both even \(\rightarrow \div 2\) (or \(\div 6\))
\(\frac{18 \div 2}{24 \div 2} = \frac{9}{12} \xrightarrow{\div 3} \frac{3}{4}\)
\(\frac{3}{4}\) (Finished!)
Case C: \(\frac{35}{60}\) Answer: \(\frac{7}{12}\)
Clue: Ends in 5 and 0 \(\rightarrow \div 5\)
\(\frac{35 \div 5}{60 \div 5} = \frac{7}{12}\)
\(\frac{7}{12}\) (7 is prime, does not divide 12)
Case D: \(\frac{27}{45}\) Answer: \(\frac{3}{5}\)
Clue: \(2+7=9\) and \(4+5=9 \rightarrow \div 3\) (or \(\div 9\))
\(\frac{27 \div 9}{45 \div 9} = \frac{3}{5}\)
\(\frac{3}{5}\) (Finished!)
Teacher Facilitation Note
Remind students that they do not need to find the Greatest Common Factor (GCF) in one single leap. If both numbers are even, taking half (\(\div 2\)) immediately reduces cognitive load and allows simpler inspection for remaining factors of 3 or 5.
Divisibility Detectives • Answer Key Page 1 of 2
Part 2: Independent Practice & Crime Scene Solutions
Divisibility Detectives
3 Part 3: Independent Cases Solutions
Case 1: \(\frac{42}{56}\) \(\mathbf{\frac{3}{4}}\)
Rules: [✓] 2 [ ] 3 [ ] 5 [ ] 10
\(\frac{42 \div 2}{56 \div 2} = \frac{21}{28}\)
\(\frac{21 \div 7}{28 \div 7} = \mathbf{\frac{3}{4}}\)
