Arrow Detectives Worksheet
Arrow Detectives
Mission: Trace the Scalar Relationships
Name:
Date:
A proportion is a claim that two ratios are equal. Your job is to find the missing value by identifying the multiplication pattern. Draw arrows indicating the direction and the multiplier (e.g., → ×3).
Part 1: Spot the Pattern
2
3
=
4
6
Operation:
10
15
=
2
3
Operation:
Part 2: Solve with Evidence
Directions: Draw arrows between numbers to show the multiplier. Then solve for x.
#01
4
5
=
12
x
x =
#02
x
9
=
4
3
x =
#03
7
2
=
x
10
x =
#04
3
12
=
1
x
x =
Part 3: The Danger Zone
Look at the arrows drawn below. One set is "illegal" because it violates the rules of proportions. Circle the incorrect pattern and explain why it doesn't work.
CASE A
2
5
=
4
10
→ ×2
→ ×2
CASE B
2
5
=
4
10
→ ×2
×2 ←
Arrow Detectives Slides
7th Grade Math
ARROW
DETECTIVES
Solving Proportions with Scalar Patterns
Case File: Equivalent Ratios
Spot the Pattern
Case #1
2
3
=
4
6
What operation connects them?
Case #2
1
5
=
3
15
What operation connects them?
Video Evidence
Embedded media
Look For:
- The definition of a proportion.
- The 4 Directions we can use for patterns.
- Why arrows shouldn't "cross."
The 4 Directions
Left to Right
Comparing the first ratio to the second.
Right to Left
Comparing the second ratio back to the first.
Top to Bottom
Comparing the numerator to the denominator.
Bottom to Top
Comparing the denominator to the numerator.
The "No-Cross" Rule
4
5
=
8
10
×2
×0.5
Arrows must point in the SAME DIRECTION.
If you go left-to-right on top, you MUST go left-to-right on bottom.
Case Debrief
1. Why did the video warn about crossing arrows?
2. What happens if you multiply the top by 2 and the bottom by 3?
Arrow Detectives Anchor Chart
MATH INVESTIGATION
Solving Proportions
Definition
"An equation where two ratios are equivalent to each other."
The 4 Directions
Across Left-to-Right
Across Right-to-Left
Vertical Top-to-Bottom
Vertical Bottom-to-Top
Example Investigation
3
4
=
x
12
Strategy: Left-to-Right
4 12 is ×3
So, 3 ×3 = 9
Strategy: Bottom-to-Top
4 3 is ×0.75
(Harder pattern, same result!)
Rule: Arrows MUST point in the same direction!
Arrow Detectives Answer Key
Arrow Detectives
Answer Key & Teacher Reference
Teacher Use Only
Part 1: Spot the Pattern Solutions
Problem 1: 2/3 = 4/6
× 2 (Left to Right)
Numerator and Denominator are both multiplied by 2.
Problem 2: 10/15 = 2/3
÷ 5 or × 1/5
Ratios are simplified by a factor of 5.
Part 2: Solve with Evidence Solutions
#01
4/5 = 12/x
Pattern: Left to Right (→)
Multiplier: ×3 (since 4 × 3 = 12)
Result: 5 × 3 = 15. So, x = 15
#02
x/9 = 4/3
Pattern: Right to Left (←)
Multiplier: ×3 (since 3 × 3 = 9)
Result: 4 × 3 = 12. So, x = 12
#03
7/2 = x/10
Pattern: Left to Right (→)
Multiplier: ×5 (since 2 × 5 = 10)
Result: 7 × 5 = 35. So, x = 35
#04
3/12 = 1/x
Pattern: Top to Bottom (↓)
Multiplier: ×4 (since 3 × 4 = 12)
Alternative: Left to Right (→) × 1/3
Result: 1 × 4 = 4. So, x = 4
Part 3: The Danger Zone Solution
Incorrect Case: B
"In Case B, the arrows point in opposite directions. For a proportion to be valid, the scalar relationship must be applied consistently to both the numerator and the denominator in the same direction. Multiplying one side by 2 and the other by 0.5 (dividing by 2) would result in two completely different, non-equivalent ratios."
Teacher Discussion Guide
- Question 1: Why did the video warn about crossing arrows?
Answer: Because proportions rely on maintaining the exact same scale between parts. Crossing arrows changes the relationship between parts, breaking the equality.
- Question 2: What happens if you multiply the top by 2 and the bottom by 3?
Answer: You create a different ratio. For example, 1/2 becomes 2/6. 1/2 is not equal to 2/6 (which is 1/3).
