Proportional Relationships

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Claims and Evidence: Building Mathematical Arguments That Stand Strong

Imagine you're trying to convince your friend that your favorite basketball player is the best in the league. You wouldn't just say "Trust me!" Instead, you'd gather stats, highlight game-winning shots, and point to championship rings. In math, we do the same thing—we make claims and back them up with solid evidence.

A claim is a statement you believe to be true that you want others to accept. In math, this might be "Triangle ABC is isosceles" or "The pattern will continue with 64 as the next term." But a claim without evidence is just an opinion floating in space.

The Evidence Foundation

Evidence is the mathematical proof that supports your claim. It could be calculations, measurements, patterns in data, or logical reasoning steps. Think of evidence as the concrete foundation that holds up your mathematical house—without it, everything collapses.

Real Example: The Mystery Pattern

Claim: "The next number in the sequence 2, 6, 18, 54, ___ is 162."

Evidence:

• 2 × 3 = 6
• 6 × 3 = 18
• 18 × 3 = 54
• Therefore: 54 × 3 = 162

The pattern shows each term is multiplied by 3 to get the next term. This mathematical evidence proves our claim is correct.

🔑 The Evidence Test

Here's something surprising: Strong evidence can actually change your original claim. Sometimes when you start gathering proof, you discover your first claim was wrong—and that's perfectly okay! Real mathematicians revise their claims when evidence points in a different direction.

Good evidence should be so clear that someone else could follow your reasoning and reach the same conclusion.

Building Unshakeable Arguments

The strongest mathematical arguments follow a simple structure: State your claim clearly, present your evidence step-by-step, and explain how each piece of evidence supports your conclusion. It's like building with blocks—each piece of evidence stacks on top of the previous one, creating something solid and convincing.

🔑 Key Takeaway

Just like that basketball argument with your friend, mathematical claims need evidence to be convincing. The difference? In math, solid evidence doesn't just win arguments—it reveals truth. When you combine a clear claim with strong mathematical evidence, you're not just solving problems; you're building knowledge that others can trust and build upon.

Sample questions

1. Which table shows a proportional relationship between x and y?

○ x: 1,2,3,4; y: 3,5,7,9

✓ x: 1,2,3,4; y: 2,4,6,8

○ x: 1,2,3,4; y: 1,4,9,16

○ x: 1,2,3,4; y: 2,3,6,8

Answer: x: 1,2,3,4; y: 2,4,6,8 — In a proportional relationship, y/x is constant. For A: 2/1=2, 4/2=2, 6/3=2, 8/4=2. All equal.

2. In a proportional relationship, what must be true about the ratio y/x?

○ It must increase each time

○ It must decrease each time

✓ It must be the same for every pair

○ It can vary as long as the numbers are whole

Answer: It must be the same for every pair — The constant of proportionality means y/x is always the same value.

3. Which table does NOT represent a proportional relationship?

✓ x: 1,2,3,4; y: 2,5,8,11

○ x: 0,2,4,6; y: 0,4,8,12

○ x: 1,3,5,7; y: 3,9,15,21

○ x: 2,4,6,8; y: 5,10,15,20

Answer: x: 1,2,3,4; y: 2,5,8,11 — D: 2/1=2, 5/2=2.5, not constant.

Skills in this topic

Identify proportional relationships from tables
Identify proportional relationships from graphs
Find the constant of proportionality (unit rate) in tables
Find the constant of proportionality from a graph
Determine if two ratios are equivalent

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