In your journal, you explained the Associative and Commutative Properties of Multiplication to a friend. You told them that they would get the same answer every time they multiplied the numbers in the problem.

Question

Imagine that your neighbor is scrolling through social media. Your cousin is considering making an in-app purchase. Your neighbor shows you advertisements for products that make bold claims:

?Get rich quick! Do you want to work fewer hours for the same amount of money per hour, and get even richer? Click here to see how you can relax and earn. You?ll work 10 fewer hours at the same wage, and you?ll still make more money!?

?Get hydrated now! Do you hate drinking water? Click here to see how drinking less water is actually the secret to having more water in your body. It?s the study the government doesn?t want you to see!?

Select one of the advertisements and advise your cousin:

Represent the advertisement?s claim with a sample mathematical statement.

Using the Associative and Commutative Properties of Multiplication, evaluate whether the claim is reasonable.

Explain your reasoning in your post.

Then read your peers? advice. Respond to two of your peers:

For each post, evaluate your peer?s sample mathematical statement and determine whether it is accurate.

Tell your peers what you agree with and disagree with in their advice.

Post a detailed response to the Multiply Fractions and Decimals Discussion prompt. Then respond to two classmates' posts.

Participate in the discussion to learn more about using multiplication to check reasonability.

Answer 1

Response to the Advertisement

I’ll choose the first advertisement, “Get rich quick!” which claims, “You’ll work 10 fewer hours at the same wage, and you’ll still make more money!”

Sample Mathematical Statement

Let's denote:

• \( H \) as the number of hours worked
• \( W \) as the wage per hour
• \( E \) as the total earnings

The original earnings can be represented as: \[ E = H \times W \]

If my cousin works 10 hours less, the earnings would be: \[ E' = (H - 10) \times W \]

According to the advertisement, the claim states that \( E' > E \), which we can write as: \[ (H - 10) \times W > H \times W \]

Evaluating the Claim

To simplify the inequality:

1. Distributing both sides, we have: \[ H \times W - 10W > H \times W \]

2. Subtracting \( H \times W \) from both sides, we get: \[ -10W > 0 \]

3. This simplifies to: \[ -10W > 0 \] This implies that \( W < 0 \), which means the wage must be negative.

Since a negative wage is not realistic in this context, the claim that working 10 fewer hours at the same wage will result in earning more money is not reasonable.

Explanation of Reasoning

The Associative and Commutative Properties of Multiplication uphold that changing the grouping or order of multiplied numbers does not affect the product. This means that no matter how the hours and wage are arranged, the product must yield the same total money earned unless the base values of hours or wage themselves change.

In this scenario, regardless of how one manipulates the numbers, earning more money while working fewer hours at the same wage violates the fundamental properties of arithmetic — it is mathematically impossible without changing the wage itself.

Responding to Peers

Once I read my classmates' posts, I will evaluate their mathematical statements and reasoning based on similar principles. I will look for clarity, accuracy, and their application of multiplication properties to determine if they matched the logical inconsistency presented in the advertisements.

For example, if a peer analyzed the second advertisement about hydration, I would assess their algebraic representation, check its validity using multiplication principles, and provide constructive feedback on their explanation, agreeing or disagreeing as necessary.

By engaging with my classmates, I hope to enhance our collective understanding of how to apply multiplication properties in evaluating real-world claims.