1. **Problem Statement:** Verify if each of the four expressions in Box 1 equals 1 by computing the expression $4 \times 4 - 5 \times 3$. 2. **Next Expression:** Find the next number expression after $5 \times 5 - 6 \times 4$ in the pattern. 3. **Algebraic Expression:** Identify which algebraic expression among the options represents the set of number expressions in Box 1. 4. **Explanation:** Justify the choice of the correct algebraic expression. 5. **Meaning of $n$:** Explain what $n$ represents in the chosen expression. --- **Step 1: Verify the expression $4 \times 4 - 5 \times 3$** Calculate each product: $$4 \times 4 = 16$$ $$5 \times 3 = 15$$ Subtract: $$16 - 15 = 1$$ So, the expression equals 1, confirming the classmate's claim for this case. **Step 2: Find the next expression after $5 \times 5 - 6 \times 4$** Calculate the current expression: $$5 \times 5 = 25$$ $$6 \times 4 = 24$$ Subtract: $$25 - 24 = 1$$ The pattern shows the first factor increasing by 1 each time, and the second factor in the first product is $n$, the second product factors are $n+1$ and $n-1$ respectively. The next expression would be: $$6 \times 6 - 7 \times 5$$ **Step 3: Identify the algebraic expression** Check each option: a. $(n)(n) - (n+3)(n+1)$ b. $(n)(n) - [(n+1)(n-1)]$ c. $(n-1)(n-1) - n(n-2)$ d. $n^2 - 3n(1)$ e. $n^2 - n - 1$ Calculate option b: $$n^2 - (n+1)(n-1) = n^2 - (n^2 - 1) = n^2 - n^2 + 1 = 1$$ This matches the pattern where each expression equals 1. **Step 4: Explanation** Option b simplifies to 1 for all $n$, matching the pattern of expressions in Box 1. The other options do not simplify consistently to 1. **Step 5: Meaning of $n$** In the chosen expression, $n$ represents the first factor in the first product of the number expressions in Box 1. It is the variable that increases by 1 in each subsequent expression. **Final answers:** 1. $4 \times 4 - 5 \times 3 = 1$ 2. Next expression: $6 \times 6 - 7 \times 5$ 3. Correct algebraic expression: b. $(n)(n) - [(n+1)(n-1)]$ 4. Because it simplifies to 1 for all $n$, matching the pattern. 5. $n$ represents the first factor in the first product, increasing by 1 each step.