Matemática Elementar

Problema 1

Evaluate each expression. Show your work step by step.

(a) $3 + 4 \times 2$

(b) $(3 + 4) \times 2$

(d) $5 \times [4 + (3 \times 2)]$

Problemas

1. Evaluate each expression. Show your work step by step.

(a) $3 + 4 \times 2$

(b) $(3 + 4) \times 2$

(d) $5 \times [4 + (3 \times 2)]$

2. A teacher writes this expression on the board:

$2 \times (15 - 7) + 3$

(a) Write a description of this expression in words, without evaluating it. For example, you might say "the sum of..." or "twice the difference of..."

(b) Now evaluate it. Show each step.

3. Sam earns $\$ 12 $per hour babysitting plus a flat$ $5 $travel fee for each job. Write an expression using parentheses that shows how much Sam earns for a$ 3$-hour babysitting job, then evaluate it.

4. Two students create patterns using different rules.

Student A starts at $0$ and uses the rule "add $3$ ."
Student B starts at $0$ and uses the rule "add $6$ ."

(a) Write the first $6$ terms of each pattern (starting with $0$ ).

(b) Look at the corresponding terms. What is the relationship between each term in Student A's pattern and the matching term in Student B's pattern?

5. Consider the expression: "Add $8$ and $7$ , then multiply by $2$ ."

(a) Write a numerical expression that matches this description.

(b) A classmate writes $8 + 7 \times 2$ instead. Is this the same expression? Explain why or why not, and evaluate both to show the difference.

Gabarito — Expressions and Patterns

1. (a) Multiplication before addition:

$3 + 4 \times 2 = 3 + 8 = 11$

(b) Parentheses first:

$(3 + 4) \times 2 = 7 \times 2 = 14$

(c) Parentheses first, then left to right:

$48 \div (2 + 6) - 1 = 48 \div 8 - 1 = 6 - 1 = 5$

(d) Innermost parentheses first, then brackets:

$5 \times [4 + (3 \times 2)] = 5 \times [4 + 6] = 5 \times 10 = 50$

2. (a) "Twice the difference of $15$ and $7$ , plus $3$ ."

(Other valid descriptions: "Multiply $2$ by the result of $15$ minus $7$ , then add $3$ .")

(b)

Step 1 — Parentheses: $15 - 7 = 8$

Step 2 — Multiply: $2 \times 8 = 16$

Step 3 — Add: $16 + 3 = 19$

$2 \times (15 - 7) + 3 = 19$

3. Sam earns $\$ 12 $per hour for$ 3 $hours, plus a$ $5$ travel fee:

$(12 \times 3) + 5$

Evaluate:

$12 \times 3 = 36$

$36 + 5 = 41$

Sam earns $\$ 41$ for the job.

4. (a)

| Term # | Student A (add 3) | Student B (add 6) | |--------|-------------------|--------------------| | 1 | $0$ | $0$ | | 2 | $3$ | $6$ | | 3 | $6$ | $12$ | | 4 | $9$ | $18$ | | 5 | $12$ | $24$ | | 6 | $15$ | $30$ |

(b) Each term in Student B's pattern is exactly $2$ times the corresponding term in Student A's pattern:

$0 = 2 \times 0, \quad 6 = 2 \times 3, \quad 12 = 2 \times 6, \quad 18 = 2 \times 9, \quad \ldots$

(c) Yes, this relationship will always hold. Student B's rule adds $6$ each time, which is exactly $2 \times 3$ (twice Student A's rule). Since both start at $0$ , every term in B is double the corresponding term in A. In general, if one pattern adds $r$ and another adds $2r$ from the same start, the second is always twice the first.

5. (a) $(8 + 7) \times 2$

(b) No, $8 + 7 \times 2$ is not the same expression.

Without parentheses, order of operations says to multiply first:

$8 + 7 \times 2 = 8 + 14 = 22$

With parentheses, we add first:

$(8 + 7) \times 2 = 15 \times 2 = 30$

The results are different ( $22 \neq 30$ ) because the grouping symbol changes which operation happens first. The parentheses are needed to match the description "add $8$ and $7$ first."