Grade 7 Math Quiz Study Notes: Integers, Order of Operations, and Expressions

Quick Rules and Core Concepts

• Grade Level Context:

  • Focuses on Grade $7$ review material covering integers, order of operations, expressions, and word problems.

• Integer Addition Rules:

  • Same Signs: Add the numbers together and keep the shared sign.
  • Different Signs: Subtract the smaller absolute value from the larger absolute value, then keep the sign of the number with the greater absolute value.

• Integer Subtraction Rule:

  • Keep the first number unchanged.
  • Change the subtraction operation to addition.
  • Change the sign of the second number to its opposite.
  • Perform the resulting addition problem.

• Multiplication and Division Rules:

  • Same Signs: Multiplying or dividing two numbers with identical signs yields a positive answer ($+$).
  • Different Signs: Multiplying or dividing two numbers with different signs yields a negative answer ($-$).

• Properties of Zero:

  • Any number multiplied by $0$ equals $0$.
  • Division by $0$ is impossible (undefined); a number cannot be divided by $0$.

• Order of Operations (Hierarchy):

  • Step 1: Parentheses and all other grouping symbols.
  • Step 2: Exponents.
  • Step 3: Multiplication and division, evaluated in order from left to right.
  • Step 4: Addition and subtraction, evaluated in order from left to right.

• Exponent Concepts and Rules:

  • An exponent indicates the exact number of times the base is multiplied by itself.
  • Concrete Example: $3^2$ represents $3 \times 3$.
  • Significance of Parentheses: Parentheses alter the base; $(-3)^2$ is fundamentally different from $-3^2$.
  • In $(-3)^2$, the base is $-3$, resulting in $(-3) \times (-3) = 9$.
  • In $-3^2$, the base is $3$, resulting in $-(3 \times 3) = -9$.

• Fraction Representation:

  • A fraction bar signifies the operation of division.
  • Example: $-\frac{24}{6}$ denotes $-24 \times \frac{1}{6}$ or $-24 \ne 0$ divided directly: $-24 \times \frac{1}{6} = -24 \ne 0$, or $-24 \times \frac{1}{6}$ rewritten as $-24 \div 6$.

• Substitution and Evaluation of Expressions:

  • Substitute each variable in the algebraic expression with its designated value.
  • Always place negative values inside parentheses upon substitution.
  • Apply the standard order of operations to solve the remaining numerical expression.

• Calculation of Mean per Hour:

  • Sum all individual hourly quantities or amounts together.
  • Divide the calculated sum by the total number of hours.

• Integer Conventions for Real-World Applications:

  • Decreases, penalties, expenditures, and financial losses are expressed as negative integers.
  • Increases, deposits, points scored, and gains are expressed as positive integers.

Practice Problems: Integer Operations

• Instructions:

  • Solve each problem without the use of a calculator.
  • Show all supporting work for each step.
  • Answers are intentionally omitted to permit self-testing.

• Problem 1:

  • $-17 + 9 = \underline{\hspace{2cm}}$

• Problem 2:

  • $-8 - (-13) = \underline{\hspace{2cm}}$

• Problem 3:

  • $(-7)(6) = \underline{\hspace{2cm}}$

• Problem 4:

  • $-96 \div (-12) = \underline{\hspace{2cm}}$

• Problem 5:

  • $-135 \div 5 = \underline{\hspace{2cm}}$

Practice Problems: Exponents, Fractions, and Order of Operations

• Instructions:

  • Solve each problem without a calculator and show all intermediate steps.

• Problem 6:

  • $(-4)^2 - 3 \times 5 = \underline{\hspace{2cm}}$

• Problem 7:

  • $36 \div (-6) + 2^3 = \underline{\hspace{2cm}}$

• Problem 8:

  • $(-18 + 6) \div 3 = \underline{\hspace{2cm}}$

• Problem 9:

  • $-\frac{42}{7} + (-3)^2 = \underline{\hspace{2cm}}$

Practice Problems: Number Patterns and Algebraic Substitution

• Instructions:

  • Complete pattern identification and evaluate the given algebraic expressions without using a calculator.

• Problem 10:

  • Write the next two numbers in the sequence:
  • $-18$, $-12$, $-6$, $0$, $\underline{\hspace{2cm}}$, $\underline{\hspace{2cm}}$

• Problem 11:

  • If $a = -5$ and $b = 4$, evaluate:
  • $3a + 2b$

• Problem 12:

  • If $x = -3$, evaluate:
  • $x^2 - 4x$

Practice Problems: Real-World Word Problems

• Instructions:

  • Model each scenario using appropriate signed numbers or expressions and compute the solution without a calculator.

• Problem 13:

  • The temperature is $-6^\circ\text{F}$ in the morning and rises $14^\circ\text{F}$.
  • Determine the new temperature.

• Problem 14:

  • An elevator starts on floor $2$, travels down $7$ floors, and then travels up $4$ floors.
  • Determine which floor the elevator is on.

• Problem 15:

  • An account has an initial amount of $\$18$ and spends $\$25$.
  • Represent the financial change using an integer.
  • Determine the final account balance.

• Problem 16:

  • Rainfall measured over a span of $4\,\text{hours}$ is recorded as $2\,\text{mm}$, $0\,\text{mm}$, $3\,\text{mm}$, and $7\,\text{mm}$.
  • Calculate the mean rainfall per hour.

• Problem 17:

  • In a mobile phone game, a player earns $35\,\text{points}$ and loses $48\,\text{points}$.
  • Calculate the total score change.

• Problem 18:

  • A game awards $6\,\text{points}$ per minute over a duration of $7\,\text{minutes}$, and subsequently subtracts a $12\,\text{point}$ penalty.
  • Formulate an expression representing the scenario.
  • Calculate the final score.

Practice Problems: Mixed Review

• Instructions:

  • Evaluate the following combined integer and operational review problems without a calculator.

• Problem 19:

  • $-5 + 3 \times (-4) = \underline{\hspace{2cm}}$

• Problem 20:

  • $(-2)^3 + 24 \div (-6) = \underline{\hspace{2cm}}$

Study Strategy and Remediation

• Study Methodology:
  • Construct and review flashcards dedicated to each arithmetic and operational rule.
  • Complete every practice problem by hand on paper before checking results.
  • Whenever an error occurs on a practice question, determine the specific rule that was misapplied prior to re-attempting the problem.