In the 6th-grade Common Core curriculum, "Operations" and "Algebra" are primarily covered under two domains: The Number System (NS) and Expressions and Equations (EE).
Below is a breakdown of these standards, organized by what students are expected to know and do.
1. The Number System (Operations)
These standards focus on computational fluency with fractions, decimals, and the introduction of negative numbers.
| Standard Category | Key Skills |
| Fraction Division | Interpret and compute quotients of fractions (e.g., $\frac{2}{3} \div \frac{3}{4}$). Solve word problems involving division of fractions by fractions. |
| Decimal Fluency | Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. |
| Factors & Multiples | Find the Greatest Common Factor (GCF) of two whole numbers $\le 100$ and the Least Common Multiple (LCM) of two whole numbers $\le 12$. |
| Rational Numbers | Understand that positive and negative numbers represent "opposite" directions (e.g., temperature, debt). Plot integers and rational numbers on a number line and coordinate plane (all four quadrants). |
| Absolute Value | Understand $ |
2. Expressions and Equations (Algebra)
This is where students transition from basic arithmetic to algebraic thinking by using variables to represent unknown numbers.
Applying Arithmetic to Algebraic Expressions
Exponents: Write and evaluate numerical expressions involving whole-number exponents (e.g., $3^4$).
Variable Expressions: Write, read, and evaluate expressions where letters stand for numbers.
Term Identification: Identify parts of an expression using mathematical terms like sum, term, product, factor, quotient, and coefficient.
Equivalent Expressions: Use the Distributive Property to produce equivalent expressions (e.g., $3(2 + x) = 6 + 3x$).
Solving One-Variable Equations & Inequalities
Solving Equations: Understand that solving an equation is a process of finding which value makes the statement true. Students solve equations in the form $x + p = q$ and $px = q$.
Inequalities: Write inequalities (e.g., $x > 5$) to represent constraints in real-world scenarios. Recognize that inequalities have infinitely many solutions and represent them on number lines.
Dependent & Independent Variables
Quantitative Relationships: Use variables to represent two quantities that change in relationship to one another (e.g., distance = speed $\times$ time).
Representations: Analyze the relationship between a dependent variable and an independent variable using graphs, tables, and equations.
3. Ratios and Proportional Relationships
While often listed separately, this domain uses "Algebraic" thinking to solve operations-based problems.
Unit Rates: Understand the concept of a unit rate $a/b$ associated with a ratio $a:b$.
Percent: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means $\frac{30}{100}$ times the quantity).
Would you like me to create a practice problem or a "cheat sheet" for one of these specific topics, such as the Distributive Property or Dividing Fractions?
In 6th grade, solving equations is taught as a process of maintaining balance. The goal is to isolate the variable (get $x$ by itself) to find the specific value that makes the mathematical sentence true.
The two primary forms required by the standard 6.EE.B.7 are addition equations ($x + p = q$) and multiplication equations ($px = q$).
1. Addition Equations ($x + p = q$)
To solve these, you use the Subtraction Property of Equality. Since $p$ is being added to $x$, you "undo" it by subtracting $p$ from both sides.
Example: Solve $x + 12 = 30$
Identify the operation: 12 is being added to $x$.
Inverse operation: Subtract 12 from both sides to keep the "scale" balanced.
Simplify:
Check: Does $18 + 12 = 30$? Yes. The statement is true.
2. Multiplication Equations ($px = q$)
When a number and a letter are touching (like $5x$), it means multiplication. To solve these, you use the Division Property of Equality.
Example: Solve $5x = 35$
Identify the operation: $x$ is being multiplied by 5.
Inverse operation: Divide both sides by 5.
Simplify:
Check: Does $5 \times 7 = 35$? Yes. The statement is true.
Key Concept: Inverse Operations
To solve any one-step equation, you must perform the opposite (inverse) operation to isolate the variable.
| If the equation has... | Use the Inverse... |
| Addition ($+$) | Subtraction ($-$) |
| Subtraction ($-$) | Addition ($+$) |
| Multiplication ($\times$ or $5x$) | Division ($\div$ or $\frac{x}{5}$) |
| Division ($\div$ or $\frac{x}{5}$) | Multiplication ($\times$) |
[Image showing a chart of inverse operations: + and -, * and /]
Verifying the Solution (Substitution)
Standard 6.EE.B.5 emphasizes that solving is a process of finding which value makes the equation true. Students are taught to check their work by substituting their answer back into the original equation:
Would you like me to generate a set of practice problems including some with decimals or fractions, which are also part of this 6th-grade standard?
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How to use this tool for 6th Grade:
Opposite Directions: Notice that if you plot (3,2) and then change it to (−3,−2), the point moves exactly the same distance from the center (the Origin) but in the opposite direction.
Four Quadrants: Use the input boxes or click directly on the grid to see how signs change in different quadrants:
Quadrant I: (+,+) - Moving right and up.
Quadrant II: (−,+) - Moving left and up.
Quadrant III: (−,−) - Moving left and down.
Quadrant IV: (+,−) - Moving right and down.
Rational Numbers: I've enabled 0.5 increments so you can practice plotting rational numbers (like −2.5) that fall between the whole-number grid lines.