6th Grade Math Lesson Plans: Ratios, Integers, Expressions, and Geometry

Sixth grade math is a major transition year. Students move from arithmetic to algebraic reasoning, from whole numbers to integers, and from simple fractions to ratios and proportional relationships. The CCSS standards for grade 6 cover six major domains — here are complete lesson plans for each.

Domain 1: Ratios and Proportional Relationships

Lesson: Introduction to Ratios (6.RP.A.1)

Objective: Students will write ratios in three forms and understand what a ratio represents.

Hook (5 min):

Show two groups: 3 red circles and 5 blue circles. "In how many ways can I describe the relationship between these groups?"

Instruction (15 min):

A ratio is a comparison of two quantities using division. It can be written three ways:

• 3 to 5
• 3:5
• 3/5

Important: ratios compare related quantities. They are not always part-to-whole.

Types of ratios:

• Part-to-part: red circles to blue circles → 3:5
• Part-to-whole: red circles to all circles → 3:8
• Whole-to-part: all circles to blue circles → 8:5

Work through 4 examples with the class. After each, ask: "Is this part-to-part, part-to-whole, or whole-to-part?"

Guided Practice (15 min):

Recipe context: A smoothie recipe calls for 2 cups of strawberries and 3 cups of mango.

• Write the ratio of strawberries to mango
• Write the ratio of mango to all fruit
• If I want 4 cups of strawberries, how many cups of mango do I need? (preview of proportional reasoning)

Independent Practice (15 min):

Students complete 12 problems: writing ratios from real-world contexts (classrooms, sports statistics, recipes, maps).

---

Domain 2: The Number System

Lesson: Negative Numbers and Absolute Value (6.NS.C.5–7)

Objective: Students will place integers on a number line, compare integers, and calculate absolute value.

Real-World Context (5 min):

Present: "The temperature in Kansas City on Monday was -8°F. On Tuesday it was -3°F. Which day was colder?"

Survey responses. Most students get this right intuitively. Now: "How do you explain WHY -8 is less than -3?"

Instruction (20 min):

Integer: any whole number, its opposite, and zero: ...−3, −2, −1, 0, 1, 2, 3...

Number line placement: Draw a horizontal number line. Positive numbers are to the right of zero; negative numbers are to the left. Order means direction — less than means to the left.

Absolute value: the distance from zero, always positive.

• |−8| = 8 (8 units from zero)
• |3| = 3 (3 units from zero)
• |0| = 0

Common misconception: absolute value does NOT change a negative number to a positive number. It tells you the distance. −(|−8|) = −8.

Comparing integers: On a number line, further left = smaller. −100 < −1.

Practice Problems (20 min):

1. Plot these on a number line: -5, 2, -1, 8, -9, 0
2. Write < or >: -7 __ -2, 4 __ -4, -1 __ 0
3. Calculate absolute value: |-15|, |6|, |-2|, |0|
4. Order from least to greatest: 3, -8, -1, 5, -12

Exit ticket: "A submarine is at -200 feet. An airplane is at 30,000 feet. Which is farther from sea level? How do you know?"

---

Domain 3: Expressions and Equations

Lesson: Writing and Evaluating Expressions (6.EE.A.2)

Objective: Students will write algebraic expressions from verbal descriptions and evaluate expressions for given values.

Key Vocabulary (10 min):

| Word | Operation |

|------|-----------|

| sum, more than, increased by | + |

| difference, less than, decreased by | − |

| product, times, multiplied by | × |

| quotient, divided by | ÷ |

Be explicit about "less than": "5 less than x" = x − 5, NOT 5 − x. This trips up many students.

Writing Expressions (15 min):

Model 6 examples aloud, then students work in pairs:

1. Three more than a number → n + 3
2. Twice a number, decreased by 7 → 2n − 7
3. The quotient of a number and 4 → n ÷ 4 or n/4
4. Five times the sum of a number and 2 → 5(n + 2)

Evaluating Expressions (15 min):

Evaluate 3x + 7 when x = 4:

• Substitute: 3(4) + 7
• Multiply: 12 + 7
• Add: 19

Common errors: forgetting to substitute everywhere, multiplication before addition (PEMDAS).

Practice: Evaluate each expression for the given value. 10 problems covering all four operations and parentheses.

---

Domain 4: Geometry

Lesson: Area of Triangles and Composite Figures (6.G.A.1)

Objective: Students will calculate the area of triangles and composite figures by decomposing them into known shapes.

Connecting to Prior Knowledge (5 min):

"You already know how to find the area of a rectangle: A = l × w. Today you'll discover how to find the area of shapes that aren't rectangles."

Triangle Area Discovery (15 min):

Give each student a pair of identical right triangles cut from colored paper. "Can you arrange these to form a shape you know?"

Students discover: two identical triangles → one parallelogram (or rectangle for right triangles). If the parallelogram has area = base × height, then one triangle has area = ½ × base × height.

Formula: A = ½bh

Key vocabulary: base = any side of the triangle; height = perpendicular distance from base to opposite vertex (must be perpendicular).

Composite Figures (20 min):

The L-shaped room: 12 ft × 8 ft minus a 4 ft × 3 ft corner cut out.

Strategies:

1. Subtract: find the big rectangle, subtract the missing piece
2. Add: decompose into two rectangles, add them

Both strategies get the same answer. The skill is recognizing which shapes the composite figure is made of.

Practice: 8 composite figure problems increasing in complexity, ending with real-world floor plans.

---

Domain 5: Statistics and Probability

Lesson: Measures of Center and Variability (6.SP.A.2–3)

Objective: Students will calculate mean, median, mode, and range, and choose the appropriate measure of center for a given data set.

Real Data Hook (5 min):

NBA player salaries from a recent team roster. List 12 salaries ranging from $1.2M (rookie minimum) to $45M (max contract). "What is the 'typical' salary on this team?"

Mean vs. Median (20 min):

Mean = sum ÷ number of values. Sensitive to outliers — one $45M max contract pulls the mean up significantly.

Median = middle value when ordered. Resistant to outliers.

Mode = most frequent value. Useful for categorical data.

Students calculate all three for the NBA salary data. Discussion: "Which is the most accurate representation of a 'typical' player's salary? Why?"

The answer: median, because the outliers (max contracts) skew the mean upward and give a false impression of typical earnings.

When to use each:

• Mean: data without extreme outliers, continuous data
• Median: data with outliers, income data, housing prices
• Mode: categorical data, most popular item

---

Sixth grade math lays the foundation for algebra. Strong conceptual understanding in ratios, integers, and expressions in 6th grade directly predicts 8th grade algebra success.