Focus Areas for Sixth Grade
- Connecting ratio and rate to multiplication and division
- Understanding division of fractions
- Extending the system of rational numbers
- Writing, interpreting, and using expressions and equations
- Developing understanding of statistical thinking
Math Standards by Domain
Ratios and Proportional Relationships (6.RP)
What Students Learn:
Students develop a foundational understanding of ratios as comparisons between two quantities. They learn to express ratios in multiple forms (a:b, a to b, a/b) and use ratio reasoning to solve real-world problems. Students discover unit rates as a special type of ratio where one quantity is compared to one unit of another quantity, such as miles per hour or price per item.
Key Skills:
- Understand ratio as an ordered pair of numbers (a:b) comparing two quantities
- Express ratios using different notations: 3:2, 3 to 2, 3/2
- Find unit rates (e.g., if 12 apples cost $6, the unit rate is $0.50 per apple)
- Use ratio tables to organize and solve problems with equivalent ratios
- Solve real-world rate problems (speed, price, density)
- Understand that a/b represents both a fraction and a ratio
- Scale ratios up and down while maintaining proportional relationships
In a class of 30 students, 18 are girls and 12 are boys. The ratio of girls to boys is 18:12, which simplifies to 3:2. For every 3 girls, there are 2 boys.
Example (Unit Rates):A car travels 150 miles in 3 hours. The unit rate is 150 ÷ 3 = 50 miles per hour.
Example (Ratio Tables):If 2 pounds of apples cost $3, complete the table:
Pounds: 2, 4, 6, 8 | Cost: $3, $6, $9, $12
The Number System (6.NS)
What Students Learn:
Students extend their understanding of division to include fractions divided by fractions. They learn that dividing by a fraction is the same as multiplying by its reciprocal and develop conceptual understanding through visual models and word problems. This skill is essential for understanding rates, ratios, and more complex algebraic concepts.
Key Skills:
- Interpret division of fractions using visual models (area models, number lines)
- Understand that dividing by a fraction means "how many groups of this size?"
- Apply the rule: a/b ÷ c/d = a/b × d/c (multiply by the reciprocal)
- Solve word problems with fraction division (e.g., "How many 1/4-cup servings in 3 cups?")
- Compute quotients of fractions by fractions fluently
- Understand that dividing by a number less than 1 results in a larger quotient
How many 1/2-cup servings are in 3 cups? 3 ÷ 1/2 = 3 × 2/1 = 6 servings
Divide: 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 = 2 2/3
Word Problem:A rope is 5/6 yard long. You need pieces that are 1/12 yard each. How many pieces can you cut? 5/6 ÷ 1/12 = 5/6 × 12/1 = 60/6 = 10 pieces
What Students Learn:
Students develop fluency with multi-digit whole number operations and extend all four operations to multi-digit decimals. They also learn to find greatest common factors (GCF) and least common multiples (LCM), which are essential for working with fractions and solving real-world problems involving cycles and patterns.
Key Skills:
- Fluently divide multi-digit whole numbers using standard algorithm
- Add, subtract, multiply, and divide multi-digit decimals (to hundredths)
- Use algorithms and strategies based on place value and properties of operations
- Find the Greatest Common Factor (GCF) of two numbers
- Find the Least Common Multiple (LCM) of two numbers
- Use GCF to simplify fractions and distribute factors
- Use LCM to find common denominators and solve scheduling problems
- Apply understanding of place value when computing with decimals
2,436 ÷ 6 = 406
Example (Decimals):3.45 + 2.8 = 6.25 | 5.6 × 3.2 = 17.92 | 12.6 ÷ 0.3 = 42
Example (GCF):GCF of 24 and 36: Factors of 24: 1,2,3,4,6,8,12,24 | Factors of 36: 1,2,3,4,6,9,12,18,36 | GCF = 12
Example (LCM):LCM of 6 and 8: Multiples of 6: 6,12,18,24,30... | Multiples of 8: 8,16,24,32... | LCM = 24
What Students Learn:
Students expand their understanding of numbers to include negative numbers, learning that the number system extends infinitely in both directions from zero. They understand rational numbers as points on a number line and develop the concept of absolute value as distance from zero. Students also learn to plot points in all four quadrants of the coordinate plane.
Key Skills:
- Understand positive and negative numbers as opposites (5 and -5)
- Recognize that negative numbers represent values less than zero (temperature, debt, elevation below sea level)
- Plot rational numbers (fractions, decimals, integers) on a number line
- Understand absolute value as distance from zero (|-5| = 5, |5| = 5)
- Compare and order rational numbers including negative numbers
- Plot points in all four quadrants of the coordinate plane using ordered pairs (x, y)
- Solve real-world problems involving rational numbers (temperature changes, account balances)
- Understand that the opposite of an opposite is the number itself (-(-3) = 3)
Temperature drops from 5°F to -3°F. It changed by 8 degrees.
Order from least to greatest: -3, -1, 0, 2, 5
Example (Absolute Value):|−8| = 8 (distance of -8 from 0 is 8 units)
|-7| < 10 because 7 < 10
Example (Coordinate Plane):Plot point A(3, -2): Start at origin, go 3 right, 2 down (Quadrant IV)
Plot point B(-4, 5): Start at origin, go 4 left, 5 up (Quadrant II)
Expressions and Equations (6.EE)
What Students Learn:
Students learn to write and interpret algebraic expressions using variables, exponents, and multiple operations. They apply properties of operations (distributive, commutative, associative) to create equivalent expressions and evaluate expressions by substituting values for variables. This is the foundation of algebraic thinking.
Key Skills:
- Write expressions with whole-number exponents (2³ = 2 × 2 × 2 = 8)
- Evaluate expressions using order of operations (PEMDAS)
- Write algebraic expressions from verbal descriptions ("5 more than x" = x + 5)
- Identify parts of expressions: terms, factors, coefficients
- Apply distributive property: 3(x + 2) = 3x + 6
- Generate equivalent expressions by combining like terms: 2x + 3x = 5x
- Understand that expressions can have multiple equivalent forms
- Substitute values for variables and evaluate expressions
Evaluate: 2⁴ + 3² = 16 + 9 = 25
Example (Order of Operations):Evaluate: 3 + 4 × 2² = 3 + 4 × 4 = 3 + 16 = 19
Example (Writing Expressions):"The product of 6 and a number n, decreased by 5" → 6n - 5
Example (Equivalent Expressions):4(x + 3) = 4x + 12 | 2x + 3x + 5 = 5x + 5
What Students Learn:
Students develop the fundamental understanding that solving an equation means finding the value(s) that make the equation true. They learn to write and solve one-step equations and inequalities, and they begin to understand the relationship between equations and real-world situations. This is a critical stepping stone to more advanced algebra.
Key Skills:
- Understand that solving x + 5 = 12 means finding what value of x makes it true
- Use substitution to check if a value is a solution to an equation
- Solve one-step equations using addition, subtraction, multiplication, or division
- Write equations to represent real-world situations with unknown quantities
- Understand and use inequality symbols (<, >, ≤, ≥)
- Solve one-step inequalities and graph solutions on a number line
- Write inequalities to model real-world constraints ("at least 5," "no more than 10")
- Recognize that variables can represent different values in different contexts
x + 7 = 15 → x = 8 (subtract 7 from both sides)
3x = 24 → x = 8 (divide both sides by 3)
x/4 = 5 → x = 20 (multiply both sides by 4)
Example (Word Problems):Maria earned $45. She earned $12 more than Tom. How much did Tom earn? Let x = Tom's earnings. x + 12 = 45, so x = 33
Example (Inequalities):x + 3 < 10 → x < 7 (Graph: open circle at 7, arrow pointing left)
What Students Learn:
Students learn to use variables to represent dependent relationships between two quantities. They write equations in two variables, create tables of values, and graph these relationships on coordinate planes. This introduces the concept of functions and prepares students for analyzing linear relationships in future grades.
Key Skills:
- Identify independent and dependent variables in real-world situations
- Write equations using two variables (e.g., y = 2x + 3)
- Create tables of values showing how one quantity depends on another
- Graph pairs of values (ordered pairs) that satisfy an equation
- Analyze graphs to understand relationships between quantities
- Understand that each input (x-value) corresponds to one output (y-value)
- Interpret points on a graph in context of the problem
A plumber charges $50 plus $30 per hour. Total cost (c) depends on hours (h).
Equation: c = 50 + 30h
Table: h = 1, 2, 3, 4 | c = 80, 110, 140, 170
Graph these points: (1, 80), (2, 110), (3, 140), (4, 170)
Another Example:The perimeter (p) of a square depends on side length (s): p = 4s
If s = 5, then p = 20. Plot points like (1, 4), (2, 8), (3, 12), (4, 16), (5, 20)
Geometry (6.G)
What Students Learn:
Students apply formulas to find areas of triangles, quadrilaterals, and other polygons by decomposing them into simpler shapes. They extend volume concepts to include fractional edge lengths and learn to use the coordinate plane to solve geometric problems. Students also learn to create nets and visualize three-dimensional figures from two-dimensional representations.
Key Skills:
- Find area of triangles using the formula A = ½bh
- Find areas of quadrilaterals (rectangles, parallelograms, trapezoids)
- Decompose complex polygons into triangles and rectangles to find total area
- Apply formula for volume of rectangular prisms: V = lwh or V = Bh
- Find volume with fractional edge lengths (e.g., 2.5 ft × 3 ft × 4.5 ft)
- Plot polygons on the coordinate plane given vertex coordinates
- Find lengths of horizontal and vertical segments using coordinates
- Draw and identify nets of three-dimensional figures (cubes, prisms, pyramids)
- Calculate surface area using nets
Triangle with base = 8 cm and height = 5 cm: A = ½ × 8 × 5 = 20 cm²
Example (Volume):Rectangular prism: length = 5 in, width = 3 in, height = 4 in
V = 5 × 3 × 4 = 60 cubic inches
Example (Coordinate Plane):Plot rectangle with vertices at (1, 2), (1, 5), (4, 5), (4, 2)
Width = 4 - 1 = 3 units, Height = 5 - 2 = 3 units, Area = 9 square units
Example (Nets):A cube's net has 6 connected squares. If each square is 4×4, surface area = 6 × 16 = 96 square units
Statistics and Probability (6.SP)
What Students Learn:
Students begin their formal study of statistics by learning to distinguish statistical questions (which anticipate variability) from non-statistical questions. They learn to analyze data distributions using measures of center (mean, median, mode) and measures of variability (range, interquartile range, mean absolute deviation). They create and interpret various data displays including dot plots, histograms, and box plots.
Key Skills:
- Recognize statistical questions as those that anticipate variability in answers
- Understand that data has a distribution that can be described by shape, center, and spread
- Calculate mean (average): sum of values ÷ number of values
- Find median (middle value when data is ordered)
- Identify mode (most frequent value)
- Calculate range (maximum - minimum)
- Understand quartiles (Q1, Q2/median, Q3) and interquartile range (IQR = Q3 - Q1)
- Display data using dot plots, histograms, and box plots
- Describe overall patterns and notable features of data (clusters, gaps, outliers)
- Calculate Mean Absolute Deviation (MAD) to measure variability
Statistical: "How tall are students in 6th grade?" (variability expected)
Not statistical: "How tall is the tallest 6th grader?" (one specific answer)
Example (Measures of Center):Data set: 12, 15, 18, 15, 20, 14, 15
Mean = (12+15+18+15+20+14+15) ÷ 7 = 109 ÷ 7 ≈ 15.6
Median: Order the data: 12, 14, 15, 15, 15, 18, 20 → Median = 15
Mode = 15 (appears most frequently)
Example (Box Plot):For data: 5, 7, 8, 9, 10, 12, 14, 15, 18, 20
Minimum = 5, Q1 = 8, Median = 11, Q3 = 15, Maximum = 20
IQR = 15 - 8 = 7 (spread of middle 50% of data)