Focus Areas for Fifth Grade
- Developing fluency with addition and subtraction of fractions
- Understanding volume
- Working with decimals to hundredths
- Extending division to 2-digit divisors
Math Standards by Domain
Operations and Algebraic Thinking (5.OA)
What Students Learn:
Students learn to use grouping symbols like parentheses ( ), brackets [ ], and braces { } to control the order of operations in numerical expressions. They interpret expressions without evaluating them and write expressions to represent real-world situations. This builds algebraic thinking and understanding of mathematical structure.
Key Skills:
- Use parentheses, brackets, or braces in numerical expressions
- Evaluate expressions with grouping symbols following order of operations
- Interpret expressions without calculating (e.g., "3 × (4 + 2)" means "3 times as large as the sum of 4 and 2")
- Write simple expressions to record calculations with numbers
- Write expressions from verbal descriptions
- Understand that different groupings can yield different results
Evaluate: 2 × (8 + 3) = 2 × 11 = 22 (not 16 + 3 = 19)
Interpret without calculating: "5 × (6 - 2)" means "5 times the difference of 6 and 2"
Write an expression: "Add 3 and 7, then multiply by 2" → 2 × (3 + 7) or (3 + 7) × 2
Compare: (5 + 3) × 2 = 16, but 5 + 3 × 2 = 11
What Students Learn:
Students learn to create and analyze numerical patterns based on given rules. They generate two patterns simultaneously, organize the data in ordered pairs, and identify relationships between corresponding terms. This develops algebraic reasoning and prepares students for working with functions and coordinate graphing.
Key Skills:
- Generate numerical patterns from given rules (e.g., "start at 0, add 3 each time")
- Create two patterns at the same time using different rules
- Identify apparent relationships between corresponding terms in two patterns
- Form ordered pairs from the two patterns
- Graph ordered pairs on a coordinate plane
- Explain patterns and relationships using properties of operations
Pattern 1 (Rule: start at 0, add 2): 0, 2, 4, 6, 8, 10...
Pattern 2 (Rule: start at 0, add 6): 0, 6, 12, 18, 24, 30...
Ordered pairs: (0,0), (2,6), (4,12), (6,18), (8,24), (10,30)
Relationship: Each term in Pattern 2 is 3 times the corresponding term in Pattern 1
Graph the ordered pairs and observe the linear pattern
Number and Operations in Base Ten (5.NBT)
What Students Learn:
Students develop a deep understanding of the base-ten place value system, extending their knowledge to include decimals to the thousandths place. They recognize that in a multi-digit number, each digit represents 10 times what it would represent in the place to its right. This understanding is crucial for operations with decimals and large numbers.
Key Skills:
- Recognize that each place value is 10 times the place to its right (6 in hundreds place = 10 times the value of 6 in tens place)
- Explain patterns in the number of zeros when multiplying by powers of 10
- Explain patterns in decimal placement when multiplying or dividing by powers of 10
- Read and write decimals to thousandths using base-ten numerals, number names, and expanded form
- Compare two decimals to thousandths using >, =, <
- Round decimals to any place (whole number, tenths, hundredths)
Place value: 4.567 = 4 ones + 5 tenths + 6 hundredths + 7 thousandths = 4 + 0.5 + 0.06 + 0.007
Powers of 10: 3.45 × 10 = 34.5 (decimal point moves 1 place right)
3.45 × 100 = 345 (decimal point moves 2 places right)
Compare: 3.456 > 3.45 (compare place by place: same ones, same tenths, 6 hundredths > 5 hundredths)
Round 3.678 to the nearest tenth: 3.7 (7 hundredths rounds up)
What Students Learn:
Students develop fluency with multiplying large numbers using the standard algorithm. They build on their understanding of place value and properties of operations from earlier grades to efficiently multiply numbers with multiple digits. This skill is essential for solving complex real-world problems and higher mathematics.
Key Skills:
- Multiply multi-digit whole numbers (e.g., 243 × 56) using the standard algorithm
- Use strategies based on place value and properties of operations
- Understand partial products in multiplication
- Accurately handle regrouping (carrying) in multi-step multiplication
- Estimate products to check reasonableness of answers
- Apply multiplication to solve real-world problems
Standard algorithm for 243 × 56:
243
× 56
-----
1458 (243 × 6)
12150 (243 × 50)
-----
13,608
Estimate to check: 240 × 60 = 14,400 (close to 13,608, so answer is reasonable)
What Students Learn:
Students extend their division skills to include two-digit divisors, one of the most challenging arithmetic operations in elementary school. They use strategies based on place value, properties of operations, and the relationship between multiplication and division. Understanding remainders and interpreting them in context is also emphasized.
Key Skills:
- Divide up to four-digit dividends by two-digit divisors (e.g., 3,456 ÷ 24)
- Use strategies based on place value, properties of operations, and area models
- Apply the standard algorithm for long division
- Interpret remainders in context (whole number, fraction, or decimal)
- Estimate quotients to assess reasonableness
- Explain the reasoning used in division strategies
Divide 2,856 ÷ 34:
Estimate: 2,800 ÷ 30 is about 90
Using long division: 2,856 ÷ 34 = 84
Check: 84 × 34 = 2,856
Real-world: 2,856 cookies divided into 34 bags = 84 cookies per bag
With remainder: 2,860 ÷ 34 = 84 R4 (84 full bags with 4 cookies left over)
What Students Learn:
Students learn to perform all four operations with decimal numbers to the hundredths place. They use concrete models, drawings, and strategies based on place value to understand decimal operations. This includes adding, subtracting, multiplying, and dividing decimals, extending their number sense to include rational numbers.
Key Skills:
- Add decimals to hundredths (e.g., 3.45 + 2.67)
- Subtract decimals to hundredths (e.g., 5.83 - 2.49)
- Multiply decimals to hundredths (e.g., 3.4 × 2.5)
- Divide decimals to hundredths using concrete models and strategies
- Use strategies based on place value, properties of operations, and the relationship between operations
- Apply decimal operations to real-world contexts like money, measurement, and data
Addition: 3.45 + 2.67 = 6.12 (line up decimal points, add like whole numbers)
Subtraction: 5.83 - 2.49 = 3.34 (line up decimal points, subtract with regrouping)
Multiplication: 3.4 × 2.5 = 8.5 (multiply as whole numbers: 34 × 25 = 850, then place decimal: 1+1=2 places → 8.50)
Division: 6.8 ÷ 0.4 = 17 (think: 68 ÷ 4 = 17)
Real-world: $12.75 + $8.50 = $21.25 (total cost)
Number and Operations—Fractions (5.NF)
What Students Learn:
Students learn to add and subtract fractions with different denominators by finding common denominators. They use visual models and equivalent fractions to understand why this process works. This skill builds on their understanding of fraction equivalence from fourth grade and is essential for working with rational numbers.
Key Skills:
- Add fractions with unlike denominators (e.g., 1/3 + 1/4)
- Subtract fractions with unlike denominators (e.g., 5/6 - 1/2)
- Find common denominators using equivalent fractions
- Add and subtract mixed numbers with unlike denominators
- Solve word problems involving addition and subtraction of fractions
- Use visual models (area models, number lines) to represent fraction operations
- Estimate fraction sums and differences to assess reasonableness
Add: 1/3 + 1/4 = 4/12 + 3/12 = 7/12 (common denominator is 12)
Subtract: 5/6 - 1/2 = 5/6 - 3/6 = 2/6 = 1/3 (simplify answer)
Mixed numbers: 2 1/4 + 1 1/3 = 2 3/12 + 1 4/12 = 3 7/12
Word problem: Sarah ran 2 3/4 miles on Monday and 3 1/2 miles on Tuesday. How far did she run total? 2 3/4 + 3 1/2 = 2 3/4 + 3 2/4 = 5 5/4 = 6 1/4 miles
What Students Learn:
Students learn that fractions represent division problems - the fraction a/b can be interpreted as a ÷ b. This deepens their understanding of fractions as numbers and connects fractions to division. They solve word problems where whole numbers are divided and the answer is expressed as a fraction or mixed number.
Key Skills:
- Interpret a/b as "a divided by b" (e.g., 3/4 means 3 ÷ 4)
- Solve word problems involving division of whole numbers with quotient expressed as a fraction
- Understand that dividing a into b equal parts means each part is a/b of the whole
- Connect division to fair-sharing situations
- Express division results as fractions or mixed numbers
- Use this understanding to solve real-world sharing problems
3/4 means 3 ÷ 4 = 0.75
Word problem: 3 pizzas are shared equally among 4 people. How much pizza does each person get? 3 ÷ 4 = 3/4 of a pizza per person
Another way: Each person gets one whole pizza divided into 4 parts, and they get 3 of those parts
Mixed number: 7 ÷ 3 = 7/3 = 2 1/3 (7 items shared by 3 people)
What Students Learn:
Students learn to multiply fractions by fractions, whole numbers by fractions, and mixed numbers. They develop conceptual understanding through area models and number lines, recognizing multiplication as scaling (resizing). They learn that multiplying by a fraction less than 1 makes the product smaller, while multiplying by a fraction greater than 1 makes it larger.
Key Skills:
- Multiply a fraction by a fraction (e.g., 2/3 × 3/4)
- Multiply a whole number by a fraction (e.g., 4 × 2/5)
- Multiply mixed numbers (e.g., 2 1/2 × 3 1/3)
- Interpret multiplication as scaling (resizing)
- Understand that a × (b/c) = (a × b)/c
- Find the area of rectangles with fractional side lengths
- Solve real-world problems involving multiplication of fractions and mixed numbers
Multiply fractions: 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2
Whole × fraction: 4 × 2/5 = 8/5 = 1 3/5
Mixed numbers: 2 1/2 × 3 = 7 1/2 (think: 2×3 + 1/2×3 = 6 + 1 1/2)
Area: A rectangle is 3 1/2 feet by 2 1/4 feet. Area = 3 1/2 × 2 1/4 = 7/2 × 9/4 = 63/8 = 7 7/8 square feet
Scaling: 1/2 × 8 = 4 (product is smaller because multiplying by fraction < 1)
What Students Learn:
Students are introduced to division with fractions in limited cases: dividing a unit fraction by a whole number (like 1/3 ÷ 4) and dividing a whole number by a unit fraction (like 4 ÷ 1/3). They use visual models to understand these situations and solve real-world problems. This lays groundwork for general fraction division in sixth grade.
Key Skills:
- Divide unit fractions by whole numbers (e.g., 1/3 ÷ 4 = 1/12)
- Divide whole numbers by unit fractions (e.g., 4 ÷ 1/3 = 12)
- Use visual fraction models and equations to represent problems
- Understand division as partitioning or measuring
- Solve real-world problems involving division with unit fractions
- Relate division to multiplication (e.g., 1/3 ÷ 4 = 1/3 × 1/4)
Unit fraction ÷ whole: 1/2 ÷ 3 = 1/6 (divide 1/2 into 3 equal parts, each is 1/6)
Word problem: A ribbon that is 1/2 yard long is cut into 3 equal pieces. How long is each piece? 1/2 ÷ 3 = 1/6 yard
Whole ÷ unit fraction: 4 ÷ 1/2 = 8 (how many 1/2s are in 4? Eight halves)
Word problem: How many 1/3-cup servings are in 4 cups of rice? 4 ÷ 1/3 = 12 servings
Visual: Draw 4 circles, divide each into thirds = 12 third-pieces total
Measurement and Data (5.MD)
What Students Learn:
Students learn to convert between different units of measurement within the same system (customary or metric). They convert lengths, weights, volumes, and time using conversion facts and multiplication/division. This skill is essential for solving multi-step real-world problems that involve different measurement units.
Key Skills:
- Convert customary length units: inches ↔ feet ↔ yards ↔ miles
- Convert customary weight units: ounces ↔ pounds ↔ tons
- Convert customary capacity units: cups ↔ pints ↔ quarts ↔ gallons
- Convert metric units: millimeters ↔ centimeters ↔ meters ↔ kilometers
- Convert metric mass: grams ↔ kilograms
- Convert metric volume: milliliters ↔ liters
- Use conversion in multi-step real-world problems
Length: 5 feet = ? inches → 5 × 12 = 60 inches (multiply by 12)
Weight: 3 pounds = ? ounces → 3 × 16 = 48 ounces
Capacity: 8 cups = ? quarts → 8 ÷ 4 = 2 quarts (4 cups in a quart)
Metric: 3.5 meters = ? centimeters → 3.5 × 100 = 350 cm
Multi-step: A recipe needs 2 quarts of milk. You have 5 pints. Do you have enough? 5 pints = 2.5 quarts (5 ÷ 2), so yes!
What Students Learn:
Students create line plots to display measurement data that involves fractions (like 1/2, 1/4, 1/8 of a unit). They then use the data in the line plot to solve problems involving addition, subtraction, and multiplication of fractions. This integrates their fraction operation skills with data analysis and real measurement contexts.
Key Skills:
- Create line plots showing measurements in fractions of a unit (halves, fourths, eighths)
- Display data on a number line with appropriate scale
- Interpret data from line plots
- Use operations on fractions to answer questions about the data
- Find total, difference, or average of fractional measurements
- Apply to real-world contexts like length, weight, or capacity measurements
Measure the lengths of 10 pencils to the nearest 1/4 inch: 5 1/4, 5 1/2, 5 3/4, 5 1/2, 5 1/4, 5 3/4, 5 1/2, 5 1/4, 5 3/4, 5 1/2
Create line plot with X's above each measurement on a number line
Questions: What is the most common length? (5 1/2 inches - appears 4 times)
What is the difference between longest and shortest? 5 3/4 - 5 1/4 = 2/4 = 1/2 inch
What is the total length of all pencils? Add all measurements
What Students Learn:
Students develop understanding of volume as the amount of space a three-dimensional object occupies. They learn that volume is measured in cubic units and practice measuring volume by counting unit cubes that fill a rectangular prism without gaps or overlaps. This is their first formal introduction to volume measurement.
Key Skills:
- Recognize volume as an attribute of 3D solid figures
- Understand that volume is measured in cubic units (cubic inches, cubic centimeters, etc.)
- Measure volumes by counting unit cubes
- Pack rectangular prisms with unit cubes without gaps or overlaps
- Understand that a 1-unit cube is a cube with edge length 1 unit
- Build rectangular prisms with specific volumes
Build a rectangular prism using 24 unit cubes. Count: The volume is 24 cubic units
A box measures 3 cubes by 4 cubes by 2 cubes. Fill it completely with unit cubes and count: 3×4×2 = 24 cubic units
Volume is measured in cubic units: cubic inches (in³), cubic feet (ft³), cubic centimeters (cm³)
A cube with edge length 1 cm has volume 1 cubic centimeter (1 cm³)
What Students Learn:
Students learn to calculate the volume of rectangular prisms using formulas rather than counting individual cubes. They understand that volume equals length × width × height (V = l × w × h) or base area × height (V = B × h). They also learn to find volumes of composite figures by adding volumes of non-overlapping parts.
Key Skills:
- Use the formula V = l × w × h to find volumes of rectangular prisms
- Use the formula V = B × h where B is the base area
- Understand why the formulas work (layers of cubes)
- Apply formulas to solve real-world problems
- Find volumes of solid figures composed of two non-overlapping rectangular prisms
- Use addition to combine volumes of composite figures
- Work with whole number edge lengths and volumes
Find volume: A box is 5 inches long, 3 inches wide, 4 inches tall
V = l × w × h = 5 × 3 × 4 = 60 cubic inches
Using base area: Base = 5 × 3 = 15 square inches, V = 15 × 4 = 60 cubic inches
Composite figure: An L-shaped figure made of two rectangular prisms
First prism: 4×2×3 = 24 cubic units, Second prism: 3×2×2 = 12 cubic units
Total volume: 24 + 12 = 36 cubic units
Geometry (5.G)
What Students Learn:
Students are introduced to the coordinate plane with a focus on the first quadrant (positive x and y values). They learn to use ordered pairs to name points, graph points, and interpret the coordinates. They apply this to solve real-world and mathematical problems, forming a foundation for algebra and graphing in later grades.
Key Skills:
- Understand the coordinate plane with perpendicular number lines (x-axis and y-axis)
- Identify the origin (0,0) and axes labels
- Use ordered pairs (x, y) to name points in the first quadrant
- Graph points given ordered pairs
- Interpret coordinate values in context of real-world situations
- Solve problems by graphing points and interpreting patterns
- Understand that the first number (x) tells horizontal distance, second (y) tells vertical distance
Plot point A at (3, 5): Move 3 units right on x-axis, then 5 units up on y-axis
What point is at (0, 4)? The point 4 units up on the y-axis
Real-world: On a map, the library is at (2, 6) and the park is at (7, 6). They have the same y-coordinate, so they're at the same "north-south" position
Pattern: Plot points (1, 2), (2, 4), (3, 6), (4, 8). What pattern do you see? y is always 2 times x
What Students Learn:
Students develop deeper understanding of how two-dimensional shapes are related to each other. They learn that shapes can be classified in a hierarchy - for example, all squares are rectangles, but not all rectangles are squares. They identify attributes like parallel sides, perpendicular sides, and angles, and use these to classify shapes into categories and subcategories.
Key Skills:
- Understand attributes of 2D figures: parallel/perpendicular sides, angles, symmetry
- Classify shapes based on properties in a hierarchy
- Recognize that shapes can belong to more than one category (a square is also a rectangle, rhombus, parallelogram, and quadrilateral)
- Identify and describe triangles based on sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse)
- Identify and describe quadrilaterals: parallelograms, rectangles, squares, rhombuses, trapezoids
- Use Venn diagrams or tree diagrams to show classification hierarchies
Hierarchy: All squares are rectangles (4 right angles, opposite sides equal) AND rhombuses (all sides equal)
A square is special because it has ALL properties: 4 equal sides, 4 right angles, opposite sides parallel
Triangle classification: An equilateral triangle has 3 equal sides AND 3 equal angles (60° each) - it's also isosceles
Quadrilateral: A parallelogram has opposite sides parallel and equal. Rectangles, squares, and rhombuses are all special types of parallelograms
A trapezoid has exactly one pair of parallel sides (in some definitions, at least one pair)