Focus Areas for Third Grade
- Developing understanding of multiplication and division
- Developing understanding of fractions
- Developing understanding of area
- Describing and analyzing two-dimensional shapes
Math Standards by Domain
Operations and Algebraic Thinking (3.OA)
What Students Learn:
Students develop a deep understanding of multiplication as repeated addition and equal groups, and division as splitting into equal groups or finding how many groups can be made. They learn to interpret and solve real-world problems involving these operations, representing situations with equations that have unknowns in different positions.
Key Skills:
- Interpret products of whole numbers (e.g., 5 x 7 as 5 groups of 7 objects)
- Interpret quotients (e.g., 56 ÷ 8 as the number in each group when 56 objects are split into 8 equal groups)
- Use multiplication and division within 100 to solve word problems
- Represent problems using drawings, equations, and arrays
- Determine the unknown number in multiplication and division equations (e.g., 8 x ? = 48, ? ÷ 3 = 9)
- Understand the relationship between multiplication and division
There are 6 bags with 4 cookies in each bag. How many cookies in all? 6 x 4 = 24 cookies
Example (Division - Partitive):24 cookies are shared equally among 6 friends. How many cookies does each friend get? 24 ÷ 6 = 4 cookies
Example (Division - Measurement):You have 24 cookies. You put 4 cookies in each bag. How many bags do you need? 24 ÷ 4 = 6 bags
What Students Learn:
Students learn to apply the commutative, associative, and distributive properties to make multiplication and division easier. They understand that division can be thought of as a missing factor problem (e.g., 32 ÷ 4 = ? is the same as 4 x ? = 32), which strengthens the connection between multiplication and division.
Key Skills:
- Apply commutative property: 6 x 3 = 3 x 6
- Apply associative property: 3 x (5 x 2) = (3 x 5) x 2
- Apply distributive property: 8 x 7 = (8 x 5) + (8 x 2)
- Understand division as an unknown-factor problem: 24 ÷ 6 = ? is the same as 6 x ? = 24
- Use properties to make computation more efficient
- Recognize and explain patterns in multiplication (multiplying by 0, 1, 5, 10)
4 groups of 7 is the same as 7 groups of 4: 4 x 7 = 7 x 4 = 28
Example (Distributive):To find 6 x 8, think: 6 x 8 = (6 x 5) + (6 x 3) = 30 + 18 = 48
Example (Unknown Factor):Find 45 ÷ 5. Think: "5 times what number equals 45?" 5 x 9 = 45, so 45 ÷ 5 = 9
What Students Learn:
Students develop fluency with multiplication and division facts through 10 x 10. By the end of third grade, they should know all single-digit multiplication facts from memory and be able to quickly derive related division facts. This fluency is essential for success in future mathematics.
Key Skills:
- Fluently (accurately and efficiently) multiply all single-digit numbers (0-10)
- Fluently divide within 100 using related multiplication facts
- Use strategies to master facts: skip counting, using known facts, patterns
- Know multiplication facts from memory by end of Grade 3
- Apply multiplication and division facts to solve problems quickly
- Understand and use fact families (e.g., 3 x 8 = 24, 8 x 3 = 24, 24 ÷ 3 = 8, 24 ÷ 8 = 3)
Students should quickly recall: 7 x 8 = 56, 9 x 6 = 54, 4 x 9 = 36
Example (Division Facts):Using multiplication facts: 72 ÷ 8 = ? Think: "8 x ? = 72" → 8 x 9 = 72, so 72 ÷ 8 = 9
Example (Fact Family):6 x 7 = 42, 7 x 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6
What Students Learn:
Students solve multi-step word problems that require using all four operations (addition, subtraction, multiplication, and division). They learn to identify which operations are needed, perform calculations in the correct order, and assess whether their answers are reasonable. They also explore patterns in arithmetic sequences and explain the rules that generate them.
Key Skills:
- Solve two-step word problems using any combination of the four operations
- Represent problems using equations with a letter for the unknown
- Assess the reasonableness of answers using mental computation and estimation
- Identify and describe arithmetic patterns (including patterns in addition and multiplication tables)
- Explain patterns using properties of operations
- Determine which operation(s) to use based on the problem context
A store has 8 shelves with 6 books on each shelf. If 12 books are sold, how many books are left? Step 1: 8 x 6 = 48 books total. Step 2: 48 - 12 = 36 books left
Example (Pattern):Pattern: 4, 8, 12, 16, 20... Rule: Add 4 each time (or count by 4s)
Multiplication table pattern: The ones digit in the 5 times table alternates between 5 and 0
Number and Operations in Base Ten (3.NBT)
What Students Learn:
Students deepen their understanding of place value to include rounding numbers for estimation. They develop fluency with addition and subtraction of three-digit numbers using standard algorithms and strategies. They also learn to multiply single-digit numbers by multiples of 10 (like 3 x 20 or 7 x 80) using place value understanding.
Key Skills:
- Round whole numbers to the nearest 10 or 100 using place value understanding
- Use rounding to estimate sums, differences, and products
- Fluently add and subtract within 1000 using strategies and algorithms
- Use standard algorithms for addition and subtraction with regrouping
- Multiply one-digit numbers by multiples of 10 (e.g., 4 x 30, 8 x 60)
- Understand that 4 x 30 means 4 x 3 tens = 12 tens = 120
Round 347 to nearest 10: Look at ones place (7 ≥ 5), so round up → 350
Round 347 to nearest 100: Look at tens place (4 < 5), so round down → 300
Example (Addition with Regrouping):456 + 378 = 834 (regroup ones: 6+8=14, regroup tens: 5+7+1=13)
Example (Multiply by 10s):5 x 40 = 5 x 4 tens = 20 tens = 200
Number and Operations—Fractions (3.NF)
What Students Learn:
Students develop a foundational understanding of fractions as numbers on the number line. They learn that fractions represent equal parts of a whole, understand unit fractions (fractions with numerator 1), and explore equivalent fractions. They learn to compare fractions with the same numerator or denominator and represent fractions visually and symbolically.
Key Skills:
- Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts
- Understand a fraction a/b as the quantity formed by a parts of size 1/b
- Represent fractions on a number line from 0 to whole numbers
- Recognize and generate simple equivalent fractions (e.g., 1/2 = 2/4, 4/6 = 2/3)
- Express whole numbers as fractions (e.g., 3 = 3/1)
- Compare two fractions with the same numerator or denominator using >, =, < symbols
- Justify comparisons using visual models and reasoning about size
3/4 means 3 pieces, where the whole is divided into 4 equal pieces. Each piece is 1/4.
Example (Number Line):Place 2/3 on a number line: Divide the distance from 0 to 1 into 3 equal parts, mark the second point
Example (Equivalent):1/2 = 2/4 = 3/6 = 4/8 (fold paper in halves, then fourths to show equivalence)
Example (Comparing):2/4 < 3/4 (same denominator, smaller numerator means fewer pieces)
1/2 > 1/3 (same numerator, larger denominator means smaller pieces)
Measurement and Data (3.MD)
What Students Learn:
Students extend their time-telling skills to the nearest minute and learn to solve elapsed time problems. They solve problems involving measurement of liquid volumes and masses using standard units (grams, kilograms, liters). Students estimate and measure to develop a sense of the size of different units.
Key Skills:
- Tell and write time to the nearest minute on analog clocks
- Read and write time using digital notation (7:46 a.m., 3:15 p.m.)
- Solve word problems involving elapsed time (how much time has passed)
- Measure and estimate liquid volumes using liters (l) and milliliters (ml)
- Measure and estimate masses using grams (g) and kilograms (kg)
- Solve one-step word problems involving masses or volumes given in the same units
- Add, subtract, multiply, or divide to solve measurement problems
Clock shows 4:37 - hour hand between 4 and 5, minute hand on 7 (which means 37 minutes after the hour)
Elapsed time: Soccer practice starts at 3:15 and ends at 4:45. How long is practice? 1 hour 30 minutes
Example (Mass):An apple has a mass of 150 grams. How many grams do 4 apples have? 4 x 150 = 600 grams
Example (Volume):A water bottle holds 500 ml. If you drink 200 ml, how much is left? 500 - 200 = 300 ml
What Students Learn:
Students learn to create and interpret scaled picture graphs and bar graphs where each symbol or unit on the scale represents more than one object (e.g., each picture = 5 objects). They collect measurement data by measuring lengths to the nearest quarter inch and display this data in line plots. This develops their ability to organize, represent, and analyze data.
Key Skills:
- Draw scaled picture graphs where symbols represent multiple units (e.g., 1 picture = 5 items)
- Draw scaled bar graphs with intervals of 1, 2, 5, or 10
- Solve one- and two-step "how many more" and "how many less" problems using information from graphs
- Measure lengths using rulers marked with halves and fourths of an inch
- Generate measurement data and show it on a line plot marked in halves and fourths
- Interpret line plots and answer questions about the data
Favorite fruits: Show data where each apple symbol = 10 votes
Apples: ●●● (30 votes), Oranges: ●●●● (40 votes), Bananas: ●● (20 votes)
Question: How many more students chose oranges than apples? 40 - 30 = 10 students
Example (Line Plot):Measure 10 pencils to nearest 1/4 inch: 5 1/4", 5 1/2", 5 1/4", 6", 5 3/4", 5 1/2", 5 1/4", 6", 5 3/4", 5 1/2"
Create line plot with X marks above each measurement. Most common length: 5 1/4 inches (3 pencils)
What Students Learn:
Students develop understanding of area as the amount of space inside a two-dimensional figure. They learn to measure area by counting unit squares and by using multiplication (for rectangles). They understand area as additive and can find areas of rectilinear figures by decomposing them into rectangles. This connects geometry, multiplication, and addition.
Key Skills:
- Recognize area as an attribute of plane figures (measured in square units)
- Measure area by counting unit squares (square cm, square m, square in, square ft)
- Relate area to multiplication: a rectangle with sides 3 and 5 has area 3 x 5 = 15 square units
- Use tiling to show that area remains the same regardless of unit square arrangement
- Find areas of rectangles with whole-number side lengths using multiplication
- Find areas of rectilinear figures by decomposing into non-overlapping rectangles and adding areas
- Understand that different rectangles can have the same area
Rectangle divided into unit squares: Count 12 unit squares → Area = 12 square units
Example (Area by Multiplication):Rectangle: 4 units wide and 6 units long → Area = 4 x 6 = 24 square units
Example (Decomposing):L-shaped figure: Break into two rectangles (3x4 and 2x5) → Area = 12 + 10 = 22 square units
Example (Same Area):24 square units can be: 1x24, 2x12, 3x8, 4x6 (all have same area, different shapes)
What Students Learn:
Students learn that perimeter is the distance around a figure. They find perimeters by adding the lengths of all sides and solve problems involving perimeters, including finding missing side lengths when perimeter is known. They distinguish between area (space inside) and perimeter (distance around) and understand that rectangles with the same perimeter can have different areas.
Key Skills:
- Understand perimeter as the distance around a figure
- Find perimeter by adding all side lengths
- Find perimeter of rectangles using formulas: P = 2l + 2w or P = 2(l + w)
- Solve problems to find unknown side lengths given the perimeter
- Distinguish between area and perimeter (area = inside, perimeter = around)
- Understand that rectangles with same perimeter can have different areas
- Create rectangles with a given perimeter
Rectangle: 5 cm long and 3 cm wide → Perimeter = 5 + 3 + 5 + 3 = 16 cm (or 2 x 5 + 2 x 3 = 16 cm)
Example (Missing Side):Triangle: Two sides are 7 inches and 5 inches. Perimeter is 20 inches. What's the third side? 7 + 5 + ? = 20, so ? = 8 inches
Example (Area vs Perimeter):Rectangle A (2x6): Area = 12, Perimeter = 16
Rectangle B (3x4): Area = 12, Perimeter = 14 (same area, different perimeter)
Geometry (3.G)
What Students Learn:
Students understand that shapes in different categories can share attributes and that shared attributes can define a larger category. For example, all squares are rectangles because they have four right angles, but not all rectangles are squares. They partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole, connecting fractions to geometry.
Key Skills:
- Understand that shapes can belong to multiple categories based on attributes
- Recognize that rhombuses, rectangles, and squares are all examples of quadrilaterals
- Draw examples of quadrilaterals that are not rectangles or rhombuses
- Partition shapes into parts with equal areas
- Express the area of each part as a unit fraction (1/2, 1/3, 1/4, 1/8) of the whole
- Understand relationships: all squares are rectangles, but not all rectangles are squares
- Identify shared attributes among shape categories
A square is a special rectangle (4 sides, 4 right angles, and all sides equal)
A rectangle is a special parallelogram (4 sides, opposite sides parallel and equal, 4 right angles)
All rhombuses are parallelograms, but not all parallelograms are rhombuses
Example (Partitioning):Divide a rectangle into 8 equal parts. Each part is 1/8 of the whole area.
If the rectangle's area is 24 square units, each part has area 24 ÷ 8 = 3 square units