FAST Math Grade 3: Number Sense and Operations

Number Sense and Operations

The Number Sense and Operations domain is the largest computational strand on FAST Math Grade 3. It tests place value through 10,000, reading and writing whole numbers in standard, expanded, and word form, comparing and ordering numbers, rounding to the nearest 10 or 100, and adding and subtracting multi-digit numbers. It also tests multiplication and division facts, the relationship between multiplication and division, multiplying one-digit numbers by multiples of 10, and using properties such as the commutative and distributive properties.

Core rules to know

In a four-digit number, each digit has a value ten times the value of the digit to its right: in 4,507, the 5 means 5 hundreds, or 500, and the 0 means 0 tens.
To compare whole numbers, compare the greatest place value first. If those digits match, move right until the digits differ.
Expanded form breaks a number into the value of each digit: 4,507 = 4,000 + 500 + 7.
To round to the nearest 10, look at the ones digit; to round to the nearest 100, look at the tens digit. If the deciding digit is 5 or greater, round up.
Multiplication and division are inverse operations: 6 × 7 = 42 means 42 ÷ 6 = 7 and 42 ÷ 7 = 6.
Multiples of 10 use place-value patterns: 4 × 30 = 4 × 3 tens = 12 tens = 120.

Worked example

Compare 4,507 and 4,570, then round 4,570 to the nearest hundred. Both numbers have 4 thousands, so compare hundreds: both have 5 hundreds. Compare tens: 4,507 has 0 tens; 4,570 has 7 tens. Since 7 > 0, 4,570 > 4,507. To round 4,570 to the nearest hundred, look at the tens digit, 7. Because 7 is 5 or more, the hundreds digit increases from 5 to 6, and the tens and ones become zeros: 4,600.

Traps that cost marks

Writing 4,507 in expanded form as 4,000 + 500 + 7 is correct, but writing 4,000 + 50 + 7 is not; the 5 is in the hundreds place.
Forgetting to regroup across a zero when subtracting, such as in 503 − 278.
Rounding 4,570 to 4,500 by looking at the hundreds digit; the tens digit decides, so it is 4,600.
Mixing up 3 × 40 = 120 with 3 × 40 = 12 by dropping the zero.
Treating the equals sign as “the answer comes next” instead of “is the same as” when checking an equation.

Fractions

The Fractions domain tests how Grade 3 students build meaning for fractions as equal parts of a whole and as numbers on a number line. Expect items on unit fractions, partitioning shapes, writing fractions, representing fractions on number lines, equivalent fractions with models, comparing fractions with the same numerator or the same denominator, and expressing whole numbers as fractions.

Core rules and concepts

A fraction such as 3/4 means 3 equal parts out of 4 equal parts in the whole. The denominator tells how many equal parts; the numerator tells how many parts are selected.
Equal parts matter: a shape split into 4 parts that are not equal does not show fourths.
On a number line from 0 to 1, the denominator tells how many equal intervals to make. For fourths, mark 0/4, 1/4, 2/4, 3/4, 4/4 = 1.
Equivalent fractions name the same point or same amount: 1/2 = 2/4 = 4/8 when the models show equal-sized wholes.
When denominators are the same, compare numerators: 3/8 < 5/8 because 3 same-size parts is fewer than 5 same-size parts.
When numerators are the same, compare denominators: 2/3 > 2/5 because thirds are larger than fifths, so 2 thirds is more than 2 fifths.
A whole number can be written as a fraction: 1 = 3/3 = 4/4; 2 = 6/3 = 8/4.

Worked example

Compare 2/3 and 2/5. Both fractions have numerator 2, so the numerators are equal. The denominators tell the size of the parts: thirds are larger than fifths because the same whole is split into fewer pieces. Two large thirds cover more than two smaller fifths. Therefore 2/3 > 2/5. A number-line check places 2/3 closer to 1 than 2/5, which is less than 1/2.

Traps that cost marks

Comparing denominators as if a larger denominator means a larger fraction. 1/8 is smaller than 1/3.
Counting intervals on a number line instead of equal parts; 2/4 must be at the second fourth-mark, not halfway between 0 and 1 unless the marks are correct.
Treating 4/4 as 0 or as 4; it equals 1.
Using unequal partitions to justify equivalence.
Writing the numerator and denominator in the wrong order when the model shows selected parts over total equal parts.

Algebraic Reasoning

The Algebraic Reasoning domain tests how Grade 3 students use operations to solve problems and describe relationships. Items ask students to find unknown factors or addends, solve one- and two-step word problems with multiplication and division, apply properties such as the distributive property, complete equations, and identify or extend patterns in tables and multiplication charts.

Core rules and concepts

An equation is a statement that two expressions have the same value. The equals sign means “is the same as,” not “the answer comes next.”
Multiplication and division are inverse operations. If 7 × □ = 56, then □ = 56 ÷ 7 = 8.
The distributive property breaks a factor into friendly parts: 6 × 8 = 6 × (5 + 3) = (6 × 5) + (6 × 3) = 30 + 18 = 48.
The commutative property lets factors switch order: 4 × 9 = 9 × 4. The identity property says any number times 1 stays that number; any number times 0 is 0.
In patterns, find the rule that moves from one term to the next, then test it on more than one pair. Rules can be additive (add 6) or multiplicative (multiply by 3).
For two-step problems, decide what must be found first, then use that result in the second step. A model or equation can show the hidden question.

Worked example

A class has 5 tables with 6 students at each table. Three students leave. How many students remain? First find the starting number: 5 × 6 = 30 students. Then subtract the 3 who leave: 30 − 3 = 27 students. As one equation: (5 × 6) − 3 = 27. If the problem asked for an unknown factor instead, such as 5 × □ = 30, the inverse operation gives □ = 30 ÷ 5 = 6.

Traps that cost marks

Solving only the first step of a two-step problem, such as reporting 30 instead of 27.
Treating the equals sign as an instruction to compute only the right side; 8 + 4 = □ + 6 requires balancing both sides, so □ = 6.
Assuming every pattern adds the same number; check whether the rule multiplies.
Breaking apart a factor incorrectly in the distributive property, such as 6 × 8 = 6 × 5 + 3 instead of 6 × 5 + 6 × 3.
Finding an unknown factor by adding instead of using the inverse operation.

Measurement

The Measurement domain tests selecting appropriate tools and units, measuring length, liquid volume, mass, and temperature, solving problems with time and elapsed time, and finding perimeter and area. Grade 3 items often include rulers, clocks, and real-world contexts such as gardening, cooking, or travel. Students must choose the right unit and keep the meaning of the measurement.

Core rules and concepts

Choose a unit that matches the object: inches, feet, yards, or miles for length; cups, pints, quarts, or gallons for liquid volume; ounces, pounds, or tons for weight; grams or kilograms for mass; degrees Fahrenheit or Celsius for temperature.
Measure length from the zero mark, not from the edge of the ruler. Read the nearest marked fraction or whole unit required by the problem.
Elapsed time is the difference between a start time and an end time. Count forward in convenient jumps, such as to the next hour, then by minutes.
Perimeter is the distance around a figure. Add all side lengths. For a rectangle, use P = 2l + 2w or P = 2(l + w).
Area is the number of unit squares that cover a figure. For a rectangle, A = length × width. Area is measured in square units; perimeter is measured in linear units.
The same rectangle can have the same perimeter but a different area from another rectangle, so do not substitute one for the other.

Worked example

A rectangle is 6 centimeters long and 4 centimeters wide. Find its perimeter and area. Perimeter: add the four sides, 6 + 4 + 6 + 4 = 20 centimeters, or use 2(6 + 4) = 20 cm. Area: multiply length by width, 6 × 4 = 24 square centimeters. The perimeter describes the distance around the rectangle; the area describes the space inside it.

Traps that cost marks

Reporting area in centimeters instead of square centimeters, or perimeter in square units.
Adding only length and width and forgetting the other two sides for perimeter.
Starting a ruler at 1 instead of 0, which makes every length one unit too short.
In elapsed time, forgetting to regroup 60 minutes into 1 hour when crossing an hour.
Choosing a unit that is too large or too small, such as measuring a pencil in miles.

Geometric Reasoning

The Geometric Reasoning domain tests Grade 3 students’ ability to identify and describe two-dimensional figures by their attributes. Items cover points, lines, line segments, rays, parallel and perpendicular lines, angles, quadrilaterals, polygons, and line symmetry. Students also compose and decompose figures and use attributes to classify shapes rather than relying on how a shape looks.

Core rules and concepts

A point names a location. A line goes straight in both directions without ending. A line segment is part of a line with two endpoints. A ray starts at one endpoint and continues in one direction.
Parallel lines never intersect and stay the same distance apart. Perpendicular lines intersect to form right angles.
A polygon is a closed plane figure with straight sides that do not cross. It is named by its number of sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6.
A quadrilateral has 4 sides and 4 angles. Special quadrilaterals include trapezoids, parallelograms, rectangles, rhombuses, and squares. A square is also a rectangle and a rhombus because it has all the required attributes.
A right angle is a square corner. Acute angles are smaller than a right angle; obtuse angles are larger.
Line symmetry means a figure can fold along a line so the two halves match exactly. A figure may have zero, one, or many lines of symmetry.

Worked example

A figure has 4 sides, 4 right angles, and opposite sides equal. What is the most specific name? The figure is a quadrilateral because it has 4 sides. The 4 right angles make it a rectangle. Opposite sides equal fits a rectangle. If all 4 sides were also equal, the most specific name would be a square. Without that extra fact, rectangle is the safest specific name. This reasoning uses attributes, not the orientation of the drawing.

Traps that cost marks

Calling any four-sided shape a rectangle; a rectangle must have four right angles.
Thinking a square is not a rectangle. In the inclusive classification used in Grade 3, a square has all the attributes of a rectangle.
Confusing a line, which continues forever, with a line segment, which has endpoints.
Counting a shape as having symmetry when the fold line does not produce matching halves.
Classifying a quadrilateral by its tilt on the page rather than by side and angle attributes.

Data Analysis and Probability

The Data Analysis and Probability domain tests how Grade 3 students collect, represent, and interpret data. Items use frequency tables, scaled picture graphs, scaled bar graphs, and line plots. Students read the scale, compare categories, find totals and differences, and answer questions that require more than simply reading one number off a graph. Probability at this grade is informal: students may describe events as likely, unlikely, certain, or impossible in simple contexts.

Core rules and concepts

A scaled graph uses each picture or bar unit to stand for more than one item. Read the key or axis scale before counting.
In a picture graph, a half symbol means half of the scale value. If one symbol = 4 books, a half symbol = 2 books.
In a bar graph, read across from the top of the bar to the matching number on the scale. Check whether the grid lines count by 1, 2, 5, or 10.
A line plot shows each data value as an X above a number line. The number of Xs above a value is its frequency.
To compare categories, subtract the smaller value from the larger value. To find a total, add the values for the categories requested.
A table must be read by matching each row or column label with the correct number; do not assume the order of rows is the order of size.

Worked example

A scaled picture graph shows favorite fruits. The key says one apple symbol = 4 votes. Apples have 3 whole symbols and 1 half symbol. How many votes do apples have? Each whole symbol is 4 votes, so 3 whole symbols are 3 × 4 = 12 votes. The half symbol is half of 4, or 2 votes. Add: 12 + 2 = 14 votes. The reasoning is: multiply whole symbols by the scale, then add the value of the partial symbol.

Traps that cost marks

Counting symbols instead of multiplying by the scale.
Reading a bar graph from the wrong grid line or ignoring that the axis starts at 0.
Mixing up “how many more” (subtract) with “how many altogether” (add).
On a line plot, counting the labels instead of the Xs, or missing that a cluster shows frequency.
Treating a half symbol as one whole symbol.