The Grade 4 Mathematics Dictionary.

Smaller than a square corner, exactly a square corner, or bigger than one.

A right angle is exactly 90° — the square corner. An acute angle is less than 90° (a small, sharp turn), and an obtuse angle is more than 90° but less than 180° (a wide, lazy turn). A straight angle is a full 180°, a flat line. Sorting angles by these names is the first step to classifying triangles.

When the bottoms match, add the tops and keep the bottom.

Adding fractions with the same denominator is just counting pieces of the same size: three eighths plus two eighths is five eighths, the same way three cats plus two cats is five cats. The denominator does not change, because the size of the piece did not change. In fourth grade you add and subtract fractions that already share a denominator — mixing different denominators comes later.

An amount of turn between two rays that share an endpoint, measured in degrees.

An angle is not about how long the lines are — it is about how much you turned. Two rays meeting at a point make an angle, and stretching those rays longer does not make the angle bigger. A full turn is 360 degrees, so one degree is 1/360 of a whole circle. That fraction is exactly why degrees work the way they do.

How much surface a shape covers — length times width for a rectangle.

Area counts how many unit squares fit inside a shape, which is why it is measured in *square* units. For a rectangle you don't have to count them one by one: rows times columns gets you there, which is exactly what length × width means. Perimeter and area answer different questions about the same shape, and two shapes can share one while differing wildly in the other.

Four-sided shapes sorted by parallel sides, equal sides, and square corners — and the categories overlap.

A quadrilateral is any shape with four straight sides. A trapezoid has at least one pair of parallel sides; a parallelogram has two pairs; a rectangle is a parallelogram with four right angles; a rhombus is a parallelogram with four equal sides; and a square is both — four right angles *and* four equal sides. The surprising part is that these are nested: every square is a rectangle, and every rectangle is a parallelogram. Sorting by properties instead of by what a shape 'looks like' is the whole lesson.

Sort triangles by their angles (right, acute, obtuse) or by their sides (equal or not).

Every triangle can be described two ways at once. By angles: a right triangle has one square corner, an obtuse triangle has one wide corner, and an acute triangle has three sharp ones. By sides: equilateral has all three equal, isosceles has two equal, scalene has none equal. A triangle can only ever have one right or obtuse angle — try drawing two and you'll see why it can't close.

Compare decimals place by place, starting from the left — and don't trust length.

The classic trap: 0.45 looks longer than 0.5, so it looks bigger. It isn't. Compare the tenths place first — 4 tenths versus 5 tenths — and 0.5 wins immediately. Longer does not mean larger, because each place to the right is worth ten times less. You can only compare two decimals fairly when they describe the same whole.

Deciding which of two fractions is bigger, even when the bottoms don't match.

You can only compare pieces fairly when the pieces are the same size. So to compare two fractions with different denominators, you rewrite them as equivalent fractions that share a denominator — then just compare the top numbers. A useful shortcut: with the same numerator, the fraction with the *bigger* denominator is smaller, because the whole got cut into more, tinier pieces.

Compare from the left: the first place where the digits differ decides it.

To compare two whole numbers, line them up and read left to right. The first place where they disagree settles the whole question — you can stop there. This works because a single thousand outweighs any number of hundreds below it. The symbols point like a hungry mouth toward the bigger number: 4,512 > 4,498.

A whole number bigger than 1 that has more than two factors.

Composite means 'made of parts.' A composite number can be built by multiplying smaller numbers, so it can be arranged into equal groups in more than the boring way. 12 is composite because 3 × 4 makes it. Every whole number over 1 is either prime or composite — and 1 is neither, which is a fact worth remembering because it shows up on tests.

Changing a measurement into a different unit — bigger unit, smaller number.

One kilometer is 1,000 meters, so 3 km is 3,000 m. The rule that keeps you from getting it backwards: going to a *smaller* unit gives a *bigger* number, because it takes more small things to make the same amount. Fourth grade sticks to converting from larger units to smaller ones, within one system (metric or customary), for length, weight, volume, and time.

A decimal is another way to write a fraction whose bottom is 10 or 100.

This is one of the best surprises in fourth grade: decimals are not a new kind of number, they are fractions in different clothing. The first place after the dot is tenths, the second is hundredths — the same place-value pattern as whole numbers, continuing to the right. So 0.7 is 7/10, and 0.07 is 7/100.

Sharing into equal groups when it doesn't come out even — and deciding what the leftover means.

Dividing 27 by 4 gives 6 with 3 left over, because 4 groups of 6 use 24 and 3 refuse to fit. The interesting part of fourth grade is that the remainder means different things in different problems. Sometimes you ignore it, sometimes you round up, sometimes the remainder *is* the answer. The arithmetic is the easy half; reading the question is the real skill.

Two fractions that look different but are the exact same amount.

If you multiply the top and the bottom of a fraction by the same number, you get a fraction that looks different but is worth exactly the same. This works because you are cutting every piece into the same number of smaller pieces — you have more pieces, but each one is smaller, so the amount never changed. This idea is the key that unlocks comparing and adding fractions.

A number that divides evenly into another number, with nothing left over.

Factors are the numbers that build a number by multiplying. Because 3 × 4 = 12, both 3 and 4 are factors of 12. Factors always come in pairs, which is why finding them is easier than it looks: check 1, 2, 3… and each time one works, you get its partner free. Every number has 1 and itself as factors.

A number that names part of a whole, written as one number over another.

A fraction tells you two things at once: how many equal pieces the whole was cut into, and how many of those pieces you have. The cut has to be into *equal* pieces — that is the rule people forget. Half a cookie only means something if both halves are the same size.

The strand about pieces of a whole — and the year fractions finally start making sense.

One of the five big parts of fourth-grade math, and the one most fourth graders remember. You learn that two different-looking fractions can be the exact same amount, how to compare fractions that don't match, how to add them, and — the surprise of the year — that a decimal like 0.7 is just another way of writing the fraction 7/10. Most people who say they are 'bad at math' got lost right here, which is why this strand is worth going slowly.

The strand about shapes, lines, corners, and the mirror line that folds a shape onto itself.

One of the five big parts of fourth-grade math. In fourth grade geometry stops being 'name the shape' and starts being 'prove it.' You learn the exact words for lines and corners, sort triangles and quadrilaterals by the properties they actually have rather than by how they look, and find the lines of symmetry that fold a shape perfectly in half.

A fraction whose top is bigger than its bottom — it's worth more than one whole.

Nothing is actually wrong with an improper fraction; the name is just old-fashioned. When the numerator is bigger than the denominator, you have more than one whole. To turn it into a mixed number, ask how many whole groups fit: 7/3 means seven thirds, and three thirds make a whole, so you get 2 wholes with 1/3 left over.

A fold line that lands a shape exactly on top of itself.

If you can fold a shape along a line and both halves match perfectly, that line is a line of symmetry. Some shapes have none, some have one, and some have many — a square has four, and a circle has infinitely many. The test is genuinely a fold: if any part sticks out, it isn't a line of symmetry, no matter how balanced the shape looks.

A graph that stacks a mark above a number line for each measurement you collected.

A line plot shows real measurements piled up so the shape of the data becomes visible — where most values cluster, and which ones are unusual. In fourth grade the measurements are often fractions, like lengths to the nearest 1/8 inch, so reading a line plot is also fraction practice.

The strand about measuring real things — length, weight, time, space, and turns.

One of the five big parts of fourth-grade math. This is math that leaves the page: converting kilometers into meters, finding how much fence goes around a garden versus how much grass grows inside it, reading a plot of measurements, and measuring an angle as an amount of *turn*. It is the strand where math becomes a tool you carry around.

Line up the center and the zero, then read the scale — and check your answer against a right angle.

Put the protractor's center dot on the angle's corner and one ray on zero, then read where the other ray crosses. Protractors have two scales running opposite ways, which is the trap: pick the one that starts at zero on your ray. The safety check is to glance at the angle first — if it is clearly smaller than a corner of a page, your answer must be under 90°. Angles can also be added: a 30° and a 40° angle side by side make 70°.

A whole number and a fraction living together, like 2 1/3.

A mixed number is how people usually talk about amounts bigger than one whole: two and a third pizzas, one and a half hours. It means the whole number plus the fraction, added together. You can always rewrite a mixed number as an improper fraction and back again — they are two ways of saying the same amount.

Adding and subtracting big numbers by working one place at a time and regrouping when needed.

The standard algorithm stacks numbers so that matching places line up, then works right to left. When a column adds past 9, you regroup — ten ones become one ten, and that ten moves next door. Subtraction does the reverse, unpacking a ten into ten ones when you need more to work with. 'Carrying' and 'borrowing' are just place value being traded, not magic.

Multiplying big numbers by breaking them into their places and multiplying the parts.

23 × 4 is hard to do all at once, so break 23 into 20 + 3, multiply each part, and add: 80 + 12 = 92. That's the area model — draw a rectangle 23 wide and 4 tall, split it, and the pieces are the partial products. The stacked algorithm is the same work written compactly. Understanding the rectangle first is what keeps the algorithm from becoming a spell you can't debug.

A problem that takes two or more steps — and where you should check if your answer makes sense.

Real questions rarely take one operation. A multi-step problem asks you to find something on the way to finding the thing you actually want. The two habits that matter: write down what each step gives you (so you don't lose track), and at the end ask whether the answer is *reasonable* — if you calculated that one person ate 400 apples, something went wrong.

What you get when you multiply a number by 1, 2, 3, and so on — its skip-counting list.

A multiple of 4 is any number in the list 4, 8, 12, 16, 20… — the numbers you land on when you skip-count. Factors and multiples are the same relationship seen from opposite ends: 4 is a factor of 12, and 12 is a multiple of 4. Every number has only a few factors but endlessly many multiples.

Multiplying a fraction by a whole number is just adding that fraction over and over.

3 × 2/5 means 2/5 + 2/5 + 2/5, which is 6/5. You multiply the top number by the whole number and leave the bottom alone, because the *size* of each piece never changes — you are just taking more of them. This is the same idea as 3 × 4 meaning 4 + 4 + 4, only the thing you are counting is a fraction.

A sequence that follows a rule — and the interesting part is the feature the rule didn't mention.

Given a rule like 'add 3 starting at 1,' you can write the pattern: 1, 4, 7, 10, 13… The fourth-grade twist is to notice things the rule never told you — that this pattern's numbers alternate odd, even, odd, even, or that adding 3 forever never lands on a multiple of 3. Finding the hidden feature is where patterns stop being busywork and start being mathematics.

The bottom number says how many pieces the whole was cut into; the top number says how many you have.

The denominator is on the bottom and it *names the piece* — it tells you the size of each part by saying how many equal parts make one whole. The numerator is on top and it *counts* those pieces. A helpful way to remember: the Denominator is Down, and it Divides the whole up.

The strand about multiplying, dividing, and finding the hidden pattern in numbers.

One of the five big parts of fourth-grade math. This strand is about using multiplication and division to solve real problems — not just knowing that 6 × 7 = 42, but knowing when to reach for it. You learn to say how many *times as many* one thing is than another, to break a number into the numbers that build it, and to spot the rule hiding inside a pattern.

Parallel lines never meet; perpendicular lines cross at a square corner.

Parallel lines run alongside each other forever without ever touching, always the same distance apart — like railroad tracks. Perpendicular lines cross and make a right angle, a perfect 90° square corner. These two relationships are what you use to sort quadrilaterals: whether the sides are parallel, and whether the corners are square.

The distance all the way around the outside of a shape.

Perimeter is a walk around the edge — add up the lengths of every side. It is measured in plain length units like centimeters or feet, because it *is* a length. For a rectangle you can take a shortcut: add the length and width, then double it, since opposite sides match.

A digit's worth depends on where it sits — and each place is ten times the one to its right.

Place value is the idea that makes our whole number system work. The digit 4 can mean four, or forty, or four hundred — the digit didn't change, its house did. Each place going left is worth ten times more, and each place going right is worth ten times less. That single rule keeps going in both directions, which is why it also explains decimals.

The strand about what each digit is worth, and how to add, subtract, multiply, and divide big numbers.

One of the five big parts of fourth-grade math. This is the year numbers get big — all the way to a million — and the year you learn *why* the old tricks work. A digit's worth depends on where it sits, and each place is ten times the one to its right. Once you believe that, multi-digit multiplication and long division stop being magic spells and start being something you can explain.

The four building blocks — a spot, a forever-line, a one-way line, and a piece with two ends.

A point is a single location with no size. A line goes on forever in both directions. A ray starts at one point and goes on forever in one direction — like a beam from a flashlight. A line segment has two endpoints, so it has a length you can measure. Fourth grade is where these get exact names, because geometry arguments fall apart without them.

A whole number bigger than 1 with exactly two factors: 1 and itself.

A prime number cannot be split into equal groups except the boring ways — one group of everything, or everything in groups of one. 7 is prime because nothing but 1 and 7 divides it evenly. Note the careful wording: 1 is *not* prime, because it has only one factor, not two. The first primes are 2, 3, 5, 7, 11, 13 — and 2 is the only even one.

Trading exactness for a number that's easier to think about — look at the digit to the right.

Rounding replaces a number with a nearby friendlier one. Find the place you are rounding to, then look at the single digit just to its right: 5 or more rounds up, 4 or less stays. Rounding is not being lazy — it is how you check whether an answer is reasonable. If you estimate 40 × 20 = 800 and your careful answer comes out 8,000, you know to look again.

Three ways to write the same number: with digits, with words, or as a sum of its places.

Standard form is the ordinary way: 3,205. Word form spells it out: three thousand, two hundred five. Expanded form breaks it into what each digit is actually worth: 3,000 + 200 + 5. Expanded form is the useful one — it shows place value directly, and it is why multi-digit multiplication works the way it does. Notice the zero: it holds the tens place open so the 5 stays in the ones.

Saying how many times bigger one amount is than another — a multiplying question, not an adding one.

There are two ways to compare: how many *more* (that's subtraction) and how many *times as many* (that's multiplication). Fourth grade is where you learn to hear the difference in a word problem. '5 more than 3' is 8. '5 times as many as 3' is 15. Getting these mixed up is the single most common word-problem mistake of the year.