Kid Einsteins · Class 9 · Fraction Concepts · Grades 3–4
Fraction Concepts — Parts of a Whole, Parts of a Set (Kid Einsteins Class 9, Grades 3–4) — 25 Practice Questions with Hidden Answers
A fraction is a number that names one or more EQUAL parts of something. The bottom number (denominator) says how many equal parts total; the top number (numerator) says how many of those parts you have. Fractions describe parts of one whole thing (a pizza, a chocolate bar, a rectangle) OR parts of a set (a group of marbles, cookies, or students). This Kid Einsteins Class 9 pack teaches grade 3–4 students what a fraction is, why the parts must be equal, unit fractions, and gives 25 practice questions with click-to-reveal step-by-step answers.
This class is part of the Kid Einsteins arc at SOMATH — School of Math on the Upper West Side. Class 9 is where fractions officially begin. Everything about fractions from here on — equivalents, comparing, adding, subtracting, multiplying, and dividing — sits on top of the two ideas in this class: fractions are EQUAL parts, and you can take those parts of one whole or of a set. It is taught in-person at 226 W 79th St. Call (646) 668-6151 or book a free 30-minute evaluation to see if your grade 3–4 child is a fit.
What’s in this class pack
- What is a fraction?
- Numerator and denominator — what each number means
- Parts of a WHOLE
- Parts of a SET
- Why parts MUST be equal
- Unit fractions — the building blocks
- Finding a fraction of a set (divide by bottom, multiply by top)
- The 5 most common grade 3–4 fraction mistakes
- 25 practice questions with hidden answers
- Answer key summary
- Frequently asked questions
1. What is a fraction?
Fraction — the definition
A fraction is a number that names one or more equal parts of something. That “something” can be:
- a single object cut into equal pieces (a pizza, a chocolate bar, a rectangle) — called parts of a whole;
- a group of separate objects arranged into equal-size groups (12 marbles, 20 students, 8 cookies) — called parts of a set.
Either way, a fraction is written the same way: a top number over a bottom number, with a bar between them.
fraction = numerator / denominator = 3/4 (read: “three-fourths”)
2. Numerator and denominator
The two numbers of every fraction
Denominator (the bottom number): says how many equal parts the whole was cut into. It tells you the SIZE of each part — the bigger the denominator, the smaller each piece.
Numerator (the top number): says HOW MANY of those parts you are counting.
Memory trick: D is Down, D is Denominator. Or: “the number under the bar tells you how the pie is cut; the number above the bar tells you how many slices you took.”
| Fraction | Denominator says… | Numerator says… | Read as |
|---|---|---|---|
| 1/2 | 2 equal parts | 1 of them | one-half |
| 1/3 | 3 equal parts | 1 of them | one-third |
| 2/3 | 3 equal parts | 2 of them | two-thirds |
| 3/4 | 4 equal parts | 3 of them | three-fourths |
| 5/8 | 8 equal parts | 5 of them | five-eighths |
| 7/10 | 10 equal parts | 7 of them | seven-tenths |
3. Parts of a WHOLE
Parts of a whole
Parts of a whole means ONE object is split into equal pieces. Classic examples: a pizza, a chocolate bar, a candy bar, a rectangle, a circle, a pie.
To write the fraction:
- Count the total number of EQUAL pieces → that’s the denominator.
- Count how many are shaded / eaten / taken / colored → that’s the numerator.
The pizza was cut into 8 equal slices, and 3 were eaten. The fraction of the pizza that was eaten is 3/8. The fraction of the pizza LEFT is 5/8. Notice that 3/8 + 5/8 = 8/8 = 1 whole pizza — all the parts together make the whole.
4. Parts of a SET
Parts of a set
Parts of a set means a GROUP of separate objects is divided by some feature (color, kind, gender, etc.). Classic examples: marbles of different colors, cookies with and without chocolate chips, boys and girls in a class, red and blue crayons.
To write the fraction:
- Count the TOTAL number of objects in the set → that’s the denominator.
- Count how many match the feature you care about → that’s the numerator.
Same rule as parts of a whole: how many you care about goes on top, how many total goes on the bottom. Also notice: 4/10 + 6/10 = 10/10 = 1 whole set of marbles. Every fraction question about a set adds up to the whole set.
5. Why the parts MUST be equal
The equal-parts rule
Fractions only work when the parts are the SAME size. If a pizza is cut into 4 pieces — but two pieces are huge and two are tiny — you CANNOT say each piece is 1/4. Each big piece is more than a fourth and each tiny piece is less than a fourth.
This is the #1 concept for grade 3. Half the class will draw a rectangle divided into 4 uneven parts and confidently label each one 1/4. Always ask: “Are these parts really the same size?” If not, redraw.
6. Unit fractions — the building blocks
Unit fraction
A unit fraction is any fraction with 1 on top: 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, and so on.
Unit fractions are the LEGO bricks of the fraction world. Every other fraction is just several copies of a unit fraction:
3/4 = 1/4 + 1/4 + 1/4 5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8
Common Core standard 3.NF.A.1 asks grade 3 students to understand a fraction a/b as “a copies of the unit fraction 1/b.” That is the entire point of this class.
7. Finding a fraction of a set
The rule: divide by the bottom, multiply by the top
To find a fraction of a whole number of objects — like 3/4 of 12 marbles — follow three steps:
- Divide the total by the DENOMINATOR → that gives you the size of one equal group.
- Multiply that answer by the NUMERATOR → that gives you how many are in the fraction.
- Write the label (marbles, cookies, students).
3/4 of 12 = (12 ÷ 4) × 3 = 3 × 3 = 9 marbles
Short version to teach the child: “Divide by the bottom, multiply by the top.” That phrase carries all the way to grade 5.
8. The 5 most common grade 3–4 fraction mistakes
- Drawing unequal parts and calling them a fraction. A rectangle cut into 4 uneven pieces is NOT fourths. Always check that the pieces are the same size.
- Flipping numerator and denominator. Writing 4/3 instead of 3/4. Fix: always ask “total pieces first” (that goes on the bottom), then “how many I care about” (that goes on top).
- Thinking a bigger denominator means bigger fraction. Kids see 1/8 and 1/4 and pick 1/8 because 8 > 4. But 1/8 is SMALLER — a pizza cut into 8 has smaller slices. Fix: draw both pizzas and compare.
- Multiplying by the bottom and dividing by the top when finding a fraction of a set. The order matters: divide by DENOMINATOR first (that finds one group), then multiply by NUMERATOR.
- Forgetting the label. Writing “9” instead of “9 marbles.” A number without a label is not an answer to a word problem.
9. Practice — 25 questions with hidden answers
Read the question with your child. Encourage them to draw the picture first, then check. Click “Show answer & work” only after they’ve tried.
Question 1 — reading a fraction
In the fraction 3/5, what does the 5 tell you? What does the 3 tell you?
Answer: 5 is the denominator — the whole was cut into 5 equal parts. 3 is the numerator — you are counting 3 of those parts.
Bottom number = how many equal parts total. Top number = how many you have. This is the pattern for every fraction.
Question 2 — parts of a whole (pizza)
A pizza is cut into 8 equal slices. Sarah eats 2 slices. What fraction of the pizza did she eat?
Answer: 2/8 of the pizza.
8 equal slices in the whole pizza → denominator = 8. She ate 2 of them → numerator = 2. So the fraction is 2/8. (Later, in Class 10, we’ll learn 2/8 = 1/4.)
Question 3 — fraction left
Same pizza as Question 2. What fraction of the pizza is LEFT (not eaten)?
Answer: 6/8 of the pizza.
If 2 slices were eaten out of 8, then 8 − 2 = 6 slices are left. So 6/8 of the pizza remains. Check: 2/8 + 6/8 = 8/8 = 1 whole pizza. Every fraction of the whole and the “left over” fraction add up to the whole.
Question 4 — parts of a whole (chocolate bar)
A chocolate bar has 12 equal squares. Mia eats 5 of them. What fraction did she eat? What fraction is left?
Answer: she ate 5/12. She left 7/12.
12 squares total → denominator = 12. Ate 5 → numerator = 5. Left: 12 − 5 = 7, so 7/12 remain. Together: 5/12 + 7/12 = 12/12 = 1 whole bar.
Question 5 — equal parts rule
A rectangle is cut into 4 pieces. Two pieces are big and two are small. Can each piece be called 1/4?
Answer: NO. The pieces are not equal, so they cannot be called fourths.
A fraction only works when the parts are the same size. To make real fourths, the rectangle must be cut into 4 pieces that are all equal to each other.
Question 6 — unit fraction
What is a unit fraction? Give three examples.
Answer: a unit fraction is any fraction with 1 on top. Examples: 1/2, 1/3, 1/4 (or any other 1/b).
Unit fractions are the building blocks. Every other fraction is made by adding copies of a unit fraction: 3/4 = 1/4 + 1/4 + 1/4.
Question 7 — breaking a fraction into unit fractions
Write 5/6 as a sum of unit fractions.
Answer: 5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 (five copies of 1/6).
The denominator 6 tells you the unit fraction is 1/6. The numerator 5 tells you there are 5 copies. This is exactly what Common Core 3.NF.A.1 asks for.
Question 8 — parts of a set (marbles)
A bag has 10 marbles: 4 are red and 6 are blue. What fraction of the marbles are red?
Answer: 4/10 of the marbles are red.
Total in the set = 10 → denominator. Red marbles = 4 → numerator. So 4/10 are red. Same rule as parts of a whole: how many I care about over how many total.
Question 9 — parts of a set (students)
A class has 20 students. 12 are girls and 8 are boys. What fraction of the class are girls? What fraction are boys?
Answer: 12/20 are girls; 8/20 are boys.
Total students = 20 → denominator. Girls = 12 → numerator for the girl fraction. Boys = 8 → numerator for the boy fraction. Check: 12/20 + 8/20 = 20/20 = 1 whole class.
Question 10 — parts of a set (cookies)
On a plate there are 6 cookies: 4 chocolate chip and 2 sugar. What fraction of the cookies are chocolate chip?
Answer: 4/6 are chocolate chip.
6 cookies total → denominator. 4 chocolate chip → numerator. So 4/6. In Class 10 we’ll simplify 4/6 to 2/3 — for now, 4/6 is the correct answer.
Question 11 — denominator = size of pieces
Would you rather have 1/4 of a pizza or 1/8 of the same pizza? Why?
Answer: 1/4 is bigger.
A pizza cut into 4 slices has BIGGER slices than a pizza cut into 8. Bigger denominator = smaller pieces. So 1/4 > 1/8. This surprises grade 3 — they see 8 > 4 and think 1/8 > 1/4. Draw both pizzas to prove it.
Question 12 — comparing unit fractions
Put these unit fractions in order from smallest to biggest: 1/6, 1/2, 1/10, 1/3.
Answer: 1/10 < 1/6 < 1/3 < 1/2.
For unit fractions (1 on top), the bigger the denominator, the smaller the fraction. 10 is biggest so 1/10 is smallest; 2 is smallest so 1/2 is biggest.
Question 13 — fraction of a whole with shading
A rectangle is split into 8 equal parts. 3 parts are shaded. Write the fraction shaded and the fraction NOT shaded.
Answer: shaded = 3/8. Not shaded = 5/8.
8 total parts → denominator. Shaded 3 → numerator = 3, so 3/8. The rest is 8 − 3 = 5, so 5/8. Together: 3/8 + 5/8 = 8/8 = 1 whole rectangle.
Question 14 — fraction of a set (fruit basket)
A basket has 15 pieces of fruit: 6 apples, 5 oranges, and 4 bananas. What fraction of the fruit are apples? What fraction are oranges?
Answer: apples = 6/15. Oranges = 5/15.
Total fruit = 6 + 5 + 4 = 15 → denominator. Apples = 6, oranges = 5. Fractions: 6/15 and 5/15. Check: 6/15 + 5/15 + 4/15 = 15/15 = 1 whole basket.
Question 15 — finding a fraction of a set (divide-multiply)
What is 1/3 of 12 marbles?
Answer: 4 marbles.
Divide by bottom, multiply by top. 12 ÷ 3 = 4 (that’s one group of thirds). Then 4 × 1 = 4 (numerator is 1). So 1/3 of 12 = 4 marbles.
Question 16 — finding 3/4 of a set
What is 3/4 of 20 cookies?
Answer: 15 cookies.
Divide by bottom: 20 ÷ 4 = 5 cookies per group. Multiply by top: 5 × 3 = 15. So 3/4 of 20 cookies = 15 cookies.
Question 17 — finding 2/5 of a set
There are 25 students in a class. 2/5 of them walk to school. How many students walk to school?
Answer: 10 students.
25 ÷ 5 = 5 students per group. 5 × 2 = 10 students walk. Always label the answer: “10 students,” not just “10.”
Question 18 — fraction equals 1 whole
A cake is cut into 6 equal pieces. All 6 pieces are eaten. What fraction of the cake was eaten?
Answer: 6/6, which equals 1 whole cake.
When the numerator equals the denominator, the fraction equals 1. So 6/6 = 1 whole. Same for 2/2, 3/3, 4/4, and so on. This is a big idea for grade 3.
Question 19 — a fraction equal to 0
Out of 10 marbles, 0 are red. What fraction are red?
Answer: 0/10, which equals 0.
When the numerator is 0, the fraction is 0 — there is none of what you’re counting. 0/10 = 0. Same for 0/5, 0/8, and so on.
Question 20 — word problem (parts of a whole)
Lucas cuts a birthday cake into 10 equal slices. His family eats 7 slices. What fraction is eaten? What fraction is left?
Answer: eaten = 7/10. Left = 3/10.
10 slices total. Eaten 7 → 7/10. Left 10 − 7 = 3 → 3/10. Check: 7/10 + 3/10 = 10/10 = 1 whole cake.
Question 21 — word problem (parts of a set)
Nora has 24 crayons. 1/4 of them are red. How many red crayons does she have?
Answer: 6 red crayons.
24 ÷ 4 = 6 crayons per group. Numerator is 1, so 6 × 1 = 6. Nora has 6 red crayons.
Question 22 — word problem (two fractions of a set)
Miguel has 30 stickers. 1/5 are gold and 2/5 are silver. How many gold stickers? How many silver stickers?
Answer: 6 gold stickers, 12 silver stickers.
One group = 30 ÷ 5 = 6 stickers. Gold: 6 × 1 = 6. Silver: 6 × 2 = 12. Together: 6 + 12 = 18 out of 30 accounted for; the other 12 are neither gold nor silver.
Question 23 — challenge: fill in the denominator
A pizza is cut into equal slices. Ryan eats 3 slices, which is 3/? of the pizza. He notices he ate exactly HALF the pizza. How many slices was the pizza cut into?
Answer: the pizza was cut into 6 slices. So Ryan ate 3/6 = 1/2.
If 3 slices is half the pizza, then the whole pizza has 3 × 2 = 6 slices. So the fraction Ryan ate is 3/6, which is the same as 1/2.
Question 24 — challenge: fractions in real life (a clock)
A clock face is a whole. If the minute hand moves from 12 to 3, what fraction of the way around the clock did it move?
Answer: 1/4 of the way around.
A clock is divided into 12 equal “hour spaces” around the full circle. From 12 to 3 is 3 spaces out of 12, so 3/12 = 1/4. This is why we say “quarter past” when the minute hand is on the 3.
Question 25 — challenge: mixed parts of a whole + set
A tray has 12 muffins arranged in 3 rows of 4. Isabel eats one whole row. What fraction of the tray did she eat — using the rows as the whole, and using the muffins as the whole? Are both answers the same amount of muffins?
Answer: using rows, she ate 1/3 (1 out of 3 rows). Using individual muffins, she ate 4/12 (4 out of 12 muffins). Both fractions represent the SAME 4 muffins — 1/3 = 4/12.
This is a first look at equivalent fractions (Class 10). Same amount can be written two ways depending on how you split the whole. 1/3 of 12 muffins = 12 ÷ 3 = 4 muffins, which matches 4/12.
10. Answer key summary
| # | Answer | # | Answer | # | Answer |
|---|---|---|---|---|---|
| 1 | 5 = parts total; 3 = parts counted | 10 | 4/6 chocolate chip | 19 | 0/10 = 0 |
| 2 | 2/8 eaten | 11 | 1/4 is bigger | 20 | 7/10 eaten; 3/10 left |
| 3 | 6/8 left | 12 | 1/10 < 1/6 < 1/3 < 1/2 | 21 | 6 red crayons |
| 4 | 5/12 eaten; 7/12 left | 13 | 3/8 shaded; 5/8 not | 22 | 6 gold; 12 silver |
| 5 | No — parts must be equal | 14 | 6/15 apples; 5/15 oranges | 23 | 6 slices; 3/6 = 1/2 |
| 6 | Fraction with 1 on top | 15 | 4 marbles | 24 | 1/4 around |
| 7 | 1/6+1/6+1/6+1/6+1/6 | 16 | 15 cookies | 25 | 1/3 rows; 4/12 muffins; same |
| 8 | 4/10 red | 17 | 10 students | ||
| 9 | 12/20 girls; 8/20 boys | 18 | 6/6 = 1 whole |
11. Frequently asked questions
What is a fraction?
A fraction is a number that names one or more EQUAL parts of something. The denominator (bottom) says how many equal parts total; the numerator (top) says how many of those parts you have. Fractions can name parts of one whole (a pizza, a rectangle) or parts of a set (a group of marbles, students, cookies).
What is the difference between the numerator and the denominator?
The denominator (bottom) says how many equal parts the whole was split into — it controls the SIZE of each piece. The numerator (top) says how many of those pieces you have. Memory: “D is Down, D is Denominator.”
What is the difference between parts of a whole and parts of a set?
Both are fractions and both use the same numerator/denominator rules. Parts of a WHOLE means one object cut into equal pieces (pizza, rectangle). Parts of a SET means a group of separate objects divided by some feature (marbles by color, students by gender). Either way: count total (bottom), count what you care about (top).
Why must the parts be equal to make a fraction?
Because a fraction only makes sense when every piece is the same size. If a rectangle is cut into 4 pieces of different sizes, you cannot call any of them 1/4. In grade 3 this is the #1 conceptual mistake — always check the parts are equal before naming a fraction.
What is a unit fraction?
A fraction with 1 on top: 1/2, 1/3, 1/4, 1/5, and so on. Unit fractions are the building blocks. Every other fraction is just several copies of a unit fraction: 3/4 = 1/4 + 1/4 + 1/4. Common Core 3.NF.A.1 asks students to understand a/b as “a copies of the unit fraction 1/b.”
How do you find a fraction of a set?
Divide by the bottom, multiply by the top. Example: 3/4 of 12 = (12 ÷ 4) × 3 = 3 × 3 = 9. That short phrase — “divide by the bottom, multiply by the top” — works all the way through elementary school.
Why does a bigger denominator make a smaller fraction?
Because the denominator controls how small each piece is. A pizza cut into 8 slices has smaller slices than the same pizza cut into 4. So 1/8 is smaller than 1/4, even though 8 is bigger than 4. This is the most surprising fact for grade 3 — always demonstrate it by drawing.
What comes after this class in the Kid Einsteins arc?
Class 9 is the foundation. Class 10 introduces equivalent fractions and comparing fractions. Class 11 covers adding and subtracting with LIKE denominators. Class 12 introduces mixed numbers and improper fractions. Class 19 (already published on the SOMATH blog) covers adding and subtracting with UNLIKE denominators.
About SOMATH — School of Math
SOMATH — School of Math is a math-focused school on the Upper West Side of Manhattan for students in grades 1–12. Class 9 is part of the Kid Einsteins arc (grades 3–4), taught in small groups by cofounder Marcelo Ambrozio (Northwestern-trained, 15+ years teaching NYC math) and the SOMATH team.
Location: 226 W 79th St, 1st Floor, New York, NY 10024 · Upper West Side
Phone: (646) 668-6151 · Email: hello@schoolofmath.us
Book: Free evaluation · Schedule · Home
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