Partition Shapes Into Equal Parts - 2nd Grade Math

1. Ms. Lopez shows her class a rectangle divided into 3 equal parts. She colors 2 parts green and leaves 1 part white. What fraction of the rectangle is white?
   Answer: One third The rectangle has 3 equal parts: 1 is white and 2 are green. The white part is one third of the rectangle. Two thirds describes the green part, and three thirds describes the whole rectangle. One half is half of the whole rectangle, which is more than the white part.
2. Circle 1 is divided into 2 equal parts with 1 part shaded. Circle 2 is divided into 4 equal parts with 2 parts shaded. Emma says both circles have the same amount shaded. Is Emma correct?
   Answer: Yes, because one half is the same amount as two fourths when the circles are the same size Emma is correct, since one half and two fourths are the same amount when the circles are the same size. Circle 1 having fewer parts shaded than Circle 2 does not by itself mean less area is shaded, since the parts are different sizes. Circles divided differently can still be compared, as long as the whole circles are the same size. Circle 1 actually has only 1 part shaded, not 2, so that reasoning does not correctly describe circle 1.
3. In the figure, two identical rectangles are each shaded to show one half. Which statement is true?
   Answer: Both rectangles show one half, even though the shaded parts have different shapes. One half means 2 equal parts. Rectangle A is cut down the middle, and one part is shaded. Rectangle B is cut corner to corner, and one triangle is shaded. Both shaded parts are half of the same-size rectangle, even though they are different shapes.
4. A rectangle is divided into 4 equal parts. Tom says, 'This rectangle shows halves.' Sarah says, 'No, this rectangle shows fourths.' Who is correct?
   Answer: Sarah is correct because there are 4 equal parts total in the rectangle Since the rectangle is divided into 4 equal parts, it shows fourths, so Sarah is correct. Choice A is wrong because counting rows doesn't tell you how many equal parts there are. Choice C is wrong because a shape divided into 4 equal parts is fourths, not halves. Choice D is wrong because there are 4 equal parts, not 3.
5. Look at the shaded rectangle below. What fraction of the whole rectangle is shaded in total?
   Answer: Half of the rectangle The rectangle is divided into 4 equal parts, and 2 of the 4 parts are shaded. Two fourths is the same as one half of the rectangle.
6. Refer to the figure. A square is divided so that one part is shaded. Which statement about the shaded part is true?
   Answer: The shaded part is one half of the square. When a shape is divided into equal parts, the name of each part depends on how many equal pieces the whole is split into. Two equal parts are called halves, three equal parts are called thirds, and four equal parts are called fourths (or quarters). So your first job with a question like this is to count the equal pieces the shape is divided into. In this figure, the square is split into 2 equal parts, and 1 of those parts is shaded. Since the square is cut into 2 equal shares and you're looking at 1 of them, the shaded part is one half of the square, making A correct. B is wrong because "one third" would require the square to be divided into 3 equal parts, not 2. C is wrong because "one fourth" would need 4 equal parts — you'd see the square split into four equal pieces with one shaded. D is wrong because the two parts of the square are the same size; they are equal shares, so this statement doesn't fit. A helpful tip: whenever you see a fraction question with a picture, count the total equal parts first (that's your bottom number, the denominator), then count how many are shaded (that's your top number, the numerator). Two equal parts → halves. Three → thirds. Four → fourths. Memorizing this pattern makes these questions quick and reliable.
7. Look at the figure. Which sentence correctly describes the shaded part of the circle?
   Answer: Half of the circle is shaded. When you see a shape split into shaded and unshaded parts, your first job is to count the total equal pieces the shape is divided into, then count how many are shaded. That gives you the fraction. Common fractions on 2nd-grade tests are halves (2 equal pieces), thirds (3 equal pieces), and fourths (4 equal pieces). For this circle, the figure shows it split into 2 equal pieces, with 1 piece shaded. That means 1 out of 2 parts is shaded, which we call one half. So B is the match. Now look at why the others don't fit. A says "a third," which would require the circle to be cut into 3 equal pieces with 1 shaded — but this circle only has 2 pieces. C says "a fourth," which needs 4 equal pieces with 1 shaded; again, the wrong number of pieces. D says "three fourths," which not only needs 4 equal pieces, but also 3 of them shaded — far more shading than the figure shows. A helpful trick: before picking your answer, whisper to yourself, "How many equal pieces in all? How many are shaded?" Write it as shaded over total. Here, that's $\frac{1}{2}$. Matching that fraction to the words — half, third, fourth — makes these questions quick and safe from tricky wording traps like "three fourths."
8. Sam cuts a circle into 3 equal pieces. He eats 2 pieces and saves 1 piece for later. Which sentence correctly describes what Sam has left?
   Answer: Sam has one third of the circle remaining for later Sam cut the circle into 3 equal parts (thirds) and saved 1 piece, so he has $\frac{1}{3}$ (one third) remaining. Choice A describes what he ate, not what remains. Choice C incorrectly describes the remaining piece as a half. Choice D would mean he has the whole circle left.
9. Shape A is a circle divided into 2 equal parts. Shape B is a rectangle divided into 2 equal parts. Maria takes $\frac{1}{2}$ from Shape A and $\frac{1}{2}$ from Shape B. Her brother says she took the same amount from both shapes. Is her brother right?
   Answer: No, because $\frac{1}{2}$ of each shape could be different amounts if the shapes are different sizes Both shapes were split into 2 equal parts, but $\frac{1}{2}$ of a shape depends on the size of that shape. If Shape A and Shape B are different sizes, then $\frac{1}{2}$ of each one is a different amount, so her brother is not right. Choice A ignores that the shapes could be different sizes. Choice B is wrong because different shapes can still be compared using fractions. Choice D focuses on the method instead of the actual amount taken.
10. In the diagram, a pizza is split among some friends. How many friends will get an equal share, and what is each share called?
    Answer: 3 friends; each share is a third. When a shape like a pizza is divided into equal parts, the number of parts tells you what to call each share. Two equal parts are called halves, three equal parts are called thirds, and four equal parts are called fourths (or quarters). The key is to count how many equal pieces the whole is split into. In this diagram, the pizza is cut into 3 equal slices, so 3 friends can each get one piece. Since the pizza is split into 3 equal parts, each share is called a third, making B correct. Choice A would only work if the pizza were cut into 2 equal pieces, but there are more than two slices here. Choice C describes a pizza cut into 4 equal parts, which doesn't match a pizza split into 3 slices. Choice D mixes things up: it correctly counts 3 friends but calls each share a "fourth" — the name of the share must match the number of equal parts, so 3 parts cannot be called fourths. A helpful tip: always match the name of the fraction to the total number of equal pieces the whole is divided into, not just to how many pieces someone takes. If you see 3 equal parts, think "thirds." If you see 4 equal parts, think "fourths." Counting the total slices first will keep you from falling for tricky answer choices that mix up the numbers and names.
11. Look at the four shapes in the figure. Which shape is NOT divided into equal shares?
    Answer: Shape 3 Equal shares are pieces that are all the same size. Shape 3 is cut into one big piece and one small piece, so its shares are not equal. Shape 1 shows 2 equal halves, Shape 2 shows 4 equal fourths, and Shape 4 shows 3 equal thirds.
12. Refer to the rectangle below. Which sentence correctly describes the whole rectangle?
    Answer: The whole is four fourths. The rectangle is divided into 4 equal parts, so each part is a fourth. The whole rectangle is made up of four fourths.
13. A rectangular paper is folded and cut into 3 equal-sized pieces. Each piece is one what?
    Answer: third Each piece is one third, since the paper was divided into 3 equal parts. A half would mean the paper was divided into 2 equal parts, not 3. A fourth would mean the paper was divided into 4 equal parts. A whole refers to the entire paper, not one of its equal pieces.
14. Jake has a rectangular pizza. He wants to share it equally among 4 people, including himself. After he cuts the pizza, Jake says, "Each person gets one fourth of the pizza, and the whole pizza is four fourths." What does Jake mean when he says the whole pizza is "four fourths"?
    Answer: The pizza was cut into 4 pieces and all 4 pieces together make the complete pizza The pizza is cut into 4 equal pieces, and putting all 4 pieces back together makes the whole pizza, which is what Jake means by 'four fourths,' matching A. B is incorrect because the pizza is not bigger than a regular pizza; it is just divided into 4 equal parts. C is incorrect because each of the 4 people gets 1 piece, not 4 pieces each. D is incorrect because the pizza is already cut into the right number of pieces; no more cuts are needed.
15. Refer to the circle in the figure. What fraction of the circle is shaded?
    Answer: One third The circle has 3 equal parts, and 1 whole part is shaded. The shaded part is one third of the circle. One half is half of the whole circle, one fourth is a quarter of it, and three thirds is the whole circle.
16. Look at the rectangle. Maya divides it into 4 equal parts by drawing one line across and one line down. Then she colors 3 of the parts blue. How much of the rectangle is NOT blue?
    Answer: one fourth of the rectangle The rectangle has 4 equal parts. 3 are blue and 1 is not blue. One part is one fourth of the rectangle. Three fourths is blue, the whole is all 4 parts, and half is 2 of the 4 parts.
17. Two identical rectangles are shown below. Rectangle 1 is split into halves by a diagonal line. Rectangle 2 is split into halves by a line down the middle. Which sentence is correct?
    Answer: Both rectangles show halves, and one half of each rectangle is the same amount. Equal shares can have different shapes. Both rectangles are split into 2 equal parts, so each shows halves, and one half of each identical rectangle is the same amount of area.
18. Refer to the figure. Sam and Tia each have a sandwich exactly the same size. Sam cuts his into 2 equal pieces and eats 1 piece. Tia cuts hers into 4 equal pieces and eats 1 piece. Who eats more sandwich?
    Answer: Sam, because a half is bigger than a fourth. When you compare fractions of the same whole, remember this key idea: the more pieces you cut something into, the smaller each piece becomes. Think about cutting a pizza — if you slice it into just 2 pieces, each piece is huge. If you slice that same pizza into 4 pieces, each slice is smaller because you're sharing the pizza among more parts. Sam cuts his sandwich into 2 equal pieces, so each piece is one-half ($\frac{1}{2}$) of the sandwich. Tia cuts hers into 4 equal pieces, so each piece is one-fourth ($\frac{1}{4}$) of the sandwich. Since both sandwiches are the same size, one-half is bigger than one-fourth. Sam eats more, which makes A correct. B falls into a common trap: thinking a bigger denominator means a bigger piece. Actually, the opposite is true — 4 pieces means each piece is smaller than if you only had 2 pieces. C is wrong because they each ate 1 piece, but the pieces were different sizes, so the amounts can't be equal. D flips the truth: fourths are actually smaller than halves, not bigger. Study tip: When comparing unit fractions (fractions with 1 on top), the fraction with the smaller bottom number is the bigger piece. A helpful trick: imagine sharing a cookie. Would you rather share it with 1 friend (halves) or 3 friends (fourths)? You get more when fewer people share!
19. Look at the shaded circle in the figure. Which sentence best describes the whole circle?
    Answer: The whole is four fourths. When you see a circle (or any shape) divided into equal parts, remember that the whole is always made up of all the equal pieces put together. The name of each piece depends on how many equal parts the whole is split into: 2 parts make halves, 3 parts make thirds, and 4 parts make fourths. Since the figure shows a circle divided into 4 equal parts, each piece is called a fourth. To make the whole circle, you need all 4 of those fourths. So the whole circle is four fourths, which matches choice C. You can also think of it as $\frac{4}{4} = 1$ whole. Choice A ("two halves") describes a circle cut into just 2 equal pieces — that's not what this figure shows. Choice B ("three thirds") would describe a circle cut into 3 equal pieces, which also doesn't match. Choice D ("four halves") is a tricky one: it mixes up the words. "Halves" means the whole was split into 2 parts, so you can't have four of them in one whole circle — that would actually be two whole circles! A helpful tip: match the number of pieces to the name of the pieces. Two pieces → halves, three pieces → thirds, four pieces → fourths. The top number (how many you have) and the bottom number (what they're called) should be the same when you're describing one whole thing.
20. In the figure, a rectangular garden is divided into equal shares. What fraction of the whole garden is one share?
    Answer: One fourth When a shape is divided into equal shares, the fraction each share represents depends on the total number of equal parts. The rule is simple: if a whole is split into $n$ equal pieces, each piece is $\frac{1}{n}$ of the whole. So your job on questions like this is to carefully count how many equal parts the figure is divided into. Since the garden in the figure is divided into 4 equal shares, one share is $\frac{1}{4}$ of the whole garden — that's "one fourth," making D correct. Now look at why the other choices don't fit. Choice A, one half, would only be right if the garden were split into 2 equal parts, but there are more than 2 pieces here. Choice B, one third, would require exactly 3 equal parts — again, not what the figure shows. Choice C, one sixth, is a trap for students who miscount or confuse "fourths" with "sixths"; it would apply only if the garden had 6 equal shares. A helpful strategy: whenever you see a fraction question with a picture, count the total equal parts first, then write "1 over that number." The bottom number (denominator) always tells you how many equal pieces make the whole, and the top number (numerator) tells you how many of those pieces you're talking about. More parts means each piece is smaller — so a garden cut into 6 pieces would have smaller shares than one cut into 4.

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