Free math worksheets

- Count the equal slices in each pizza. - Each pizza has exactly one shaded slice. - Compare how the size of one slice changes when a same-size whole is divided into more equal pieces.

1. Pizza A is divided into \(4\) equal slices, so its shaded slice is \(\frac{1}{4}\). 2. Pizza B is divided into \(6\) equal slices, so its shaded slice is \(\frac{1}{6}\). 3. Because the pizzas are the same size, dividing one into fewer equal parts makes each part larger. 4. Therefore, \(\frac{1}{4}>\frac{1}{6}\).

Pizza A's shaded slice is greater: \(\frac{1}{4}>\frac{1}{6}\).

- Which bar has the longer shaded section? - For unit fractions, a larger denominator means each equal part is smaller.

1. Figure A shows \(\frac{1}{2}\), and Figure B shows \(\frac{1}{6}\). 2. The shaded part in Figure A is one-half of the whole, while the shaded part in Figure B is only one-sixth. 3. Therefore, \(\frac{1}{2} > \frac{1}{6}\).

Figure A shows the greater fraction, \(\frac{1}{2}\).

- Count the equal slices in each circle. - Each circle has one shaded slice, so compare the size of that one slice. - More equal parts in the same-size whole make each part smaller.

1. Circle A has \(1\) of \(4\) equal parts shaded, so it shows \(\frac{1}{4}\). 2. Circle B has \(1\) of \(8\) equal parts shaded, so it shows \(\frac{1}{8}\). 3. For unit fractions of the same-size whole, more equal parts make each part smaller. 4. Therefore, \(\frac{1}{8}<\frac{1}{4}\).

Circle B shows the smaller fraction: \(\frac{1}{8}<\frac{1}{4}\).

- Imagine sharing the same-size candy bar with many friends or with only a few friends. In which case is each share larger? - What does the denominator tell you about the number of equal parts? - What happens to the size of each part when the whole is divided into fewer parts?

1. One slice of Pizza A is \(\frac{1}{6}\) of the pizza, and one slice of Pizza B is \(\frac{1}{3}\) of the pizza. 2. The pizzas are the same size, but Pizza B is divided into fewer equal parts. 3. Fewer equal parts make each part larger, so \(\frac{1}{3} > \frac{1}{6}\).

One slice of Pizza B is larger. A slice of Pizza A is \(\frac{1}{6}\), and a slice of Pizza B is \(\frac{1}{3}\). Because the same-size whole is divided into fewer equal parts, each part is larger.

- All five fractions name one equal part, so focus on the denominators. - Think about how the size of one part changes when the same whole is split into more equal parts. - Decide which denominator gives the smallest unit fraction first, then continue in order.

1. Every fraction has numerator \(1\), so each is one equal part of a whole. 2. For unit fractions, dividing the same-size whole into more equal parts makes each part smaller. 3. From smallest part to largest part, the denominators go \(8, 6, 4, 3, 2\). 4. Therefore, \(\frac{1}{8} < \frac{1}{6} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2}\).

\(\frac{1}{8} < \frac{1}{6} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2}\)

- Count the equal parts and shaded parts in each circle. - Both shaded fractions represent one equal part of a same-size whole. - Compare the size of one part when the whole is divided into fewer or more equal pieces.

1. Circle a) has \(1\) of \(3\) equal parts shaded, so it shows \(\frac{1}{3}\). 2. Circle b) has \(1\) of \(8\) equal parts shaded, so it shows \(\frac{1}{8}\). 3. Both fractions have numerator \(1\). In the same-size whole, thirds are larger parts than eighths. 4. Therefore, \(\frac{1}{3}>\frac{1}{8}\).

\(\frac{1}{3}>\frac{1}{8}\)

- Count the equal parts in each same-size bar. - Each model shades exactly one part, so compare the sizes of those single parts. - Dividing the same-size whole into fewer equal parts makes each part larger.

1. Figure a) shows \(\frac{1}{4}\). 2. Figure b) shows \(\frac{1}{3}\). 3. Figure c) shows \(\frac{1}{6}\). 4. The same-size whole in b) is divided into the fewest equal parts, so one part is largest. Therefore, \(\frac{1}{3}>\frac{1}{4}>\frac{1}{6}\), and figure b) is greatest.

b), because \(\frac{1}{3}\) is greater than \(\frac{1}{4}\) and \(\frac{1}{6}\) for the same-size whole.

- Count the equal parts and shaded parts in each cake. - Both people eat the same number of pieces, so compare the sizes of those pieces. - In same-size wholes, fewer equal parts make each part larger.

1. Figure a) has \(2\) of \(3\) equal parts shaded, so Lucas eats \(\frac{2}{3}\). 2. Figure b) has \(2\) of \(6\) equal parts shaded, so Mia eats \(\frac{2}{6}\). 3. The fractions have the same numerator. Thirds are larger pieces than sixths in same-size wholes. 4. Therefore, \(\frac{2}{3}>\frac{2}{6}\), so Lucas eats more cake.

Lucas eats more: \(\frac{2}{3}>\frac{2}{6}\).

- Count the total equal cells and shaded cells in each grid. - Both models shade the same number of cells, so compare the size of one cell in each partition. - Fewer equal parts in a same-size whole make each part larger.

1. Figure a) has \(4\) of \(6\) equal cells shaded, so it shows \(\frac{4}{6}\). 2. Figure b) has \(4\) of \(8\) equal cells shaded, so it shows \(\frac{4}{8}\). 3. Both fractions have numerator \(4\). Sixths are larger parts than eighths in same-size wholes. 4. Therefore, \(\frac{4}{6}>\frac{4}{8}\), so figure a) is greater.

Figure a): \(\frac{4}{6}>\frac{4}{8}\).

- Count the equal parts and shaded parts in each bar. - Notice that the same number of parts is shaded in both models. - Compare the size of one part in the fourths partition with one part in the sixths partition.

1. Bar a) has \(3\) of \(4\) equal parts shaded, so it shows \(\frac{3}{4}\). 2. Bar b) has \(3\) of \(6\) equal parts shaded, so it shows \(\frac{3}{6}\). 3. Both fractions have numerator \(3\). Fourths are larger pieces than sixths in same-size wholes. 4. Therefore, \(\frac{3}{4}>\frac{3}{6}\).

\(\frac{3}{4}>\frac{3}{6}\)

- Count the equal parts and shaded parts in each circle. - Both circles shade the same number of parts. - Compare the size of one part in each same-size whole.

1. Figure a) has \(3\) of \(6\) equal parts shaded, so it shows \(\frac{3}{6}\). 2. Figure b) has \(3\) of \(8\) equal parts shaded, so it shows \(\frac{3}{8}\). 3. Both fractions have numerator \(3\). Sixths are larger pieces than eighths in same-size wholes. 4. Therefore, \(\frac{3}{6}>\frac{3}{8}\), so figure a) is greater.

Figure a): \(\frac{3}{6}>\frac{3}{8}\).

- Count the equal parts and shaded parts in each strip. - The strips represent same-size wholes and shade the same number of parts. - Compare the size of one eighth with the size of one sixth.

1. Strip a) has \(3\) of \(8\) equal parts shaded, so it shows \(\frac{3}{8}\). 2. Strip b) has \(3\) of \(6\) equal parts shaded, so it shows \(\frac{3}{6}\). 3. The strips are the same size and the fractions have the same numerator. Eighths are smaller parts than sixths. 4. Therefore, \(\frac{3}{8}<\frac{3}{6}\), so strip a) has the smaller shaded fraction.

Strip a): \(\frac{3}{8}<\frac{3}{6}\).

- Count the equal parts and shaded parts in each grid. - Both models shade the same number of parts. - Compare the size of one part when the same-size whole is divided into six parts versus eight parts.

1. Figure a) has \(2\) of \(6\) equal parts shaded, so it shows \(\frac{2}{6}\). 2. Figure b) has \(2\) of \(8\) equal parts shaded, so it shows \(\frac{2}{8}\). 3. Both fractions have numerator \(2\). Eighths are smaller pieces than sixths in same-size wholes. 4. Therefore, \(\frac{2}{8}<\frac{2}{6}\).

Figure b): \(\frac{2}{8}<\frac{2}{6}\).

- Which shaded section looks smaller? - For unit fractions, more equal parts means each part is smaller.

1. Triangle A is divided into \(3\) equal parts, with \(1\) shaded: \(\frac{1}{3}\). 2. Triangle B is divided into \(4\) equal parts, with \(1\) shaded: \(\frac{1}{4}\). 3. Since one-fourth is less than one-third, \(\frac{1}{4}\) is the smaller fraction.

Figure B shows the smaller fraction, \(\frac{1}{4}\).

- Both fractions count the same number of equal parts. - Compare the size of one fourth with the size of one sixth in same-size wholes. - Decide what happens when you take five copies of the larger unit fraction.

1. Both fractions have numerator \(5\), so each amount uses \(5\) equal parts. 2. A fourth is larger than a sixth when the wholes are the same size. 3. Five fourths therefore cover more than five sixths. 4. Thus, \(\frac{5}{4}>\frac{5}{6}\).

\(\frac{5}{4}>\frac{5}{6}\) because fourths are larger than sixths in same-size wholes.

- Count the equal parts and shaded parts in each bar. - Compare the length of one part in the thirds partition with one part in the sixths partition. - Use the partition boundaries to see how many smaller parts fit exactly in the larger shaded part.

1. Model a) has \(1\) of \(3\) equal parts shaded, so it shows \(\frac{1}{3}\). 2. Model b) has \(1\) of \(6\) equal parts shaded, so it shows \(\frac{1}{6}\). 3. A third is larger than a sixth, so \(\frac{1}{3}>\frac{1}{6}\). 4. Two sixth-size parts have the same total length as one third-size part.

\(\frac{1}{3}>\frac{1}{6}\), and \(2\) sixth-size parts have the same total length as one third-size part.

- Start with the point that is exactly at \(1\). - Compare the other two locations with the labeled \(\frac{1}{2}\) and \(1\) positions. - Once the fractions are matched, farther right on the number line means greater.

1. Point \(A\) is at \(1\), so \(A=\frac{4}{4}\). 2. Point \(B\) is at \(\frac{2}{3}\), which is \(\frac{4}{6}\). 3. Point \(C\) is at \(\frac{1}{2}\), which is \(\frac{4}{8}\). 4. Reading the points from right to left gives \(\frac{4}{4}>\frac{4}{6}>\frac{4}{8}\).

\(A=\frac{4}{4}\), \(B=\frac{4}{6}\), \(C=\frac{4}{8}\), and \(\frac{4}{4}>\frac{4}{6}>\frac{4}{8}\).

- For fractions with the same numerator, compare the sizes of the equal parts. - Decide which group has already completed more. - Completing less means having more left. - You can also compare each completed fraction with \(\frac{1}{2}\).

1. The completed fractions have the same numerator. With the same numerator, the fraction with the smaller denominator is greater, so \(\frac{4}{6}>\frac{4}{8}\). 2. Group A has completed more of the trail, so Group B must have more left. In fact, Group A has \(\frac{2}{6}=\frac{1}{3}\) left and Group B has \(\frac{4}{8}=\frac{1}{2}\) left.

Group B has a greater fraction of the trail left because \(\frac{4}{8}<\frac{4}{6}\), so Group B has completed less of the trail.

- For unit fractions, relate the denominator to the size of one equal part of the same whole. - In each part, decide whether the missing unit fraction must be larger or smaller than the given one. - Test only the listed Grade 3 denominators and keep every choice that satisfies the full comparison.

1. a) The fraction on the right must be smaller than \(\frac{1}{4}\), so its denominator must be larger than \(4\). From the choices, \(6\) and \(8\) work. 2. b) The fraction on the right must be larger than \(\frac{1}{6}\), so its denominator must be smaller than \(6\). From the choices, \(2\), \(3\), and \(4\) work. 3. c) The unit fraction must be smaller than \(\frac{1}{3}\) but larger than \(\frac{1}{8}\). From the choices, \(4\) and \(6\) work.

a) \(6\) or \(8\) b) \(2\), \(3\), or \(4\) c) \(4\) or \(6\)

- Each fraction has \(2\) parts, so compare the size of one part. - The middle fraction needs parts smaller than thirds but larger than sixths. - Try both choices in both comparison signs.

1. All three fractions have numerator \(2\), so compare the sizes of their equal parts. 2. Fourths are smaller than thirds but larger than sixths. 3. Therefore, \(\frac{2}{6}<\frac{2}{4}<\frac{2}{3}\). 4. Eighths are smaller than sixths, so \(\frac{2}{8}\) would not make the left comparison true.

The missing denominator is \(4\).

- Each fraction has numerator \(5\), so each uses \(5\) equal parts. - Compare the size of one third, one sixth, and one eighth. - Five larger parts make a larger fraction.

1. With the same numerator, smaller denominators make larger equal parts. 2. Third-size parts are larger than sixth-size parts, so \(\frac{5}{3}>\frac{5}{6}\). 3. Eighth-size parts are smaller than sixth-size parts, so \(\frac{5}{8}<\frac{5}{6}\).

Replacing \(6\) with \(3\) makes a greater fraction. Replacing \(6\) with \(8\) makes a smaller fraction.

- Each fraction has \(2\) equal parts. - Order the size of one third, one fourth, and one eighth. - Look for two part sizes where one is exactly twice the other.

1. All three fractions have numerator \(2\). 2. Thirds are largest, fourths are next, and eighths are smallest. 3. Therefore, \(\frac{2}{3}>\frac{2}{4}>\frac{2}{8}\). 4. One fourth is twice one eighth, so two fourths are twice two eighths. Thus, \(\frac{2}{4}\) is twice \(\frac{2}{8}\).

\(\frac{2}{3}>\frac{2}{4}>\frac{2}{8}\). The pair with an exact double relationship is \(\frac{2}{4}\) and \(\frac{2}{8}\).

- Count the equal parts and shaded parts in each model. - Notice that the same number of parts is shaded in both models. - Compare the size of one part in the fourths partition with one part in the eighths partition.

1. Model a) shows \(3\) of \(8\) equal parts shaded, so it represents \(\frac{3}{8}\). 2. Model b) shows \(3\) of \(4\) equal parts shaded, so it represents \(\frac{3}{4}\). 3. Both fractions have numerator \(3\). Fourths are larger parts than eighths in same-size wholes. 4. Therefore, \(\frac{3}{4}>\frac{3}{8}\).

Model a) shows \(\frac{3}{8}\), model b) shows \(\frac{3}{4}\), and \(\frac{3}{4}>\frac{3}{8}\).

- Each fraction uses \(4\) equal parts. - Compare the size of one third, one fourth, and one sixth. - Compare each numerator with its denominator to decide whether the fraction is below, at, or above \(1\).

1. All three fractions have numerator \(4\). 2. Thirds are larger than fourths, and fourths are larger than sixths. 3. Therefore, \(\frac{4}{3}>\frac{4}{4}>\frac{4}{6}\). 4. Also, \(\frac{4}{3}>1\), \(\frac{4}{4}=1\), and \(\frac{4}{6}<1\).

\(\frac{4}{3}>\frac{4}{4}>\frac{4}{6}\). Also, \(\frac{4}{3}>1\), \(\frac{4}{4}=1\), and \(\frac{4}{6}<1\).

- Keep the numerator \(1\). - The fraction must be smaller than one third but larger than one sixth. - Test the allowed denominators against both comparisons.

1. With numerator \(1\), a smaller denominator makes a greater fraction. 2. The denominator must be greater than \(3\) to make the fraction less than \(\frac{1}{3}\). 3. It must also be less than \(6\) to make the fraction greater than \(\frac{1}{6}\). 4. The only allowed denominator between \(3\) and \(6\) is \(4\), so the fraction is \(\frac{1}{4}\).

\(\frac{1}{4}\)

- Read the painted fraction from each banner model. - Compare the painted amounts using the common numerator. - For same-size whole banners, think about how painted and unpainted amounts are related.

1. Banner a) has \(2\) of \(3\) equal parts painted, so it shows \(\frac{2}{3}\). 2. Banner b) has \(2\) of \(6\) equal parts painted, so it shows \(\frac{2}{6}\). 3. Thirds are larger than sixths, so \(\frac{2}{3}>\frac{2}{6}\); banner a) has more painted. 4. Because the banners are the same size, the banner with less painted has more unpainted. Thus, banner b) has more unpainted.

Banner a) has more painted: \(\frac{2}{3}>\frac{2}{6}\). Banner b) has more unpainted.

- Use the longer fourth-size tick spacing to identify one point. - Use the smaller eighth-size spaces to identify the other point. - On the interval from \(0\) to \(1\), compare the left-right positions to the endpoints.

1. Point \(A\) is at the third fourth-size step from \(0\), so \(A=\frac{3}{4}\). 2. Point \(B\) is at the third eighth-size step from \(0\), so \(B=\frac{3}{8}\). 3. Since \(\frac{3}{8}<\frac{3}{4}\), point \(B\) is closer to \(0\). 4. Point \(A\) is farther right on the interval from \(0\) to \(1\), so it is closer to \(1\).

\(A=\frac{3}{4}\) and \(B=\frac{3}{8}\). Point \(B\) is closer to \(0\), and point \(A\) is closer to \(1\).

- Count how many equal spaces make one whole on each line. - The numerator is the same because both points are two spaces from \(0\). - Compare the physical size of one space in each partition.

1. In line a), one whole is divided into \(3\) equal spaces, so point \(A\) is \(\frac{2}{3}\). 2. In line b), one whole is divided into \(6\) equal spaces, so point \(B\) is \(\frac{2}{6}\). 3. Both fractions count \(2\) spaces, but a third-size space is larger than a sixth-size space. 4. Therefore, \(\frac{2}{3}>\frac{2}{6}\).

\(A=\frac{2}{3}\), \(B=\frac{2}{6}\), and \(\frac{2}{3}>\frac{2}{6}\).

- Read the shaded fraction in each bar. - Compare the length of one equal part in the fourths and eighths partitions. - Decide whether the size relationship for one part remains true when the same number of parts is shaded.

1. Bar a) shows \(\frac{3}{4}\), and bar b) shows \(\frac{3}{8}\). 2. Both fractions have \(3\) shaded parts, but one fourth is twice as long as one eighth in same-size bars. 3. Therefore, \(\frac{3}{4}>\frac{3}{8}\). 4. Because each fourth is twice one eighth, three fourths have exactly twice the shaded length of three eighths.

\(\frac{3}{4}>\frac{3}{8}\). Yes, the shaded length in bar a) is exactly twice the shaded length in bar b).

- Keep the numerator \(2\). - A smaller denominator makes each equal part larger. - Choose the smallest listed denominator that does not make the fraction equal to \(1\).

1. With numerator \(2\), a smaller denominator makes larger equal parts and a fraction closer to \(1\). 2. The smallest allowed denominator greater than \(2\) is \(3\). 3. Therefore, the fraction is \(\frac{2}{3}\).

\(\frac{2}{3}\)

- First compare the two fraction numbers using the fact that they have the same numerator. - For each strip, use its own total length to determine the size of one equal fraction part. - After finding the two shaded lengths, compare them with your fraction-number comparison. - In the explanation, pay attention to whether the two fractions refer to the same-size whole.

1. a) The fractions have the same numerator. Thirds are larger fraction parts than sixths, so \(\frac{2}{3} > \frac{2}{6}\). 2. b) Strip A is divided into thirds, so each third is \(6 \div 3 = 2\,\text{in.}\). Two thirds is \(2 \times 2 = 4\,\text{in.}\). 3. Strip B is divided into sixths, so each sixth is \(12 \div 6 = 2\,\text{in.}\). Two sixths is \(2 \times 2 = 4\,\text{in.}\). 4. c) The fraction numbers compare parts of a same-size whole, but these strips are different lengths. Here the shaded physical lengths are equal even though \(\frac{2}{3} > \frac{2}{6}\).

a) \(\frac{2}{3} > \frac{2}{6}\) b) Strip A: \(4\,\text{in.}\); Strip B: \(4\,\text{in.}\) c) The wholes are different lengths. The same-numerator fraction comparison tells which fraction number is greater, but it does not by itself compare physical shaded lengths from different-size wholes.