Free Math Worksheets

- Think about sharing the same-size cake among more or fewer people. - The more equal parts there are, the smaller each part is. - Which denominator represents fewer, larger parts?

1. Pizza A is divided into \(3\) equal parts, while Pizza B is divided into \(5\) equal parts. 2. Because the pizzas are the same size, dividing one into fewer equal parts makes each part larger. 3. Therefore, \(\frac{1}{3} > \frac{1}{5}\).

\(\frac{1}{3}\) is greater.

- 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, \(\frac{1}{2}\).

- Look at the sizes of the shaded slices. Which one is narrower? - The more equal parts a whole is divided into, the smaller each part becomes.

1. Compare the denominators, \(4\) and \(8\). 2. The whole for \(\frac{1}{8}\) is divided into twice as many equal parts as the whole for \(\frac{1}{4}\). 3. Each eighth is half the size of each fourth. 4. Therefore, \(\frac{1}{8} < \frac{1}{4}\).

\(\frac{1}{8}\) is the smaller fraction.

- 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.

- Which fraction represents the smallest piece of one whole? - Compare the denominators. - Think about what happens to each piece when a whole is divided into more equal parts.

1. Each fraction has the same numerator, \(1\). 2. For unit fractions, a larger denominator means the whole is divided into more equal parts, so each part is smaller. 3. Order the denominators from greatest to least: \(100, 10, 8, 5, 2\). 4. Therefore, \(\frac{1}{100} < \frac{1}{10} < \frac{1}{8} < \frac{1}{5} < \frac{1}{2}\).

\(\frac{1}{100} < \frac{1}{10} < \frac{1}{8} < \frac{1}{5} < \frac{1}{2}\)

- Both fractions represent one piece. - Compare how many equal pieces each whole is divided into. - Fewer equal pieces means each piece is larger.

1. Each circle has one shaded part, so the fractions have the same numerator. 2. When unit fractions have the same numerator, the fraction with the smaller denominator is greater because the whole is divided into fewer, larger pieces. 3. Since \(3<5\), \(\frac{1}{3}>\frac{1}{5}\).

\(\frac{1}{3}>\frac{1}{5}\)

- Write the shaded fraction for each figure. - All three fractions have numerator \(1\). - For unit fractions, fewer equal parts means a larger piece.

1. Figure a) shows \(\frac{1}{4}\). 2. Figure b) shows \(\frac{1}{3}\). 3. Figure c) shows \(\frac{1}{6}\). 4. These are unit fractions. The fraction with the smallest denominator is greatest, so \(\frac{1}{3}>\frac{1}{4}>\frac{1}{6}\). Figure b) has the greatest shaded fraction.

b)

- Both people eat two equal pieces from their own cake. - Compare the size of one third with the size of one sixth. - With equal numerators, the smaller denominator gives the greater fraction.

1. The fractions have the same numerator, \(2\). 2. Thirds are larger pieces than sixths because the whole is divided into fewer equal parts. 3. Therefore, \(\frac{2}{3}>\frac{2}{6}\), so Lucas eats more cake.

Lucas

- Both figures have four shaded parts. - Compare how many equal parts are in each whole. - With equal numerators, the larger denominator gives the smaller fraction.

1. Both fractions have numerator \(4\). 2. Sixths are larger pieces than twelfths because the whole is divided into fewer equal parts. 3. Therefore, \(\frac{4}{6}>\frac{4}{12}\), so figure a) shows the greater fraction.

Figure a), \(\frac{4}{6}\)

- Both figures have three shaded parts. - Compare the size of one fourth with the size of one tenth. - With equal numerators, the smaller denominator gives the greater fraction.

1. Both fractions have numerator \(3\). 2. Fourths are larger pieces than tenths because the whole is divided into fewer equal parts. 3. Therefore, \(\frac{3}{4}>\frac{3}{10}\).

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

- Both circles have four shaded parts. - Compare the size of one seventh with the size of one ninth. - With equal numerators, the smaller denominator gives the greater fraction.

1. Both fractions have numerator \(4\). 2. Sevenths are larger pieces than ninths because the whole is divided into fewer equal parts. 3. Therefore, \(\frac{4}{7}>\frac{4}{9}\), so figure a) has the greater shaded fraction.

Figure a), \(\frac{4}{7}\)

- Write the selected students over the total students in each club. - The fractions have the same numerator. - Three students are a smaller part of a larger group.

1. The selected fraction in the theater club is \(\frac{3}{12}\). 2. The selected fraction in the art club is \(\frac{3}{6}\). 3. The fractions have the same numerator. Since \(12>6\), twelfths are smaller than sixths, so \(\frac{3}{12}<\frac{3}{6}\). 4. The selected fraction is smaller in the theater club.

The theater club

- Both figures have two shaded parts. - Compare how many equal parts are in each whole. - With equal numerators, the larger denominator gives the smaller fraction.

1. Both fractions have numerator \(2\). 2. Eighths are smaller pieces than fifths because the whole is divided into more equal parts. 3. Therefore, \(\frac{2}{8}<\frac{2}{5}\).

\(\frac{2}{8}\), shown in figure b)

- 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, \(\frac{1}{4}\).

- 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.

- In each part, decide whether the whole must be divided into more or fewer equal parts. - If a unit fraction must become smaller, should its denominator become greater or less? - In part c), look for a denominator between two given values.

1. a) To make \(\frac{1}{n}\) less than \(\frac{1}{4}\), the denominator must be greater than \(4\). Any positive whole number \(n > 4\) works. 2. b) To make \(\frac{1}{n}\) greater than \(\frac{1}{10}\), the denominator must be less than \(10\). Any whole number from \(1\) through \(9\) works. 3. c) The denominator must be greater than \(6\) and less than \(12\). The possible values are \(7, 8, 9, 10,\) and \(11\).

Possible answers are: a) \(5\) (any positive whole number greater than \(4\)) b) \(1\) (any whole number from \(1\) through \(9\)) c) \(10\) (any whole number from \(7\) through \(11\))