Free math worksheets

- Count the equal parts and shaded parts in diagram a). - Do the same count in diagram b) instead of judging by which way the rectangle points. - Decide whether a turn changes the size of the whole or any of its parts.

1. In diagram a), the rectangle is divided into \(4\) equal parts and \(3\) are shaded, so it represents \(\frac{3}{4}\). 2. In diagram b), the same rectangle is still divided into the same \(4\) equal parts and the same \(3\) parts are shaded, so it also represents \(\frac{3}{4}\). 3. A quarter-turn changes only the direction of the paper. It does not change the whole, the equal parts, or which parts are shaded. 4. Therefore, both diagrams represent \(\frac{3}{4}\).

Both diagrams represent \(\frac{3}{4}\). Turning the paper changes its direction but not the whole, its \(4\) equal parts, or the \(3\) shaded parts.

- Count the equal parts and shaded parts in each same-size bar. - Compare where the shaded regions end. - Explain how one partition could be refined to make the parts line up with the other partition.

1. Bar a) shows \(1\) of \(3\) equal parts shaded, or \(\frac{1}{3}\). 2. Bar b) shows \(2\) of \(6\) equal parts shaded, or \(\frac{2}{6}\). 3. Splitting one third into \(2\) equal pieces makes two sixths without changing the shaded length. 4. Therefore, \(\frac{1}{3}=\frac{2}{6}\).

One third can be split into \(2\) equal sixths, so \(\frac{1}{3}=\frac{2}{6}\).

- Read the shaded fraction in each model. - Compare the shaded lengths of the same-size bars. - Explain how the larger parts in one model can be split to match the smaller parts in the other model.

1. Bar a) shows \(3\) of \(4\) equal parts shaded, or \(\frac{3}{4}\). 2. Bar b) shows \(6\) of \(8\) equal parts shaded, or \(\frac{6}{8}\). 3. Splitting every fourth into \(2\) equal pieces makes \(8\) equal parts and \(6\) shaded parts without changing the shaded amount. 4. Therefore, \(\frac{3}{4}=\frac{6}{8}\).

Splitting every fourth into \(2\) equal pieces makes \(8\) equal parts and \(6\) shaded parts. The shaded amount stays the same, so \(\frac{3}{4}=\frac{6}{8}\).

- Compare the unshaded part in each same-size whole. - One model leaves one third unshaded; the other leaves one fourth unshaded. - Different unshaded amounts mean different shaded amounts.

1. Two thirds leave one third of the first whole unshaded. 2. Three fourths leave one fourth of the second same-size whole unshaded. 3. One third is larger than one fourth, so the shaded amounts are different. 4. Therefore, \(\frac{2}{3}\ne\frac{3}{4}\).

The rule is false. The models show different shaded amounts, so \(\frac{2}{3}\ne\frac{3}{4}\).

- Count the total parts and shaded parts in each model. - Compare the endpoint of the shaded region across the three same-size bars. - Ask what happens to the part size when the same whole is divided into more equal pieces.

1. Model a) shows \(1\) of \(2\) equal parts shaded, so it represents \(\frac{1}{2}\). 2. Model b) shows \(2\) of \(4\) equal parts shaded, so it represents \(\frac{2}{4}\). 3. Model c) shows \(3\) of \(6\) equal parts shaded, so it represents \(\frac{3}{6}\). 4. Each bar has the same shaded length. Dividing the same half into smaller equal pieces increases both the number of total parts and the number of shaded parts without changing the shaded amount. Therefore, \(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}\).

The models show \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{3}{6}\). They are equivalent because each shades the same half of a same-size whole even though that half is partitioned into different numbers of equal pieces.

- Count the equal parts in each whole. - Count the shaded parts in each model. - Decide whether both models show one half.

1. Each shape is divided into equal parts. 2. One of two equal parts is shaded in the circle, so it shows one half. 3. Two of four equal parts are shaded in the card, so it also shows one half. 4. The part shapes are different, but both models show the same fraction: \(\frac{1}{2}=\frac{2}{4}\).

Both models show one half of their whole, so \(\frac{1}{2}=\frac{2}{4}\). The shapes of the parts do not change the fraction.

- Compare the length of two eighth-size spaces with one fourth-size space. - Both fractions are measured left from the same point, \(1\). - Equal distances from \(1\) land at the same point.

1. Two eighth-size spaces have the same total length as one fourth-size space. 2. Both points are therefore the same distance to the left of \(1\). 3. A single number line has only one point at that distance from \(1\). 4. Thus, \(\frac{3}{4}=\frac{6}{8}\).

Two eighths equal one fourth, so both fractions are one fourth below \(1\). Therefore, \(\frac{3}{4}=\frac{6}{8}\).

- First read the shaded fraction from the model. - Ask whether “two parts” means pieces of the same size in both fraction names. - Look for a way to group all of the sixth-size parts into larger equal groups.

1. The model has \(4\) of \(6\) equal parts shaded, so it shows \(\frac{4}{6}\). 2. Eli's reason is not correct: the model leaves \(2\) sixths unshaded, while \(\frac{2}{3}\) leaves \(1\) third unshaded. 3. Group the \(6\) equal parts into \(3\) equal pairs. The \(4\) shaded sixths form \(2\) complete pairs. 4. Therefore, \(\frac{4}{6}=\frac{2}{3}\), but the correct reason is the equal regrouping, not simply that “two parts” are involved.

Eli's explanation is not correct because the parts being counted are different sizes. A correct explanation is that the \(6\) equal parts can be grouped into \(3\) equal pairs, with \(2\) pairs shaded, so \(\frac{4}{6}=\frac{2}{3}\).

- Count the equal parts covered in each grid. - Focus on the physical region covered, not only on the number of pieces. - Explain why splitting each column into smaller equal parts does not change the covered area.

1. In grid a), the cover spans \(2\) of \(3\) equal columns, so it covers \(\frac{2}{3}\) of the whole. 2. In grid b), each column is split into \(2\) equal cells. The same cover spans \(4\) of the \(6\) equal cells, so it covers \(\frac{4}{6}\). 3. The cover has the same size and position on two same-size wholes, so it covers the same amount in both grids. 4. Therefore, \(\frac{2}{3}=\frac{4}{6}\).

The grids show \(\frac{2}{3}\) and \(\frac{4}{6}\). The same cover occupies the same region of each same-size whole, so \(\frac{2}{3}=\frac{4}{6}\).

- Read the fraction shown by each model. - Track which model is known to match model b). - If two amounts each match the same amount, decide what that tells you about those two amounts.

1. The three models show \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{4}{8}\). 2. Models a) and b) shade the same amount of a same-size whole. 3. Models b) and c) also shade the same amount of a same-size whole. 4. Therefore, models a) and c) both match the same shaded amount, so \(\frac{1}{2}=\frac{4}{8}\).

Both \(\frac{1}{2}\) and \(\frac{4}{8}\) represent the same shaded amount as \(\frac{2}{4}\). Therefore, \(\frac{1}{2}=\frac{4}{8}\).

- Count the shaded unit-fraction parts across each pair of bars. - Compare the partially shaded second bar in each pair. - Think about how many fourth-size pieces cover the same amount as one half-size piece.

1. Bars a) and b) contain \(3\) shaded halves altogether, so they show \(\frac{3}{2}\). 2. Bars c) and d) contain \(6\) shaded fourths altogether, so they show \(\frac{6}{4}\). 3. In each pair, one whole bar is fully shaded and half of the second bar is shaded. Two fourths cover the same amount as one half. 4. Therefore, the two amounts are equal and \(\frac{3}{2}=\frac{6}{4}\).

The models show \(\frac{3}{2}\) and \(\frac{6}{4}\). They are equivalent because both show one whole plus the same half-whole amount, so \(\frac{3}{2}=\frac{6}{4}\).

- Count the small equal spaces from \(0\) to \(1\) to identify the unit fraction. - Count how many of those spaces reach point \(X\). - Think about how the small spaces can be grouped equally to name the same distance in thirds.

1. From \(0\) to \(1\), the number line has \(6\) equal spaces, so each small space is one sixth. 2. Point \(X\) is \(8\) small spaces from \(0\), so \(X=\frac{8}{6}\). 3. Two sixth-size spaces have the same length as one third-size space. Thus, \(8\) sixths make \(4\) thirds. 4. Therefore, \(X=\frac{8}{6}=\frac{4}{3}\).

\(X=\frac{8}{6}=\frac{4}{3}\). Two sixth-size spaces make one third-size space, so 8 sixths make 4 thirds. Therefore, \(\frac{8}{6}\) and \(\frac{4}{3}\) name the same point.

- Count the shaded unit-fraction parts in each model. - Compare the size of one part in model a) with the smaller parts in model b). - Explain what changes in the part count and what stays the same in the total amount.

1. Model a) shows \(5\) half-size parts, so it represents \(\frac{5}{2}\). 2. Model b) shows \(10\) fourth-size parts, so it represents \(\frac{10}{4}\). 3. Each half-size part has the same size as \(2\) fourth-size parts. Five halves therefore have the same size as \(10\) fourths. 4. Thus, \(\frac{5}{2}=\frac{10}{4}\).

Model a) shows \(\frac{5}{2}\), and model b) shows \(\frac{10}{4}\). Every half can be split into two fourths, so \(\frac{5}{2}=\frac{10}{4}\).

- Count the total equal parts in each same-size bar. - One unshaded piece does not necessarily have the same size in two different partitions. - Compare the unshaded fractions before deciding whether the shaded amounts match.

1. Model a) has \(3\) of \(4\) equal parts shaded, so its unshaded part is \(\frac{1}{4}\) of the whole. 2. Model b) has \(5\) of \(6\) equal parts shaded, so its unshaded part is \(\frac{1}{6}\) of the whole. 3. One fourth is larger than one sixth, so the models leave different amounts unshaded. 4. Therefore, the shaded fractions \(\frac{3}{4}\) and \(\frac{5}{6}\) are not equivalent.

The single unshaded parts are different sizes. One model leaves \(\frac{1}{4}\) unshaded and the other leaves \(\frac{1}{6}\), so the shaded fractions are not equivalent.

- Check whether each method splits every original part in the same way. - A fraction model must divide the whole into equal parts. - Count the new equal parts and shaded parts in Method A.

1. Method A creates \(6\) equal parts across the whole bar, with \(4\) shaded, so it shows \(\frac{4}{6}=\frac{2}{3}\). 2. Method B creates smaller shaded pieces but leaves one larger unshaded piece. 3. Method B does not divide the whole into equal parts. 4. Only Method A makes a valid equivalent-fraction model.

Method A is valid because it divides the entire whole into \(6\) equal parts and shows \(\frac{2}{3}=\frac{4}{6}\). Method B creates unequal parts.

- First read the shaded fraction from the strip. - After the proposed change, check whether every visible part of the whole would have the same size. - A valid new denominator must describe equal parts across the entire whole, not only the shaded region.

1. The strip has \(2\) of \(8\) equal parts shaded, so it begins as \(\frac{2}{8}\). 2. Removing only the divider between the shaded parts creates one larger shaded piece but leaves six separate unshaded eighths. 3. The visible pieces are then unequal, so they do not form fourths. Every pair of eighths across the whole strip must be grouped in the same way. 4. Then the whole has \(4\) equal parts and \(1\) is shaded, showing \(\frac{2}{8}=\frac{1}{4}\).

The student must group every pair of eighths across the entire strip, not only the shaded pair. Then the whole has \(4\) equal parts and \(1\) is shaded, so \(\frac{2}{8}=\frac{1}{4}\).

- Check whether every visible part in each final bar has the same width. - A denominator of \(6\) must describe six equal parts of the entire whole. - The shaded amount may stay the same even when the new partition is not a valid fraction partition.

1. Diagram a) divides the entire bar into \(6\) equal parts. 2. Four of those \(6\) equal parts are shaded, so diagram a) shows \(\frac{4}{6}\), the same shaded amount as \(\frac{2}{3}\). 3. Diagram b) splits only the originally shaded thirds. Its unshaded section remains twice as wide as each smaller shaded section. 4. Because diagram b) does not divide the whole into equal parts, it cannot be read as sixths. Diagram a) is the valid model.

Diagram a) is valid and shows \(\frac{2}{3}=\frac{4}{6}\). Diagram b) is invalid because its visible parts are not all equal.