Free math worksheets

- Compare each numerator with its denominator. - Notice that every part is shaded in each model. - A fully shaded model represents one whole.

1. In each fraction, the numerator equals the denominator. 2. This means every equal part of the whole is shaded. 3. Shading all the parts makes one complete whole, whether it is divided into \(2\), \(4\), or \(6\) parts. 4. Therefore, \(\frac{2}{2}=\frac{4}{4}=\frac{6}{6}=1\).

When the numerator equals the denominator, every equal part is shaded, so the fraction equals one whole.

- Notice what changes when the model is turned. - Check whether the whole, the equal parts, or the shaded parts changed. - Turning a shape does not change how much is shaded.

1. The model still has the same whole and the same \(4\) equal parts. 2. The same \(3\) parts are still shaded. 3. Turning the model changes its direction, not the shaded amount. 4. Therefore, the model still represents \(\frac{3}{4}\).

The model still shows \(3\) of the same \(4\) equal parts shaded, so it remains \(\frac{3}{4}\).

- Compare the shaded lengths in the two same-size bars. - Think about splitting one third into \(2\) equal pieces. - Name the two smaller pieces in sixths.

1. Split one third into \(2\) equal pieces. 2. Each new piece is one sixth of the whole, so the original third is the same size as two sixths. 3. Therefore, \(\frac{1}{3}=\frac{2}{6}\).

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

- Look at what happens when each fourth is split into \(2\) equal pieces. - Count all the new equal parts. - Count the new shaded parts and compare the shaded amount.

1. Start with \(4\) equal parts and \(3\) shaded parts. 2. Split every fourth into \(2\) equal pieces. 3. The whole now has \(8\) equal parts, and \(6\) of them are shaded. 4. The shaded amount did not change, so \(\frac{3}{4}=\frac{6}{8}\).

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 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.

- Check whether the unshaded parts in the two fraction names are the same size. - Try regrouping all of the smallest parts into equal groups. - Describe both the shaded and total numbers of new groups.

1. The sixths model leaves \(2\) sixths unshaded, not \(2\) thirds. 2. Group the \(6\) equal parts into \(3\) equal pairs. 3. The \(4\) shaded sixths form \(2\) complete pairs, and the \(2\) unshaded sixths form \(1\) pair. 4. Therefore, \(\frac{4}{6}=\frac{2}{3}\).

Eli's explanation is not correct because the word “parts” refers to different-sized parts. A correct explanation is that the sixths can be grouped into three equal pairs, with two pairs shaded, so \(\frac{4}{6}=\frac{2}{3}\).

- Focus on the amount covered, not only the number of grid parts. - The two whole grids are the same size. - Name the covered amount using thirds and then sixths.

1. The transparent cover has the same size and position on both same-size wholes. 2. It covers the same amount of each whole. 3. The first grid names that amount \(\frac{2}{3}\), and the second names it \(\frac{4}{6}\). 4. Therefore, \(\frac{2}{3}=\frac{4}{6}\).

The same cover takes up the same amount of both same-size wholes. That amount is \(\frac{2}{3}\) on one grid and \(\frac{4}{6}\) on the other, so \(\frac{2}{3}=\frac{4}{6}\).

- The middle model shows the same amount as the first model. - It also shows the same amount as the third model. - Two amounts that match the same shaded amount match each other.

1. The first and second models shade the same amount of a same-size whole. 2. The second and third models also shade the same amount of a same-size whole. 3. Therefore, the first and third shaded amounts both match the second shaded amount. 4. Thus, \(\frac{1}{2}=\frac{4}{8}\).

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

- Determine how many unit-fraction parts make one whole in each model. - Group each numerator into complete-whole sets. - Compare the number of complete groups rather than the part sizes.

1. Four fourths make one whole, so \(8\) fourths make \(2\) wholes. 2. Two halves make one whole, so \(4\) halves make \(2\) wholes. 3. Both models therefore represent the same amount of \(2\) complete wholes. 4. Thus, \(\frac{8}{4}=\frac{4}{2}=2\).

Eight fourths form two groups of four fourths, and four halves form two groups of two halves. Both make \(2\) wholes, so \(\frac{8}{4}=\frac{4}{2}=2\).

- Compare one half with fourth-size parts. - Use that relationship for all \(5\) halves. - Splitting parts changes their number, not the total amount.

1. Each half-size part has the same size as \(2\) fourth-size parts. 2. Five half-size parts therefore have the same size as \(5\times2=10\) fourth-size parts. 3. Splitting every half into \(2\) equal fourths changes the part count but not the total amount. 4. Therefore, \(\frac{5}{2}=\frac{10}{4}\).

Every half can be split into two fourths. Five halves therefore make ten fourths without changing the total amount, so \(\frac{5}{2}=\frac{10}{4}\).

- Do not compare only the number of unshaded pieces. - Name the unshaded fraction in each model. - Different remainders in same-size wholes lead to different shaded amounts.

1. The unshaded part in the fourths model is \(\frac{1}{4}\) of the whole. 2. The unshaded part in the sixths model is \(\frac{1}{6}\) of the same-size whole. 3. One fourth is larger than one sixth, so the models leave different amounts unshaded. 4. Their shaded amounts are different; in fact, \(\frac{3}{4}<\frac{5}{6}\).

The single unshaded parts are different sizes. A fourth is larger than a sixth, 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.

- Check whether all visible pieces have the same size after the change. - Apply the regrouping pattern across the entire whole. - Count the new equal groups and the shaded groups.

1. Removing only the shaded divider creates one larger shaded piece but leaves six separate unshaded eighths. 2. The visible pieces are then unequal, so they do not form fourths. 3. All \(8\) eighths must be grouped into \(4\) equal pairs by removing the matching dividers across the whole strip. 4. One of the \(4\) equal pairs is shaded, so \(\frac{2}{8}=\frac{1}{4}\).

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