- Start by comparing the whole bars. - Count equal parts and shaded parts in each one. - Check how far the shading reaches across each bar.

1. Bar a) has \(1\) of \(2\) equal parts shaded: \(\frac{1}{2}\). 2. Bar b) has \(2\) of \(4\) equal parts shaded: \(\frac{2}{4}\). 3. The shaded lengths match, so \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent.

Yes. a) shows \(\frac{1}{2}\), and b) shows \(\frac{2}{4}\).

- Are the wholes the same size? - How many equal parts make each whole? - Is one of six parts the same size as one of three?

1. Circle a) has \(1\) of \(3\) parts shaded, or \(\frac{1}{3}\). 2. Circle b) has \(1\) of \(6\) parts shaded, or \(\frac{1}{6}\). 3. One sixth is smaller than one third, so the fractions are not equivalent.

No. Circle a) shows \(\frac{1}{3}\), while circle b) shows \(\frac{1}{6}\). One third of the same-size circle is larger than one sixth.

- Count the small equal parts and the shaded parts in the model. - Think about making two equal groups from all of the small parts. - Decide how many of those two groups are completely shaded.

1. The bar has \(6\) equal parts, with \(3\) shaded. 2. Regrouping the \(6\) parts into \(2\) equal large parts puts \(3\) small parts in each half. 3. The shaded \(3\) sixths fill \(1\) of the \(2\) large parts. 4. Therefore, \(\frac{3}{6}=\frac{1}{2}\).

\(\frac{1}{2}\) of the bar is shaded.

- Count the equal parts and shaded parts in each bar. - Compare where the shaded region ends in the two same-size bars. - Write the two fractions as an equality if the shaded lengths match.

1. Bar a) has \(1\) of \(4\) equal parts shaded, so it shows \(\frac{1}{4}\). 2. Bar b) has \(2\) of \(8\) equal parts shaded, so it shows \(\frac{2}{8}\). 3. The shaded lengths are the same because splitting one fourth into \(2\) equal pieces makes two eighths. 4. Therefore, \(\frac{1}{4}=\frac{2}{8}\).

Yes. \(\frac{1}{4}=\frac{2}{8}\).

- Name the shaded part before any new cuts. - Would making more cuts change how far the blue part reaches? - Check that the new pieces would all have the same length.

1. The bar has \(3\) equal sections, with \(1\) shaded, so the original fraction is \(\frac{1}{3}\). 2. Splitting every section into \(2\) parts makes \(3\times2=6\) equal small parts. 3. The shaded third becomes \(2\) shaded small parts. 4. The new fraction is \(\frac{2}{6}\), so \(\frac{1}{3}=\frac{2}{6}\).

There would be \(6\) small parts, with \(2\) shaded. \(\frac{1}{3}=\frac{2}{6}\).

- Read the shaded fraction in model a) from its equal parts. - Compare how the same-size whole is partitioned in model b). - Think about how many smaller parts fit inside each shaded part of model a).

1. Model a) shows \(2\) of \(3\) equal parts shaded, or \(\frac{2}{3}\). 2. Model b) divides the same-size whole into \(6\) equal parts. 3. Splitting each third into \(2\) equal pieces makes \(4\) shaded sixths. 4. Therefore, \(\frac{2}{3}=\frac{4}{6}\).

Shade \(4\) parts of model b). \(\frac{2}{3}=\frac{4}{6}\).

- A whole contains two equal halves. - The second half needs the same number of tiles as the first. - Use the \(4\) covered tiles over the total number of tiles.

1. The other half needs the same number of tiles. 2. The whole strip needs \(4+4=8\) tiles. 3. Four of the \(8\) tiles cover one half, so the fraction is \(\frac{4}{8}\).

The whole strip takes \(8\) tiles, and the covered half is \(\frac{4}{8}\).

- Keep the whole bar the same length even if you group its pieces differently. - Look for a way to join neighboring small parts into equal larger parts. - Does your new way of grouping change the amount that is blue?

1. Pairing the \(6\) small parts gives \(3\) equal larger parts. 2. The \(2\) blue small parts form \(1\) complete larger part. 3. The blue part is \(\frac{1}{3}\) of the bar, the same amount as \(\frac{2}{6}\).

\(\frac{1}{3}\). The \(6\) small parts can be joined in pairs to make \(3\) equal larger parts. The \(2\) blue parts make one of these larger parts.

- Find the position of P relative to \(0\) and \(1\). - How many fourth-size steps reach P? - How many eighth-size steps reach that same place?

1. P is \(2\) fourths from \(0\), so one name is \(\frac{2}{4}\). 2. P is also \(4\) eighths from \(0\), so another name is \(\frac{4}{8}\). 3. Both fractions name the same point halfway along the same unit interval.

\(\frac{2}{4}\) and \(\frac{4}{8}\); both name the halfway point.

- Use each bookmark's own full length as its whole. - Check the fraction question separately from the question about inches.

1. The shorter bookmark has \(\frac{2}{4}\) of its length colored, or one half. 2. The longer bookmark has \(\frac{4}{8}\) of its length colored, also one half. 3. The fractions match, but the colored lengths of \(2\) inches and \(4\) inches do not.

Yes, the same fraction is colored: \(\frac{2}{4}=\frac{4}{8}=\frac{1}{2}\) of each bookmark. No, the colored lengths differ: \(2\) inches and \(4\) inches.

- For each figure, count the equal parts and the shaded parts. - Compare the actual shaded lengths because all three bars represent the same-size whole. - Look for two partitions whose shaded portions end at the same place.

1. Figure a) shows \(\frac{1}{4}\). 2. Figure b) shows \(\frac{2}{8}\), which is the same amount as \(\frac{1}{4}\). 3. Figure c) shows \(\frac{2}{6}\), which is the same amount as \(\frac{1}{3}\). 4. Figures a) and b) show the same shaded amount, while figure c) shows a different amount.

Figure c). Figures a) and b) both show \(\frac{1}{4}\) of the same-size whole, while figure c) shows \(\frac{1}{3}\).

- Name the shaded fraction in each model before changing anything. - Compare where the shaded region ends in the two same-size circles. - Think about how many sixth-size parts cover the same amount as the shaded thirds in model a).

1. Model a) has \(2\) of \(3\) equal parts shaded, so it shows \(\frac{2}{3}\). 2. Model b) has \(3\) of \(6\) equal parts shaded. 3. The shaded region in model a) reaches through \(4\) sixth-size parts of the same-size circle, so model b) needs \(1\) additional shaded part. 4. After that change, model b) shows \(\frac{4}{6}\), so \(\frac{2}{3}=\frac{4}{6}\).

One additional part of model b) must be shaded; \(\frac{2}{3}=\frac{4}{6}\).

- Count all equal cells and the shaded cells. - Then look at each vertical column as a larger equal part of the same rectangle. - Check whether both fraction names refer to exactly the same shaded region.

1. The grid has \(6\) equal cells, with \(4\) shaded, so the cell fraction is \(\frac{4}{6}\). 2. The grid has \(3\) equal columns, with \(2\) complete columns shaded, so the column fraction is \(\frac{2}{3}\). 3. Both fractions describe exactly the same shaded region, so \(\frac{4}{6}=\frac{2}{3}\).

By cells: \(\frac{4}{6}\) By columns: \(\frac{2}{3}\) \(\frac{4}{6}=\frac{2}{3}\)

- Compare the shaded parts with the whole bar in each picture. - Count the equal pieces and the shaded pieces in each bar. - Would moving a shaded piece change how much is shaded altogether?

1. One fourth has the same total size as two eighths. 2. Both model b) and model c) shade exactly \(2\) of \(8\) equal parts. 3. The positions of the shaded parts do not change their total area. 4. Both models show \(\frac{2}{8}=\frac{1}{4}\).

Both b) and c) show \(\frac{2}{8}=\frac{1}{4}\). The shaded parts do not need to touch.

- Use the longer tick marks first to name \(Q\) in fourths. - Then count all of the smaller equal spaces in one whole unit. - Count how many of those smaller spaces reach from \(0\) to \(Q\).

1. Point \(Q\) is at the third fourth-size step from \(0\), so \(Q=\frac{3}{4}\). 2. There are \(6\) small equal spaces from \(0\) to \(Q\) and \(8\) such spaces in one whole unit. 3. Each small space is \(\frac{1}{8}\) of one unit, so \(Q=\frac{6}{8}\). 4. Therefore, \(\frac{3}{4}=\frac{6}{8}\).

\(Q=\frac{3}{4}=\frac{6}{8}\), and each small space is \(\frac{1}{8}\) of one unit.

- Read the shaded fraction in each model. - Compare where each shaded region ends on the same-size bars. - For a pair that appears to match, explain how one partition can be split into the other partition without changing the shaded amount.

1. Model a) shows \(\frac{2}{3}\), model b) shows \(\frac{4}{6}\), and model c) shows \(\frac{5}{8}\). 2. The shaded region in models a) and b) ends at the same place on the same-size whole. 3. Each third in model a) can be split into \(2\) equal sixths, so \(2\) shaded thirds become \(4\) shaded sixths. 4. Therefore, \(\frac{2}{3}=\frac{4}{6}\); model c) shows a different shaded amount.

Models a) and b) are equivalent: \(\frac{2}{3}=\frac{4}{6}\).

- Read the shaded fraction in each model. - Ask what changes, and what stays the same, when a shaded part is only moved. - Compare the total shaded lengths of the same-size bars rather than the positions of individual shaded pieces.

1. Model a) shows \(\frac{2}{3}\), and model b) shows \(\frac{3}{6}\). 2. Moving a shaded sixth changes its position but not the number of shaded sixths, so model b) still shows \(\frac{3}{6}\). 3. The same-size bars show different shaded lengths: \(\frac{2}{3}\) reaches farther than \(\frac{3}{6}\). 4. Therefore, moving one shaded part cannot make the models equivalent.

No. Moving a shaded part does not change the total shaded amount, so model b) still shows \(\frac{3}{6}\), which is not equivalent to \(\frac{2}{3}\).

- First read the shaded fraction from the model. - The numerator must change from the model's shaded-part count to \(6\) without changing the shaded amount. - Whatever happens to each shaded part must also happen to every part of the whole.

1. The model has \(3\) of \(4\) equal parts shaded, so it shows \(\frac{3}{4}\). 2. To change \(3\) shaded parts into \(6\) shaded parts without changing the amount, split every fourth into \(2\) equal pieces. 3. The whole then has \(8\) equal parts, and \(6\) are shaded. 4. Therefore, \(\frac{3}{4}=\frac{6}{8}\), so the denominator is \(8\).

The denominator is \(8\), giving \(\frac{6}{8}\).

- Identify the shaded fraction of the strip from its equal parts. - Find how many equal spaces divide the unit on the number line. - Compare the two positions with three fourths of the whole.

1. The strip has \(3\) out of \(4\) equal parts shaded, so it represents \(\frac{3}{4}\). 2. The number line from \(0\) to \(1\) has \(8\) equal spaces. P is at \(\frac{5}{8}\). 3. Q is at \(\frac{6}{8}\). 4. Six eighths cover three fourths of the same whole, so \(\frac{3}{4}=\frac{6}{8}\). Point Q matches.

The strip shows \(\frac{3}{4}\). P is at \(\frac{5}{8}\), and Q is at \(\frac{6}{8}\). Q matches because \(\frac{3}{4}=\frac{6}{8}\).

- Find the total represented by each chart before naming any slice as a fraction. - Write each color's value over the total for its own chart. - Compare corresponding slices by asking whether they cover the same share of their wholes.

1. Red is the largest slice in both charts. 2. In a), the total is \(2 + 1 + 1 = 4\). The red, blue, and green slices are \(\frac{2}{4}\), \(\frac{1}{4}\), and \(\frac{1}{4}\). 3. In b), the total is \(4 + 2 + 2 = 8\). The corresponding slices are \(\frac{4}{8}\), \(\frac{2}{8}\), and \(\frac{2}{8}\). 4. The red slices both represent one half of their chart, and each smaller slice in b) is two eighths, the same amount of its whole as one fourth in a). Therefore, the charts have the same proportions.

1) Red is largest in both charts. 2) Yes. The corresponding slices represent the same fractions of their wholes: \(\frac{2}{4}=\frac{4}{8}\) and \(\frac{1}{4}=\frac{2}{8}\).

- Read the shaded fraction from the bar first. - Group the eighth-size parts into pairs. - Check whether every shaded eighth belongs to a completely shaded pair.

1. The bar has \(5\) of \(8\) equal parts shaded, so it shows \(\frac{5}{8}\). 2. Each pair of eighths makes one fourth of the bar. 3. Five shaded eighths make \(2\) complete shaded pairs, with \(1\) shaded eighth left over. 4. Therefore, the shaded amount cannot be named as a whole number of fourths.

No. Five shaded eighths make \(2\) complete pairs with \(1\) eighth left over, so \(\frac{5}{8}\) cannot be written as an equivalent fraction with denominator \(4\).

- Read the shaded and unshaded parts of model a). - Compare the size of one part in model a) with the smaller parts in model b). - Preserve the same unshaded length, then count how many parts of model b) are left shaded.

1. Model a) has \(3\) of \(4\) equal parts shaded, so \(1\) fourth is unshaded. 2. Model b) is divided into \(8\) equal parts. The same unshaded length occupies \(2\) of those eighths. 3. Therefore, \(2\) eighths should remain unshaded and \(6\) eighths should be shaded. 4. The shaded fractions are \(\frac{3}{4}=\frac{6}{8}\).

Model b) should have \(2\) eighths unshaded; the shaded equivalence is \(\frac{3}{4}=\frac{6}{8}\).

- First read the shaded fraction in each model. - Consider how each proposed one-part change affects the shaded length. - The correct change must make the two same-size bars end at the same shaded point.

1. Model a) shows \(\frac{2}{3}\), while model b) shows \(\frac{3}{6}\). 2. Adding one shaded part to model b) makes \(\frac{4}{6}\). 3. The shaded length of \(\frac{4}{6}\) matches the shaded length of \(\frac{2}{3}\), so choice a) works. 4. Removing one shaded third from model a) makes \(\frac{1}{3}\), which does not match \(\frac{3}{6}\).

a) Add one shaded part to model b), giving \(\frac{2}{3}=\frac{4}{6}\).

- Count the shaded fourth-size parts across bars a) and b). - Count the shaded eighth-size parts across bars c) and d). - Compare how many eighth-size parts fit in one fourth-size part of the same-size whole.

1. In Model A, bar a) has \(4\) shaded fourths and bar b) has \(1\) shaded fourth, for \(\frac{5}{4}\). 2. In Model B, bar c) has \(8\) shaded eighths and bar d) has \(2\) shaded eighths, for \(\frac{10}{8}\). 3. Each fourth has the same size as \(2\) eighths. 4. Therefore, \(5\) fourths and \(10\) eighths have the same total size, so \(\frac{5}{4}=\frac{10}{8}\).

Model A shows \(\frac{5}{4}\), and Model B shows \(\frac{10}{8}\). They are equivalent: \(\frac{5}{4}=\frac{10}{8}\).