- Remember to count the white sections as part of the whole bar. - Compare how much of the bar is green with how much is still white.

1. The bar has \(3\) shaded and \(3\) unshaded sections of equal size. 2. The shaded sections and the unshaded sections each make one of two equal halves. 3. Thus \(\frac{3}{6}\) covers one half of the bar.

\(\frac{3}{6}\) is green. It is also \(\frac{1}{2}\), because the green and white parts cover equal lengths.

- Count the equal pieces in the whole bar and the blue pieces. - Would bringing the blue pieces together change their total length? - Compare the total blue length with the total white length.

1. The bar has \(4\) equal parts, with \(2\) blue parts, giving \(\frac{2}{4}\). 2. The two blue fourths make the same length as two white fourths, so half the bar is blue. 3. The blue parts need not touch. Therefore \(\frac{2}{4}=\frac{1}{2}\), and Sam is wrong.

No. The blue fraction is \(\frac{2}{4}=\frac{1}{2}\). The blue parts together cover half the bar even though they do not touch.

- Compare the length of each whole bar. - Are the parts in bar b) as large as those in a)? - How far does the shading reach across each bar?

1. Bar a) has \(1\) of \(3\) equal parts shaded. 2. Bar b) has \(2\) of \(6\) equal parts shaded. 3. Each shaded part in b) is half as large as a shaded part in a), so the total shaded lengths are equal.

No. Each sixth in bar b) is half as large as each third in bar a). So the two shaded sixths cover the same length as one shaded third: \(\frac{1}{3}=\frac{2}{6}\).

- Imagine putting the two little shaded pieces back together. Did the shaded length change? - Think about what a new cut changes and what it does not change. - Check whether Ana is still talking about the same whole strip.

1. Ana now has two small shaded pieces and two larger unshaded pieces. 2. The four pieces are not all the same size, so they are not fourths. 3. The shaded amount is still \(\frac{1}{3}\) of the strip.

No. The four pieces are not equal. The shaded fraction is still \(\frac{1}{3}\).

- Count the unshaded parts, not just the colored ones. - Compare each unshaded length with its whole bar. - Could the different numbers of white pieces still cover the same length?

1. Bar a) has \(2\) of \(4\) equal parts unshaded. 2. Bar b) has \(3\) of \(6\) equal parts unshaded. 3. In each bar, the unshaded part is exactly half of the whole bar, so the fractions are equal.

Yes. \(\frac{2}{4}=\frac{3}{6}=\frac{1}{2}\) of each bar is unshaded.

- Make sure you are comparing the same-sized whole bars. - Where does the shading end in each one? - Would equivalent fractions cover the same length?

1. Bar a) has \(1\) of \(2\) equal parts shaded, or \(\frac{1}{2}\). 2. Bar b) has \(2\) of \(6\) equal parts shaded, or \(\frac{2}{6}\). 3. The shading in b) ends before the halfway point, so the fractions are not equivalent.

No. \(\frac{1}{2}\) is larger than \(\frac{2}{6}\); the shaded lengths differ.

- Count the spaces between 0 and 1, not the tick marks. - How many spaces are there from 0 to each letter? - Can you group the eighths into fourths to compare the fractions?

1. Line a) has 4 equal spaces from 0 to 1. A is 3 spaces from 0, so A shows \(\frac{3}{4}\). 2. Line b) has 8 equal spaces from 0 to 1. B is 6 spaces from 0, so B shows \(\frac{6}{8}\). 3. Each fourth is worth 2 eighths. So 3 fourths equal 6 eighths. 4. The fractions are equal: \(\frac{3}{4}=\frac{6}{8}\).

A: \(\frac{3}{4}\) B: \(\frac{6}{8}\) Yes, the fractions are equal. Each fourth equals 2 eighths, so 3 fourths equal 6 eighths.

- 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.

- Look for the longer and shorter tick marks between \(0\) and \(1\). - Can you count the distance to X using either size of step? - Why does a different number of steps reach the same point?

1. The longer tick marks divide the unit into fourths. X is at the third fourth-size space, or \(\frac{3}{4}\). 2. The smaller tick marks divide the unit into eighths. X is at the sixth eighth-size space, or \(\frac{6}{8}\). 3. Two eighth-size spaces make one fourth-size space, so the two counts reach the same point. 4. Therefore, \(\frac{3}{4}=\frac{6}{8}\).

\(\frac{3}{4}=\frac{6}{8}\). Both counts of equal steps reach point X.

- Can two fractions have even numerators but different shaded amounts? - Look at the whole bar and its six equal parts. - Can you join neighboring small pieces without changing the green length?

1. Having even numerators does not prove that two fractions are equal. The sizes and numbers of equal parts also matter. 2. The bar has \(4\) of \(6\) equal parts shaded, or \(\frac{4}{6}\). 3. Group its \(6\) parts into \(3\) equal pairs. Exactly \(2\) pairs are fully shaded. 4. The shaded fraction is therefore \(\frac{2}{3}\), showing \(\frac{4}{6}=\frac{2}{3}\) for the right reason.

No. Even numerators alone do not show equality. The six small parts can be grouped into three equal pairs, with two shaded pairs, 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 \(\frac{2}{8}=\frac{1}{4}\).

Harper 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.