- Count the shaded parts in the complete bar. - Keep counting the same-size parts in the next bar. - How many fourths have been counted?

1. One whole bar has \(4\) fourth-size parts. 2. There is \(1\) more shaded fourth-size part in the next bar. 3. That makes \(5\) fourths, or \(\frac{5}{4}\).

\(5\) fourth-size parts; \(\frac{5}{4}\).

- Use the labeled whole numbers to work out the size of one step. - Keep counting from zero even after you pass \(1\).

1. There are \(2\) equal spaces from \(0\) to \(1\), so each small space is one half. 2. P is \(3\) half-size spaces from \(0\), so it marks \(\frac{3}{2}\).

\(\frac{3}{2}\)

- Use the picture to count the complete wholes and the remaining sixth-size pieces. - Think about how many one-sixth pieces make one whole waffle. - Combine the sixth-size pieces from all of the wholes and the partial waffle before writing the fraction.

1. The picture shows \(2\) whole waffles and \(5\) sixth-size pieces of another waffle. 2. Two whole waffles contain \(2 \times 6=12\) one-sixth pieces. 3. Including the other \(5\) pieces gives \(12+5=17\) one-sixth pieces. 4. Seventeen one-sixth pieces are \(\frac{17}{6}\) of a waffle.

a) \(17\) one-sixth pieces b) \(\frac{17}{6}\)

- Count the equal intervals from \(0\) to \(1\). - Use that count to name the size of one step. - Count steps from \(0\) to each marker.

1. The interval from \(0\) to \(1\) is divided into \(4\) equal parts, so one step is \(\frac{1}{4}\). 2. Point \(A\) is two steps from \(0\), so \(A=\frac{2}{4}\). 3. Point \(B\) is five steps from \(0\), so \(B=\frac{5}{4}\). 4. Point \(C\) is seven steps from \(0\), so \(C=\frac{7}{4}\).

\(A=\frac{2}{4}\), \(B=\frac{5}{4}\), \(C=\frac{7}{4}\)

- Count the completely shaded squares. - Find the fraction shaded in the last square. - A mixed number has a whole-number part and a fractional part.

1. Two squares are completely shaded. 2. The last square is divided into \(4\) equal parts, and \(1\) part is shaded. This is \(\frac{1}{4}\). 3. The mixed number is \(2\frac{1}{4}\).

\(2\frac{1}{4}\)

- Count the complete circles. - Decide whether the last circle is complete. - Compare the number of complete wholes with \(3\).

1. The picture shows \(2\) whole circles and \(\frac{3}{4}\) of another circle, so the value is \(2\frac{3}{4}\). 2. Since it has only \(2\) wholes and part of a third whole, \(2\frac{3}{4}<3\).

The value is less than \(3\) because the picture shows \(2\) whole circles and only \(\frac{3}{4}\) of a third circle.

- Count how many thirds make one complete bar. - Count all of the shaded thirds in the model. - Use the total number of shaded thirds as the numerator.

1. Each complete bar contains \(3\) thirds. 2. Two complete bars contain \(6\) thirds. 3. One additional shaded third gives \(7\) thirds altogether. 4. The total shaded amount is \(\frac{7}{3}\).

\(\frac{7}{3}\)

- How many tiles make one full square card? - How many complete cards can you cover with the tiles? - Count the same pieces as fourths of a card.

1. Four tiles cover one whole card. 2. Eight tiles cover two cards, with two tile-size parts left. 3. That is \(2\) wholes and \(2\) fourths of another whole. 4. Counting all \(10\) tiles in fourths gives \(\frac{10}{4}\) of a card. Both students are correct.

Both students are correct. The amount is \(\frac{10}{4}\) of a card, or \(2\) whole cards and \(2\) fourths of another.

- Count how many complete bars are fully shaded in the model. - Each complete bar is made of \(4\) fourth-size parts. - After counting the fourths in the complete bars, include the shaded part of the last bar.

1. Each complete bar contains \(4\) fourths. 2. Two complete bars contain \(8\) fourths. 3. One more shaded fourth makes \(9\) fourths in all. 4. The total shaded amount is \(\frac{9}{4}\).

\(\frac{9}{4}\)

- One whole contains \(6\) sixths. - Make groups of \(6\) from the \(12\) sixths. - Count the complete groups.

1. The fraction \(\frac{12}{6}\) means \(12\) sixth-size parts. 2. Six sixths make \(1\) whole. 3. Twelve sixths make \(2\) complete groups of \(6\). 4. Therefore, \(\frac{12}{6}=2\).

\(\frac{12}{6}=2\) because \(12\) sixths make \(2\) wholes.

- One whole contains \(4\) fourths. - Make as many complete groups of \(4\) as possible. - Count the fourths left over.

1. Four fourths make \(1\) whole. 2. Eight fourths make \(2\) complete wholes. 3. From \(11\) fourths, \(11-8=3\) fourths remain. 4. Therefore, \(\frac{11}{4}\) is \(2\) wholes and \(3\) fourths.

\(2\) complete wholes with \(3\) fourths remaining.

- Count the third-size sections in all three panels. - Subtract the one section that is not filled. - Make complete groups of \(3\) thirds.

1. Three complete panels contain \(3\times3=9\) third-size sections. 2. One section is empty, so \(9-1=8\) sections are filled. 3. The filled amount is \(\frac{8}{3}\). 4. Six thirds make \(2\) wholes, leaving \(2\) thirds.

\(\frac{8}{3}\), which is \(2\) whole panels and \(2\) thirds of another panel.

- Write one whole as fourths. - Add the \(3\) extra fourths. - Use the total number of fourths as the numerator.

1. One whole is \(\frac{4}{4}\). 2. Add the \(3\) extra fourths: \(4+3=7\) fourths. 3. The amount is \(\frac{7}{4}\).

\(\frac{7}{4}\)

- Write \(1\) as sixths. - Write \(2\) as sixths. - Compare the numerator \(7\) with those two numerators.

1. One whole is \(\frac{6}{6}\). 2. Two wholes are \(\frac{12}{6}\). 3. Since \(6<7<12\), \(\frac{7}{6}\) lies between \(1\) and \(2\).

\(\frac{7}{6}\) lies between \(1\) and \(2\).

- The denominator stays \(4\). - Undo adding \(2\) fourths. - Use the starting number of fourths as the numerator.

1. Work backward by removing \(2\) fourth-size parts from \(7\) fourths. 2. \(7-2=5\). 3. The original fraction was \(\frac{5}{4}\).

\(\frac{5}{4}\)

- Count the complete shaded bars and the shaded part of the last bar. - Each complete bar contains \(3\) thirds. - Compare the total number of shaded thirds with the number of thirds in \(3\) complete wholes.

1. The model shows \(2\) complete wholes, which contain \(6\) thirds. 2. One additional third makes \(7\) thirds, so the amount is \(\frac{7}{3}\). 3. Three wholes contain \(9\) thirds. 4. Since \(9 - 7 = 2\), \(2\) more thirds are needed.

The amount is \(\frac{7}{3}\). It needs \(2\) more thirds to make \(3\) wholes.

- Count the equal spaces between the two labeled whole numbers. - Name the size of one space as a unit fraction. - Start from one whole and count unit-fraction spaces to each interior tick.

1. The number line divides the interval from \(1\) to \(2\) into \(3\) equal spaces, so each space is \(\frac{1}{3}\). 2. One whole is \(\frac{3}{3}\), so the first interior tick is \(\frac{3}{3}+\frac{1}{3}=\frac{4}{3}\). 3. The second interior tick is \(\frac{3}{3}+\frac{2}{3}=\frac{5}{3}\).

The first interior tick is \(\frac{4}{3}\), and the second is \(\frac{5}{3}\).

- Think of a cup as a part of one full pitcher, not as a whole pitcher. - Count the cups in the full pitcher before you count the extra cups.

1. One cup is \(\frac{1}{3}\) of a pitcher. 2. One full pitcher holds \(3\) thirds, and the extra \(2\) cups make \(2\) more thirds. 3. Priya served \(5\) thirds of a pitcher, or \(\frac{5}{3}\) of one pitcher.

\(\frac{5}{3}\) of a pitcher. One pitcher is \(3\) thirds, and the other \(2\) cups add \(2\) more thirds.

- Count how many thirds make one whole bar. - How many thirds make two whole bars? - Compare with the parts already shaded.

1. Two whole bars contain \(3+3=6\) third-size parts. 2. The model already has \(5\) shaded thirds. 3. One more third must be shaded to make both wholes complete.

\(1\) more third-size part

- Imagine filling a measuring cup with these same scoops. - How could you check the distance from the amount in the bowl to each whole-cup mark?

1. The scoops make \(7\) fourths of a cup, or \(\frac{7}{4}\) cup. 2. One cup is \(4\) scoops and two cups are \(8\) scoops. 3. Seven scoops are \(3\) scoops past one cup but only \(1\) scoop short of two cups, so the amount is closer to \(2\) cups.

Closer to \(2\) cups. The \(7\) scoops make \(\frac{7}{4}\) cup, which is \(3\) scoops more than \(1\) cup but only \(1\) scoop less than \(2\) cups.

- Count the equal intervals from \(1\) to \(2\). - Think of \(1\) as a complete set of eighth-size steps from \(0\). - Count additional eighth-size steps from \(1\) to each marked point.

1. The interval from \(1\) to \(2\) is divided into \(8\) equal parts, so each small interval represents \(\frac{1}{8}\). 2. Point \(A\) is one eighth past \(1=\frac{8}{8}\), so \(A=\frac{9}{8}\). 3. Point \(B\) is five eighths past \(1\), so \(B=\frac{13}{8}\). 4. Point \(C\) is seven eighths past \(1\), so \(C=\frac{15}{8}\).

a) \(8\) equal parts; one interval is \(\frac{1}{8}\) b) \(A=\frac{9}{8}\), \(B=\frac{13}{8}\), \(C=\frac{15}{8}\)

- Count the equal intervals from \(4\) to \(5\) to determine the step size. - Count how many eighth-size steps the marker lies to the right of \(4\). - Think of each whole as \(8\) eighth-size parts when naming the point with denominator \(8\).

1. The interval from \(4\) to \(5\) is divided into \(8\) equal parts, so each step is \(\frac{1}{8}\). 2. The marker is \(3\) eighth-size steps to the right of \(4\). 3. Four wholes are \(\frac{32}{8}\), so the marked point is \(\frac{32}{8}+\frac{3}{8}=\frac{35}{8}\).

\(\frac{35}{8}\)

- Count how many thirds make one whole in the model. - Count all of the shaded thirds across the complete and partial wholes. - Compare that total number of thirds with the numerator in \(\frac{8}{3}\).

1. Each whole in the model is divided into \(3\) equal parts. 2. The two complete wholes contain \(6\) thirds, and the last part contains \(2\) more thirds. 3. Altogether the model shows \(8\) thirds, or \(\frac{8}{3}\). 4. Therefore, the two values are equal.

The values are equal because the model shows \(8\) thirds, which is \(\frac{8}{3}\).

- Treat each step as one copy of the same unit fraction. - Group the thirds into complete sets for whole yards. - Compare the leftover distance to the neighboring whole numbers.

1. Eight steps of \(\frac{1}{3}\,\text{yard}\) make \(\frac{8}{3}\,\text{yards}\). 2. Six thirds equal \(2\) wholes, so \(\frac{8}{3}\) is \(2\) wholes and \(2\) thirds. 3. Nine thirds equal \(3\) wholes. 4. The distance is \(1\) third less than \(3\,\text{yards}\), so it is closer to \(3\,\text{yards}\) and is \(\frac{1}{3}\,\text{yard}\) away.

The rover travels \(\frac{8}{3}\,\text{yards}\). It is closer to \(3\,\text{yards}\) and is \(\frac{1}{3}\,\text{yard}\) less than \(3\,\text{yards}\).

- First identify the fraction named by the marked point. - Count fourth-size spaces from the marker to each whole number. - The smaller number of spaces shows which whole number is closer.

1. The marked point is at \(\frac{7}{4}\). 2. Two wholes equal \(\frac{8}{4}\), so reaching \(2\) requires \(8-7=1\) fourth-size space to the right. 3. One whole equals \(\frac{4}{4}\), so reaching \(1\) requires \(7-4=3\) fourth-size spaces to the left. 4. Since \(1<3\), the marked point is closer to \(2\).

Move \(1\) fourth-size space right to reach \(2\), or \(3\) fourth-size spaces left to reach \(1\). The point is closer to \(2\).

- How many eighths make a full lap? - Count the extra distance using the same-sized parts. - Think of two laps in eighths before finding what is missing.

1. One full lap contains \(8\) eighths. 2. With \(5\) more eighths, Mia has walked \(8+5=13\) eighths, or \(\frac{13}{8}\) laps. 3. Two full laps contain \(16\) eighths. 4. She needs \(16-13=3\) more eighths to finish both laps.

\(\frac{13}{8}\) laps altogether; \(3\) more eighths of a lap.

- Think about how much of a box one marker would fill. - After one box is filled, where do the extra markers go? - Compare how many markers separate Paige's amount from one box and from two boxes.

1. One marker is \(\frac{1}{8}\) of a box, so \(11\) markers are \(\frac{11}{8}\) of a box. 2. Eight markers fill one box, leaving \(11-8=3\) markers. 3. Two boxes hold \(16\) markers. Paige has \(3\) more than one box and needs \(5\) more to fill two boxes, so her amount is closer to one box.

a) \(\frac{11}{8}\) of a box b) \(1\) full box, with \(3\) markers left over c) Closer to \(1\) box, because it is \(3\) markers away instead of \(5\).

- Count half-size spaces from \(0\) mentally, using the labeled whole numbers as anchors. - A difference between numerators counts denominator-sized parts. - Decide how many halves make one complete whole before judging the two statements.

1. Point \(A\) is at \(1\frac{1}{2}\), which is \(\frac{3}{2}\). 2. Point \(B\) is at \(2\frac{1}{2}\), which is \(\frac{5}{2}\). 3. The numerator difference is \(5-3=2\), but those \(2\) counted parts are halves, not wholes. 4. Two halves make \(1\) whole, so \(B\) is \(1\) whole greater than \(A\). Malik is correct.

\(A=\frac{3}{2}\) and \(B=\frac{5}{2}\). Malik is correct because the points are \(2\) halves, or \(1\) whole, apart.