Free math worksheets

- Count the equal spaces between \(0\) and \(1\) on the displayed line. - Count how many of those spaces lie between \(0\) and \(M\). - Connect the total-space count and the spaces-to-\(M\) count to the two parts of a fraction.

1. The interval from \(0\) to \(1\) is divided into \(4\) equal tick-spaces, so each space is one fourth. 2. Point \(M\) is \(3\) tick-spaces from \(0\), so the numerator is \(3\). 3. Therefore, point \(M\) is at \(\frac{3}{4}\).

Point \(M\) is at \(\frac{3}{4}\). There are \(4\) equal spaces in the whole interval and \(M\) is \(3\) of those spaces from \(0\).

- Count the equal spaces from \(0\) to \(1\). - Count the spaces from \(0\) to point \(A\). - Use the total number of spaces as the denominator.

1. The unit interval has \(6\) equal spaces, so the denominator is \(6\). 2. Point \(A\) is \(5\) spaces from \(0\), so it is at \(\frac{5}{6}\).

Point \(A\) is at \(\frac{5}{6}\).

- Count the equal intervals from \(0\) to \(1\) to determine the fraction size of one step. - Count equal eighth-steps from \(0\) to each labeled point. - Keep each answer in eighths so it matches the partition shown on the number line.

1. The interval from \(0\) to \(1\) is divided into \(8\) equal parts, so each step is \(\frac{1}{8}\). 2. Point \(A\) is \(3\) eighth-steps from \(0\), so \(A=\frac{3}{8}\). 3. Point \(B\) is \(6\) eighth-steps from \(0\), so \(B=\frac{6}{8}\). 4. Point \(C\) is \(9\) eighth-steps from \(0\), so \(C=\frac{9}{8}\). 5. Point \(D\) is \(14\) eighth-steps from \(0\), so \(D=\frac{14}{8}\).

A) \(\frac{3}{8}\) B) \(\frac{6}{8}\) C) \(\frac{9}{8}\) D) \(\frac{14}{8}\)

- Count the equal intervals from \(0\) to \(1\). - Use that interval count to name the size of one step. - Count eighth-steps from \(0\) to each labeled point and keep the answers in eighths.

1. The interval from \(0\) to \(1\) is divided into \(8\) equal parts, so one step represents \(\frac{1}{8}\). 2. Point \(P\) is \(2\) eighth-steps from \(0\), so \(P=\frac{2}{8}\). 3. Point \(Q\) is \(7\) eighth-steps from \(0\), so \(Q=\frac{7}{8}\). 4. Point \(R\) is \(11\) eighth-steps from \(0\), so \(R=\frac{11}{8}\).

a) \(8\) equal parts; one step is \(\frac{1}{8}\) b) \(P=\frac{2}{8}\), \(Q=\frac{7}{8}\), \(R=\frac{11}{8}\)

- Count the equal intervals from \(0\) to \(\frac{2}{3}\). - Rewrite \(\frac{2}{3}\) as an equivalent fraction whose numerator matches that interval count. - Count equal fraction steps from \(0\) to each marker and simplify.

1. There are \(4\) equal intervals from \(0\) to \(\frac{2}{3}\). 2. Rewrite \(\frac{2}{3}\) as \(\frac{4}{6}\). Therefore, each interval represents \(\frac{1}{6}\). 3. Thus, \(U=\frac{1}{6}\), \(V=\frac{2}{6}=\frac{1}{3}\), and \(W=\frac{3}{6}=\frac{1}{2}\).

\(U=\frac{1}{6}\), \(V=\frac{1}{3}\), \(W=\frac{1}{2}\)

- Count the equal tick-spaces across the whole interval from \(0\) to \(1\). - Count how many of those spaces the displayed segment crosses. - After locating the landing point, count the remaining equal spaces to \(1\).

1. The interval from \(0\) to \(1\) has \(6\) equal tick-spaces, so each jump is one sixth. 2. The displayed segment crosses \(4\) tick-spaces, so the frog has made \(4\) jumps and lands at \(\frac{4}{6}\). 3. There are \(2\) tick-spaces from that landing point to \(1\), so \(2\) more same-size jumps are needed.

a) \(4\) jumps b) \(\frac{4}{6}\) c) \(2\) more jumps

- Count all equal spaces between \(0\) and \(1\) to determine the unit fraction. - Count only the spaces covered by the segment. - Use the covered-space count over the total equal-space count.

1. The interval from \(0\) to \(1\) has \(8\) equal tick-spaces, so each space represents \(\frac{1}{8}\). 2. The displayed segment spans \(5\) of those equal spaces. 3. Therefore, the segment is \(5\) tick-spaces long and its length is \(\frac{5}{8}\) of the whole interval.

The segment is \(5\) equal tick-spaces long, so its length is \(\frac{5}{8}\).

- The denominator tells how many equal spaces make the interval from \(0\) to \(1\). - Compare the number of spaces shown on the two lines. - Then count to the third tick on the line whose partition matches the denominator.

1. The denominator \(4\) means the interval from \(0\) to \(1\) must be divided into \(4\) equal spaces. 2. Line a) has \(4\) equal spaces, so its third tick after \(0\) represents \(\frac{3}{4}\). 3. Line b) has \(3\) equal spaces, so its third tick after \(0\) represents \(\frac{3}{3}=1\).

Line a) can correctly show \(\frac{3}{4}\).

- Eight equal spaces need boundaries between neighboring spaces. - Each space is one eighth of the unit interval. - Count \(5\) spaces from \(0\) to name the requested tick.

1. Eight equal spaces require \(7\) interior tick marks between the endpoints. 2. Each space has length \(\frac{1}{8}\). 3. The fifth tick after \(0\) is the endpoint of \(5\) eighth-size spaces. 4. That tick represents \(\frac{5}{8}\).

\(7\) interior tick marks are needed. The fifth tick after \(0\) is \(\frac{5}{8}\).

- 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 \(\frac{5}{3}\).

- Count the equal intervals from \(0\) to \(\frac{3}{4}\). - Rewrite \(\frac{3}{4}\) as an equivalent fraction whose numerator matches that interval count. - Count equal fraction steps from \(0\) to each marker.

1. There are \(6\) equal intervals from \(0\) to \(\frac{3}{4}\). 2. Rewrite \(\frac{3}{4}\) as \(\frac{6}{8}\). Therefore, each of those six equal intervals represents \(\frac{1}{8}\). 3. Point \(A\) is \(3\) intervals from \(0\), so \(A=\frac{3}{8}\). 4. Point \(B\) is \(10\) intervals from \(0\), so \(B=\frac{10}{8}=\frac{5}{4}=1\frac{1}{4}\). 5. Point \(C\) is \(14\) intervals from \(0\), so \(C=\frac{14}{8}=\frac{7}{4}=1\frac{3}{4}\).

\(A=\frac{3}{8}\), \(B=\frac{5}{4}=1\frac{1}{4}\), \(C=\frac{7}{4}=1\frac{3}{4}\)

- Count the equal intervals from \(0\) to \(1\) on each number line. - Use the interval count to determine the fraction size of one step on each line. - Count steps from \(0\) to each letter and keep the denominator that matches its line.

1. In a), the interval from \(0\) to \(1\) is divided into eighths, so \(A=\frac{2}{8}\), \(B=\frac{5}{8}\), \(C=\frac{7}{8}\), and \(D=\frac{9}{8}\). 2. In b), the interval from \(0\) to \(1\) is divided into sixths, so \(E=\frac{2}{6}\), \(F=\frac{3}{6}\), \(G=\frac{5}{6}\), and \(H=\frac{7}{6}\).

a) \(A=\frac{2}{8}\), \(B=\frac{5}{8}\), \(C=\frac{7}{8}\), \(D=\frac{9}{8}\) b) \(E=\frac{2}{6}\), \(F=\frac{3}{6}\), \(G=\frac{5}{6}\), \(H=\frac{7}{6}\)

- Count the equal intervals between consecutive whole numbers on each line. - That count gives the denominator for one step. - Count steps to each marker and combine them with the whole number when needed.

1. In a), each step is \(\frac{1}{4}\), so \(A=1\frac{1}{4}\). 2. In b), each step is \(\frac{1}{3}\), so \(B=3\frac{2}{3}\) and \(C=4\frac{1}{3}\). 3. In c), each step is \(\frac{1}{6}\), so \(D=10\frac{5}{6}\). 4. In d), each step is \(\frac{1}{8}\), so \(E=\frac{5}{8}\).

a) \(A=1\frac{1}{4}\) b) \(B=3\frac{2}{3}\), \(C=4\frac{1}{3}\) c) \(D=10\frac{5}{6}\) d) \(E=\frac{5}{8}\)

- First count the equal spaces from \(0\) to \(1\) on the displayed line. - Use that partition to name the first interior tick. - Compare the size of \(\frac{1}{6}\) with one of those equal tick-spaces.

1. The displayed interval from \(0\) to \(1\) has \(3\) equal spaces, so the first interior tick is at \(\frac{1}{3}\). 2. One sixth is half of one third, so \(\frac{1}{6}\) lies halfway from \(0\) to \(\frac{1}{3}\). 3. Therefore, \(\frac{1}{6}\) is not on a shown tick. It lies halfway between \(0\) and the first interior tick, \(\frac{1}{3}\).

The point would not land on a shown tick. It would be halfway between \(0\) and the first interior tick, \(\frac{1}{3}\).

- First determine the value of one tick-step on the number line. - Read the two endpoints of the gray segment from the figure. - Shift both endpoints by the same one-tick distance.

1. The interval from \(0\) to \(1\) is divided into \(3\) equal spaces, so one tick-step is \(\frac{1}{3}\). 2. The gray segment begins at \(0\) and ends at \(\frac{2}{3}\). 3. Shifting both endpoints one tick right moves the start to \(\frac{1}{3}\) and the end to \(\frac{3}{3}=1\).

The new starting point is \(\frac{1}{3}\), and the new ending point is \(1\).

- Recall what the denominator requires of the unit interval. - Compare the physical lengths of the spaces described. - Equal fraction steps must represent equal distances.

1. A sixth on a number line must be one of \(6\) equal lengths in the unit interval. 2. The given spaces have different lengths. 3. Equal increases of \(\frac{1}{6}\) cannot be shown by unequal spaces. 4. Therefore, the tick labels are not valid sixths locations.

The labels are invalid because the interval from \(0\) to \(1\) was not partitioned into \(6\) equal spaces.

- 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, \(\frac{3}{4}=\frac{6}{8}\).

- Read the location of \(K\) from the supplied number line. - Notice which marked point lies farther to the right. - Count the equal eighth-size spaces between the two marked locations.

1. The number line is divided into eighths, and \(K\) is at the sixth eighth-step from \(0\), or \(\frac{6}{8}\). 2. The marked point \(\frac{1}{2}\) is to the left of \(K\). 3. There are \(2\) eighth-size spaces from \(\frac{1}{2}\) to \(K\).

Point \(K\) is to the right of \(\frac{1}{2}\) by \(2\) eighth-size spaces.

- Count the equal spaces from \(0\) to \(1\) on the figure. - Use that count to name the size of one space. - Count equal spaces from \(0\) to point \(P\).

1. From \(0\) to \(1\), the number line has \(4\) equal spaces, so each space is \(\frac{1}{4}\) unit. 2. Point \(P\) is at the fifth fourth-size step from \(0\). 3. Therefore, \(P=\frac{5}{4}\).

Each space is \(\frac{1}{4}\) unit long. Point \(P\) is at \(\frac{5}{4}\).

- Read the two labeled fractions from the number line. - Count the equal fourth-size spaces between those locations. - Check whether the tick for \(1\) lies between the starting and ending points.

1. The labeled endpoints are \(\frac{9}{4}\) and \(\frac{3}{4}\). 2. Counting fourth-size spaces from \(\frac{9}{4}\) left to \(\frac{3}{4}\) gives \(6\) spaces. 3. One whole is \(\frac{4}{4}\), which lies between the two endpoints. 4. Therefore, the path crosses \(1\).

The point moves \(6\) fourth-size spaces left, and the path crosses \(1\).

- Read the endpoints of each labeled gap from the number line. - Count how many equal tick-spaces make each gap. - Compare the sizes of the unit-fraction steps represented by those spaces.

1. Gap A runs from \(\frac{1}{4}\) to \(\frac{1}{2}\), so its length is one fourth-size step, or \(\frac{1}{4}\). 2. Gap B runs from \(\frac{1}{8}\) to \(\frac{1}{4}\), so its length is one eighth-size step, or \(\frac{1}{8}\). 3. A fourth-size step is larger than an eighth-size step. 4. Therefore, Gap A is larger.

Gap A is larger. It has size \(\frac{1}{4}\), while Gap B has size \(\frac{1}{8}\).

- Read the starting marker from the number line and notice the size of each equal step. - Move exactly \(3\) equal spaces to the right. - After locating the endpoint, count the remaining equal space or spaces to \(1\).

1. The number line is divided into eighths, and marker \(S\) is at \(\frac{4}{8}\). 2. Moving \(3\) eighth-size spaces right reaches \(\frac{7}{8}\). 3. From \(\frac{7}{8}\) to \(1=\frac{8}{8}\) is one eighth-size space, so the distance is \(\frac{1}{8}\).

The final location is \(\frac{7}{8}\), which is \(\frac{1}{8}\) from \(1\).

- Read the two marked endpoints and count the equal spaces between them. - Share that total number of spaces equally among the \(3\) shorter segments. - Step by the resulting equal length from the left endpoint to find the two interior division points.

1. The marked endpoints are \(\frac{1}{8}\) and \(\frac{7}{8}\), which are \(6\) eighth-size spaces apart. 2. Dividing \(6\) spaces into \(3\) equal segments gives \(2\) eighth-size spaces per segment. 3. Two spaces right of \(\frac{1}{8}\) is \(\frac{3}{8}\). 4. Two more spaces right is \(\frac{5}{8}\).

The new division points are \(\frac{3}{8}\) and \(\frac{5}{8}\).

- 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.

- Use the sixth-size spacing to read point \(L\)'s location with denominator \(6\). - Reverse the motion by moving the stated number of equal spaces to the left. - Compare the starting point's position with the tick for \(1\).

1. Reading the sixth-size spacing on the number line, point \(L\) is at \(\frac{8}{6}\). 2. Work backward by moving \(3\) sixth-size spaces left. 3. The starting point is \(\frac{5}{6}\). 4. Since \(\frac{5}{6}\) is one sixth-size space left of \(1\), \(\frac{5}{6}<1\).

The point started at \(\frac{5}{6}\), and \(\frac{5}{6}<1\).

- Read the locations of \(A\) and \(B\) from the number line. - Move one equal fourth-size space left from \(B\) to locate \(C\). - Count the equal spaces between \(A\) and \(C\).

1. From the number line, \(A=\frac{5}{4}\) and \(B=\frac{9}{4}\). 2. One fourth-size space left of \(\frac{9}{4}\) is \(\frac{8}{4}\), so \(C=\frac{8}{4}=2\). 3. From \(\frac{5}{4}\) to \(\frac{8}{4}\) there are \(3\) fourth-size spaces.

\(C=\frac{8}{4}=2\), and \(A\) and \(C\) are \(3\) fourth-size spaces apart.

- Read marker \(S\)'s starting location from the number line and follow the right move first. - To make the left move on this line, determine how many sixth-size spaces have the same length as \(\frac{1}{3}\) unit. - Check the final position against the labeled whole-number tick.

1. From the number line, \(S\) starts at \(\frac{5}{6}\). 2. Moving \(3\) sixth-size spaces right reaches \(\frac{8}{6}\). 3. One third of a unit has the same length as \(2\) sixth-size spaces. 4. Moving \(2\) sixth-size spaces left from \(\frac{8}{6}\) reaches \(\frac{6}{6}=1\).

Marker \(S\) finishes at \(1\).

- Think about what the denominator says about the whole interval. - After deciding the size of one step, use the numerator as a count of those steps. - Your plan should specify both the partition and the point's position.

1. The denominator \(6\) means the whole interval from \(0\) to \(1\) must be divided into \(6\) equal spaces. 2. Each space then represents \(\frac{1}{6}\). 3. The numerator \(5\) means count \(5\) of those spaces from \(0\). 4. Place the point at the fifth division point from \(0\).

Divide the interval into \(6\) equal spaces and place the point after \(5\) spaces from \(0\).

- Read the starting point and the equal tick size from the number line. - Count how many equal tick-spaces lie between \(S\) and \(1\). - Share that distance equally among the \(3\) jumps, then follow the landings in order.

1. The interval from \(0\) to \(1\) is divided into fourths, and \(S\) is at \(\frac{1}{4}\). 2. The distance from \(\frac{1}{4}\) to \(1=\frac{4}{4}\) is \(3\) fourth-size spaces. 3. Three equal jumps share those \(3\) spaces, so each jump is \(\frac{1}{4}\) unit. 4. After the first jump the marker is at \(\frac{2}{4}=\frac{1}{2}\), and after the second it is at \(\frac{3}{4}\).

Each jump is \(\frac{1}{4}\) unit. After the first jump the marker is at \(\frac{1}{2}\), and after the second it is at \(\frac{3}{4}\).

- Split the three equal spaces into a two-to-one arrangement. - Decide how many spaces must lie on each side of the point. - Use the spaces from zero to name the location.

1. The unit interval has \(3\) equal third-size spaces. 2. To make the left distance twice the right distance, place \(2\) spaces on the left and \(1\) on the right. 3. Point \(A\) is \(2\) third-size spaces from \(0\), so it is at \(\frac{2}{3}\).

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

- 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 point \(A\) and the step size from the number line. - Think about how many eighth-size spaces make one whole. - Halfway from \(A\) to a point one whole away uses half of that many spaces.

1. The number line is divided into eighths, and point \(A\) is at \(\frac{3}{8}\). 2. One whole is \(\frac{8}{8}\), so \(B=\frac{3}{8}+\frac{8}{8}=\frac{11}{8}\). 3. Points \(A\) and \(B\) are \(8\) eighth-size spaces apart, so halfway is \(4\) spaces from \(A\). 4. Four spaces right of \(\frac{3}{8}\) is \(\frac{7}{8}\), so \(C=\frac{7}{8}\).

\(B=\frac{11}{8}\) and \(C=\frac{7}{8}\).

- Read point \(M\)'s location and the sixth-size spacing from the number line. - Move two equal spaces left to locate \(L\) and two equal spaces right to locate \(R\). - Use the left-to-right positions to write the final order.

1. The line is partitioned in sixth-size spaces, and \(M\) is at \(\frac{9}{6}\). 2. Two sixth-size spaces left of \(M\) is \(\frac{7}{6}\), so \(L=\frac{7}{6}\). 3. Two sixth-size spaces right of \(M\) is \(\frac{11}{6}\), so \(R=\frac{11}{6}\). 4. From left to right, \(\frac{7}{6}<\frac{9}{6}<\frac{11}{6}\), so the order is \(L,M,R\).

\(L=\frac{7}{6}\) and \(R=\frac{11}{6}\). From left to right: \(L,M,R\), or \(\frac{7}{6}<\frac{9}{6}<\frac{11}{6}\).

- Read the locations of \(A\) and \(B\) from the number line. - Count the equal spaces from \(A\) to \(B\). - Because \(B\) is halfway, repeat that same distance to the right of \(B\).

1. From the number line, \(A=\frac{1}{4}\) and \(B=\frac{3}{4}\). 2. The distance from \(A\) to \(B\) is \(2\) fourth-size spaces. 3. Because \(B\) is halfway, the distance from \(B\) to \(C\) is also \(2\) fourth-size spaces. 4. Moving \(2\) fourth-size spaces right from \(\frac{3}{4}\) reaches \(\frac{5}{4}\), so \(C=\frac{5}{4}\).

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

- Read the locations of \(A\) and \(B\) from the number line and count the spaces between them. - Split that total distance into two whole-number space counts whose difference is \(2\). - Use the longer of those two distances starting from \(A\).

1. From the number line, \(A=\frac{2}{8}\) and \(B=\frac{6}{8}\), so they are \(4\) eighth-size spaces apart. 2. The two distances total \(4\) spaces and differ by \(2\) spaces. 3. They must be \(3\) spaces from \(A\) to \(C\) and \(1\) space from \(C\) to \(B\). 4. Three eighth-size spaces right of \(\frac{2}{8}\) is \(\frac{5}{8}\), so \(C=\frac{5}{8}\).

\(C=\frac{5}{8}\)