Free math worksheets

- Read each measured length exactly as written. - Match each measurement to the same value on the line-plot scale. - Use one X for each board length.

1. Each X must be placed above its exact measurement. 2. Place one X above \(3\frac{1}{4}\). 3. Place the other X above \(3\frac{3}{4}\).

Place one X at \(3\frac{1}{4}\) feet and one X at \(3\frac{3}{4}\) feet.

- Each X represents one measurement. - Compare the numbers of Xs, not the distance between the labels. - Subtract the shorter stack from the taller stack.

1. Compare the two stack heights: \(5\) Xs and \(2\) Xs. 2. Subtract: \(5 - 2 = 3\).

There are \(3\) more measurements at \(2\frac{1}{4}\) inches.

- Compare the two stack heights. - The taller stack must be reduced to match the shorter stack.

1. Find the difference in the stack heights: \(5 - 3 = 2\). 2. Removing \(2\) Xs from the taller stack leaves \(3\) Xs in each stack.

Remove \(2\) Xs from the stack at \(1\frac{1}{2}\) inches.

- Each measured leaf contributes one X. - Match each leaf length to the same value on the scale. - Count how many leaves have each length before placing the Xs.

1. Each X represents one measured leaf. 2. Place \(2\) Xs above \(1\frac{1}{4}\) inches. 3. Place \(2\) Xs above \(1\frac{1}{2}\) inches. 4. Place \(2\) Xs above \(1\frac{3}{4}\) inches.

Place \(2\) Xs at each of the three lengths: \(1\frac{1}{4}\), \(1\frac{1}{2}\), and \(1\frac{3}{4}\) inches.

- One added pencil measurement adds one X to the plot. - Check whether the X creates a new scale position or joins an existing stack.

1. Adding one X changes the total number of measurements from \(8\) to \(9\). 2. The new X joins an existing stack, so the number of different measured values stays \(3\).

The line plot now shows \(9\) measurements at \(3\) different values.

- List the possible quarter-inch marks from the shortest length to the longest length. - Tally each measurement before deciding how many Xs belong above a mark.

1. Count how often each length appears. 2. \(3\) appears once; \(3\frac{1}{4}\) appears three times; \(3\frac{1}{2}\) appears twice; \(3\frac{3}{4}\) appears once; \(4\) appears once.

\(3\): \(1\) X \(3\frac{1}{4}\): \(3\) Xs \(3\frac{1}{2}\): \(2\) Xs \(3\frac{3}{4}\): \(1\) X \(4\): \(1\) X

- Compare the stack heights above the three measured lengths. - A taller stack represents more shells of that length. - Add the Xs in all the stacks to find the total number of shells.

1. The tallest stacks each have \(4\) Xs. 2. The stacks at \(2\frac{1}{4}\) inches and \(2\frac{3}{4}\) inches each have \(4\) Xs. 3. Count all the shells: \(4 + 2 + 4 = 10\).

The lengths \(2\frac{1}{4}\) inches and \(2\frac{3}{4}\) inches have the most shells. The line plot shows \(10\) shells altogether.

- Look at the smallest fractional step used by the measurements. - List the quarter-inch values in order from \(4\) to \(5\).

1. The data use quarter-inch measurements between \(4\) and \(5\). 2. The missing quarter-inch labels are \(4\frac{1}{4}\) and \(4\frac{3}{4}\).

Add \(4\frac{1}{4}\) and \(4\frac{3}{4}\).

- Look for the smallest fractional step that appears in the data. - Make sure equal spaces on the scale represent equal numerical changes.

1. The measurements change by halves of an inch. 2. A half-inch interval shows every value exactly. 3. The labels are \(6\), \(6\frac{1}{2}\), \(7\), and \(7\frac{1}{2}\).

Use \(\frac{1}{2}\)-inch intervals. Label the scale \(6\), \(6\frac{1}{2}\), \(7\), and \(7\frac{1}{2}\).

- Each X stands for one measurement equal to the value below it. - Keep repeated measurements in the list rather than writing each value only once.

1. Repeat each measurement once for each X above it. 2. The ordered data are \(2\frac{1}{2}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), \(3\), \(3\), and \(3\) inches.

\(2\frac{1}{2}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), \(3\), \(3\), \(3\) inches

- Move along the scale one quarter inch at a time. - Include both endpoints when listing values, but count the spaces between them. - Do not count the labeled values themselves as intervals.

1. List the scale values from shortest to longest: \(2\frac{1}{4}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), and \(3\). 2. Count the spaces between consecutive values: there are \(3\) quarter-inch intervals.

There are \(3\) quarter-inch intervals.

- A duplicate is removed rather than moved to another value. - Recount the total and compare the two corrected stack heights.

1. Remove the duplicate X from \(1\frac{1}{4}\): \(3 - 1 = 2\) Xs. 2. The \(1\frac{1}{2}\)-inch stack remains at \(2\) Xs. 3. The corrected plot has \(2 + 2 = 4\) measurements, and the stack heights are equal.

Each stack has \(2\) Xs, \(4\) measurements remain, and the stacks are tied.

- Count forward by equal fourths. - “Strictly between” means neither endpoint is included.

1. Count by quarters after \(1\frac{1}{2}\): \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), then \(2\frac{1}{2}\). 2. Excluding both endpoints leaves \(3\) values.

There are \(3\) values: \(1\frac{3}{4}\), \(2\), and \(2\frac{1}{4}\) inches.

- Separate the scale’s structure from the locations that happen to have measurements. - A labeled value does not need to have an X above it. - Consider what removing one value would do to equal spacing.

1. A line-plot scale must keep equal quarter-inch intervals across the entire range. 2. A scale value can have no Xs above it and still remain labeled. 3. Removing the label and closing the gap would make the spacing or sequence misleading.

No. The \(1\frac{1}{2}\)-inch label should remain so the scale keeps equal quarter-inch intervals.

- List every quarter-inch value from the first endpoint to the last. - Include both endpoints and the value between them. - Add the Xs at all included values.

1. Include the stacks at \(1\frac{1}{4}\), \(1\frac{1}{2}\), and \(1\frac{3}{4}\) inches. 2. Add the Xs: \(2 + 3 + 1 = 6\).

There are \(6\) measurements in the interval.

- Tally each repeated measurement in the data list. - Compare every tally with the corresponding line-plot stack.

1. The data contain \(3\) measurements of \(2\frac{3}{4}\) inches. 2. The plot shows only \(2\) Xs at that value. 3. Add one X above \(2\frac{3}{4}\) inches.

One X is missing at \(2\frac{3}{4}\) inches. The corrected stack should have \(3\) Xs.

- Read the scale from least to greatest. - Check that every adjacent pair differs by one quarter inch.

1. Equal spaces must increase by \(\frac{1}{4}\) inch each time. 2. The labels \(3\frac{1}{2}\) and \(3\frac{3}{4}\) are reversed. 3. The correct order is \(3\), \(3\frac{1}{4}\), \(3\frac{1}{2}\), \(3\frac{3}{4}\), \(4\).

The middle two labels are out of order. The correct scale is \(3\), \(3\frac{1}{4}\), \(3\frac{1}{2}\), \(3\frac{3}{4}\), \(4\).

- Compare the numerical step from one label to the next. - A graph scale must represent equal numerical changes with equal physical spacing.

1. Each numerical change is \(\frac{1}{4}\), so each physical interval must have the same length. 2. Unequal spacing would misrepresent the measurement distances. 3. The labels should be placed at equally spaced marks in the same order.

No. The scale is not valid until the quarter-inch labels are placed at equal intervals.

- Intervals are the spaces between neighboring labels. - Including both endpoints makes the number of labels one more than the number of intervals.

1. The intervals are \(3\) to \(3\frac{1}{4}\), to \(3\frac{1}{2}\), to \(3\frac{3}{4}\), to \(4\). 2. There are \(4\) intervals and \(5\) labels.

There are \(4\) equal intervals and \(5\) scale labels.

- Decide whether the new measurement uses a previously empty scale value. - Compare the new stack height with the two existing stack heights. - Use Grade 3 language: count the Xs in the tallest stack.

1. The new X creates a third measured length, so \(3\) different lengths are shown. 2. The original stacks still each have \(3\) Xs, while the new stack has \(1\) X. 3. The greatest number of Xs in any stack remains \(3\).

The plot shows \(3\) different lengths, and the tallest stacks have \(3\) Xs.

- Each X represents one measured button. - Add the Xs in all four stacks for the total. - Find the measured width with the tallest stack.

1. Count all the Xs: \(1 + 2 + 3 + 4 = 10\). 2. The tallest stack has \(4\) Xs at \(2\) inches.

\(10\) buttons were measured, and \(2\) inches is the width that appears most often.

- Remove one X from the old measurement value. - Add one X to the new measurement value. - Compare the three updated stack heights.

1. The first stack decreases from \(3\) to \(2\) Xs. 2. The second stack increases from \(2\) to \(3\) Xs. 3. The third stack stays at \(1\) X. 4. The length \(1\frac{1}{2}\) inches now appears most often.

The new stack heights are \(2\), \(3\), and \(1\) Xs. The length \(1\frac{1}{2}\) inches appears most often.

- Update the stack that receives the new X. - Count the Xs above each ribbon length. - Find the length with the smallest number of Xs.

1. The number of Xs at \(3\frac{1}{4}\) inches becomes \(1 + 1 = 2\). 2. The other numbers of Xs are \(3\) and \(1\). 3. The length \(3\frac{3}{4}\) inches appears only once, so it appears the fewest times.

\(3\frac{3}{4}\) inches appears the fewest times.

- Compare the greatest data value with the current greatest scale label. - Keep the same quarter-inch interval when extending the scale.

1. The current scale ends before the value \(3\frac{1}{4}\). 2. Extend the scale one quarter-inch interval and add the label \(3\frac{1}{4}\).

Extend the scale to \(3\frac{1}{4}\) inches and add that label at the next equally spaced mark.

- Start at \(1\frac{1}{2}\) inches on the scale. - Include the stack at that value and every stack to its right. - Add the Xs in the included stacks.

1. “At least \(1\frac{1}{2}\) inches” includes \(1\frac{1}{2}\), \(1\frac{3}{4}\), and \(2\) inches. 2. Add the Xs at those lengths: \(5 + 3 + 1 = 9\).

There are \(9\) craft sticks that are at least \(1\frac{1}{2}\) inches long.

- Read the quarter-inch labels one mark at a time. - Start at \(2\frac{1}{2}\) inches and count two marks to the right. - Reverse that movement to correct the X.

1. One mark to the right of \(2\frac{1}{2}\) inches is \(2\frac{3}{4}\) inches. 2. Two marks to the right is \(3\) inches. 3. The X must move two marks to the left to return to \(2\frac{1}{2}\) inches.

The X was placed at \(3\) inches. It must move \(2\) marks to the left.

- Each X represents one measured ribbon. - Add every stack to find the total number measured. - For part b), count only stacks at lengths greater than \(1\frac{1}{2}\) inches.

1. For part a), add all the Xs: \(2 + 1 + 3 + 2 + 1 = 9\). 2. For part b), use the stacks to the right of \(1\frac{1}{2}\) inches: \(3 + 2 + 1 = 6\).

a) \(9\) ribbons b) \(6\) ribbons

- Each X represents one measured crayon. - Add the stack heights to find the total. - The length with the tallest stack appears most often.

1. For part a), add all the Xs: \(1 + 2 + 4 + 3 = 10\). 2. For part b), the tallest stack has \(4\) Xs at \(1\frac{1}{2}\) inches.

a) \(10\) crayons b) \(1\frac{1}{2}\) inches

- Identify which labels are whole numbers. - Count the Xs at the whole-number label. - Add the Xs at all remaining labels.

1. Only \(1\) inch is a whole-number length, and it has \(2\) Xs. 2. The other lengths are not whole numbers and have \(1 + 3 + 4 = 8\) Xs altogether.

There are \(2\) measurements at whole-number lengths and \(8\) at lengths that are not whole numbers.

- Do not include either endpoint named in the question. - Identify the labels strictly between \(1\frac{1}{4}\) and \(2\) inches. - Add the Xs at those labels.

1. The included lengths are \(1\frac{1}{2}\) inches and \(1\frac{3}{4}\) inches. 2. Add the Xs at those lengths: \(3 + 4 = 7\).

There are \(7\) leaves in the stated length range.

- Do not count the middle stack at \(2\) inches. - Use the stacks at \(1\frac{1}{2}\) and \(2\frac{1}{2}\) inches. - Add the Xs in those two stacks.

1. Strings that are not exactly \(2\) inches long are in the two outside stacks. 2. Add those Xs: \(4 + 4 = 8\).

There are \(8\) strings that are not exactly \(2\) inches long.

- Include both endpoint lengths named in the question. - Identify every quarter-inch label between the endpoints. - Add the Xs at all included labels.

1. Include the stacks at \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), and \(2\frac{1}{4}\) inches. 2. Add the Xs: \(2 + 3 + 2 + 1 = 8\).

There are \(8\) nails in the stated length interval.

- Compare each ribbon length with \(1\frac{1}{2}\) inches. - Keep lengths equal to the reference value in their own group. - Count the values in each of the three groups.

1. The values below \(1\frac{1}{2}\) are \(1\), \(1\frac{1}{4}\), and \(1\frac{1}{4}\), giving \(3\) measurements. 2. There is \(1\) measurement exactly at \(1\frac{1}{2}\). 3. The values above \(1\frac{1}{2}\) are \(1\frac{3}{4}\), \(2\), and \(2\), giving \(3\) measurements.

There are \(3\) ribbon lengths below, \(1\) exactly equal to, and \(3\) above \(1\frac{1}{2}\) inches.

- Combine the Xs at matching measurement values from the two plots. - Keep the two length values separate while adding their stack heights. - Compare the new stack heights to see which length appears more often.

1. At \(2\frac{1}{4}\) inches, combine \(2 + 1 = 3\) Xs. 2. At \(2\frac{1}{2}\) inches, combine \(1 + 3 = 4\) Xs. 3. Since \(4 > 3\), \(2\frac{1}{2}\) inches appears more often.

The combined plot has \(3\) Xs at \(2\frac{1}{4}\) inches and \(4\) Xs at \(2\frac{1}{2}\) inches. The length \(2\frac{1}{2}\) inches appears more often.

- Count all the Xs before removing any strips. - Remove the entire stack at \(2\frac{1}{4}\) inches. - Then count both the remaining Xs and the lengths that still have at least one X.

1. The original plot has \(2 + 3 + 1 + 2 = 8\) Xs. 2. Removing the \(3\) Xs at \(2\frac{1}{4}\) inches leaves \(8 - 3 = 5\) Xs. 3. Xs remain at \(2\), \(2\frac{1}{2}\), and \(2\frac{3}{4}\) inches, so \(3\) different lengths remain.

There are \(5\) strips left at \(3\) different lengths.

- Compare the number of points with the number of spaces between adjacent points. - Work backward from the endpoint using the correct interval count.

1. Nine marks have only \(8\) spaces between them, so there are \(8\) intervals. 2. Eight quarter-inch intervals total \(8 \times \frac{1}{4} = 2\) inches. 3. The first label is \(3 - 2 = 1\) inch.

Theo counted marks instead of intervals. The first label is \(1\) inch.

- Form the “at least” group using the two greatest labels. - Form the “at most” group using the two least labels. - Compare the two group totals by subtraction.

1. At least \(1\frac{1}{2}\) inches includes \(8 + 3 = 11\) pieces. 2. At most \(1\frac{1}{4}\) inches includes \(3 + 4 = 7\) pieces. 3. Find the difference: \(11 - 7 = 4\).

There are \(4\) more pieces in the group that is at least \(1\frac{1}{2}\) inches long.

- Skipping one quarter-inch mark means moving two quarter-inch intervals each time. - Continue the pattern from the first endpoint to the last endpoint. - Count the listed labels, including both endpoints.

1. Every other quarter-inch mark is separated by \(\frac{1}{2}\) inch. 2. Starting at \(2\), the labels are \(2\), \(2\frac{1}{2}\), \(3\), \(3\frac{1}{2}\), and \(4\). 3. There are \(5\) such labels.

The labels are \(2\), \(2\frac{1}{2}\), \(3\), \(3\frac{1}{2}\), and \(4\) inches. There are \(5\) labels.

- List every quarter-inch mark between the endpoints, including both endpoints. - Remove only the two labels stated to have no Xs. - Each remaining X represents one measured stem.

1. The quarter-inch labels are \(1\), \(1\frac{1}{4}\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), \(2\frac{3}{4}\), and \(3\). 2. There are \(9\) scale marks, and \(2\) have no X, so the plot has \(9 - 2 = 7\) Xs. 3. The represented lengths are \(1\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), and \(3\) inches.

The plot shows \(7\) stems. Their lengths are \(1\), \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(2\), \(2\frac{1}{4}\), \(2\frac{1}{2}\), and \(3\) inches.

- Add all the X counts to find the total. - The length with the tallest stack appears most often. - “At least \(1\frac{1}{2}\) yards” includes that length and all greater lengths.

1. For part a), add the X counts: \(1 + 2 + 2 + 4 + 1 + 2 = 12\). 2. For part b), the tallest stack has \(4\) Xs at \(1\frac{1}{2}\) yards. 3. For part c), add the counts at \(1\frac{1}{2}\), \(1\frac{3}{4}\), and \(2\) yards: \(4 + 1 + 2 = 7\).

a) \(12\) ribbons b) \(1\frac{1}{2}\) yards c) \(7\) ribbons