Free Math Worksheets

- Use the scale along the horizontal axis. - Compare the lengths of the bars.

1. Use the scale to read the four bars. 2. The longest bar is the chickadee bar, which shows \(8\) birds.

Chickadees were seen most often: \(8\) birds.

- Use the scale on the vertical axis. - Compare the heights of the bars.

1. Use the scale to read the bars. 2. The shortest bar is the Tuesday bar, which shows \(4\) books.

The fewest books were checked out on Tuesday: \(4\) books.

- Break the task into axis labels and bar heights. - Check that each axis label describes the information shown on that axis.

1. The categories are the four classes, so label the horizontal axis “Class.” 2. The values count sheets of paper, so label the vertical axis “Sheets of paper.” 3. In table order, the bar heights are \(45, 35, 50, 40\).

Horizontal axis: Class. Vertical axis: Sheets of paper. Bar heights: \(45, 35, 50, 40\).

- Break the task into the zero value and the positive values. - Check that every bar height matches the value in the table.

1. Match each day to its rainfall value. 2. Tuesday’s bar has height \(0\,\text{mm}\), so no bar rises above the axis, but the Tuesday category label remains. 3. The other bar heights are \(4\,\text{mm}\), \(7\,\text{mm}\), and \(3\,\text{mm}\).

Tuesday has a bar height of \(0\,\text{mm}\), with its category label still shown. The other heights are \(4\,\text{mm}\), \(7\,\text{mm}\), and \(3\,\text{mm}\).

- Check the title and both axis labels. - Match the categories and measured quantity with the correct axes.

1. The categories are the students, so “Students” belongs on the horizontal axis. 2. The measured quantity is reading time, so “Reading time in minutes” belongs on the vertical axis. 3. The title “Reading Time of Four Students” accurately describes the graph. Therefore, choice B) is correct.

Choice B) is correct.

- Compare each title with the information represented by the graph. - Choose the title that is specific rather than vague.

1. A useful title should identify both the data and the event. 2. “Field Day Points” describes exactly what the graph shows.

“Field Day Points” fits best because it names the event and the quantity shown.

- Make each label specific to the data. - Check that a reader can tell both what is counted and where the counts come from.

1. The title should name the animal and the locations being compared. 2. The vertical axis should name the quantity being counted.

For example, title: “Frogs at Four Ponds”; vertical axis: “Number of frogs.”

- Read each bar length using the horizontal scale. - Add the two requested categories for part b). - Add all five categories for the total number of participants.

1. The longest bar is Cat, with \(11\) votes, so Cat is the most popular. 2. Add the Dog and Cat votes: \(8 + 11 = 19\) students. 3. Add all five categories: \(8 + 11 + 4 + 6 + 3 = 32\) students.

a) Cat b) \(19\) students c) \(32\) students

- Read each bar height using the vertical axis. - Identify the month with the tallest bar. - Add all four monthly values to find the total.

1. Read the bar values: March, \(50\,\text{mm}\); April, \(65\,\text{mm}\); May, \(80\,\text{mm}\); June, \(45\,\text{mm}\). 2. The greatest value is \(80\,\text{mm}\), in May. 3. Add the four values: \(50 + 65 + 80 + 45 = 240\,\text{mm}\).

a) May b) \(240\,\text{mm}\)

- Use the horizontal-axis scale to read each bar length. - Find the longest bar for part a). - Find the shortest bar before subtracting. - Subtract the smaller distance from the larger distance.

1. Read the values from the graph: Lena, \(12\,\text{ft}\); Felix, \(28\,\text{ft}\); Jonah, \(32\,\text{ft}\); Maria, \(16\,\text{ft}\); Toby, \(24\,\text{ft}\). 2. Jonah has the farthest throw, \(32\,\text{ft}\). 3. Lena has the shortest throw, \(12\,\text{ft}\). 4. Find the difference: \(32 - 12 = 20\,\text{ft}\).

a) Jonah, \(32\,\text{ft}\) b) \(20\,\text{ft}\)

- Read the count represented by each bar. - Decide whether the category of exactly \(2\) books belongs in “more than \(2\).” - Identify the categories included in “fewer than \(2\).” - Add the counts for the relevant categories.

1. “More than \(2\)” includes the categories \(3\), \(4\), and \(5+\) books. Add their counts: \(10 + 5 + 2 = 17\) students. 2. “Fewer than \(2\)” includes the categories \(0\) and \(1\) book. Add their counts: \(3 + 8 = 11\) students.

a) \(17\) students b) \(11\) students

- Use the horizontal scale to read each bar endpoint. - The second-fastest animal has the second-longest bar. - Identify the greatest and least values before subtracting. - Subtract the smaller speed from the larger speed.

1. Read the values: Hare, \(35\,\text{mph}\); Greyhound, \(45\,\text{mph}\); Ostrich, \(40\,\text{mph}\); Lion, \(50\,\text{mph}\); Pronghorn, \(55\,\text{mph}\); Cheetah, \(70\,\text{mph}\). 2. The Cheetah is fastest at \(70\,\text{mph}\), and the Pronghorn is second fastest at \(55\,\text{mph}\). 3. The Hare is slowest at \(35\,\text{mph}\). The difference is \(70 - 35 = 35\,\text{mph}\).

a) Hare: \(35\,\text{mph}\); Greyhound: \(45\,\text{mph}\); Ostrich: \(40\,\text{mph}\); Lion: \(50\,\text{mph}\); Pronghorn: \(55\,\text{mph}\); Cheetah: \(70\,\text{mph}\) b) Pronghorn, \(55\,\text{mph}\) c) \(35\,\text{mph}\)

- Compare every bar with the vertical scale. - Find the tallest bar for part a). - Add only Monday through Friday for part b). - For part c), subtract the least daily value from the greatest daily value.

1. Read the daily values: Monday, \(4\); Tuesday, \(6\); Wednesday, \(3\); Thursday, \(8\); Friday, \(5\); Saturday, \(10\); Sunday, \(7\) hours. 2. The greatest value is \(10\) hours on Saturday. 3. Add Monday through Friday: \(4 + 6 + 3 + 8 + 5 = 26\) hours. 4. The least value is \(3\) hours on Wednesday. The difference is \(10 - 3 = 7\) hours.

a) Saturday b) \(26\) hours c) \(7\) hours

- Read the exact value for each city from the vertical axis. - Use the axis label to include the correct unit. - Subtract to compare two rainfall amounts. - Compare each city’s value with \(70\,\text{mm}\).

1. Read the values from the graph: Seattle, \(75\,\text{mm}\); Denver, \(50\,\text{mm}\); Miami, \(95\,\text{mm}\); Chicago, \(60\,\text{mm}\); Phoenix, \(45\,\text{mm}\). 2. The greatest value is \(95\,\text{mm}\) in Miami, and the least is \(45\,\text{mm}\) in Phoenix. 3. Find the difference between Chicago and Phoenix: \(60 - 45 = 15\,\text{mm}\). 4. The cities below \(70\,\text{mm}\) are Denver, Chicago, and Phoenix.

a) Most: Miami, \(95\,\text{mm}\); least: Phoenix, \(45\,\text{mm}\) b) \(15\,\text{mm}\) c) Denver, Chicago, and Phoenix

- Examine the axis scale before reading the bar values. - Find the longest bar for part a). - Read the two requested landmark heights. - Subtract the smaller height from the larger height.

1. Read the approximate heights: Washington Monument, \(550\,\text{ft}\); Space Needle, \(600\,\text{ft}\); Gateway Arch, \(650\,\text{ft}\); Statue of Liberty, \(300\,\text{ft}\). 2. The Gateway Arch is the tallest at about \(650\,\text{ft}\). 3. Find the difference between the Space Needle and Washington Monument: \(600 - 550 = 50\,\text{ft}\).

a) The Gateway Arch, about \(650\,\text{ft}\) b) About \(50\,\text{ft}\)

- Read each bar height using the vertical axis. - Add all five values for the total. - Subtract Noah’s value from Emma’s value to find how many more pages she read.

1. The tallest bar belongs to Emma, with \(140\) pages. 2. Add all five values: \(80 + 120 + 60 + 140 + 100 = 500\) pages. 3. Find the difference between Emma and Noah: \(140 - 60 = 80\) pages.

a) Emma, \(140\) pages b) \(500\) pages c) \(80\) pages

- Read the exact value of each bar. - Check each statement separately using the values you recorded. - Add when a statement asks for a total. Use multiplication to check whether one value is twice another.

1. Read the values from the graph: dogs, \(8\); cats, \(6\); rabbits, \(3\); birds, \(2\); hamsters, \(4\). 2. For a), add the dog and bird votes: \(8 + 2 = 10\), so the statement is true. 3. For b), compare the cat and rabbit votes: \(2 \times 3 = 6\), so the statement is true. 4. For c), add all the votes: \(8 + 6 + 3 + 2 + 4 = 23\), so the statement is false.

a) **True**, because \(8 + 2 = 10\). b) **True**, because \(2 \times 3 = 6\). c) **False**, because there were \(23\) votes altogether.

- Compare the lengths of the bars with the scale. - Find the bar that extends farthest to the right. - Add the values for Monday, Tuesday, and Wednesday.

1. Read the values: Monday, \(12\); Tuesday, \(18\); Wednesday, \(15\); Thursday, \(22\); Friday, \(10\). 2. Thursday has the greatest value, \(22\) books. 3. Add the first three days: \(12 + 18 + 15 = 45\) books.

a) Thursday, \(22\) books b) \(45\) books

- Add the values of all the bars to find the total. - Subtract to find a difference. - To test whether one value is twice another, multiply the smaller value by \(2\).

1. Add all four values: \(12 + 8 + 10 + 6 = 36\) students. 2. Subtract the music value from the sports value: \(12 - 6 = 6\) students. 3. Check Tim’s statement: \(2 \times 6 = 12\). Therefore, Tim is correct.

a) \(36\) students b) \(6\) students c) Yes. Sports received \(12\) votes and music received \(6\) votes, and \(12 = 2 \times 6\).

- Read each bar using the scale on the vertical axis. - Find the tallest bar for part a). - Add all four weekly values for part b).

1. The tallest bar is Week 4, with \(50\) bottles. 2. Add the four weekly values: \(30 + 45 + 25 + 50 = 150\) bottles.

a) Week 4, \(50\) bottles b) \(150\) bottles

- Use the vertical-axis scale to read each bar. - Find the tallest bar for part a). - Add the values for Monday, Tuesday, and Wednesday for part b).

1. Read the values: Monday, \(250\); Tuesday, \(300\); Wednesday, \(150\); Thursday, \(400\); Friday, \(450\). 2. Friday has the greatest value, \(450\) visitors. 3. Add the first three days: \(250 + 300 + 150 = 700\) visitors.

a) Friday, \(450\) visitors b) \(700\) visitors

- Read the height of each bar. - Find the tallest bar for part a). - Add every bar value for part b). - Subtract the smaller value from the larger value to find the difference.

1. Read the values: dogs, \(8\); cats, \(10\); rabbits, \(5\); birds, \(3\); fish, \(2\). 2. Cats have the greatest value, \(10\). 3. Add all the votes: \(8 + 10 + 5 + 3 + 2 = 28\). 4. Subtract the bird votes from the cat votes: \(10 - 3 = 7\).

a) Cats, \(10\) students b) \(28\) students c) \(7\) students

- Read each bar using the vertical-axis scale. - Find the tallest bar for part a). - For part b), first add Anna’s and David’s values, and then subtract Ben’s value.

1. Read the values: Anna, \(8\); Ben, \(5\); Clara, \(11\); David, \(4\) books. 2. Clara has the greatest value, \(11\) books. 3. Anna and David read \(8 + 4 = 12\) books together. Compare their total with Ben’s value: \(12 - 5 = 7\) books.

a) Clara, \(11\) books b) \(7\) more books

- Read each bar using the scale, which increases by \(5\). - Add the two values for part b). - Subtract Clara’s value from David’s value for part c).

1. The tallest bar belongs to Ben, with \(50\) points. 2. Add Anna’s and Clara’s values: \(35 + 20 = 55\) points. 3. Subtract Clara’s value from David’s value: \(45 - 20 = 25\) points.

a) Ben b) \(55\) points c) \(25\) points

- Read the height of each bar. - Add all the bar values for part a). - Double the strawberry value for part b). - For part c), subtract the chocolate value from the total.

1. Add all the votes: \(8 + 6 + 4 + 2 + 5 = 25\) students. 2. Strawberry received \(4\) votes. Twice that number is \(2 \times 4 = 8\). Chocolate received \(8\) votes. 3. Subtract the chocolate votes from the total: \(25 - 8 = 17\) students.

a) \(25\) students b) Chocolate c) \(17\) students

- Read each bar using the scale, which increases by \(2\). - Find the tallest bar for part a). - Add for part b) and subtract to find the difference in part c).

1. Read the values: May, \(12\); June, \(18\); July, \(22\); August, \(20\); September, \(16\) days. 2. July has the greatest value, \(22\) sunny days. 3. Add June and July: \(18 + 22 = 40\) sunny days. 4. Subtract the May value from the August value: \(20 - 12 = 8\) days.

a) July, \(22\) sunny days b) \(40\) sunny days c) \(8\) days

- The horizontal labels show the number of siblings, and each bar height shows the number of students. - “More than two” includes the categories \(3\) and \(4+\). - Add every bar height to find the class total.

1. The bar for one sibling has a height of \(12\), so \(12\) students have exactly one sibling. 2. More than two siblings includes the bars for \(3\) and \(4+\). Add them: \(2 + 1 = 3\) students. 3. Add all the bar heights: \(6 + 12 + 4 + 2 + 1 = 25\) students.

a) \(12\) students b) \(3\) students c) \(25\) students

- Read each bar using the vertical-axis scale. - Look for two bars that end at the same height. - Add the Saturday and Sunday values for the weekend total.

1. The Friday bar shows \(300\) visitors. 2. The Monday and Thursday bars both show \(150\) visitors. 3. Add the Saturday and Sunday values: \(450 + 550 = 1000\) visitors.

1. \(300\) visitors 2. Monday and Thursday 3. \(1000\) visitors

- Use the scale on the vertical axis to read each bar. - Subtract the birch height from the spruce height.

1. The oak bar reaches \(14\), so the oak is \(14\,\text{ft}\) tall. 2. The tallest bar is the spruce bar at \(16\,\text{ft}\). 3. The height difference is \(16 - 8 = 8\,\text{ft}\).

a) \(14\,\text{ft}\) b) The spruce is tallest. c) \(8\,\text{ft}\)

- Read the end of each bar using the horizontal scale. - Add the two weights for question 2. - Subtract the smaller weight from the larger weight to find the difference.

1. Other trash has the longest bar, with a weight of \(50\,\text{lb}\). 2. Add the glass and metal weights: \(30 + 15 = 45\,\text{lb}\). 3. Subtract the plastic weight from the paper weight: \(45 - 25 = 20\,\text{lb}\).

1. Other trash, \(50\,\text{lb}\) 2. \(45\,\text{lb}\) 3. \(20\,\text{lb}\)

- Read each bar height using the vertical axis. - Add all five bar values to find the total. - For part b), first find Wednesday’s value and then double it.

1. Read the bar values: Monday, \(150\); Tuesday, \(200\); Wednesday, \(100\); Thursday, \(250\); Friday, \(300\). 2. Add the five values: \(150 + 200 + 100 + 250 + 300 = 1000\) visitors. 3. Twice Wednesday’s attendance is \(2 \times 100 = 200\). Tuesday has a bar value of \(200\). 4. Find the difference between Friday and Monday: \(300 - 150 = 150\) visitors.

a) \(1000\) visitors b) Tuesday c) \(150\) visitors

- Use the scale to read the two bars named in the question. - Subtract the smaller value from the larger value.

1. Team B has \(10\) stamps, and Team C has \(4\) stamps. 2. The difference is \(10 - 4 = 6\).

Team B has \(6\) more stamps than Team C.

- Use the scale to read only the apple and cherry bars. - Add the two values.

1. The graph shows \(6\) apples and \(4\) cherries. 2. Altogether, there are \(6 + 4 = 10\) pieces of fruit.

There are \(10\) apples and cherries altogether.

- Use the scale to read each bar. - Add all four values. Pairing two values at a time may help.

1. The four bars show \(4\), \(8\), \(2\), and \(6\) boats. 2. Pair the counts: \(4 + 8 = 12\) and \(2 + 6 = 8\). 3. The total is \(12 + 8 = 20\) boats.

The class made \(20\) paper boats in all.

- Use the scale to read each bar. - Order the bars from longest to shortest.

1. The graph shows lake: \(8\), forest: \(6\), mountain: \(4\), and fort: \(2\). 2. From greatest to least, the order is lake, forest, mountain, fort.

Lake, forest, mountain, fort.

- Use the scale to read each student's bar. - Find bars that reach \(8\) or higher, then count them.

1. Noah and Ben each make \(10\) jumps, which is at least \(8\). 2. The other students make fewer than \(8\) jumps. 3. Therefore, \(2\) students make at least \(8\) jumps.

a) Noah and Ben make at least \(8\) jumps. b) That is \(2\) students.

- Read the table headings and category labels carefully. - Keep the categories in the same order as the table.

1. Keep the bird categories in the order shown in the table. 2. Match each category to its value. 3. Use these bar heights: Duck, \(8\); Robin, \(5\); Sparrow, \(14\); Heron, \(3\).

Duck: \(8\), Robin: \(5\), Sparrow: \(14\), Heron: \(3\).

- Think about what equal spacing on the number axis represents. - Make sure the axis maximum is at least as great as every data value.

1. Match each weekday to its value in the table. 2. The bar heights, from Monday through Friday, are \(6, 9, 4, 11, 7\). 3. A maximum of \(12\) is reasonable because it is just above the greatest value and allows an even scale.

The bar heights are \(6, 9, 4, 11, 7\). A reasonable axis maximum is \(12\).

- Decide which information belongs on each axis. - Check that each shortened label still identifies exactly one stop.

1. Shorten each stop name without changing its number. 2. One clear set of labels is S1, S2, S3, and S4. 3. Match the labels to the values: S1, \(15\); S2, \(32\); S3, \(26\); S4, \(9\).

For example: S1: \(15\), S2: \(32\), S3: \(26\), S4: \(9\).

- Consider both the bar values and the scale interval. - Make sure the axis starts at \(0\) and reaches the greatest value.

1. The greatest bar height is \(21\), which is already a multiple of \(3\), so the smallest possible maximum is \(21\). 2. Count by threes from \(0\): \(0, 3, 6, 9, 12, 15, 18, 21\). 3. In table order, the bar heights are \(12, 18, 15, 21\).

Axis labels: \(0, 3, 6, 9, 12, 15, 18, 21\). Bar heights: \(12, 18, 15, 21\).

- Check whether all data values are multiples of each possible interval size. - Then compare the number of intervals needed to reach \(45\).

1. With intervals of \(1\), the axis would have \(45\) intervals. With intervals of \(3\), it would have \(15\) intervals. 2. Every data value is a multiple of \(9\), so intervals of \(9\) place every bar endpoint on a gridline. 3. The axis then has \(45 \div 9 = 5\) equal intervals, making \(9\) the clearest suitable choice.

Use intervals of \(9\). The axis has \(5\) equal intervals from \(0\) to \(45\).

- First describe the pattern in the axis labels. - Then compare each bar value with the size of one interval.

1. The axis increases by \(40\) each time, so the missing value is \(120\). 2. Divide each bar value by \(40\): \(80 \div 40 = 2\), \(120 \div 40 = 3\), \(200 \div 40 = 5\), and \(160 \div 40 = 4\).

The missing value is \(120\). The bars extend across \(2, 3, 5, 4\) intervals, respectively.

- Use both the scale maximum and the number of equal intervals. - Check that the interval size reaches from \(0\) to \(40\) in exactly \(5\) steps.

1. Divide the full scale by the number of intervals: \(40 \div 5 = 8\). Each interval represents \(8\). 2. Divide each bar height by \(8\). The bars extend across \(2, 3, 4, 5\) intervals, respectively.

Each interval represents \(8\). The bars extend across \(2, 3, 4, 5\) intervals, respectively.

- First identify the value of one grid interval. - Check that the number of intervals matches the size of each bar value.

1. Each grid interval represents \(16\). 2. Compute each number of intervals: \(32 \div 16 = 2\), \(48 \div 16 = 3\), \(64 \div 16 = 4\), and \(80 \div 16 = 5\).

\(32\): \(2\) intervals, \(48\): \(3\) intervals, \(64\): \(4\) intervals, \(80\): \(5\) intervals.

- Describe what the graph is measuring. - Then compare that quantity with the current axis label.

1. The values count laps, not units of time. 2. The vertical axis should be labeled “Laps” instead of “Minutes.”

Change the vertical-axis label from “Minutes” to “Laps.”

- Compare every name in the table with the names in the graph. - Use the table value to determine the height of the missing bar.

1. The graph includes bars for Ira, Leo, and Sam. 2. Comparing the graph with the table shows that Nia is missing. Her bar must have a height of \(17\).

Add a bar for Nia with a height of \(17\).

- List the weekdays in chronological order. - Compare that order with the order of the bars in the graph.

1. In the graph, Tuesday appears before Monday. 2. Monday and Tuesday must switch places. Wednesday and Thursday are already in the correct order.

The bars labeled Mon and Tue must switch places.

- Check each student’s graph value against the table. - Identify the only mismatch and state both the displayed and correct values.

1. Compare each bar with the corresponding table value. 2. Eli, Kim, and Noa match the table. Luz does not. 3. Luz’s bar shows \(70\), but the table gives \(60\). Lower the bar to \(60\).

Luz’s bar is too high. It shows \(70\), but it should show \(60\).

- Compare the greatest data value with the planned axis maximum. - Make sure the new maximum belongs to the scale pattern.

1. The value \(28\) is greater than the planned maximum of \(24\), so that bar would extend beyond the axis. 2. Since \(28\) is a multiple of \(4\), the smallest suitable maximum is \(28\).

The axis is too short. With intervals of \(4\), it must extend to at least \(28\).

- Read the title and labels carefully. - Update bar B, then compare its new height with each other bar.

1. Bar B increases to \(9 + 3 = 12\). 2. The other bar heights are \(6\), \(4\), and \(8\), and none is greater than \(12\).

Bar B has height \(12\) and is the tallest bar.

- First identify the original Green value. - After subtracting, compare the new value with the other bars.

1. Subtract the removed cards from the Green value: \(20 - 2 = 18\). 2. The Red bar also has length \(18\).

The Green bar has length \(18\) and is then the same length as the Red bar.

- Add \(5\) to Jon’s original value. - Compare all four values and order them from greatest to least.

1. Jon’s new value is \(9 + 5 = 14\) laps. 2. Order the values from greatest to least: Jon, \(14\); Tao, \(12\); Pia, \(11\); Mia, \(8\).

Jon: \(14\); Tao: \(12\); Pia: \(11\); Mia: \(8\)

- Update the Bay value first. - Then add all four values, using the updated Bay value.

1. The Bay value becomes \(10 + 3 = 13\) boats. 2. Add the four updated values: \(9 + 12 + 7 + 13 = 41\) boats.

The Bay bar will have a length of \(13\), and the sum will be \(41\).

- Check whether both graphs match the same categories with the same values. - Think about which orientation leaves more room for long labels.

1. Both graphs match Monday, Tuesday, and Wednesday with the values \(12, 8, 16\), respectively. 2. Therefore, both graphs represent the data correctly. 3. A horizontal bar graph can provide more space for long category labels.

Both graphs are correct. Horizontal bars can be more practical when category labels are long.

- Compare the grid interval used in each graph. - Locate where \(11\) would fall on each scale.

1. Graph a) uses intervals of \(1\), so every whole-number value lies on a gridline. 2. Graph b) uses intervals of \(2\). The value \(11\) lies between the gridlines at \(10\) and \(12\). 3. Therefore, all three values can be read directly at gridlines in graph a).

Graph a). It uses intervals of \(1\), while \(11\) falls between \(10\) and \(12\) in graph b).

- Identify the greatest and least values first. - Then check whether every animal is matched with its correct bar height.

1. Chicken must have the tallest bar, with a value of \(20\). 2. Pony must have the shortest bar, with a value of \(4\), and Sheep must have a value of \(9\). 3. Only graph b) matches all three category-value pairs.

Graph b) is correct. Chicken has the greatest value, \(20\), and Pony has the least value, \(4\).

- Identify what quantity the graph represents. - Compare that quantity with the meaning of each possible label.

1. The values represent counts of stones. 2. Therefore, the missing label should be “Number of stones.” The label “Weight” would describe a different quantity that was not measured.

The missing label is “Number of stones.” “Weight” is incorrect because the stones were counted, not weighed.

- Separate the categories from the measured quantity. - A length needs an appropriate unit of length.

1. The measured quantity is length, and its unit is feet. 2. The categories are the four paths.

Vertical axis: “Feet”; horizontal axis: “Paths.”

- Decide which information is categorical and which is numerical. - Check that each axis label matches the information shown on that axis.

1. The categories below the bars are the days. 2. The bar heights represent numbers of child visits.

Horizontal axis: “Days”; vertical axis: “Child visits.”

- Identify what the letters stand for. - Make sure the vertical-axis label names the quantity being counted.

1. Add a key or note explaining that A through D are garden plots. 2. The vertical axis represents a count of sunflowers.

Add “A–D: Garden plots.” Label the vertical axis “Number of sunflowers.”

- Decide which information belongs in the axis label. - Avoid repeating the same unit at every tick mark.

1. The unit does not need to be repeated beside every tick value. 2. Label the vertical axis once as “Bicycles repaired” and use only numbers at the tick marks.

Label the vertical axis “Bicycles repaired” once, and label the tick marks with numbers only.

- Count the existing categories and the new category. - Keep every original value and add the new value in the stated category.

1. Adding Pine to the four existing categories creates a fifth category. 2. The graph will have \(5\) bars. 3. The heights are Linden, \(22\); Oak, \(18\); Maple, \(26\); Birch, \(14\); Pine, \(20\).

There will be \(5\) bars: Linden: \(22\), Oak: \(18\), Maple: \(26\), Birch: \(14\), Pine: \(20\).

- Determine what equal spacing means for the numerical pattern. - Check that the difference between consecutive labels is constant.

1. Equal spacing from \(0\) to \(20\) across four intervals means the labels should increase by \(5\). 2. The middle label should therefore be \(10\), not \(12\).

Replace \(12\) with \(10\).

- Separate the categories from the numerical quantity. - Check that each axis label describes the information shown there.

1. The horizontal axis lists the categories, which are the days. 2. The vertical axis shows how many pages were read.

Horizontal axis: “Days”; vertical axis: “Pages read.”

- Decide whether the values represent separate categories or connected intervals. - Think about how spacing helps readers distinguish the categories.

1. The bars represent separate categories rather than connected numerical intervals. 2. Visible spaces help show that the categories are distinct.

The bars should have visible spaces because they represent separate categories.

- Determine what equal physical spacing should mean on a vertical axis. - Check whether the numerical differences stay constant.

1. The numerical interval changes even though the physical grid spacing stays the same. 2. Every equal grid interval must represent the same numerical increase.

The scale interval is inconsistent. Every equal grid interval must represent the same numerical increase.

- Use the vertical-axis scale to determine the value of each interval. - Find the bar for the requested day and read its height. - Use subtraction to compare two bars. - Add all five bar values to find the total.

1. Read the values from the vertical axis: Monday, \(150\); Tuesday, \(250\); Wednesday, \(200\); Thursday, \(350\); Friday, \(450\). 2. Thursday’s bar shows \(350\) visitors. 3. Find the difference between Friday and Monday: \(450 - 150 = 300\) visitors. 4. Add all five values: \(150 + 250 + 200 + 350 + 450 = 1400\) visitors.

a) \(350\) visitors b) \(300\) more visitors c) \(1400\) visitors

- This bar graph is horizontal, so bar length represents the number of votes. - Use the horizontal axis to read the value of each bar. - Subtract the two bar values to find how many more votes one fruit received. - Add all four bar values to find the total number of votes.

1. Read each bar using the horizontal scale: strawberries, \(10\) votes; apples, \(8\) votes; bananas, \(6\) votes; pears, \(4\) votes. 2. Strawberries have the longest bar, so they were most popular with \(10\) votes. 3. The banana bar shows \(6\) votes. 4. Find the difference between apples and pears: \(8 - 4 = 4\) votes. 5. Add all four values: \(10 + 8 + 6 + 4 = 28\) students.

a) Strawberries, \(10\) votes b) \(6\) votes c) \(4\) more votes d) \(28\) students

- Use the horizontal-axis scale to read each bar endpoint. - Identify the longest bar for part a). - Add all five values for the total. - “Three times as many” means multiplying by \(3\).

1. Read the values from the graph: Ethan, \(10\); Noah, \(5\); Mia, \(15\); Leo, \(8\); Julia, \(12\). 2. Mia has the greatest value, \(15\) books. 3. Add the five values: \(10 + 5 + 15 + 8 + 12 = 50\) books. 4. Check Ethan’s statement: \(3 \times 5 = 15\). Since Mia read \(15\) books, the statement is correct.

a) Mia, \(15\) books b) \(50\) books c) Yes. Noah read \(5\) books, and \(3 \times 5 = 15\), which is Mia’s total.

- Use the scale on the vertical axis to read each bar. - Add only the bars named in each question.

1. Read the bars using the scale: soccer: \(6\), swimming: \(4\), gymnastics: \(2\), and biking: \(2\). 2. For a), the soccer bar shows \(6\) students. 3. For b), \(2 + 4 = 6\) students chose gymnastics or swimming. 4. For c), \(6 + 4 + 2 + 2 = 14\) students took the survey.

a) \(6\) students b) \(6\) students c) \(14\) students

- Read each weekly height from the vertical axis. - Subtract consecutive weekly heights to find each week’s growth. - Compare the increases to identify the greatest one.

1. Read the Week 3 bar: the sunflower was \(45\,\text{cm}\) tall. 2. Find each weekly increase: Week 2, \(25 - 10 = 15\,\text{cm}\); Week 3, \(45 - 25 = 20\,\text{cm}\); Week 4, \(60 - 45 = 15\,\text{cm}\); Week 5, \(70 - 60 = 10\,\text{cm}\). 3. The greatest increase was \(20\,\text{cm}\), during Week 3.

a) \(45\,\text{cm}\) b) Week 3; \(20\,\text{cm}\)

- Read the value for every day from the vertical axis. - Add the daily values to find the weekly total. - Compare each bar with \(15\,\text{kWh}\) and count only the bars above it. - Think about what affects solar-panel output.

1. Read the daily values: Monday, \(18\,\text{kWh}\); Tuesday, \(22\,\text{kWh}\); Wednesday, \(5\,\text{kWh}\); Thursday, \(7\,\text{kWh}\); Friday, \(12\,\text{kWh}\); Saturday, \(20\,\text{kWh}\); Sunday, \(21\,\text{kWh}\). 2. Add all seven values: \(18 + 22 + 5 + 7 + 12 + 20 + 21 = 105\,\text{kWh}\). 3. The values above \(15\,\text{kWh}\) occur on Monday, Tuesday, Saturday, and Sunday, for a total of \(4\) days. 4. One possible reason for the low values on Wednesday and Thursday is cloudy or rainy weather that reduced sunlight.

a) \(105\,\text{kWh}\) b) \(4\) days c) For example, cloudy or rainy weather reduced the sunlight reaching the panels.

- Read each student’s bar value carefully. - Add all four values for part b). - For part c), first combine Mia’s and Leo’s values, and then compare that total with Jonah’s value.

1. Read the values: Mia, \(12\); Leo, \(8\); Sophia, \(15\); Jonah, \(6\) books. 2. Sophia has the greatest value, \(15\), and Jonah has the least, \(6\). 3. Add all four values: \(12 + 8 + 15 + 6 = 41\) books. 4. Mia and Leo read \(12 + 8 = 20\) books. Jonah needs \(20 - 6 = 14\) more books.

a) Sophia read the most, \(15\) books; Jonah read the fewest, \(6\) books. b) \(41\) books c) \(14\) more books

- Use the horizontal axis to find the number of books and the vertical axis to read the number of students. - Add every bar height for the total. - “More than \(3\)” does not include \(3\). - Find the tallest bar for part d).

1. The bar at \(3\) books has a height of \(6\), so \(6\) students read exactly \(3\) books. 2. Add all the bar heights: \(2 + 5 + 8 + 6 + 4 + 3 = 28\) students. 3. More than \(3\) books means \(4\) or \(5\) books. Add those bar heights: \(4 + 3 = 7\) students. 4. The tallest bar is at \(2\) books and has a height of \(8\).

a) \(6\) students b) \(28\) students c) \(7\) students d) \(2\) books, reported by \(8\) students

- Read each bar using the horizontal-axis scale. - Add all five values for part a). - Add the two named categories for part c). - Subtract the track value from the soccer value for part d).

1. Add all the votes: \(16 + 12 + 8 + 6 + 4 = 46\) students. 2. Soccer has the longest bar, with \(16\) votes. 3. Add the swimming and gymnastics votes: \(12 + 8 = 20\) students. 4. Subtract the track votes from the soccer votes: \(16 - 6 = 10\) students.

a) \(46\) students b) Soccer, \(16\) votes c) \(20\) students d) \(10\) more students

- Read each bar using the vertical-axis scale. - Add all four values for part b). - Add the two named categories for part c). - Find the difference between the soccer and riding values for part d).

1. Read the values: soccer, \(14\); swimming, \(7\); dancing, \(9\); riding, \(4\). 2. Soccer has the greatest value. 3. Add all four values: \(14 + 7 + 9 + 4 = 34\) students. 4. Add swimming and dancing: \(7 + 9 = 16\) students. 5. Find the difference between soccer and riding: \(14 - 4 = 10\) votes.

a) Soccer b) \(34\) students c) \(16\) students d) \(10\) additional votes

- Read each height using the scale, which increases by \(5\). - Find twice the birch height for part b). - Add the fir and birch heights, and then look for another pair with the same sum. - Multiply the birch height by \(3\) to check the claim.

1. Read the heights: oak, \(35\,\text{ft}\); beech, \(25\,\text{ft}\); fir, \(45\,\text{ft}\); birch, \(15\,\text{ft}\); pine, \(30\,\text{ft}\). 2. Twice the birch height is \(2 \times 15 = 30\,\text{ft}\), which is the pine’s height. 3. The fir and birch have a combined height of \(45 + 15 = 60\,\text{ft}\). The oak and beech also total \(35 + 25 = 60\,\text{ft}\). 4. Three times the birch height is \(3 \times 15 = 45\,\text{ft}\), which is the fir’s height. The gardener is correct.

a) Oak: \(35\,\text{ft}\); beech: \(25\,\text{ft}\); fir: \(45\,\text{ft}\); birch: \(15\,\text{ft}\); pine: \(30\,\text{ft}\) b) The pine c) The oak and the beech d) Yes, because \(3 \times 15 = 45\).

- Read each bar using the vertical-axis scale. - Growth between two weeks is the later height minus the earlier height. - Find each week-to-week increase and compare them.

1. The Week 4 bar shows \(75\,\text{cm}\). 2. From Week 2 to Week 3, the growth was \(55 - 30 = 25\,\text{cm}\). 3. The weekly increases were \(30 - 15 = 15\,\text{cm}\), \(55 - 30 = 25\,\text{cm}\), \(75 - 55 = 20\,\text{cm}\), and \(90 - 75 = 15\,\text{cm}\). The greatest increase was from Week 2 to Week 3.

a) \(75\,\text{cm}\) b) \(25\,\text{cm}\) c) From Week 2 to Week 3

- Read the vertical scale carefully. Each grid step represents \(5\) students. - For the first statement, find half of the \(50\) students. - Use the values you read from the graph to check the statements about “three times” and “half as many.” - Compare the bar heights to identify the most and least common activities.

1. Read the values from the bar graph: playing sports, \(35\); spending time with friends, \(45\); reading, \(15\); listening to music, \(25\); crafts, \(10\). 2. Organize the values in a table: <table><tr><th>Activity</th><th>Number of students</th></tr><tr><td>Playing sports</td><td>\(35\)</td></tr><tr><td>Spending time with friends</td><td>\(45\)</td></tr><tr><td>Reading</td><td>\(15\)</td></tr><tr><td>Listening to music</td><td>\(25\)</td></tr><tr><td>Crafts</td><td>\(10\)</td></tr></table> 3. The first statement is true because half of \(50\) is \(25\), and \(35 > 25\). 4. The second statement is true because \(10\) is the smallest value. 5. The third statement is true because \(3 \times 15 = 45\). 6. The fourth statement is false because twice \(25\) is \(50\), not \(35\).

a) <table><tr><th>Activity</th><th>Number of students</th></tr><tr><td>Playing sports</td><td>\(35\)</td></tr><tr><td>Spending time with friends</td><td>\(45\)</td></tr><tr><td>Reading</td><td>\(15\)</td></tr><tr><td>Listening to music</td><td>\(25\)</td></tr><tr><td>Crafts</td><td>\(10\)</td></tr></table> b) Statement 1 is true because \(35 > 25\). Statement 2 is true because \(10\) is the smallest value. Statement 3 is true because \(3 \times 15 = 45\). Statement 4 is false because twice \(25\) is \(50\), not \(35\).

- Read each bar using the scale, which increases by \(10\). - Order the scores from greatest to least for the ranking. - Add the Class A and Class C scores for part c).

1. Class B has the tallest bar, with \(150\) points. 2. Order the scores from greatest to least: \(150 > 120 > 110 > 90\). The ranking is Class B, Class A, Class D, Class C. 3. Add the scores for Classes A and C: \(120 + 90 = 210\) points.

a) Class B b) 1st: Class B; 2nd: Class A; 3rd: Class D; 4th: Class C c) \(210\) points

- The horizontal scale increases by \(1\), so each bar endpoint can be read directly. - Count only values greater than \(10\) for part c). - Add the Portland and Denver values for part d).

1. Read the values: Portland, \(14\); Denver, \(8\); Boston, \(12\); Chicago, \(11\); Atlanta, \(9\) days. 2. Boston had \(12\) rainy days. 3. Denver has the least value, \(8\) days. 4. Portland, Boston, and Chicago each had more than \(10\) rainy days, so the count is \(3\) cities. 5. Portland and Denver had \(14 + 8 = 22\) rainy days altogether.

a) \(12\) rainy days b) Denver c) \(3\) cities d) \(22\) rainy days

- The horizontal axis shows the number of hits, and the vertical axis shows the number of students. - For part a, find \(5\) on the horizontal axis and read the bar height. - For part b, add the frequencies for all hit counts of at least \(7\). - For part c, add the heights of all visible bars.

1. The bar above \(5\) has height \(8\), so \(8\) students got exactly \(5\) hits. 2. Students with at least \(7\) hits are represented by the bars at \(7\), \(8\), \(9\), and \(10\). The bars at \(9\) and \(10\) are missing, so they represent \(0\) students. There are \(3 + 1 + 0 + 0 = 4\) prize winners. 3. Add all bar heights: \(1 + 2 + 4 + 6 + 8 + 5 + 3 + 1 = 30\). Therefore, \(30\) students participated.

a) \(8\) students b) \(4\) students c) \(30\) students

- Read each bar using the horizontal scale, which increases by \(5\). - Subtract to find the difference in question 3. - Add all five values for the total.

1. Mystery has the longest bar, with \(120\) checkouts. 2. The adventure bar shows \(85\) books. 3. Subtract the nonfiction value from the comics value: \(110 - 60 = 50\) books. 4. Add all five values: \(120 + 110 + 85 + 60 + 45 = 420\) books.

1. Mystery 2. \(85\) books 3. \(50\) books 4. \(420\) books

- First describe how the data values would appear on the graph. - Then check each value against the multiples of \(5\).

1. Match each student to the value in the table. 2. The bar lengths are Mia, \(24\); Owen, \(18\); Maya, \(31\); Ben, \(27\). 3. Gridlines occur at multiples of \(5\). None of the four values is a multiple of \(5\), so all four bars end between gridlines.

Mia: \(24\), Owen: \(18\), Maya: \(31\), Ben: \(27\). All four bars end between gridlines.

- Decide which information belongs on each axis. - Check whether every category and value would be easy to read with your scale.

1. Compare the data values with each possible scale interval. 2. An interval of \(10\) places every value on a gridline but creates many gridlines. An interval of \(50\) does not place most values on gridlines. 3. An interval of \(20\) places every value exactly on a gridline without using too many gridlines, so it is the clearest choice.

Use \(20\) visitors per gridline because every value lies on a gridline and the graph will not need too many gridlines.

- Determine what one equal interval on the vertical axis represents. - Make sure the scale is evenly spaced and reaches above the greatest data value.

1. The maximum must be at least \(550\). 2. Using intervals of \(100\) gives the labeled values \(0, 100, 200, 300, 400, 500, 600\). 3. This scale has \(6\) equal intervals and includes every data value, so an interval size of \(100\) and a maximum of \(600\) work.

Use intervals of \(100\) and a maximum of \(600\).

- Think about what one equal interval on the vertical axis represents. - Check whether every data value is a multiple of the interval size.

1. The value \(21\) is not a multiple of \(2\), so intervals of \(2\) do not work for every bar. 2. Several values are not multiples of \(10\), so intervals of \(10\) do not work. 3. All four values are multiples of \(7\). Since \(42 \div 7 = 6\), the scale needs \(6\) intervals to reach \(42\).

Use intervals of \(7\). The scale needs \(6\) intervals to reach \(42\).

- Read every value and every possible interval size carefully. - Test each value against each interval size instead of checking only one bar.

1. All four values are multiples of \(5\), so intervals of \(5\) work. 2. The value \(35\) is not a multiple of \(10\), so intervals of \(10\) do not work. 3. The value \(50\) is not a multiple of \(15\), so intervals of \(15\) do not work.

Only intervals of \(5\) work for all four values. Counterexamples are \(35\) for intervals of \(10\) and \(50\) for intervals of \(15\).

- Consider what each equal interval represents. - Check both correctness and how easy each scale is to read.

1. Both interval sizes divide every data value evenly, so both graphs can show the data correctly. 2. Graph A has \(100 \div 25 = 4\) intervals. 3. Graph B has \(100 \div 5 = 20\) intervals. 4. Graph A is quicker to read because it uses far fewer gridlines.

Both graphs are correct. Graph A has \(4\) intervals, and Graph B has \(20\) intervals. Graph A is quicker to read.

- Compare each bar height with the corresponding value in the table. - First match every height to the correct tree type, and then check the displayed labels.

1. The second bar has height \(12\), so it should be labeled Maple, but it is labeled Birch. 2. The third bar has height \(15\), so it should be labeled Birch, but it is labeled Maple. 3. Therefore, the Maple and Birch labels must be switched.

The labels Maple and Birch must be switched.

- Compare each bar with both the scale and the table. - Check every category before deciding which bar is incorrect.

1. Compare each bar value with the matching table value. 2. Only the Green bar does not match. 3. The Green bar shows \(32\), but the table gives \(28\). Its correct length is \(28\).

The Green bar is too long. It shows \(32\) instead of \(28\), so its correct length is \(28\).

- First read all four starting heights from the graph. - Change only Mia’s and Tom’s values, and then report all four heights.

1. Mia’s new value is \(12 + 4 = 16\). 2. Tom’s corrected value is \(15 - 3 = 12\). 3. Eli and Zoe stay at \(10\) and \(14\), respectively.

Mia: \(16\), Tom: \(12\), Eli: \(10\), Zoe: \(14\).

- Identify which two categories must be combined. - Keep the values of all categories that are not being combined.

1. Combine the South and East values: \(30 + 20 = 50\) bags for Southeast. 2. The North and West values stay at \(25\) and \(35\), respectively.

North: \(25\), Southeast: \(50\), West: \(35\).

- Read each starting value from the graph. - Change only the two named categories, and then compare all four new values.

1. Yogurt changes from \(20\) to \(20 + 8 = 28\). 2. Fruit changes from \(24\) to \(24 - 4 = 20\). 3. Pretzels stays at \(32\), and Juice stays at \(28\). Therefore, Juice and Yogurt have equal heights.

Pretzels: \(32\), Juice: \(28\), Fruit: \(20\), Yogurt: \(28\). Juice and Yogurt are equal in height.

- Identify the total and the two known categories. - Subtract the known values before checking the graphs.

1. Find the missing number: \(24 - 11 - 5 = 8\). 2. Only graph a) shows a volleyball value of \(8\).

Graph a) is correct; the volleyball value is \(8\).

- Decide whether each bar represents a count or a monetary value. - Match the axis label to the quantity actually measured.

1. The bar values \(25\), \(18\), and \(9\) are counts of coins. 2. The total value in cents would require a separate calculation using each coin’s value. 3. Therefore, the axis should be labeled “Number of coins.”

The correct label is “Number of coins,” not “Value in cents.”

- Add the three known bar heights first. - Subtract their sum from the stated total.

1. Add the known heights: \(18 + 24 + 16 = 58\). 2. Subtract from the total: \(80 - 58 = 22\).

The East bar must have height \(22\).

- Consider the length of the category labels. - Check that every data value is represented by the correct bar length.

1. The category names are long. 2. A horizontal bar graph provides more room for long labels. 3. In table order, the bar lengths are \(28, 35, 22, 31\).

Use a horizontal bar graph because the category labels are long. The bar lengths are \(28, 35, 22, 31\).

- Use the comparisons with A to find B and C. - Choose a scale maximum that is a multiple of \(5\) and at least as great as every bar length.

1. Find B: \(26 + 9 = 35\). 2. Find C: \(26 - 4 = 22\). 3. The greatest value is \(35\). With intervals of \(5\), a suitable maximum above it is \(40\).

A: \(26\), B: \(35\), C: \(22\); use an axis maximum of \(40\).

- Check the suggested scale one value at a time. - Look for an interval size that divides every bar value evenly.

1. With intervals of \(50\), only \(150\) is a multiple of \(50\). The bars at \(60, 90, 120\) would end between gridlines. 2. All four values are multiples of \(30\), so intervals of \(30\) place every bar endpoint on a gridline.

With intervals of \(50\), the bars at \(60, 90, 120\) end between gridlines. Use intervals of \(30\) to place all four endpoints on gridlines.

- Compare the scale size with the largest value in each data set. - Check that each graph is accurate without using an unnecessary number of gridlines.

1. Intervals of \(4\) work well for data set A because the scale reaches \(16\) in only \(4\) intervals. 2. For data set B, intervals of \(4\) would require \(160 \div 4 = 40\) intervals, making the graph crowded. 3. Intervals of \(40\) work well for data set B because the scale then reaches \(160\) in \(4\) intervals.

No. Use intervals of \(4\) for data set A and, for example, intervals of \(40\) for data set B.

- Test each data value against each possible interval size. - Count how many values are exact multiples of each interval size.

1. With intervals of \(2\), none of the four odd values lies on a gridline. 2. With intervals of \(3\), the values \(21, 27, 33\) lie on gridlines, while \(17\) does not. 3. With intervals of \(6\), none of the four values is a multiple of \(6\). 4. Therefore, intervals of \(3\) place the most endpoints on gridlines.

Use intervals of \(3\). The values \(21, 27, 33\) lie on gridlines, and \(17\) lies between gridlines.

- Compare both the bar values and the axis scales. - Base your choice on readability, not on whether one graph is mathematically correct.

1. In both graphs, categories A, B, C, and D have values \(20, 30, 10, 40\). 2. Graph a) uses intervals of \(10\) and extends to \(50\). Graph b) uses intervals of \(5\) and extends to \(40\). 3. Both graphs are correct. Graph a) is a reasonable choice for a quick overview because it has fewer gridlines, though graph b) is also acceptable with a clear justification.

Both graphs are correct. Graph a) has a coarser scale, and graph b) has a finer scale. A justified choice of either graph is acceptable; graph a) is quicker to scan because it has fewer gridlines.