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 x-axis. 2. The measured quantity is reading time, so “Reading time in minutes” belongs on the y-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-highest rating belongs to the second-longest bar. - Identify the greatest and least values before subtracting.

1. Read the ratings: Hare, \(35\); Greyhound, \(45\); Ostrich, \(40\); Lion, \(50\); Pronghorn, \(55\); Cheetah, \(70\). 2. Cheetah has the greatest rating, \(70\), and Pronghorn has the second-highest rating, \(55\). 3. Hare has the least rating, \(35\). The difference is \(70 - 35 = 35\) rating points.

a) Hare: \(35\); Greyhound: \(45\); Ostrich: \(40\); Lion: \(50\); Pronghorn: \(55\); Cheetah: \(70\) b) Pronghorn, \(55\) c) \(35\) rating points

- 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 each bar endpoint using the vertical scale. - Compare the five values to find the greatest and least. - Subtract the Red Rock value from the Lakeside value. - Check each town against \(70\,\text{mm}\).

1. Read the values: Cedar Bay, \(75\,\text{mm}\); Mesa Ridge, \(50\,\text{mm}\); Palm Cove, \(95\,\text{mm}\); Lakeside, \(60\,\text{mm}\); Red Rock, \(45\,\text{mm}\). 2. Palm Cove has the greatest value, \(95\,\text{mm}\), and Red Rock has the least, \(45\,\text{mm}\). 3. The difference between Lakeside and Red Rock is \(60 - 45 = 15\,\text{mm}\). 4. The towns below \(70\,\text{mm}\) are Mesa Ridge, Lakeside, and Red Rock.

a) Most: Palm Cove, \(95\,\text{mm}\); least: Red Rock, \(45\,\text{mm}\) b) \(15\,\text{mm}\) c) Mesa Ridge, Lakeside, and Red Rock

- Examine the horizontal scale before reading the bars. - Find the longest bar for part a). - Read the two requested heights and subtract the smaller from the larger.

1. Read the approximate heights: Bay Lantern, \(300\,\text{ft}\); Cedar Column, \(550\,\text{ft}\); Sky Needle, \(600\,\text{ft}\); River Arch, \(650\,\text{ft}\). 2. River Arch is tallest at about \(650\,\text{ft}\). 3. The difference between River Arch and Sky Needle is \(650 - 600 = 50\,\text{ft}\).

a) River 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 the requested categories when a statement gives a combined total.

1. Read the values: dogs, \(8\); cats, \(6\); rabbits, \(3\); birds, \(2\); hamsters, \(4\). 2. For a), \(8 + 2 = 10\), so the statement is true. 3. For b), \(6 + 3 = 9\), so the statement is true. 4. For c), \(8 + 6 + 3 + 2 + 4 = 23\), so the statement is false.

a) **True**, because \(8 + 2 = 10\). b) **True**, because \(6 + 3 = 9\). 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 all four bar values for part a). - Subtract the smaller requested value from the larger one for part b). - For part c), compare the sports and reading bars directly.

1. Add all four values: \(12 + 8 + 10 + 6 = 36\) students. 2. The difference between sports and music is \(12 - 6 = 6\) students. 3. Compare sports and reading: \(12 - 8 = 4\), so Tim is correct.

a) \(36\) students b) \(6\) students c) Yes. Sports received \(12\) votes and reading received \(8\), a difference of \(4\).

- 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). - For part b), compare each flavor with strawberry using subtraction. - 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. Chocolate received \(8\), which is \(8 - 4 = 4\) more. 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), compare each day with Wednesday using subtraction. - For part c), subtract Monday's value from Friday's.

1. Read the values: Monday, \(150\); Tuesday, \(200\); Wednesday, \(100\); Thursday, \(250\); Friday, \(300\). 2. Add the five values: \(150 + 200 + 100 + 250 + 300 = 1000\) visitors. 3. Tuesday had \(200 - 100 = 100\) more visitors than Wednesday. 4. Friday had \(300 - 150 = 150\) more visitors than Monday.

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 Thursday value from the graph. - Compare that value with the goal. - Subtract to find how much is still needed.

1. The Thursday bar shows \(40\,\text{gal}\). 2. The amount still needed is \(70 - 40 = 30\,\text{gal}\).

\(30\,\text{gal}\) more are needed.

- Read the two team scores from the graph. - Find the difference between the scores. - Check that adding your answer to Team Beaver's score makes the scores equal.

1. Team Beaver has \(15\) points, and Team Otter has \(20\) points. 2. The number needed is \(20 - 15 = 5\) points.

Team Beaver needs \(5\) more points.

- Read the Tuesday and Thursday bars. - Remember that the scale increases by \(10\). - Add the two values.

1. Tuesday shows \(40\) minutes, and Thursday shows \(50\) minutes. 2. Altogether, \(40 + 50 = 90\,\text{min}\).

\(90\,\text{min}\) of practice were recorded altogether.

- Increase only the number of rings. - Find the new number of rings. - Compare that number with the other bars.

1. The number of rings changes from \(12\) to \(12 + 6 = 18\). 2. The graph shows \(18\) balls. 3. Therefore, there will be the same number of rings and balls.

There will be the same number of rings and balls.

- Read the first and last times named in the question. - Subtract the earlier value from the later value. - State whether the number increased or decreased.

1. At \(8{:}00\) a.m., \(10\) chairs are open. At \(4{:}00\) p.m., \(15\) chairs are open. 2. The change is \(15 - 10 = 5\) chairs, an increase.

At \(4{:}00\) p.m., there are \(5\) more open chairs than at \(8{:}00\) a.m.

- Read the value of each bar. - Look for two bars with the same height. - Name both matching days.

1. The Tuesday and Thursday bars both have a value of \(15\). 2. The other days have different values.

On Tuesday and Thursday, \(15\) tickets were sold each day.

- Read each bar value. - “At least \(15\)” includes exactly \(15\). - Select every supply with a value of \(15\) or more.

1. Scissors have \(15\), and markers have \(20\), so both meet the requirement. 2. Tape and paintbrushes each have only \(10\), so they do not meet the requirement.

There are enough scissors and markers.

- Read only the Monday, Wednesday, and Friday bars. - Add the three values. - Check that your total is greater than each single value.

1. The three values are \(6\), \(8\), and \(10\). 2. Their sum is \(6 + 8 + 10 = 24\).

\(24\) packages were picked up altogether.

- Read the lengths of Routes \(2\) and \(4\). - Subtract the shorter route from the longer route. - Include the unit in your answer.

1. Route \(2\) is \(180\,\text{yd}\) long, and Route \(4\) is \(90\,\text{yd}\) long. 2. The difference is \(180 - 90 = 90\,\text{yd}\).

Route \(2\) is \(90\,\text{yd}\) longer.

- Compare one color at a time. - Read its value in both graphs. - Look for an exact match.

1. Blue has a value of \(20\) in Graph a). 2. Blue also has a value of \(20\) in Graph b). 3. The other colors have different values in the two graphs.

Blue has the same value in both graphs.

- Find the tallest bar in each graph. - Match each tallest bar to its team. - Compare the two team names.

1. In a), Team Owl has the greatest score, \(30\). 2. In b), Team Fox has the greatest score, \(35\). 3. Therefore, the leader changed.

1) Team Owl leads in Graph a). 2) Team Fox leads in Graph b). 3) Yes, the leader changed.

- Match equal values across the two graphs. - Follow each starting bar named in the question. - Record all four value moves.

1. A in a) and C in b) both have \(12\). 2. B in a) and D in b) both have \(18\). 3. C in a) and A in b) both have \(24\). 4. D in a) and B in b) both have \(30\).

1) A to C: \(12\) 2) B to D: \(18\) 3) C to A: \(24\) 4) D to B: \(30\)

- Find Wednesday in both graphs. - Read the two values. - Add them.

1. Wednesday has a value of \(15\) in a) and \(25\) in b). 2. Altogether, \(15 + 25 = 40\) snails were found.

\(40\) snails were found on the two Wednesdays altogether.

- 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: \(12, 18, 15, 21\).

- 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.

- Trace the labeled axis from \(0\) upward one interval at a time. - Count spaces between neighboring gridlines, not the number of labels. - Match each bar height to the number of equal jumps needed to reach it from \(0\).

1. Each pair of neighboring labels marks one grid interval. 2. From \(0\) to \(32\) there are \(2\) intervals: \(0\) to \(16\), then \(16\) to \(32\). 3. Continuing the same count, \(48\) spans \(3\) intervals, \(64\) spans \(4\), and \(80\) spans \(5\).

\(32\): \(2\) 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 bar B's starting value from the graph. - Increase only bar B by the stated amount. - Compare the new bar B height with the other displayed bars.

1. Read bar B's current height from the graph: \(9\). 2. Add the new birdhouses: \(9 + 3 = 12\). 3. The other bar heights are \(6\), \(4\), and \(8\), so 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 scale to read the height of the Postcards bar. - Read the height of the Magnets bar in the same way. - Subtract the smaller count from the larger count to find “how many more.”

1. Read the bars: \(24\) magnets and \(16\) postcards were sold. 2. Find the difference: \(24 - 16 = 8\).

The shop sold \(8\) more magnets than postcards.

- Read each bar using the scale that counts by twos. - Add all three bar values for the total. - Compare Class C with Class A by subtraction.

1. Read the bars: Class A read \(6\) books, Class B read \(8\), and Class C read \(10\). 2. For part a), add: \(6 + 8 + 10 = 24\). 3. For part b), subtract: \(10 - 6 = 4\).

a) \(24\) books b) \(4\) more books

- Use the vertical scale, which counts by threes, to read each bar. - Record all three bar values before adding. - Combine the three counts to find the total.

1. Read the bars: Ava read \(12\) books, Ben read \(9\), and Cora read \(6\). 2. Add the values: \(12 + 9 + 6 = 27\).

They read \(27\) books altogether.

- Use the horizontal scale, which counts by fours, to read each bar. - Add the two bar values for the total. - Subtract the Tide Pool value from the Sandy Beach value to find “how many more.”

1. Read the bars: Tide Pool has \(12\) shells, and Sandy Beach has \(20\) shells. 2. For part a), add: \(12 + 20 = 32\). 3. For part b), subtract: \(20 - 12 = 8\).

a) \(32\) shells b) \(8\) more shells

- Use the table for the category names, but use the stated scale to determine the numerical axis labels. - Translate each student count into a number of equal \(3\)-student grid intervals. - Keep the zero category on the horizontal axis even though its bar has height \(0\).

1. The horizontal axis should list the four activity categories: Board games, Cards, Puzzles, and Reading. 2. The vertical axis represents the number of students. With \(3\) students per interval and a greatest value of \(9\), its labels are \(0, 3, 6, 9\). 3. Board games has \(6\) students, so its bar is \(2\) grid intervals high. 4. Cards has \(0\) students, so its bar is \(0\) intervals high while the category label remains. 5. Puzzles has \(9\) students, so its bar is \(3\) intervals high. 6. Reading has \(3\) students, so its bar is \(1\) interval high.

Horizontal-axis labels: Board games, Cards, Puzzles, Reading Vertical axis: number of students, labeled \(0, 3, 6, 9\) Bar heights in grid intervals: Board games \(2\), Cards \(0\), Puzzles \(3\), Reading \(1\)

- Use the picture-graph key to convert each row into a data value. - Those data values become the bar heights. - Test each candidate scale interval against all three heights, then choose the largest one that works.

1. Butterflies represent \(3 \times 2 = 6\), bees represent \(2 \times 2 = 4\), and ladybugs represent \(4 \times 2 = 8\). 2. Intervals of \(1\) work for all three values. 3. Intervals of \(2\) also work for \(6, 4, 8\). 4. Intervals of \(4\) do not work for \(6\). 5. Therefore, \(2\) is the largest listed interval that works.

Bar heights: butterflies \(6\), bees \(4\), ladybugs \(8\). Use an interval of \(2\).

- 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. - For part c), combine Noah's and Ethan's values before comparing with Mia's.

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. Noah and Ethan read \(5 + 10 = 15\) books altogether, which matches Mia's \(15\) books.

a) Mia, \(15\) books b) \(50\) books c) Yes. Noah and Ethan read \(15\) books altogether, the same as Mia.

- 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 \(5\). - Compare the birch with each other tree for part b). - Add the fir and birch heights, then find another pair with the same sum. - Use subtraction to check the gardener's 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. The pine is \(30 - 15 = 15\,\text{ft}\) taller than the birch. 3. The fir and birch total \(45 + 15 = 60\,\text{ft}\). The oak and beech also total \(35 + 25 = 60\,\text{ft}\). 4. The fir is \(45 - 25 = 20\,\text{ft}\) taller than the beech, so 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. \(45 - 25 = 20\,\text{ft}\).

- 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, compare the sports value with half of \(50\). - Use addition for the combined-response statement. - Use subtraction to check the “fewer” statement.

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 \(45 + 15 = 60\). 6. The fourth statement is true because \(35 - 25 = 10\).

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 \(45 + 15 = 60\). Statement 4 is true because \(35 - 25 = 10\).

- 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 Pine Ridge and Stone Creek values for part d).

1. Read the values: Pine Ridge, \(14\); Stone Creek, \(8\); Maple Harbor, \(12\); Lake Bend, \(11\); Sunny Glen, \(9\) days. 2. Maple Harbor had \(12\) rainy days. 3. Stone Creek has the least value, \(8\) days. 4. Pine Ridge, Maple Harbor, and Lake Bend each had more than \(10\) rainy days, so the count is \(3\) towns. 5. Pine Ridge and Stone Creek had \(14 + 8 = 22\) rainy days altogether.

a) \(12\) rainy days b) Stone Creek c) \(3\) towns 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

- Read the values for Team Fox and Team Badger. - Decide the smallest total that is greater than Team Badger's total. - Subtract Team Fox's current total from that target.

1. Team Fox has \(15\) cards, and Team Badger has \(20\) cards. 2. To have more than \(20\), Team Fox needs at least \(21\) cards. 3. The number still needed is \(21 - 15 = 6\) cards.

Team Fox needs at least \(6\) more cards.

- Write down the four values from the graph. - Test pairs in an organized way. - Check that you found every possible pair.

1. Read the four values: North: \(15\), South: \(35\), East: \(25\), and West: \(10\). 2. Test pairs. North and East give \(15 + 25 = 40\). 3. No other pair has a total of \(40\).

The North and East entrances have exactly \(40\) visitors altogether.

- Read the Tuesday and Wednesday values. - Pay attention to the word “more.” - Subtract to state the corrected comparison exactly.

1. The graph shows \(15\) loaves on Wednesday and \(18\) loaves on Tuesday. 2. Since \(15 < 18\), Jonah's statement is false. 3. The difference is \(18 - 15 = 3\).

The statement is false. On Wednesday, \(3\) fewer loaves were sold than on Tuesday.

- Check each statement separately against the graph. - Pay attention to words such as “same,” “more,” and “most.” - For the total, add all four bar values.

1. Dogs and rabbits each received \(12\) votes, so A is true. 2. Cats received \(16 - 12 = 4\) more votes than dogs, so C is true. 3. B is false because cats received the most votes. 4. D is false because \(12 + 16 + 8 + 12 = 48\), which is not fewer than \(40\).

Statements A and C are true.

- Update Team Green's score first. - Keep the other teams' scores unchanged. - Compare all four scores after the change.

1. Team Green's score changes from \(30\) to \(30 + 6 = 36\). 2. Compare \(36\) with the other scores: \(25\), \(35\), and \(20\). 3. Since \(36\) is greatest, Team Green is in the lead.

Team Green is in the lead with \(36\) points.

- Change only Team Star's score. - Compare the new score with all three other scores. - Order the four scores from greatest to least.

1. Team Star's new score is \(40 - 8 = 32\) points. 2. The other teams have \(35\), \(40\), and \(45\) points. 3. Since \(32\) is the least score, Team Star is in fourth place.

Team Star is in fourth place.

- Add the bike and walking bars first. - Compare that sum with the car bar. - Subtract to find how much greater or smaller it is.

1. By bike and on foot together, \(12 + 8 = 20\) students travel to the field. 2. By car, \(16\) students travel to the field. 3. The difference is \(20 - 16 = 4\), so the bike-and-walking total is greater.

\(4\) more students travel by bike and on foot together than by car.

- Read each team's value from the graph. - “At least \(10\)” includes \(10\). - “No more than \(15\)” includes \(15\).

1. The values from \(10\) through \(15\), including both endpoints, are \(10\), \(15\), and \(15\). 2. Those values belong to Teams B, C, and F.

Teams B, C, and F.

- Read the cocoa and iced-tea values first. - Look for a value greater than \(15\) but less than \(25\). - Check that only one drink fits both conditions.

1. Cocoa received \(15\) votes, and iced tea received \(25\) votes. 2. Juice is the only drink between them, with \(20\) votes.

The mystery drink is juice.

- Leave out the Wednesday bar. - Add the other four bar values. - Pair numbers to make the addition easier.

1. The values for Monday, Tuesday, Thursday, and Friday are \(15\), \(20\), \(25\), and \(30\). 2. Add them: \(15 + 20 + 25 + 30 = 90\).

\(90\) pretzels were baked on all days except Wednesday.

- Find how far each estimate is from \(55\). - Use a positive difference whether the estimate is above or below \(55\). - Choose the least difference.

1. Find how far each estimate is from \(55\). 2. Group A: \(55 - 40 = 15\); Group B: \(55 - 50 = 5\); Group C: \(55 - 30 = 25\); Group D: \(70 - 55 = 15\). 3. The least difference is \(5\), for Group B.

Group B is closest to \(55\) steps.

- Add Lina's and Mina's amounts first. - Compare their total with \(45\). - Subtract to find how many are still needed.

1. Lina and Mina have \(20 + 15 = 35\) stickers altogether. 2. The number still needed is \(45 - 35 = 10\) stickers.

They need \(10\) more stickers.

- First determine how much each tick interval represents. - Match the top of Bar C to a scale value. - Check your reading by counting in fives.

1. The scale increases by \(5\) at each tick mark, not by \(1\). 2. Bar C ends at the ninth interval above \(0\), so its value is \(9 \times 5 = 45\).

Mara counted tick intervals instead of scale values. The correct value is \(45\).

- Add the orange and green values. - Subtract their sum from the total of \(60\). - Compare your result with the purple bar.

1. The orange and green strips total \(20 + 15 = 35\). 2. The remaining number is \(60 - 35 = 25\). 3. This matches the purple bar, which has a value of \(25\).

There were \(25\) purple paper strips.

- Identify the two shortest bars and add their values. - Identify the tallest bar. - Compare the sum with the tallest bar by subtraction.

1. The two least values are \(10\) and \(15\), with a sum of \(10 + 15 = 25\). 2. The greatest value is \(30\). 3. Since \(30 - 25 = 5\), the sum of the two least values is less by \(5\).

The two shortest bars together are \(5\) less than the tallest bar.

- Find the greatest and second-greatest bar values. - The second-greatest value is second place. - Subtract the two values to find the gap.

1. The two greatest values are Team Dune with \(40\) keys and Team Meadow with \(35\) keys. 2. Team Meadow is in second place. 3. The gap is \(40 - 35 = 5\) keys.

Team Meadow is in second place, \(5\) keys behind Team Dune.

- Double only the stone value. - Keep the wood and leaf values unchanged. - Add the three new values.

1. The number of stones changes from \(10\) to \(2 \times 10 = 20\). 2. The new total is \(15 + 20 + 20 = 55\) pieces.

The collage will have \(55\) pieces altogether.

- Identify the largest and second-largest flocks. - Find the difference between their sizes. - Check that removing that number makes the flock sizes equal.

1. The largest flock is the cranes with \(40\) birds. The second-largest flock is the geese with \(35\) birds. 2. The difference is \(40 - 35 = 5\) birds.

\(5\) cranes must leave the flock.

- Add the amounts in all four boxes first. - Compare that total with \(70\). - Subtract to find how many items must be added.

1. The current total is \(10 + 15 + 20 + 20 = 65\) items. 2. The number needed is \(70 - 65 = 5\) items.

\(5\) items must be added to Box S4.

- Find how many points each group below \(20\) needs. - Do not add points to groups already at or above \(20\). - Add the needed amounts.

1. Group A needs \(20 - 15 = 5\) points. 2. Group C needs \(20 - 10 = 10\) points. 3. Groups B and D already meet the goal. 4. Altogether, \(5 + 10 = 15\) points must be distributed.

\(15\) points must be distributed altogether.

- Check the total claim by addition. - Check the comparison claim by subtraction. - Decide whether each claim is true.

1. The total is \(24 + 16 = 40\), so the first claim is true. 2. The difference is \(24 - 16 = 8\), so the second claim is true. 3. Both claims match the graph.

Yes. Both claims match the graph.

- Subtract \(5\) from Group A and add \(5\) to Group B. - Keep Group C's value unchanged. - Compare the three new values.

1. Group A will have \(30 - 5 = 25\) cards. 2. Group B will have \(20 + 5 = 25\) cards. 3. Group C already has \(25\) cards.

All three groups will have \(25\) cards each.

- Read the spruce and pine values from the scale. - Find the bar whose value lies strictly between those two values. - Check the final condition against the shortest bar.

1. The spruce has a value of \(15\), and the pine has a value of \(25\). 2. The only bar with a value between \(15\) and \(25\) is the fir at \(20\). 3. The fir is not the shortest bar, so it satisfies every condition.

The mystery tree is the fir.

- Read all four bar values. - Add the values in convenient pairs. - Check that the total is reasonable compared with the bars.

1. The four values are \(120\), \(200\), \(160\), and \(160\). 2. Add them: \(120 + 200 + 160 + 160 = 640\).

The graph shows \(640\) rolls altogether.

- Check each part of the description separately. - Find the difference between D and C. - Find the difference between B and D.

1. Bar C has a value of \(18\), Bar D has \(21\), and Bar B has \(24\). 2. Since \(21 - 18 = 3\) and \(24 - 21 = 3\), both parts of the description are correct.

Yes. Nuri's description is correct.

- Write down all four values. - Add possible pairs. - Look for a single bar with the same value as a pair sum.

1. Bars A and B have a combined value of \(5 + 15 = 20\). 2. Bar C also has a value of \(20\).

Bar C is as tall as Bars A and B combined.

- Read all three bar values. - Add the values. - Compare the sum with \(200\).

1. The bars show \(70\), \(50\), and \(90\) tickets. 2. Their sum is \(70 + 50 + 90 = 210\). 3. Since \(210 \ne 200\), Leah is not correct.

No. The three booths sold \(210\) raffle tickets altogether.

- Identify and leave out the greatest bar. - Add the other three values. - Check that you used exactly three days.

1. Tuesday has the greatest value, \(25\). 2. The remaining values are \(15\), \(10\), and \(20\). 3. Their sum is \(15 + 10 + 20 = 45\).

The remaining three days have a sum of \(45\).

- Add the values of A and B. - Compare each other bar with that sum. - Remember that equal to is not the same as greater than.

1. Bars A and B have a combined value of \(10 + 15 = 25\). 2. Bar C has a value of \(20\), and Bar D has a value of \(25\). 3. Neither value is greater than \(25\).

No individual bar is taller than Bars A and B combined.

- Find each pair total separately. - Compare the two totals. - Subtract to find the difference.

1. Bars A and B total \(10 + 20 = 30\). 2. Bars C and D total \(25 + 10 = 35\). 3. The difference is \(35 - 30 = 5\).

Bars C and D have the greater total, by \(5\).

- Compare the same group in both graphs. - Find each change from a) to b). - Choose the greatest positive change.

1. Group A increased by \(15 - 10 = 5\) points. 2. Group B changed by \(20 - 20 = 0\) points. 3. Group C increased by \(25 - 15 = 10\) points. 4. Group D decreased by \(25 - 20 = 5\) points. The greatest increase is \(10\), for Group C.

Group C increased its score the most, by \(10\) points.

- Compare the same weekday in both graphs. - Find each decrease from a) to b). - Choose the greatest decrease.

1. The decreases are Monday: \(30 - 25 = 5\), Tuesday: \(35 - 20 = 15\), Wednesday: \(25 - 25 = 0\), and Thursday: \(30 - 20 = 10\). 2. The greatest decrease is \(15\), on Tuesday.

The number decreased the most on Tuesday, by \(15\) visitors.

- Find the total for each graph separately. - Compare the two totals. - Subtract to find how many more.

1. Week \(1\) total: \(15 + 25 + 20 = 60\) bags. 2. Week \(2\) total: \(20 + 15 + 30 = 65\) bags. 3. The difference is \(65 - 60 = 5\) bags.

Week \(2\) had \(5\) more bags collected.

- Add all four values in Graph a). - Add all four values in Graph b). - Compare the totals.

1. Graph a) has a total of \(10 + 20 + 15 + 25 = 70\). 2. Graph b) has a total of \(15 + 10 + 25 + 20 = 70\). 3. The totals are equal.

Yes. Both graphs have a total of \(70\).

- Compare the same region in both graphs. - Mark each change as an increase or a decrease. - Check all four regions.

1. South increased from \(25\) to \(30\), and West increased from \(20\) to \(25\). 2. North decreased from \(30\) to \(25\), and East decreased from \(35\) to \(30\).

1) South and West increased. 2) North and East decreased.

- Find the A–B gap in each graph. - Compare the two gaps. - State whether the gap became larger or smaller.

1. In a), the gap is \(40 - 30 = 10\) points. 2. In b), the gap is \(45 - 40 = 5\) points. 3. The gap becomes \(10 - 5 = 5\) points smaller.

The gap becomes \(5\) points smaller.

- Add the bike and walking values in each graph. - Compare each sum with the bus value in the same graph. - State the result for both a) and b).

1. In a), bike and walking total \(15 + 10 = 25\), the same as the bus value of \(25\). 2. In b), bike and walking total \(20 + 15 = 35\), while the bus value is \(20\). 3. In b), the combined total is \(35 - 20 = 15\) greater.

1) In Graph a), the numbers are equal. 2) In Graph b), \(15\) more students travel by bike and on foot together than by bus.

- Compare each product in the two graphs. - Find each increase or decrease. - Choose the greatest positive change.

1. Apple crates increase by \(25 - 20 = 5\). 2. Pear crates increase by \(25 - 15 = 10\). 3. Nut crates decrease by \(25 - 20 = 5\). 4. The greatest increase is \(10\), for pears.

Pears contribute the most, with an increase of \(10\) crates.

- Add the two values for one group at a time. - Record all four sums. - Compare the sums.

1. Group A: \(5 + 20 = 25\). 2. Group B: \(10 + 15 = 25\). 3. Group C: \(15 + 10 = 25\). 4. Group D: \(20 + 5 = 25\). All four sums are equal.

All four groups have the same sum: \(25\) each.

- Find the greatest and least value in each graph. - Subtract to find each difference. - Compare the two differences.

1. In a), the difference between the greatest and least values is \(25 - 10 = 15\). 2. In b), the difference between the greatest and least values is \(20 - 15 = 5\). 3. Since \(15 > 5\), the values are farther apart in a).

The greatest and least values are farther apart in Graph a).

- Order the three bars in each graph from least to greatest. - Compare the two orders. - Check whether any bar keeps the same value.

1. In a), \(A < B < C\). 2. In b), \(A > B > C\). 3. The order is reversed, and B stays at \(20\) in both graphs.

1) The order is reversed. 2) Bar B stays at \(20\).

- Find the total in each graph first. - Add the two graph totals. - Check that all six bars were included.

1. Graph a) has \(15 + 20 + 10 = 45\) objects. 2. Graph b) has \(20 + 15 + 15 = 50\) objects. 3. Together, the graphs show \(45 + 50 = 95\) objects.

The two graphs show \(95\) objects altogether.

- Check the value in a) first. - Then check the same group in b). - Apply both conditions to every group.

1. Group A increases from \(15\) to \(20\), so it meets both conditions. 2. Group C remains below \(20\). 3. Groups B and D were already above \(20\) in a).

Only Group A meets both conditions.

- Add each group's two values. - Record the four totals. - Choose the greatest total.

1. Group A: \(10 + 20 = 30\). 2. Group B: \(15 + 10 = 25\). 3. Group C: \(20 + 15 = 35\). 4. Group D: \(25 + 20 = 45\). The greatest total is \(45\).

Group D has the greatest combined total, \(45\).

- Find the total of each graph. - Find the difference between the totals. - Only Bar C changes.

1. Graph a) has a total of \(20 + 15 + 25 = 60\). 2. Graph b) has a total of \(15 + 20 + 20 = 55\). 3. The difference is \(60 - 55 = 5\), so C must increase by \(5\).

Bar C in b) must increase by \(5\).

- 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.

- Test each listed interval against all four values. - Eliminate any interval that would put even one bar endpoint between gridlines. - Among the intervals that work, choose the largest one.

1. Intervals of \(10\) work because all four values land on gridlines. 2. Intervals of \(20\) also work: \(80, 100, 120, 160\) all land on gridlines. 3. Intervals of \(50\) do not work because, for example, \(120\) lies between two \(50\)-visitor gridlines. 4. Of the working choices, \(20\) is the largest interval.

Use \(20\) visitors per gridline. It is the largest listed interval that places every value on a gridline.

- 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 \(15\).

- Check whether both scales can show each data value exactly. - Count equal jumps from \(0\) to \(100\) on each scale. - Compare the two interval counts.

1. Both interval sizes place every data value on a gridline, so both graphs can show the data correctly. 2. Counting by \(25\) from \(0\) to \(100\) gives \(4\) intervals. 3. Counting by \(5\) from \(0\) to \(100\) gives \(20\) intervals. 4. Graph A uses fewer intervals.

Both graphs are correct. Graph A has \(4\) intervals, Graph B has \(20\) intervals, and Graph A uses fewer intervals.

- 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.

- Read the soccer and basketball values from one of the candidate graphs. - Subtract those two known categories from the total of \(24\). - Choose the graph whose volleyball bar matches the remaining number.

1. Read the fixed bars from the candidate graphs: soccer, \(11\); basketball, \(5\). 2. Find the missing number of volleyballs: \(24 - 11 - 5 = 8\). 3. 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.”

- One category loses \(4\) artworks and another category gains \(4\). - Check that the total number of artworks stays the same.

1. For Wood, subtract \(4\): \(19 - 4 = 15\). 2. For Paper, add \(4\): \(17 + 4 = 21\). 3. The Clay and Fabric values stay the same.

Clay: \(13\) Wood: \(15\) Fabric: \(11\) Paper: \(21\)

- Read each shown bar height from the vertical scale before doing any arithmetic. - Add the three displayed counts. - Compare their sum with the stated total of \(80\).

1. Read the graph: North is \(18\), South is \(24\), and West is \(16\). 2. Add the shown bars: \(18 + 24 + 16 = 58\). 3. 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 \(40\).

- 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.

- Use the vertical scale to read each bar exactly. - For the total, combine all three category values. - For “how many more,” subtract the Mystery value from the Humor value.

1. Read the bars: Mystery has \(8\) votes, Adventure has \(16\) votes, and Humor has \(24\) votes. 2. For part a), add the votes: \(8 + 16 + 24 = 48\). 3. For part b), subtract: \(24 - 8 = 16\).

a) \(48\) votes b) \(16\) more votes

- Match the categories in order before comparing heights. - A zero category should still be labeled but should have no bar above the baseline. - Use the scale of \(5\) to check each nonzero endpoint.

1. A correct graph needs bar heights \(5, 0, 10, 15\) in the order pencils, erasers, rulers, markers. 2. Graph a) has those values and keeps the erasers category with no visible bar above zero. 3. Graph b) incorrectly gives erasers a height of \(5\). 4. Graph c) incorrectly gives rulers a height of \(5\).

Graph a) is correct.

- Write down the four values. - Find the difference for each possible pair in an organized way. - Make sure you list every pair with a difference of \(8\).

1. Hedgehog and Mouse differ by \(20 - 12 = 8\). 2. Frog and Owl differ by \(16 - 8 = 8\). 3. Checking the other pairs shows that none of their differences is \(8\).

Hedgehog and Mouse; Frog and Owl.

- Write down all four scores. - Compare the differences between possible pairs. - Choose the least positive difference.

1. Compare the score differences for all pairs. 2. The least difference is between Group B with \(30\) points and Group C with \(25\) points. 3. Their difference is \(30 - 25 = 5\) points.

Groups B and C have the closest scores. Their scores differ by \(5\) points.

- Write down the four point values. - Test ways to split the groups into two pairs. - Add both pair totals to check that they are equal.

1. Groups A and B have \(10 + 25 = 35\) points. 2. Groups C and D have \(15 + 20 = 35\) points. 3. The two pair totals are equal.

Pair A with B and pair C with D. Each pair has \(35\) points.

- Find the current greatest score. - Determine the least score that is greater than it. - Subtract Group C's current score from that target.

1. Group C has \(20\) points, and the current greatest score is \(25\). 2. To be alone in first place, Group C must have at least \(26\) points. 3. It needs \(26 - 20 = 6\) more points.

Group C needs at least \(6\) more points.

- Write down all five day values. - Test pairs in an organized way. - List every pair with a sum of \(36\).

1. Monday and Friday give \(12 + 24 = 36\). 2. Tuesday and Thursday give \(15 + 21 = 36\). 3. Checking the other day pairs shows that none has a sum of \(36\).

Monday and Friday; Tuesday and Thursday.

- Check the \(50\)-cap scale one value at a time. - Test each candidate interval against all four bar values. - Choose the largest candidate that makes every endpoint land exactly on a gridline.

1. With intervals of \(50\), only \(150\) lands on a gridline. The bars at \(60, 90, 120\) end between gridlines. 2. Intervals of \(10\) work for all four values. 3. Intervals of \(20\) do not work for \(90\) or \(150\). 4. Intervals of \(30\) work for all four values, and \(50\) does not. 5. Therefore, \(30\) is the largest listed interval that works.

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.