Line Graphs, Time Graphs and Tables

A line graph shows how an amount changes. Each reading becomes a point. Straight pieces of line join the points in order. When time runs along the bottom, you can also call it a time graph. You may hear the names "broken-line graph," "trend graph," or "time-series graph."

Start with the title. "Temperature in Our Schoolyard on May 3" tells you the place, date, and measurement. Read the two axes, the lines along the bottom and side. The horizontal axis is also called the x-axis. It usually shows time. The vertical axis, or y-axis, shows the measured or counted amount.

The labels tell you what each axis shows. The unit tells you how an amount is measured. A label such as "Height (cm)" means centimeters. "Temperature (°F)" means degrees Fahrenheit. A point is like an ordered pair. On a plant graph, $(3,9)$ means week 3 and a height of 9 cm. Read across first, then up.

Work out one small step

The scale is the set of numbers on an axis. Equal distances stand for equal amounts. A scale can count by 2, 5, 10, 25, or 100. A small square does not always mean 1.

Suppose two nearby labels are 40 and 50. There are five equal spaces between them.

1. Find the difference: $50-40=10$.
2. Share that difference among the five spaces: $10÷5=2$.
3. Count up from 40: 42, 44, 46, 48, 50.

Count the spaces, not the gridlines around them. Five spaces have six boundary lines, including both ends.

Now use a scale labeled 0, 20, 40, 60, and 80. Four spaces separate each pair of labels. Each space is worth $20÷4=5$. The third gridline above 40 is 45, then 50, then 55. A point there shows 55, even though the number 55 is not printed.

Check where the scale begins. A temperature scale might begin at 40 °F. A small zigzag, called an axis break, shows the skipped part below it. The bottom of that graph stands for 40 °F, not 0 °F.

Read a scale with bigger numbers

Suppose a visitor graph has labels 0, 100, and 200. Five small spaces lie between 100 and 200. The difference is 100 visitors, so each space stands for $100÷5=20$ visitors.

A point on the third gridline above 100 shows 160 visitors. Count 120, 140, then 160. You count three spaces from a known label. Starting at 0 and counting three spaces would land at 60, a different height.

The two axes can use different step sizes. On this graph, suppose two horizontal squares stand for one hour. One horizontal square then means half an hour, while one vertical square means 20 visitors. A square's meaning comes from its own axis.

You can mark the small steps beside the axis before reading. Above 100, write 120 at the next gridline and 140 at the next. Check that five steps really reach 200. If your count misses that label, check the number of spaces or the division before reading any points.

A label belongs beside a line, not inside a square. That is why the spaces between two labeled lines are the parts you divide.

Imagine drawing a line graph from these schoolyard readings. Time goes across the bottom. Temperature goes up the side, from 40 °F to 72 °F. Each small vertical step is 2 °F.

Each row gives one point. Join neighboring points from left to right.

Start with a time

To answer "How warm was it at 10 a.m.?", find 10 a.m. along the bottom. Go straight up until you reach its point. Then move straight across to the temperature scale. You reach 58 °F. You can check the last few gridlines by counting 52, 54, 56, 58.

A ruler or the edge of paper can help you follow a straight path. Slanting across to the scale can land you on the wrong number. Write "58 °F" with the unit, so your answer tells the whole reading.

Start with an amount

You can follow the path backward. For "When was it 70 °F?", start at 70 on the vertical axis. Move across to the line, then down to the time. The point is above 2 p.m.

Watch what the question asks. "When was it warmest?" needs a time: 2 p.m. "What was the highest temperature?" needs an amount: 70 °F. Both use the highest point, but their answers mean different things.

The lowest point gives the smallest reading. Here that is 48 °F at 6 a.m. The bottom of the scale is 40 °F, but no reading was that low. Read the plotted points when you look for the lowest recorded amount.

An amount may appear more than once. The line reaches 66 °F at noon. It reaches that height again while cooling after 4 p.m. Trace across the whole graph when you look for every time. A later meeting between plotted points gives an estimated time.

A point does not have to sit on a numbered gridline. Look at how far it lies between the lines. On a scale marked every 10, halfway between 50 and 60 is about 55. A point just above 50 could be about 51 or 52. Use "about" for a reading you judge by eye.

You can also read the connecting line between two measured points. The schoolyard was 58 °F at 10 a.m. and 66 °F at noon. What does the line suggest for 11 a.m.?

1. Eleven is halfway through the two-hour gap.
2. The full rise is $66-58=8$ degrees.
3. Half the rise is $8÷2=4$ degrees.
4. Add it to the first reading: $58+4=62$.

The estimate is about 62 °F.

Find an in-between time

For 60 °F, start on the temperature axis and look across. The meeting lies between 10 a.m. and noon. The rise from 58 to 60 is 2 degrees. That is one quarter of the full 8-degree rise. One quarter of two hours is 30 minutes. So the line reaches 60 °F at about 10:30 a.m.

Now look for 66 °F again in the afternoon. The readings fall from 68 °F at 4 p.m. to 62 °F at 6 p.m. To reach 66, the line falls 2 of the full 6 degrees. That is one third of the time gap. One third of two hours is 40 minutes. The second time is about 4:40 p.m.

Know what the connecting line means

The straight line treats the change as steady between readings. The real air could have warmed or cooled a little along the way.

Temperature, height, and distance are continuous measurements. They can have values between readings. A monthly book count is different. A point halfway between October and November does not give a real monthly count. Use the plotted points for those amounts.

Past the last point, you are making a prediction. A plant measuring 18 cm in week 5 and 20 cm in week 6 might grow more slowly later. Its graph cannot tell you its week 20 height. Extend a pattern only with a reason to expect it to continue, and call the result a prediction.

Follow a falling segment

The same halfway method works when an amount falls. At 4 p.m. the schoolyard is 68 °F; at 6 p.m. it is 62 °F. To estimate 5 p.m., begin at the earlier reading.

1. The full fall is $68-62=6$ degrees.
2. Five o'clock is halfway through the two-hour gap.
3. Half the fall is $6÷2=3$ degrees.
4. Subtract from the starting reading: $68-3=65$.

The estimate is about 65 °F. Adding 3 would give 71 °F, above both endpoints, even though this segment falls. A useful check is whether your in-between estimate lies between the two endpoint amounts.

Read the time limits too

The last schoolyard reading is at 6 p.m. A question about 5 p.m. falls inside the recorded time span. A question about 7 p.m. lies outside it, even if there is room on the paper to draw farther.

For the bean plant, continuing the last 2 cm of weekly growth would suggest 22 cm in week 7. That is a prediction based on the growth continuing. The measured week 6 point is still 20 cm; extending its pattern does not add a recorded week 7 measurement.

Some straight-line graphs link two amounts without showing time. Think about apples costing $2 per pound. Put pounds across the bottom and cost in dollars up the side.

These points lie on one straight line. Values between the points have meaning too. You can buy 3.5 pounds. Go up from 3.5 pounds to the line, then across: the cost is $7. The price rule checks it:

To find how much $9 buys, start at 9 on the cost axis. Go across to the line, then down. You reach 4.5 pounds. The rule gives $4.5\times 2=9$. Start on the axis for the amount you know.

Change the unit

A conversion graph connects two units for the same amount. Five miles is about 8 kilometers. Draw miles horizontally and kilometers vertically. The straight line goes through $(0,0)$, $(5,8)$, and $(10,16)$.

From 3 miles, read up and across to about 4.8 km. From 12 km, read across and down to about 7.5 miles. Keep "about": this graph uses an approximate conversion.

You can scale up beyond its edge because the conversion stays the same. Ten miles is about 16 km. Fifty miles is five times as far, so it is about $5\times 16=80$ km.

Look for a starting charge

A straight line can begin above zero. A taxi charges $3 to start, then $2 for each mile. At 0 miles its cost is already $3. At 1 mile it costs $5. At 3 miles it costs $9.

For 6 miles, first find the distance charge: $6\times 2=12$. Add the starting charge once: $12+3=15$. The ride costs $15.

Doubling the 3-mile cost would give $18. That counts the starting charge twice. Only double the whole cost when the straight-line relationship starts at zero and keeps the same price per unit. Check where the line meets the vertical axis before scaling up.

A graph gives a quick reading. When you know an exact price rule, calculate with it to check a reading between gridlines.

Compare one ride with two rides

A starting charge also explains the cost of splitting a journey. With the same taxi prices, one 6-mile ride costs $15. Two separate 3-mile rides each cost $9, so together they cost $9+9=18$ dollars.

The total distance is 6 miles in both cases. The difference is the number of starts. Two rides include two starting charges, while one ride includes one. Their $3 difference is one extra starting charge.

Read the point at 0 miles to find this fixed part of the cost. The line begins at $3, even before any miles are added. On the apple graph, the point at 0 pounds has cost $0 because no apples have been bought.

Scale a conversion beyond the drawing

Suppose your miles-to-kilometers graph ends at 10 miles and about 16 km. You want a distance of 30 miles.

1. Work out the scale-up: $30÷10=3$.
2. Multiply the matching kilometer amount: $3\times 16=48$.
3. Report about 48 km, keeping the approximate conversion.

The units must stay on their own sides of the comparison. You scaled 10 miles to 30 miles, so you must scale 16 km by the same factor. You are not adding miles to kilometers.

Read a time graph from left to right. An upward line means the amount increases. A downward line means it decreases. A flat line means the amount stays the same. A trend is the general way the amount changes over time.

A steeper piece shows faster change on the same graph. A gentle upward piece still shows an increase, but a slower one. A gentle downward piece still shows a decrease. Check the direction as well as the steepness.

Watch a bean plant grow

The line rises every week because every height is larger. Its steepest part is from week 3 to week 4. The plant grew $14-9=5$ cm in that week. After week 4 it still grows, but its weekly growth gets smaller. A line that becomes flatter can still be going up.

These gaps are all one week long. That lets you compare the changes directly. If the time gaps differ, a bigger increase might simply have had more time. Use the steepness on correctly spaced axes to compare how fast the change happened.

Follow a bike trip

Sam leaves home at 10:00 a.m. and reaches a park at 10:30 a.m. The park is 3 miles from home. Sam stays there until 11:00 a.m., then rides home, arriving at 11:20 a.m.

On a distance-from-home graph, the line rises from 0 to 3 miles. It stays flat at 3 miles during the park stop. Then it falls to 0 miles. The flat section is not zero: Sam is still 3 miles from home. The falling line shows Sam getting closer to home; it says nothing about hills.

This sketch uses minutes after 10:00 a.m. along the bottom:

Distance from home (miles)
3 |      ●─────●
2 |    ╱        ╲
1 |  ╱           ╲
0 |●               ●
  +------------------
   0     30    60  80
Minutes after 10:00 a.m.

Sam rides the same 3 miles each way. The outward trip takes 30 minutes; the return takes 20. The return line is steeper, showing the faster ride.

When you describe a graph, use its amounts and times. For example: "The plant grew fastest between weeks 3 and 4, gaining 5 cm." Someone can check that sentence against the points.

Give a large change the right amount of time

From week 4 to week 6, the plant grows from 14 to 20 cm. That is a 6 cm increase, larger than the 5 cm increase from week 3 to week 4. But the 6 cm increase takes two weeks.

Look at those two weeks separately. Week 4 to week 5 adds 4 cm. Week 5 to week 6 adds 2 cm. Neither one-week section is as steep as the 5 cm rise from week 3 to week 4. A bigger total across a longer time does not by itself identify the fastest change.

Keep track of what distance means

Sam's last point is at 0 miles from home. That does not mean Sam traveled 0 miles. Sam rode 3 miles out and 3 miles back, traveling 6 miles altogether.

The graph tracks distance from home, not all the miles already traveled. On the return, this distance gets smaller while the number of miles ridden keeps growing. Read the axis label before telling the trip's story. Two points at the same height can happen at different stages: Sam is at home at the start and at home again at the end.

A graph question often needs a reading and a calculation. First write down the amounts you need. Then decide whether the question asks for a difference or a sum.

"How much more?" and "How much did it change?" ask for subtraction. The schoolyard warmed from 48 °F at 6 a.m. to 70 °F at 2 p.m. The rise was $70-48=22$ °F. Later it cooled to 62 °F. That fall was $70-62=8$ °F.

For a change over several readings, use the first and last amounts. The plant was 5 cm tall in week 2 and 18 cm tall in week 5. It grew $18-5=13$ cm. Adding the heights would count the same plant's earlier height again. You can also add its separate weekly growth amounts: $4+5+4=13$ cm.

Find a useful total

"How many altogether?" asks for addition when the amounts belong together. Here are books borrowed in separate months.

October and November together give $180+210=390$ books. To total all five months, add in steps: $140+180=320$, then $320+210=530$, then $530+150=680$, then $680+190=870$ books.

These monthly counts make a useful total. Adding 48 °F and 52 °F would not give the temperature of anything. Think about what a total would mean before adding.

Compare months

November has the highest point: 210 books. It has $210-150=60$ more books than December. November to December is the only downward piece, so it is the only month-to-month drop.

Keep the units attached. A temperature change is measured in degrees. Plant growth is measured in centimeters. Subtracting week numbers gives elapsed weeks, not growth. Counting three vertical squares only gives a change after you multiply by the scale's step size.

Check the direction of your result against the line. Over equal time gaps, a steeper rise should give a larger increase. If the line rises but your written answer says "fell," reread the two points.

Decide which readings belong in the calculation

A question might compare two groups of months. For example, how many more books were borrowed in November through January than in September and October?

First sort the readings into the two groups:

• November, December, January: 210, 150, 190 books.
• September, October: 140, 180 books.

Find each subtotal before comparing them. The first is $210+150+190=550$ books. The second is $140+180=320$ books. The difference is $550-320=230$ books. Mixing a month into both groups would change the answer.

What if a reading in the middle is missing?

The plant's week 2 and week 5 heights still tell you the total growth between them. You can subtract 5 cm from 18 cm even without a week 3 reading. Both endpoint amounts are known.

A different question, such as which individual week had the greatest growth, needs the weekly readings. The two endpoints tell you the total change across the interval; they do not show how that growth was shared among its weeks.

Two lines can share one pair of axes. A key, or legend, tells you which line belongs to which group. A solid line might show Hillview. A dashed line might show Bayside. Use the key before following either line.

Imagine a graph of the towns' average high temperatures each month.

To compare one month, keep your finger above that month. Read both lines there. In March, Bayside is warmer by $56-45=11$ °F. In June, Hillview is warmer by $78-68=10$ °F. Comparing Hillview in March with Bayside in June would mix two different times.

The vertical gap shows the difference. Look for the widest gap, then check the numbers. January's gap is $50-30=20$ °F. That is the largest gap in this set of readings.

When the lines cross

Hillview is lower in April: 58 °F compared with 60 °F. In May it is higher: 69 °F compared with 64 °F. The connecting lines cross between those month labels. At their crossing, both drawn lines have the same value.

Before that crossing, Bayside's line is higher. Afterward, Hillview's is higher. These are monthly average highs. The crossing does not give a measured temperature on a particular day.

Higher and steeper mean different things

A higher point means a larger amount at that time. A steeper line shows faster change. Hillview starts lower, but rises much more from January to June:

• Hillview: $78-30=48$ °F.
• Bayside: $68-50=18$ °F.

Both changes cover the same months. Hillview's line climbs more steeply overall. A line can be lower at the start and still have the greater increase.

If you lose track at a crossing, return to a part where the lines are easy to separate. Check the color or line style there, then follow the same line to the month you need.

Watch how the gap changes

Both lines can rise while the gap gets smaller. From January to February, Hillview rises from 30 to 34 °F. Bayside rises from 50 to 52 °F. Hillview gains 4 degrees while Bayside gains 2, so Hillview catches up a little.

The gap changes from $50-30=20$ °F to $52-34=18$ °F. Bayside is still warmer, but only by 18 degrees. A shrinking gap does not mean both towns are cooling.

Later, from May to June, both towns warm again. Hillview rises from 69 to 78 °F, while Bayside rises from 64 to 68 °F. The gap grows from 5 to 10 degrees because the already warmer town has the larger rise.

To check the closest pair of recorded readings, compare the monthly gaps:

April's points are closest, just 2 degrees apart. That is different from a crossing between month labels, where the connecting lines meet. A question about recorded months uses the month points.

Plan the graph before plotting. Read the table to find the earliest and latest times and the largest amount. Time goes on the horizontal axis. The measurement goes on the vertical axis.

For the bean plant, time runs from week 0 to week 6. The greatest height is 20 cm. Suppose your grid is ten squares tall. A vertical step of 2 cm per square reaches 20 cm at the top:

A step of 1 cm would reach only 10 cm. You would run out of paper. A step of 5 cm would fit, but 20 cm would use only four squares. A small, squashed graph makes nearby values harder to read. Choose a simple step that fits and uses most of your space.

Another table has a largest value of 47. With ten squares, steps of 5 reach 50 and work well. Steps of 4 reach only 40. Steps of 10 fit, but leave the upper half mostly empty. Useful steps include 1, 2, 5, 10, 20, 25, 50, and 100.

Keep each step equal

Choose a step and keep it along the whole axis. Labels 0, 5, 10, 20, 40 cannot be equally spaced. The last jump is much larger than the first. Giving those jumps the same space would change the line's shape.

The horizontal scale needs equal steps too. Suppose readings were taken on days 1, 2, 4, and 7. With one square per day, use this spacing:

• Day 2 is one square after day 1.
• Day 4 is two squares after day 2.
• Day 7 is three squares after day 4.

Leave room for days 3, 5, and 6, even though no readings were taken then. Four readings do not mean four equally spaced times.

Finish the labels

Start the vertical scale at 0 for an ordinary graph. If the values are far above 0, you can begin higher and show a small zigzag break. The schoolyard's 48–70 °F readings fit on a scale starting at 40 °F. Readers must be able to see that skipped part.

Write "Height (cm)" beside the plant's vertical axis and "Time (weeks)" below the horizontal axis. Put scale numbers beside the gridlines, not halfway between them. That shows exactly which height or time each line stands for.

Plan the width as well as the height

You have a grid twelve squares wide and ten squares tall. For the bean plant, six weeks must fit across and 20 cm must fit up the side.

Use two horizontal squares for each week. The six week-spaces then use $6\times 2=12$ squares. Vertically, use 2 cm per square, reaching 20 cm in ten squares. The two directions have their own meanings: two squares across represent one week; two squares up represent 4 cm.

Before plotting, write the horizontal labels 0, 1, 2, 3, 4, 5, 6 under every second gridline. Check that week 6 reaches the last vertical gridline. Label the height scale separately.

For days 1, 2, 4, and 7, try a different plan. An axis from day 0 to day 7 needs seven day-spaces. With ten squares available, one square per day fits. Two squares per day would need fourteen squares and would run out of room.

Count the full time span when planning. Counting only the four readings would leave too little room for the unmeasured days between them. Leave those time positions in place before making any dots.

Use the bean plant readings to make your graph. Its heights in weeks 0–6 are 0, 2, 5, 9, 14, 18, and 20 cm. Use one horizontal square per week and one vertical square per 2 cm.

Plot one row at a time

Take week 3, when the plant is 9 cm tall.

1. Find 3 on the horizontal axis.
2. Move straight up to 9 cm.
3. Make a small dot or cross there.

Nine is halfway between the 8 cm and 10 cm gridlines. Put the point halfway between them. It is four and a half squares above 0 because each square stands for 2 cm. Putting it nine squares up would show 18 cm.

Do the same for week 2 at 5 cm. Its point lies halfway between 4 cm and 6 cm. Week 4 at 14 cm sits exactly on a gridline. Small points help you see their exact positions; a thick blob can cover several possible readings.

The first point is week 0 at 0 cm. The last is week 6 at 20 cm. Count your plotted points: seven rows should give seven points. A missing point can hide a change that matters.

Join neighboring points

Use a ruler to draw a straight segment from week 0 to week 1. Then join week 1 to week 2, continuing through week 6. Visit every point in time order. A single line from the first point to the last would miss the weeks when growth was faster or slower.

Add the title "Height of Our Bean Plant." The title names what grew. The labels "Time (weeks)" and "Height (cm)" give the meaning of the numbers.

Read your graph backward

Check each point against its row in the table. Above week 3, read across: do you get 9 cm? Above week 5, do you get 18 cm? If a point is wrong, check the scale and move that point before redrawing its two joining segments.

The plant grew every week, so every segment should rise. A downward segment signals a plotting problem for these readings. This shape check helps, but it does not replace checking every number.

You can also make a graph in a spreadsheet. Check its title, labels, units, and chosen scale. With uneven measurement times, make sure the horizontal spacing matches the actual time gaps.

A two-way table sorts the same people or things in two ways. Here one way is grade. The other is how students travel to school.

Each box is a cell. To find Grade 5 bus riders, follow the Grade 5 row across to the Bus column. Their cell contains 6. Both headings matter: these students are in Grade 5 and ride the bus.

A question about all walkers uses the Total row: 27. A question about Grade 4 walkers uses one grade's cell: 12. Look for whether the question names one group or everyone.

Read and check totals

The Total column adds across each grade. For Grade 4, $12+9+7=28$. The Total row adds down each travel type. For walkers, $12+15=27$.

The bottom-right cell is the grand total, 57 students. You can check it in two ways:

• Add the grade totals: $28+29=57$.
• Add the travel totals: $27+15+15=57$.

Both count the whole group. Adding the row totals and column totals together would count everyone twice.

Use a total to fill a gap

Suppose Grade 4's Car cell is missing. Its row total is 28. Walk and Bus already account for $12+9=21$ students. Subtract: $28-21=7$. So 7 Grade 4 students travel by car.

For several gaps, find a row or column with one unknown first. Suppose Grade 5 Walk, the Bus total, and the Car total are missing. The Walk total is still 27.

1. Grade 5 Walk: $27-12=15$.
2. Bus total: $9+6=15$.
3. Car total: $7+8=15$.
4. Check the grand total: $27+15+15=57$.

Each filled cell can help with the next one.

Build a table from a list

Draw a row for each grade and a column for each travel type. Read one student's two facts. A Grade 4 bus rider gets one tally in the Grade 4–Bus cell. Continue through the list, counting each student once. Turn the tallies into numbers and add the totals. The grand total should match the number of students in your list.

Fill a row total before another cell

Suppose the Grade 5 total and its Walk cell are missing. You still know the grand total is 57 and the Grade 4 total is 28.

1. Find Grade 5's total: $57-28=29$.
2. Grade 5 already has 6 bus riders and 8 car riders.
3. Those known cells contain $6+8=14$ students.
4. The missing Walk cell is $29-14=15$ students.

You can check that result down the Walk column: $12+15=27$, which agrees with its total. A row and a column can check the same cell from different directions.

Try a short list

Here is a fresh four-student list: Mia is in Grade 4 and walks; Evan is in Grade 5 and goes by car; Luca is in Grade 4 and takes the bus; Sara is in Grade 4 and walks.

Start a new, small table for this list. Mia and Sara add two tallies to Grade 4–Walk. Luca adds one to Grade 4–Bus. Evan adds one to Grade 5–Car. All other cells have zero.

Grade 4's row total is 3 and Grade 5's is 1. Walk totals 2, Bus totals 1, and Car totals 1. Both routes give a grand total of 4, matching the four names.

A timetable, or schedule, is a table of times. Read the headings first. In this timetable, a row is a stop and a column is one bus trip. All times are a.m.

To follow Bus 2, stay in its column: Main Street at 8:15, Library at 8:22, Park Gate at 8:34, School at 8:45. Reading across the Library row would switch between different buses.

The dash means Bus 3 does not stop at the Library. It is not a missing time to work out. You cannot board or leave that bus there.

Find how long a trip takes

Bus 2 leaves Main Street at 8:15 and reaches School at 8:45. That takes 30 minutes. Keep both readings in the same bus column.

For Bus 1 from Library to Park Gate, count through the hour:

1. From 7:52 to 8:00 is 8 minutes.
2. From 8:00 to 8:04 is 4 minutes.
3. Add: $8+4=12$ minutes.

Bus 3 takes 19 minutes from Main Street to Park Gate: 10 minutes from 8:50 to 9:00, then 9 minutes to 9:09. Subtracting $909-850=59$ treats an hour like 100 minutes. Clock hours have 60 minutes, so count on through the hour.

Choose a bus and find the wait

You need to reach School by 8:40 a.m. Start in the School row. Bus 1 arrives at 8:15, which is early enough. Bus 2 arrives at 8:45, which is too late. Choose Bus 1, then read up its column for your boarding time.

If you reach the Library at 8:10, Bus 1 has already gone. Bus 2 stops there at 8:22. Your wait is 12 minutes.

If you reach the Library at 8:30, neither Bus 1 nor Bus 2 can help. Bus 3 is later, but its dash means it skips your stop. No later bus in this timetable stops there.

Some timetables put trips in rows and stops in columns. The headings, not the direction on the page, tell you how to follow one journey.

Find the latest trip that still works

Suppose you reach Main Street at 7:40 a.m. and need to reach School by 8:50 a.m. You want the latest possible departure.

Bus 1 leaves at 7:45 and arrives at 8:15. Bus 2 leaves at 8:15 and arrives at 8:45. Both can be boarded and both arrive early enough. Bus 3 arrives at 9:20, too late.

Choose Bus 2 for the latest departure. Your wait is 20 minutes from 7:40 to 8:00, then 15 more minutes: 35 minutes. The ride itself lasts 30 minutes. From reaching the stop to reaching School, the total is $35+30=65$ minutes. A ride time and a waiting time answer different parts of the journey.

Read a timetable turned sideways

The same times could be arranged this way, all a.m.:

Now follow a row to stay with one bus. To compare buses at the Library, read down that column. The trip has not changed; the headings have moved. Find the named trip and stop each time instead of remembering only a direction on the page.

A table and a line graph can show the same readings. A table puts each time beside its amount. A line graph places those pairs on axes, making the shape of the change easier to see.

To turn the schoolyard graph into a table, make headings "Time" and "Temperature (°F)." Read each plotted point, then write its pair in a row. The first rows are 6 a.m. with 48 °F, 8 a.m. with 52 °F, and 10 a.m. with 58 °F. Continue through noon, 2 p.m., 4 p.m., and 6 p.m.

Read the points taken as measurements. Do not quietly add an 11 a.m. row as though someone measured it. The line can suggest an estimate there, but that row would need an "estimated" label.

Turn rows into points

Going the other way, take each table row and find its time horizontally and amount vertically. Plot one point for each pair. Put the rows in time order before joining. If noon is written after 2 p.m. in a table, noon's point still belongs to the left of 2 p.m.

Check the table and graph one pair at a time. Suppose the table says noon was 66 °F, but the noon point is level with 64 °F. Follow the point across again. Check the scale's step size. Once you know which display has the error, correct it using the recorded reading.

A graph can have the right general shape and still contain a wrong point. Moving the noon point down by 2 degrees might leave the line rising. Checking only whether it rises would miss that error.

Choose the view for your question

For 11 a.m., the schoolyard table has no recorded reading. The connecting line suggests about 62 °F. The graph adds a way to estimate between measured times; it does not turn that estimate into an exact observation.

A plant can grow through 9 cm, 9.5 cm, and 9.51 cm on its way to 10 cm. Height is continuous data: a measurement can take any value in a range. Temperature, time, and distance are continuous too.

A book count works in separate steps. A library can lend 180 books or 181 books during a month. It cannot lend 180.5 whole books. Counts like books, people, and goals are discrete data.

Join measurements over time

A plant measured in week 3 and week 4 also has a height between those times. Joining its points helps you estimate that height. The line treats its growth as steady between readings. A point halfway along gives an estimate halfway through the change.

How often you measure does not decide the data type. Reading the temperature every two hours still gives continuous data. Counting visitors every hour still gives discrete data. Think about what the amount measures or counts.

Show a trend in counts

You can join monthly library counts to make a trend easy to follow. The line can show a rise from October to November and a fall afterward. But a point halfway between those months is not a monthly count. Read the real monthly amounts at their plotted points.

The order matters. Months have an order, so the line helps you follow time. Favorite fruits have no such order. Putting apples before bananas is a choice, and another order would change the joining line. Use a bar graph to compare those separate categories.

Before estimating between points, ask:

1. Is the amount a continuous measurement, such as height or temperature?
2. Does the horizontal axis have a real order, such as time?

When both answers are yes, an in-between estimate makes sense. Keep the word "about," because a straight segment fills a gap between measurements.

Choose a display to answer your question. Use a line graph to follow temperature through a day. Use bars to compare favorite sports. A two-way table shows counts sorted in two ways, such as grade and travel type. A table is useful for looking up exact recorded values.

Sometimes more than one display works. Monthly book counts can use bars for comparing months or a line for following the trend. Explain your choice by naming the data and purpose: "I chose a line graph to show how the book count changed from month to month."

Check a dramatic-looking rise

A lemonade stand sells 200 cups in week 1 and 210 cups in week 2. Its graph starts the vertical axis at 195. The first point is 5 cups above the bottom. The second is 15 cups above it. The second height above the bottom is three times the first.

The sales did not triple. The actual increase is $210-200=10$ cups. The starting value makes that small change look large. Read the scale before judging the shape. Compare the 10 extra cups with the original 200 cups.

A scale starting above zero needs a visible break. Even with a break, you still need the numbers to judge the change. A taller or narrower graph can also make the same rise look steeper. Graphs with different shapes on the page may show the same readings.

Check the labels and the data

Look for a title, both axis labels, and units. Without "Temperature (°F)," a reader cannot tell what 60 means. Check that equally spaced gridlines stand for equal steps. A scale labeled 0, 10, 20, 50 at equal distances changes the shape unfairly.

Ask where the numbers came from and how much was measured. One day's temperature graph tells you about that day. It cannot tell you the weather on every May 3. More days would be needed for that question.

When drawing your own graph, give equal amounts equal spaces. Use a title and labeled axes with units. Start the vertical scale at 0, or mark a break clearly when starting higher. Let the readings show the change, with labels that help others judge its size.