Organising data in a table and reading it in both directions, building and interpreting bar graphs and line graphs, choosing between them, and recognising how a broken or uneven vertical scale can make a truthful set of numbers tell a misleading story.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 1 — Connections to Algebra
Tables and Graphs
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 42-47 — the lesson these objectives are drawn from
Warm-up
You read tables and graphs constantly. What this lesson adds is suspicion.
Discussion prompt
Think of a graph you have seen recently in an advertisement or a news story. What was the first thing you looked at — the bars, the numbers, or the axis? Be honest.
Hint: Almost nobody looks at the axis first, and that fact is the reason this lesson exists.
Answer:
Almost everyone reads the shape of the bars first and the axis last, if at all. That is a completely reasonable way to read a graph, and it is exactly what a misleading graph exploits: the numbers can all be correct and the axis correctly labelled while the picture still tells a story the data does not support.
The habit this lesson builds is small and unnatural: look at the vertical scale before you look at the bars.
Concept
Data are information, facts or numbers that describe something. A list of numbers hides its patterns; a table or a graph reveals them. But a graph can also invent a pattern that is not in the numbers, which makes reading one a skill rather than a glance.
data — Information, facts or numbers that describe something. Organising data into a table or a graph is what makes its patterns visible.
The same seven numbers can look like a flat line or like explosive growth, depending entirely on the vertical scale.
Figure (svg): A three-row data table of dairy, vegetable and fruit consumption per person for seven years from 1970 to 2000
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 42-42
Section
Section 1
Concept
A data table has two directions and they answer different questions. Reading across a row follows a single category through the years. Reading down a column compares different categories in a single year.
Figure (svg): A three-row data table of dairy, vegetable and fruit consumption per person for seven years from 1970 to 2000
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 42-42 — Example 1 and the Study Tip on reading the table
Picture it
Three foods, seven years, twenty-one numbers.
Figure (svg): A three-row data table of dairy, vegetable and fruit consumption per person for seven years from 1970 to 2000
Twenty-one numbers in a list would hide everything. Arranged like this, the fact that vegetables rose steadily while dairy wandered is visible without any calculation at all.
Worked example
Example 1's table. Practise the crossing motion before doing anything harder with it.
\[ \text{Find dairy in } 1980, \text{ vegetables in } 1995, \text{ and fruit in } 1970. \]
Find the Dairy row, then the 1980 column
Why: Go across the row until you reach the column heading 1980, and read the entry where they meet.
\[ 543.2 \]
Find the Vegetables row, then the 1995 column
Why: Same motion, one row down and three columns across.
\[ 405.0 \]
Find the Fruit row, then the 1970 column
Why: The leftmost data column, in the bottom row.
\[ 237.7 \]
Attach the units to all three
Why: The table's heading says pounds per person per year, so every entry carries that unit.
Figure (svg): The solution to Worked example read three values from the table shown as a ladder of expressions, one row per algebraic move
\[ 543.2, \quad 405.0, \quad 237.7 \text{ pounds per person per year} \]
Verify: sanity-check the sizes against each other
Why: Dairy is the largest of the three foods in every year and fruit the smallest, so a dairy figure in the five hundreds and a fruit figure in the two hundreds sits exactly where it should. If a fruit reading had come out above a dairy reading, a row had been misread.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 42-42
Matching
Four questions about the same table. Each one is answered by moving a different way.
Match the pairs
Why: Questions about change over time are answered along a row, because a row holds one food across many years. Questions comparing foods are answered down a column, because a column holds one year across all the foods. Recognising which kind of question you have been asked is what tells you which way to move, and it is the whole skill of reading a table.
Worked example
Example 1's actual question. Make a table of total dairy and vegetables per year, and find the least and greatest.
\[ \text{Add the dairy and vegetable figures for each year, then find the least and greatest totals.} \]
Add the two entries in each column
Why: For 1970, 563.8 plus 335.4 is 899.2. Repeat down every column.
\[ 899.2, 876.1, 879.6, 951.8, 951.2, 989.4, 1000.0 \]
Scan the new row for the smallest value
Why: 876.1 in 1975 is the lowest of the seven.
\[ \text{least in } 1975 \]
Scan for the largest value
Why: 1000.0 in 2000 is the highest.
\[ \text{greatest in } 2000 \]
State both answers with their years
Why: The question asked in which year, so the answer is a year and not a number of pounds.
\[ 1975\text{ and } 2000 \]
Figure (svg): The solution to Worked example build a new row from two old ones shown as a ladder of expressions, one row per algebraic move
\[ \text{least: } 876.1 \text{ in } 1975 \qquad \text{greatest: } 1000.0 \text{ in } 2000 \]
Verify: check one total against its two parts
Why: For 1985 the total is 951.8, and 593.7 plus 358.1 is indeed 951.8. Checking a single column confirms the addition was set up correctly, and checking that every total exceeds both of its parts confirms none of them was mis-added.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 42-42
Trap
Looking for dairy in 1980, the eye starts on the Dairy label, drifts down a row while moving right, and lands on 336.4.
Track across a wide table by eye alone
Why: Seven columns is wide enough that a row drifts, especially when the rows are close together.
336.4 is the vegetables figure. It is a real number from the table, which is why nothing about it looks wrong.
Put a finger or a ruler on the Dairy row, slide it across to the 1980 column, and read 543.2.
Constrain one direction physically before moving in the other
Why: A straight edge removes the drift entirely, which is why every table in a reference book is designed to be read with one.
Then sanity-check: dairy figures in this table are in the five hundreds. A reading in the three hundreds is the wrong row, and knowing the rough size of each row catches the slip immediately.
Sorting
Decide which direction each question requires.
Sort into buckets
Sort each question by the direction it is answered in.
Notice how similar items c and e look on the page and how differently they are answered. One asks for a value and the other for a year, but both scan a single row — while items b and f, which look similar to each other, both scan a column.
Faded example
Two columns of the totals row are given. Supply two more.
Fill in the blanks
1970: \; 563.8 + 335.4 = 899.2 \qquad 1985: \; 593.7 + 358.1 = 951.8 \qquad 2000: \; 590.0 + 410.0 = 1000
Why: Each total is the sum of the dairy and vegetable entries in that column. Every total has to be larger than either of its parts, which is a fast check on all seven at once: if any total in the row is smaller than a number above it, that column was mis-added.
Prediction
Read along the two rows before you answer.
Predict first
Between 1970 and 2000, which rose more steadily: dairy or vegetables?
Correct: Vegetables, which rose in almost every period.
\[ \text{vegetables: } 335.4 \rightarrow 410.0, \text{ a rise of } 74.6 \]
\[ \text{dairy: } 563.8 \rightarrow 590.0, \text{ a rise of } 26.2 \text{ with several falls} \]
Why: Vegetables go 335.4, 337.0, 336.4, 358.1, 382.8, 405.0, 410.0 — one tiny dip and otherwise a steady climb of about seventy-five pounds. Dairy goes 563.8, 539.1, 543.2, 593.7, 568.4, 584.4, 590.0, wandering up and down and ending only twenty-six pounds above where it started. Reading along a row is what makes the difference between the two obvious.
Section
Section 2
Concept
A bar graph represents the data in a table by height. The bars may be vertical or horizontal. For a reader to judge quantities by comparing bar heights, the tick marks must be evenly spaced and each must represent the same amount.
bar graph — A graph that represents data with bars whose lengths stand for the quantities. Vertical bars go straight up and down; horizontal bars go across.
Those two conditions on the scale are what make the picture trustworthy, and they are exactly what a misleading graph breaks.
Figure (svg): A bar graph of total consumption from 1970 to 2000 with the vertical axis starting at zero
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-43 — the Bar Graphs paragraph and Example 2
Picture it
Vertical axis starting at zero, evenly spaced ticks, each worth the same amount.
Figure (svg): A bar graph of total consumption from 1970 to 2000 with the vertical axis starting at zero
The bars are similar heights because the quantities really are similar: about 899 in 1970 and about 1000 in 2000, a rise of roughly eleven percent over thirty years. The picture and the numbers agree.
Worked example
Guided Practice 2. The totals of dairy, vegetables and fruit for each year, drawn so as not to mislead.
\[ \text{Draw a bar graph of the totals } 1136.9, \; 1128.2, \; 1142.0, \; 1221.2, \; 1224.7, \; 1274.8, \; 1290.0. \]
Choose a vertical scale starting at zero
Why: Starting anywhere else breaks the link between height and quantity, which is the whole point of a bar graph.
\[ \text{start at } 0 \]
Choose an interval that reaches the largest value
Why: The largest total is 1290, so ticks every 200 up to 1400 give a scale that fits with room to spare.
\[ 0\text{ to } 1400\text{ by } 200 \]
Draw the bars to their heights
Why: Each bar's height is its total measured against the scale, and all bars have the same width.
Label both axes
Why: Year along the bottom, pounds per person up the side, with the unit stated.
Describe the pattern
Why: A gentle, fairly steady rise across the thirty years, with one small dip in 1975.
Figure (svg): The solution to Worked example build a bar graph from the totals shown as a ladder of expressions, one row per algebraic move
\[ 1136.9 \rightarrow 1290.0, \text{ a rise of about } 13\% \]
Verify: compare the tallest bar with the shortest
Why: The 2000 bar is 1290 and the 1975 bar is 1128, so the tallest is about 1.14 times the shortest — and on a graph starting at zero it should look about fourteen percent taller, which is a small visible difference. If the drawn bars look dramatically different, the scale is not starting at zero.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-43
Estimation
On a graph whose ticks are 0, 250, 500, 750 and 1000, a bar reaches slightly above the third tick.
Predict first
Roughly what value does that bar represent?
Correct: About 550.
\[ \text{tick } 3 = 500, \quad \text{interval} = 250, \quad \text{reading} \approx 550 \]
Why: The third tick is 500 and each interval is 250, so slightly above the third tick is a little over 500. Option B is the classic error of counting ticks instead of reading their values, and it is worth naming because it is what happens when the scale is ignored entirely.
Worked example
Reading a graph is the reverse of drawing one, and it needs the same attention to the scale.
\[ \text{On a graph with ticks at } 0, 250, 500, 750, 1000, \text{ a bar reaches just past the fourth tick. What is its value?} \]
Identify what one tick is worth
Why: The ticks run 0, 250, 500, 750, 1000, so each interval is 250.
\[ \text{one interval } = 250 \]
Locate the top of the bar between two ticks
Why: Just past the fourth tick means just above 750.
\[ \text{between } 750\text{ and } 1000 \]
Estimate the fraction of the interval used
Why: About three fifths of the way from 750 to 1000 is about 150 more.
\[ 750 + 150 \]
State the reading with an appropriate precision
Why: About 900 — reading a graph gives an estimate, not an exact figure.
\[ \text{about } 900 \]
Figure (svg): The solution to Worked example read a quantity off a bar graph shown as a ladder of expressions, one row per algebraic move
\[ 750 + \tfrac{3}{5}(250) = 900 \text{ approximately} \]
Verify: check the reading against the table it came from
Why: The table's 1970 total for dairy and vegetables is 899.2, so a reading of about 900 is right. A graph is for seeing patterns and a table is for exact values, and using each for what it is good at is the point of having both.
Error analysis
The student described the honest bar graph of totals. Two of the three claims are wrong.
Annotate
On: \( \begin{aligned} &\text{1. The 2000 bar is about five times the 1970 bar.} \\ &\text{2. Every bar is taller than the one before it.} \\ &\text{3. The graph shows a gradual rise across the thirty years.} \end{aligned} \)
The first claim is exactly what the misleading version of this graph appears to show. Holding the picture up against the table is the only reliable defence.
Elimination
Three of these are requirements and one is not.
Eliminate the wrong options
Which of these is NOT required for a bar graph to be read honestly?
Survives elimination: A
Why: Colour is presentation, not measurement — different colours may help a reader tell categories apart but they carry no quantitative information. The other three are all conditions on the scale, and each one, if broken, changes what a bar's height actually means.
Translation
On a scale from 0 to 1000 with ticks every 250, match each value to how its bar looks.
Match the pairs
Why: On a scale starting at zero, a bar's fraction of the full height equals its fraction of the top value, which is exactly what makes heights comparable. A value of 500 is half of 1000 and its bar is half the height; on a scale starting at 1125 that relationship would not hold for any of these values.
Socratic
It is the single most important thing about a bar graph's scale.
Discussion prompt
Explain what a bar's height measures when the axis starts at zero, and what it measures when the axis starts somewhere else. Use the consumption totals to say how the visual impression changes.
Hint: Ask what quantity the bottom of the bar is sitting on.
Answer:
When the axis starts at zero, the height of a bar measures the quantity itself, so a bar twice as tall represents twice as much. That is the assumption every reader makes without thinking about it, and it is what makes a bar graph readable at a glance.
When the axis starts at 1125, a bar's height measures only the amount above 1125. The 1970 total of 1136.9 becomes a stub of 11.9 and the 2000 total of 1290 becomes a bar of 165 — nearly fourteen times taller, for quantities that differ by thirteen percent. Nothing has been falsified and the impression is completely wrong.
Section
Section 3
Concept
A graph can be built entirely from correct numbers and still create a false impression. The usual mechanism is a vertical scale that does not start at zero, marked with a zigzag break to show that part of the scale has been cut away.
To make a bar graph that cannot be misinterpreted, space the tick marks evenly and make each one represent the same amount, starting from zero.
Figure (svg): The same bar graph with the vertical axis starting at 1125 and a zigzag break, making the differences look enormous
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-43 — Example 2, on why the graph could be misinterpreted
Picture it
The same seven totals as the honest graph, drawn on a scale beginning at 1125.
Figure (svg): The same bar graph with the vertical axis starting at 1125 and a zigzag break, making the differences look enormous
It appears that Americans ate about five times as much in 1995 as in 1970. They did not: the 1970 figure is about eighty-eight percent of the 1995 one. Everything on this graph is factually correct.
Worked example
Example 2 from the textbook. The claim is that consumption in 1995 was about five times that in 1970.
\[ \text{Explain why the bar graph could be misinterpreted.} \]
Look at the vertical scale before the bars
Why: The zigzag near the bottom shows a break where part of the scale is not shown.
Work out what the first tick actually represents
Why: Because of the break, the first tick mark stands for 1125 pounds per person, not zero.
\[ \text{first tick } = 1125 \]
Work out what each later tick represents
Why: The remaining ticks are 25 pounds apart, which is a very fine interval on quantities above a thousand.
\[ 25\text{ pounds per tick} \]
Compare the bar heights with the real figures
Why: The 1970 total is about 1137 and the 1995 total about 1275, a difference of about twelve percent — but as heights above 1125 they are 12 and 150, a ratio of more than twelve to one.
State the fix
Why: Space the tick marks evenly, make each represent the same amount, and start the scale at zero.
Figure (svg): A close-up of the zigzag break symbol on a vertical axis, with an explanation of what it hides
\[ \text{first tick} = 1125, \text{ so bar height} = \text{value} - 1125 \]
Verify: compute the real ratio and compare it with the visual one
Why: The real ratio of the 1995 total to the 1970 total is about 1275 over 1137, which is 1.12. The visual ratio of the drawn heights is about 150 over 12, which is 12.5. The picture overstates the difference by a factor of more than ten, and that gap is entirely produced by the scale.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-43
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Axis starts at 0 | Axis starts at 1125 | |
|---|---|---|
| What a bar's height measures | the quantity itself | the amount above 1125 |
| Visual ratio, 1995 to 1970 | about 1.1 to 1 | about 12 to 1 |
| Impression given | a gentle rise | explosive growth |
Every number in the underlying table is identical between the two columns. The entire difference is a decision about where the axis begins.
Worked example
Guided Practice 2. The fix is a scale decision, not a change to any number.
\[ \text{Choose a scale for the totals } 1128 \text{ to } 1290 \text{ that cannot mislead.} \]
Start the scale at zero
Why: This is the decision that restores the link between height and quantity.
\[ \text{start at } 0 \]
Pick an interval that reaches above the largest value without too many ticks
Why: The largest total is 1290, so intervals of 200 give eight ticks up to 1400.
\[ 0\text{ to } 1400\text{ by } 200 \]
Check that every interval is the same size
Why: Two hundred each, with no break anywhere.
Describe what the redrawn graph shows
Why: Seven bars of similar height rising gently, which is what a thirteen percent change over thirty years looks like.
Figure (svg): A bar graph of total consumption from 1970 to 2000 with the vertical axis starting at zero
\[ \text{scale } 0 \text{ to } 1400, \text{ ticks every } 200 \]
Verify: check that the pattern description still matches the numbers
Why: The redrawn graph shows a gentle rise with a small dip in 1975, and the table confirms both: the totals climb from 1136.9 to 1290.0 with one fall at 1128.2. The honest graph tells the same story the numbers tell, which is the test any graph has to pass.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-43
Trap
The axis is labelled, the break is drawn, and the numbers are all correct — so the graph is fine.
Judge a graph by whether it contains any false statements
Why: Nothing on the misleading graph is untrue, so by that standard it passes.
The graph still leaves almost every reader believing consumption rose fivefold when it rose by about thirteen percent.
A graph is judged by the impression it creates, not only by whether its labels are accurate.
Compare the visual ratio of the bars with the real ratio of the numbers
Why: If the two disagree, the graph is misleading whatever its labels say.
This is why the fix is to change the scale rather than to add a warning. A reader who has to do arithmetic to avoid being misled is being misled.
Anomaly
A report states that food consumption rose fivefold between 1970 and 1995.
Predict first
What is the quickest way to see that this claim cannot be right, without looking at any graph?
Correct: Work out what fivefold would mean in pounds of food per day.
\[ \tfrac{1137}{365} \approx 3.1 \text{ lb per day} \qquad 5 \times \text{ that} \approx 15.6 \text{ lb per day} \]
Why: The 1970 total is about 1137 pounds per person per year, which is roughly three pounds a day. Fivefold would be about fifteen pounds of dairy, vegetables and fruit per person per day, which no population eats. A quick conversion into an everyday unit is often the fastest way to reject a claim, and it needs no access to the original data at all.
Socratic
Breaking an axis is not always dishonest. Decide when it is not.
Discussion prompt
Describe a situation where starting a vertical axis above zero is genuinely the right choice, and say what the graph's author owes the reader when they do it.
Hint: Think about data where the interesting variation is tiny compared with the values themselves.
Answer:
Body temperature is a good case. Healthy readings sit between about 36.1 and 37.2 degrees, and a graph starting at zero would show a row of near-identical bars with all the medically important variation invisible. Here the variation, not the total, is the subject.
What the author owes the reader is a clear break marker, an explicit statement that the axis does not start at zero, and — most importantly — the use of a line or point graph rather than bars. Bars carry an implicit promise that height means quantity; lines and points do not, which is why a broken axis is far less misleading on a line graph.
Edge cases
See how far a scale alone can distort a fixed set of numbers.
Discussion prompt
Take the 1970 total of 1137 and the 2000 total of 1290. Choose a starting value for the vertical axis that makes the 2000 bar look about ten times the 1970 bar, and say what starting value would make them look almost identical. What does this tell you about reading bar graphs?
Hint: The height of each bar is its value minus the starting value.
Answer:
\[ \text{start at } 1120: \quad \tfrac{1290 - 1120}{1137 - 1120} = \tfrac{170}{17} = 10 \]
\[ \text{start at } 0: \quad \tfrac{1290}{1137} \approx 1.13 \]
Starting at 1120 makes the ratio ten to one; starting at zero makes it about 1.13 to one; and starting at a large negative number would make them look almost identical. The author of a bar graph can produce almost any visual ratio they like from the same two numbers by choosing the axis, which is why the scale is the first thing to read and the bars the second.
Section
Section 4
Concept
A line graph plots the data as points and joins them. Joining the points is itself a claim: that the quantity existed and changed smoothly between the plotted moments. That claim is reasonable for a quantity measured over time and false for separate categories.
line graph — A graph in which data points are plotted and joined by line segments, used when the horizontal axis has a natural order such as time.
Ask one question to choose: does the horizontal axis have a natural order? If it does, a line is defensible; if it does not, joining the points asserts something untrue.
Figure (svg): A line graph of total consumption rising from about 899 in 1970 to about 1000 in 2000
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-44
Picture it
The same seven numbers, plotted and joined.
Figure (svg): A line graph of total consumption rising from about 899 in 1970 to about 1000 in 2000
The dip in 1975 and the jump between 1980 and 1985 are much easier to see here than in the bar version. A line graph is better at showing change; a bar graph is better at comparing sizes.
Worked example
The choice is decided by one question about the horizontal axis.
\[ \text{Choose bar or line for: dairy vs vegetables vs fruit in 1990; dairy from 1970 to 2000; the three foods' totals for each year.} \]
Three foods in one year has no natural order
Why: Dairy, vegetables and fruit could be listed in any order, so joining them would suggest a progression that does not exist.
Dairy from 1970 to 2000 has a natural order
Why: Years come in a fixed sequence and the quantity existed at every moment between them.
Totals for each year also have a natural order
Why: Same reasoning as the previous one: time is the horizontal axis.
State the deciding question
Why: Does the horizontal axis have a natural order? That single question settles all three.
Figure (svg): The solution to Worked example choose the right graph three times shown as a ladder of expressions, one row per algebraic move
\[ \text{categories} \rightarrow \text{bar} \qquad \text{time} \rightarrow \text{line} \]
Verify: test the rule on a case where it says no
Why: A graph of favourite fruits in a class has no natural order — nothing sits between apples and bananas — so a line joining them would claim a smooth transition from apples to bananas, which is meaningless. The rule correctly rejects it.
Sorting
Decide which graph type each data set calls for.
Sort into buckets
Sort each data set by the graph it should use.
The test is always the same and it is about the horizontal axis, never about which graph looks better. If reordering the horizontal categories would be equally valid, use bars.
Worked example
Line graphs are read for direction and change rather than for exact values.
\[ \text{Describe the trend in the totals from } 1970 \text{ to } 2000 \text{ and locate the one fall.} \]
Look at the overall direction from the first point to the last
Why: The line ends higher than it started, so the overall trend is upward.
Look for any segment that goes down
Why: The segment from 1970 to 1975 falls, from 899.2 to 876.1.
Look for the steepest rise
Why: The segment from 1980 to 1985 climbs from 879.6 to 951.8, the largest single jump.
\[ \text{steepest } 1980\text{ to } 1985 \]
Summarise in one sentence
Why: An overall rise of about a hundred pounds per person, mostly achieved in a single five-year jump after 1980.
Figure (svg): The solution to Worked example read a trend off a line graph shown as a ladder of expressions, one row per algebraic move
\[ 899.2 \rightarrow 1000.0, \text{ dip in } 1975, \text{ steepest rise } 1980 \text{ to } 1985 \]
Verify: confirm the steepest segment against the table
Why: The differences are -23.1, +3.5, +72.2, -0.6, +38.2, +10.6, and the largest of those is indeed the +72.2 between 1980 and 1985. The eye picked the same segment the arithmetic does, which is what a well-drawn line graph is for.
Trap
A line graph of dairy, vegetables and fruit consumption in 1990, with the three points joined.
Use a line graph because it looks tidier and shows the differences clearly
Why: Line graphs feel more sophisticated, so they get chosen for presentation reasons.
The line between dairy and vegetables implies there is something halfway between them that was measured. There is not, and the picture asserts a quantity that does not exist.
A bar graph of the three foods in 1990, with a gap between the bars.
Use bars when the categories have no order, so the gaps between them stay honest
Why: The gap says these are separate things, which is exactly the truth about three different foods.
Ask the one question: does the horizontal axis have a natural order? If the answer is no, the gaps between bars are carrying real information and must not be filled in.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Bar graph | Line graph | |
|---|---|---|
| Horizontal axis holds | separate categories | an ordered sequence, usually time |
| Best at showing | comparing sizes | change and trend |
| What the gaps mean | these are separate things | nothing — the line fills them |
The bottom row is the reason the choice matters. A gap in a bar graph is information, and a line graph deliberately removes it — which is right for time and wrong for categories.
Elimination
A class records the number of students whose favourite colour is red, blue, green or yellow.
Eliminate the wrong options
Which representation would be misleading?
Survives elimination: A
Why: The line would suggest a continuous transition from red through blue to green, and it would also imply that the order of the colours means something — reorder them and the line's shape changes completely, while the underlying data does not. Any representation whose appearance depends on an arbitrary ordering is misleading.
Socratic
The segments are not decoration. They are assertions.
Discussion prompt
The line graph of totals joins the 1990 point to the 1995 point. State precisely what that segment claims about 1993, and say whether the claim is justified here.
Hint: Read a value off the middle of the segment and ask where it came from.
Answer:
The segment claims that in 1993 the total was about 967 pounds per person — roughly halfway between the 951.2 of 1990 and the 989.4 of 1995 — and more generally that the quantity moved steadily from one to the other.
Nobody measured 1993 in this table, so the claim is an interpolation rather than data. It is reasonable here because food consumption is a real quantity that existed continuously and changes slowly, so a straight-line estimate is likely to be close. It would not be reasonable for something that jumps, such as a population immediately after a border change, and it would be meaningless for categories, where there is no in-between at all.
Section
Section 5
Concept
Reading a graph well means checking a short list before looking at its shape. The list takes about ten seconds and catches almost every misleading graph you will meet.
Only after all four have been answered is it safe to look at the bars.
Figure (svg): Two columns contrasting when a bar graph is the right choice with when a line graph is
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-44
Picture it
The decision rests on a single property of the horizontal axis.
Figure (svg): Two columns contrasting when a bar graph is the right choice with when a line graph is
Both columns describe honest graphs. The dishonest ones come from mixing them up, or from a scale that breaks the link between height and quantity.
Worked example
Applying the four questions to the graph from Example 2.
\[ \text{Run the four checks on the bar graph whose axis starts at } 1125. \]
Where does the vertical axis start?
Why: At 1125, with a zigzag break below it. The first check already fails.
\[ \text{starts at } 1125 \]
Are the ticks even and equal?
Why: Above the break they are, at 25 pounds each — but the interval from the origin to the first tick is 1125, so overall they are not.
What are the units?
Why: Pounds per person per year, a per-person rate rather than a national total.
Does the graph type match the axis?
Why: The horizontal axis is time, which is ordered, so a line graph would have been the better choice — and would have been far less misleading with a broken axis.
Figure (svg): The solution to Worked example run the checklist on the misleading graph shown as a ladder of expressions, one row per algebraic move
\[ \text{axis starts at } 1125 \;\rightarrow\; \text{heights not proportional} \]
Verify: confirm that the honest version passes all four
Why: Starting at zero, ticks every 200, the same per-person units, and either a bar or a line graph would be acceptable once the scale is fixed. The honest version passes every check, which is what the checklist is calibrated to detect.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 43-43
Ranking
Four things to establish before believing a graph's shape.
Put in order
Why: The starting value comes first because it determines what a height means at all. Even spacing comes second, because uneven ticks break the link between distance and quantity. The units come third, since a total and a per-person rate can trend in opposite directions. Only when all three are settled is it safe to read the shape.
Worked example
Not every problem with a graph is about the scale. Sometimes it is about what was measured.
\[ \text{A graph shows total national food consumption rising steadily over thirty years. What should you ask?} \]
Check the axis and the ticks
Why: Suppose both are fine: it starts at zero and the ticks are even.
Check the units carefully
Why: This graph plots a national total, not a per-person figure.
Ask what else changed over the same period
Why: The population also grew over those thirty years, so a rising total is expected even if each person ate exactly the same amount.
State what the graph can and cannot support
Why: It supports a claim about the country's total consumption and says nothing at all about individual eating habits.
Figure (svg): The solution to Worked example a graph that passes but still needs care shown as a ladder of expressions, one row per algebraic move
\[ \text{total} = \text{per person} \times \text{population} \]
Verify: check the direction of the possible error
Why: If population rose by twenty percent and the total rose by twenty percent, the per-person figure did not change at all. The table in this lesson gives per-person figures precisely so that this ambiguity does not arise, and noticing which of the two a graph plots is part of reading its units.
Trap
Glance at the graph, form an impression from the shape, then read the axis to confirm it.
Read a graph the way it was designed to be read
Why: The shape is the largest, brightest thing on the page, so the eye starts there.
Once an impression has formed it is very hard to dislodge, and a misleading graph has already done its work by the time the axis is read.
Read the vertical axis first: where does it start, and are the ticks even? Then look at the bars.
Establish what a height means before letting any height mean anything
Why: The axis is the key that decodes the picture, and reading a message before its key is how you get the wrong message.
Ten seconds on the axis, every time. It is a small, unnatural habit, and it is the whole defence.
Missing information
A graph can be perfectly well drawn and still leave you unable to answer.
Discussion prompt
A bar graph shows seven bars of increasing height, labelled 1970 to 2000, with no vertical axis labels at all. Say exactly what you can and cannot conclude, and what one addition would make the graph usable.
Hint: Think about what you know about the order of the values compared with their sizes.
Answer:
You can conclude the order: each year's value is at least as large as the one before, since the bars increase. You cannot conclude anything about size — not the values, not the ratios, not whether the rise is one percent or one thousand percent.
The one addition that fixes it is a labelled vertical axis starting at zero with evenly spaced ticks. That single element converts a picture of an ordering into a picture of quantities, which is what a bar graph is supposed to be.
Elimination
The totals rose from about 1137 in 1970 to about 1290 in 2000.
Eliminate the wrong options
Which claim is supported?
Survives elimination: A
Why: The rise from 1137 to 1290 is about 153 pounds, which is roughly thirteen percent of the starting value. The three wrong claims are the three characteristic overreaches: exaggerating the size, generalising to every year, and inferring something the units cannot support.
Socratic
Tables and graphs sit in Chapter 1 for a reason.
Discussion prompt
Explain what a table of inputs and outputs has in common with the tables in this lesson, and predict what Lesson 1.8 is likely to do with them.
Hint: Think about what the two rows of a table could stand for.
Answer:
Both are two-row structures where each entry in the top row is paired with exactly one entry in the bottom row. In this lesson the top row is a year and the bottom a quantity; the pairing is what makes the table readable in the first place.
Lesson 1.8 gives that pairing a name — a function — and adds one requirement: each input must have exactly one output. It then shows the same pairing in four different forms: a table, a rule, a graph, and a description in words. Everything you have just done with tables and graphs is about to be reused, which is why these two lessons sit next to each other.
Comparison
Fill the blanks from memory before you scroll back. The deciding question is in the first row.
Comparison matrix
| Bar graph | Line graph | |
|---|---|---|
| Horizontal axis has a natural order? | No | Yes |
| What joining points would claim | something false — there is no in-between | the quantity changed continuously |
| Best at | comparing sizes | showing change and trend |
| Ruined by a broken axis? | Yes, badly | Less so, but still misleading |
The last row is worth noting: a broken axis damages a bar graph more than a line graph, because a bar's height carries an implicit promise about quantity that a plotted point does not.
Pattern
Whether you are building a graph or reading someone else's, the same five moves cover it.
Step three is the one that separates an honest graph from a misleading one, and step five is what catches a misleading graph drawn by somebody else.
OpenStax Elementary Algebra 2e, §4.1 Use the Rectangular Coordinate System §4.1
Check
Reading a table. Use a straight edge if you need one.
Check your understanding
In the consumption table, the Dairy row reads 563.8, 539.1, 543.2, 593.7, 568.4, 584.4, 590.0 for 1970 to 2000. In which year was dairy consumption highest?
Answer: A
Why: Scanning across the row, 593.7 in 1985 is the largest of the seven values, just ahead of 590.0 in 2000. Finding a maximum means comparing every entry in the row rather than assuming the last one is largest.
Check
Reading a scale. Look at the axis before the bar.
Check your understanding
A bar graph has a zigzag break and its first tick mark represents 1125, with later ticks 25 apart. A bar reaches the third tick above the break. What value does it represent?
Answer: A
Why: The first tick is 1125 and each further tick adds 25, so three ticks above it is 1125 plus 75, which is 1200. Reading a broken scale means starting from the value of the first tick rather than from zero.
Check
Choosing a graph. Ask the one question about the horizontal axis.
Check your understanding
You want to show how one plant's height changed over ten weeks. Which graph should you use, and why?
Answer: A
Why: The horizontal axis holds weeks, which come in a fixed sequence, and the plant had a height at every moment between the measurements. Joining the points therefore claims something true, and the line makes the growth pattern visible in a way separate bars would not.
Real world
An advertisement shows a bar chart comparing its product's battery life with two competitors. Its bar is roughly three times the height of the others, and the vertical axis is labelled starting at 18 hours.
Discussion prompt
Given that the axis starts at 18, and the three bars appear at heights in the ratio 3 to 1 to 1, work out what the actual battery lives might be if the shortest bar represents 20 hours. Then say by what factor the picture exaggerates the advantage, and what one change would make the chart honest.
Hint: The height of each bar is its value minus 18.
Answer:
\[ \text{shortest bar: } 20 - 18 = 2 \quad \text{tallest bar: } 3 \times 2 = 6 \;\rightarrow\; 18 + 6 = 24 \]
So the product lasts about 24 hours against 20 for the competitors — an advantage of about twenty percent, drawn to look like an advantage of two hundred percent. The picture overstates the real difference by roughly a factor of ten.
The single change that fixes it is starting the vertical axis at zero. The bars would then be in the ratio 24 to 20 to 20, and the product's genuine advantage would still be visible — just honestly, as a bar about a fifth taller rather than three times taller.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Can a graph in which every number is correct and every axis is labelled still be misleading?
Correct: Yes, because the scale decides what the shape means.
\[ \text{real ratio } \tfrac{1275}{1137} \approx 1.12 \qquad \text{visual ratio } \tfrac{150}{12} \approx 12.5 \]
Why: The Example 2 graph in this lesson contains no false statement anywhere: the totals are right, the axis is labelled, and the break is drawn. It still leaves a reader believing consumption rose about fivefold when it rose about thirteen percent. Readers judge a bar graph by the shape of the bars, and the author of the graph chooses the scale that produces that shape — which is why the scale, not the labels, is what has to be checked.
Explain it
They can read a bar graph confidently and have never questioned one.
Discussion prompt
In no more than four sentences, explain how a graph can be truthful and misleading at the same time. Then give them one habit that protects against it, and say why the habit feels unnatural.
Hint: The habit is about what you look at first.
Answer:
A usable answer: a bar graph works because you read height as quantity, and that only holds when the vertical axis starts at zero. If it starts somewhere else, a bar's height measures only the part above that starting point, so a small real difference can be drawn to look enormous. Every number on such a graph can be correct while the picture tells a false story.
The habit is to read the vertical axis before looking at the bars. It feels unnatural because the bars are the biggest thing on the page and the axis is small grey text at the side — which is exactly why a misleading graph works on almost everybody.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Table reading is fixed by using a straight edge and by knowing the rough size of each row so a misread jumps out. Combining rows is fixed by checking one column against its parts and confirming every total exceeds both. Broken scales are fixed by computing the real ratio and the visual ratio and comparing them. The graph choice is fixed by one question: does the horizontal axis have a natural order? Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page, copy any three columns of the consumption table, keeping the row and column headings. Underneath, draw the same three values twice: once as a bar graph with the vertical axis starting at zero, and once with the axis starting just below the smallest value. Beside each version write the ratio of the tallest bar to the shortest as it appears to the eye, and write the true ratio of the two numbers underneath. In the lower third of the page, write the four questions to ask before believing a graph, in the order you would ask them. Finally, in the margin, write the one question that decides between a bar graph and a line graph.
Your two visual ratios should differ substantially while the true ratio stays the same. If they came out similar, your second axis did not start close enough to the smallest value to show the effect.
Recap
Five things, and the third one is the one you will use most often outside this classroom.
| If the question says | Your first move is |
|---|---|
| How much dairy in 1980 | Cross the Dairy row with the 1980 column |
| Make a table showing the total | Add down each column, one year at a time |
| Explain why the graph could be misinterpreted | Find where the vertical axis starts |
| Draw a graph that is not misleading | Start the scale at zero with even ticks |
| Which graph should you use | Ask whether the horizontal axis is ordered |
Lesson 1.8 takes the two-row table from this lesson, adds one requirement — each input has exactly one output — and calls the result a function, which is the idea the rest of the book is built around.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.7 Tables and Graphs §1.7, pp. 42-47 — everything on these slides traces back here
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