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A map supports claims only within its date, scale, source and extent; percent change and the yearly rate measure how far a map has fallen out of date.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to name a map's limits and measure change since its date with percent change and a yearly rate.
You know that a map answers a question for an audience at a chosen scale. Every map is also evidence from one source on one date. This lesson asks what a map cannot tell you, and measures the most common limit, time, with percent change.
| Term | What it means |
|---|---|
| Limit | A claim a map cannot support because of its date, scale, source or extent. |
| Currency | How up to date a map's data are. |
| Extent | The area a map covers. |
| Percent change | The change between two dates divided by the earlier value, times one hundred. |
| Average rate of change | The whole change divided by the years it took. |
| Metadata | Information about a map's data: source, date, scale and method. |
A map can support a claim only within its limits.
To measure change over time, use percent change:
$$\text{percent change} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100,$$
and the average change per year, the whole change divided by the years.
Another way: picture
Picture a photograph of a crowd taken ten years ago. It is a true record of that moment, but nobody would use it to count who is standing there today. A map is the same kind of snapshot, and its date tells you how far to trust it now.
Another way: steps
Every map has limits of four kinds.
| Limit | Example | Question to ask |
|---|---|---|
| date | a new subdivision missing | When were the data collected? |
| scale | a footpath missing from a state map | What is the smallest feature shown? |
| source | a flood zone from a model | How were the data made? |
| extent | a town off the sheet | Does the map cover the place? |
Good maps state all four in their metadata, often in the margin or legend.
Percent change is always measured from the starting value. A town that grew from $40000$ to $50000$ gained $10000$ people, which is $10000 \div 40000$, a $25$ percent increase.
Dividing by the new value instead gives $20$ percent, a smaller and wrong answer. A decrease gives a negative percent: a county that fell from $50000$ to $45000$ changed by $-10$ percent.
Raw changes favor big places. A large county that gained $80000$ people and a small one that gained $3000$ may have grown at similar rates. Percent change puts places of every size on one scale.
That is why census maps of growth usually map percent change, not the number of people added.
Dividing a change by the years between two dates gives the average change per year. A district that went from $1500$ to $1900$ homes in $8$ years added $50$ homes a year on average.
An average hides booms and pauses, but it lets a planner estimate how far a map has fallen behind: a map ten years old, in a district adding fifty homes a year, is missing about five hundred homes.
If a quantity has been growing steadily, you can project it: the mapped value plus the yearly change times the years. Paving that covered $1200$ acres and grows $40$ acres a year covers about $1200 + 40t$ acres after $t$ years.
A projection is not a measurement. It assumes the past rate continues, which a new road or a recession can overturn.
The U.S. Census Bureau counts everyone every ten years. The country grew from about $308.7$ million people in 2010 to $331.4$ million in 2020, a change of about $7.4$ percent, the second-slowest decade of growth in its history.
Maps built from the 2010 count guided a decade of decisions about schools, roads and congressional districts. By 2020 some counties had grown by a quarter and others had shrunk, which is why each new census redraws so many maps.
Metadata is data about data: who made a map, from what source, on what date, at what scale, with what method and accuracy. Federal geographic data in the United States follow published metadata standards so that users can judge fitness for use.
A map without metadata can still be useful, but it cannot be checked, and a reader must treat its claims with more caution.
Checking an answer. A growth gives a positive percent, a loss a negative one, and a percent change of more than one hundred means the value more than doubled.
Dividing by the old value is required because percent change asks how large the change is compared with where things started. Dividing by the years gives a fair average because each year counts once.
Projecting with a straight line is allowed as an estimate over short periods, when the yearly change has been steady.
Some maps are updated constantly: road navigation maps change daily, and satellite images of a fire or flood may be hours old. Others, such as geological maps, change slowly because the ground they describe changes slowly.
Whether a map is out of date depends on how fast its subject changes, not on its age alone.
Comparing a place over time assumes the place stayed the same shape. Counties, tracts and city limits change: cities annex suburbs, tracts are split as they grow, and in 2022 the Census Bureau replaced Connecticut's eight counties with nine planning regions for its statistics.
A city whose population jumps after annexing a suburb did not grow by births and moves alone; its borders grew. Careful analysts recompute old figures on today's boundaries, or compare only areas whose borders held still, and say which they did. A percent change across a boundary change measures the map as much as the place.
The most common slip is dividing by the new value instead of the old. Another is reporting the raw difference as if it were a percent.
A third is treating an old map as a current one. A fourth is projecting a trend far into the future as if it were certain.
Geographers speak of a map's fitness for use: whether its limits allow it to answer a particular question. A ten-year-old map may be perfectly fit for showing mountains and rivers, and unfit for counting homes.
Asking about fitness for use before drawing a conclusion is the habit this whole course builds.
After every census, states redraw the boundaries of their congressional and legislative districts so that each holds about the same number of people. The maps drawn after the 2010 census were based on the 2010 counts, and for ten years they stayed in place while people moved.
By 2020 some districts had grown far faster than others. Fast-growing counties around Austin, Phoenix and Raleigh gained a fifth or a quarter of their population, while some rural and older industrial counties lost people. Districts that were equal in 2010 were no longer equal by the end of the decade.
Percent change is how analysts measure that drift, county by county, and it is one of the reasons the Constitution requires a count every ten years. The maps were accurate when drawn; their limit was time.
The U.S. Geological Survey's National Land Cover Database classifies every thirty-meter square of the lower forty-eight states as forest, farmland, water, wetland, developed land and other classes, and releases updated editions every few years.
Comparing editions shows where suburbs have spread. A county whose developed land grew from $1200$ to $1500$ acres over a few years has grown its paved area by a quarter, which matters for flooding, wildlife and water supply.
A planner using an older edition to judge flood risk would miss the newest parking lots and rooftops. Checking the edition's date, and the percent change since, tells the planner whether the map is still fit for the question.
A common slip is to divide a change by the later value, or to report the raw number of people gained as if it were a percent. Percent change always compares the change with the starting value, so that growth and decline are measured on the same footing.
A second slip is to treat any map as current. Every map records its date, and the question is whether the place has changed enough since then to matter for the question being asked.
A map shows $60000$ people; today there are $75000$. Find the change.
$75000 - 60000 = 15000$
New minus old.
Divide by the old value.
$\dfrac{15000}{60000} = 0.25$
From the start.
Convert to a percent.
$25\%$
Times one hundred.
Name the wrong answer to avoid.
$\dfrac{15000}{75000} = 20\%$
Dividing by the new value.
A county fell from $90000$ to $81000$ people. Find the change.
$81000 - 90000 = -9000$
A loss.
Divide by the old value.
$\dfrac{-9000}{90000} = -0.1$
From the start.
Convert to a percent.
$-10\%$
A ten percent decline.
Find the yearly change over $10$ years.
$\dfrac{-9000}{10} = -900$
People a year.
Say what the map now overstates.
$\text{the county's population}$
By nine thousand.
A district had $3200$ homes on a map; $12$ years later it has $3800$. Find the change.
$3800 - 3200 = 600$
Homes added.
Find the yearly change.
$\dfrac{600}{12} = 50$
Homes a year.
Find the percent change.
$\dfrac{600}{3200} \times 100 = 18.75\%$
From the map's value.
Project $4$ more years.
$3800 + 50 \times 4 = 4000$
If the rate holds.
Say what the projection assumes.
$\text{a steady rate}$
Not guaranteed.
Decide on the map.
$\text{update before planning schools}$
Too many homes missing.
Find the change.
$30000 - 25000 = 5000$
New minus old.
Divide by the old value.
$\dfrac{5000}{25000} = 0.2$
From the start.
Convert to a percent.
A map drawn years ago shows a town of $80000$ people. Today it has $100000$. By what percent has the population changed since the map was drawn?
Complete the worked solution: a map shows $400$ homes in a neighborhood; $20$ years later there are $500$. Find the change, the percent change, and the average change per year.
Find the change.
$\text{today} - \text{mapped} =$ c
Homes added.
Find the percent change.
$\dfrac{\text{change}}{\text{mapped}} \times \text{a hundred} =$ p
From the starting value.
Find the yearly change.
$\dfrac{\text{change}}{\text{years}} =$ r
Homes a year.
Judge the map.
$\text{update it before planning}$
A large share of homes is missing.
Match each problem with a map to the kind of limit it shows.
| a limit of date | a limit of scale | a limit of source | a limit of extent | |
|---|---|---|---|---|
| a school opened last year is not on the map | ||||
| a footpath is missing from a statewide road map | ||||
| a flood zone was drawn from a computer model, not surveyed | ||||
| the town you asked about lies beyond the map's edge |
A population map drawn from one census is compared with the next. County A went from $50000$ to $54000$, county B from $40000$ to $38000$, and county C from $20000$ to $26000$. Fill in each county's percent change.
| percent change | |
|---|---|
| county A (% change) | |
| county B (% change) | |
| county C (% change) |
A land-cover map shows $2400$ acres of paved surface in a county. Paving has been growing by about $60$ acres a year. Write the paved acres as a function of the years $t$ since the map's date.
Answer:
A map showed $3200$ homes in a district; $12$ years later there are $3800$. What was the average change per year?
Answer: homes a year
A planner's map uses the 2010 census count for Wake County, North Carolina, about $901000$ people. The 2020 census counted about $1129000$. By what percent did the population change? Round to one decimal place.
Answer: %
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A land-cover map shows $600$ acres of paved surface in a county. Paving has been growing by about $15$ acres a year. Write the paved acres as a function of the years $t$ since the map's date.
Answer:
You can judge a map's limits. Explain why percent change divides by the starting value.
24. Your turn: a county grew from $25000$ to $30000$ people. What is the percent change?, step 3
$20\%$
Times one hundred.