Read counts with their units. A percentage divides a part by its named whole and multiplies by 100. A model assumption is not an observed fact.
Aggregation
Combining smaller geographic units into a larger unit.
Assumption
A condition accepted for a model or argument; it needs justification before applying the result elsewhere.
01 · TRY
Can two accurate maps tell different stories?
Draw two block rectangles, label counts and denominators, then make a merged district. Write a caption that stays true at both scales.
Draw two block rectangles, label counts and denominators, then make a merged district. Write a caption that stays true at both scales. Use the supplied records and show one intermediate step.
Make a provisional claim
Loading saved evidence…
Inspect evidence and test an explanation
The investigation controls appear when your local evidence is ready.
04 · UNDERSTAND
Connect evidence and mechanism
How boundaries change the map
Start with the question
Can two accurate maps tell different stories?
Inspect the teaching record
Fictional blocks: North has 100 residents and 20 library users; South has 900 residents and 90 users. Both counts cover the same month and define a user as one distinct resident.
This is fictional teaching evidence. It is not an observation about an actual community or economy.
Explain the mechanism
A count and a population rate answer different questions. Aggregation can hide local differences. Boundary changes alter the people included in a statistic; they do not by themselves change individual behavior. Map scale is the relationship between map distance and ground distance, while analysis scale names the geographical level being compared.
Follow the worked example
North: 20/100 = 20%; South: 90/900 = 10%. Combined: 110/1000 = 11%, not the unweighted average 15%. On a 1:10000 map, 1 cm represents 100 m.
Challenge the first explanation
A district report merges both blocks and reports 110 users among 1000 residents. A map maker instead shades by raw user count and colors South darkest.
Make and revise your evidence
Draw two block rectangles, label counts and denominators, then make a merged district. Write a caption that stays true at both scales.
Revise your answer to “Can two accurate maps tell different stories?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.
Name the idea
Aggregation: Combining smaller geographic units into a larger unit.
Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.
Extend the investigation
Independently extend the investigation “How boundaries change the map” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.
Observed counties: a different summary, the same residents
Before changing the map, predict whether the county with the largest population has the highest population per square kilometre of land. This is a deliberate six-county selection, not a representative sample.
Temporary experiment: controls reset on reload/navigation. Copy the analysis JSON below to retain a comparison; this does not change saved lesson work.
U.S. Census Bureau 2020 generalized boundaries; local equirectangular display centered on 41°N. Distances are approximate and areas are distorted. The 12 cm maximum CSS width gives a nominal scale around 1:667,000 at ordinary print sizing; screen/zoom scaling varies. Do not enlarge this into a detailed navigation/property map. Source geometry is for 1:500,000 or smaller-scale display only. Unshown surrounding land is not labeled as water.
Fixed legend: people per km² of land. Bins stay fixed as the group changes.
0–1,000
over 1,000–5,000
over 5,000–15,000
over 15,000
The total-population map answers “where are more residents in this selected set?” Larger colored areas do not imply higher density. The table supplies exact totals and numerical comparisons without relying on color.
April 1, 2020 residents and archived 2020 Gazetteer land area; displayed derived values rounded
Map / county
In group
Residents
Land km²
People/km²
1. Bronx County
No
1,472,654
109.225
13,482.8
2. Kings County
No
2,736,074
179.684
15,227.1
3. New York County
Yes
1,694,251
58.679
28,873
4. Queens County
No
2,405,464
281.594
8,542.3
5. Richmond County
No
495,747
148.983
3,327.5
6. Westchester County
Yes
1,004,457
1,115.568
900.4
Your group calculation
2 counties selected. Sum of residents: 2,698,708. Sum of published land area: 1,174.247714 km².
Combined density = 2,698,708 ÷ 1,174.247714 = 2,298.2 people/km². Unweighted mean of selected county densities: 14,886.7 people/km².
An unweighted mean gives each county equal influence. Combined density weights by land area. The map retains each county's own value; selecting a group changes its summary, not county values or borders. An empty group has no defined density.
Sources, date differences and reproducible analysis
Population: Census TIGERweb 2020 county table, POP100, original April 1, 2020 counts. The enclosing URL includes acs26 but the table explicitly identifies the 2020 Census view. These are not current estimates or revised estimate bases.
Land/water attributes: 2020 Gazetteer, matching the archived 2020 county cartographic file. TIGERweb's area fields differ slightly and are not used. Density uses integer square metres, never polygon area or the rounded table values. All sources retrieved 28 October 2026.
Copy this read-only JSON into a local text file to compare initial and revised groupings. It includes exact measures, the selected IDs, formulas and source links. No import into existing saved lesson records is implied.
Explain the observed comparison
Start with New York County and Westchester, the experiment’s initial group. A square kilometre is the area of a square one kilometre on each side; it equals 1,000,000 square metres. New York County’s calculation uses 1,694,251 residents divided by 58.679457 km², about 28,873 people per km². These are county averages over land, not counts of people inside each mapped square.
Predict which mean will be larger for this two-county group: the combined land density or the unweighted mean of the two county densities. Inspect the displayed areas before calculating. Explain why a large, less dense county contributes more land to the combined denominator. Neither calculation is a mistake if it is correctly labeled; they answer different questions.
Change the group and repair a conclusion
Before changing the controls, record the initial group’s residents, land area and combined density. Replace Westchester with Bronx County and predict the direction of the density change. Use the unrounded JSON measures to reconstruct the new result. Then select all six counties. Write a caption explaining why a changed summary does not establish that anyone moved or became more crowded.
Repair this claim: “The map became darker when I switched from total population to density, so more people arrived.” Your correction must name the changed measure, its units and a county whose rank differs between the two measures. Explain why a fair comparison uses the appropriate fixed legend rather than interpreting colors across different units as comparable quantities.
Review the grouping and measure interpretation
Kings County has the greatest population in the selected set; New York County has the highest land density. The default New York–Westchester grouping has about 2,298.2 people/km², while the unweighted mean of those counties is about 14,886.7. Westchester contributes much more land, so its lower density has more influence in the combined measure. Replacing it with Bronx increases the combined density. Selecting all six gives about 5,179.5 people/km². These differences come from changed membership or measure, not evidence of a population change over time.
Your selection is an artificial grouping of existing county observations. It is not a new government district, an annexation, or a reconstructed historical boundary change. Keep the original borders visible. Density alone does not measure housing crowding or service access: those questions require different denominators and evidence.
Transfer and continue
Create a different two-county group before opening feedback again. Predict which county has more influence in its combined density and justify that prediction from land areas. Capture your initial and revised JSON in separately named local files, alongside a short explanation of what changed. An empty selection is not a zero-density place.
High school: explain the difference between residents and residents per km² using one county and one artificial group. AP practice: distinguish map scale, county aggregation and population distribution, and identify a claim these county averages cannot support. College: audit the source versions and explain why substituting TIGERweb’s different area fields without disclosure would change reproducibility. Public feedback supports practice; independent assessment and broader geographic coverage remain separate.
Worked example
North: 20/100 = 20%; South: 90/900 = 10%. Combined: 110/1000 = 11%, not the unweighted average 15%. On a 1:10000 map, 1 cm represents 100 m.
