The
Math of
Government
Yes, and maths explains how. Count ballots three different ways, compare votes with seats, read a city budget, work out taxes, judge a poll’s margin of error and see how drawing boundaries can change who wins.
- 00First Principlesp.2
- 01Three ways to count votesp.4
- 02Votes vs seatsp.7
- 03Budgets & per capitap.8
- 04How taxes are calculatedp.10
- 05Polls & margin of errorp.11
- 06Boundaries & gerrymanderingp.12
- ✦Self-check · Make it · 5 Questions · Side Questp.13
You'll need: pencil, grid paper, a calculator and coloured pencils.
Principles
A first principle is a basic truth you can build on. Civic maths rests on four: the counting rule shapes the result, shares need a whole (“30% of what?”), per person makes fair comparisons and every sample has uncertainty.
What do I already know?
What is necessary?
Everything in this module is built from four bricks. Rate yourself on each one, then check the example to see if your rating holds up.
What is the objective?
By the end, you’ll be able to count an election three ways, compare vote shares with seat shares, read a budget and work out per-capita figures, calculate simple taxes, judge a poll’s margin of error and explain how boundaries can be drawn fairly or unfairly.
- 01Count first-past-the-post and ranked ballots.
- 02Allocate seats proportionally.
- 03Compare vote shares and seat shares.
- 04Read a budget and find per-capita figures.
- 05Calculate simple taxes and rates.
- 06Judge polls and riding boundaries.
The list on the left is ours. Now make it yours. What do you want out of this?
How much do you know about the math of government right now? Shade the boxes in pencil. You'll come back at the end (p.13) and shade it again in pen.
Why start this way? When you know what you already have (1), what you need (2) and where you're going (3), you learn faster and you notice the gaps. Every module on the Government Path starts with these same three questions.
First-past-the-post
The simplest count: the most votes wins. Simple doesn’t mean the result is simple.
Most votes wins the riding
In first-past-the-post (FPTP), each riding (electoral district) elects one person: whoever gets the most votes, even if it’s less than half. That’s a plurality. More than half is a majority.
Canada uses FPTP for federal elections, and so does Alberta for provincial ones.
Riverbend’s five ridings (fictional)
| Riding | Lantern | River | Oak | Winner |
|---|---|---|---|---|
| 1 | 400 | 350 | 250 | Lantern |
| 2 | 380 | 340 | 280 | Lantern |
| 3 | 420 | 300 | 280 | Lantern |
| 4 | 200 | 500 | 300 | River |
| 5 | 300 | 250 | 450 | Oak |
Different people value different things
| Supporters say | Critics say |
|---|---|
| simple to count and understand | winners can have far less than 50% |
| one clear local representative | seat totals can differ a lot from vote totals |
| often produces stable single-party governments | voters may vote “strategically” instead of for their favourite |
Run it yourself
Hold a class or family vote with three options. Count it FPTP. Did the winner get a majority or a plurality?In riding 1, what share of voters did not vote for the winner?
Ranked ballots
Let voters say who they like second and third, and the winner can change.
Number your choices 1, 2, 3
Voters rank candidates. If no one has a majority of first choices, the last-place candidate is dropped and their ballots move to each voter’s next choice. Repeat until someone has more than half. This is often called instant-runoff or preferential voting.
Australia uses it for its House of Representatives; some cities and party leadership races use it too.
Student council (fictional, 100 voters)
Weigh it up
| Supporters say | Critics say |
|---|---|
| winner needs majority support in the final round | harder to count and explain |
| less “vote splitting” between similar candidates | a majority of remaining ballots, not of all voters, if some ballots run out of rankings |
| you can vote for your favourite first | results can take longer |
Rank it
Ask 10 people to rank 4 pizza toppings (or books, or parks). Count first choices, then run the rounds. Did the winner change?Why is ranked voting said to reduce “vote splitting”?
Proportional representation
Count the same votes a third way, and the seat totals change again.
Seats match votes (roughly)
In proportional representation (PR), parties win seats in proportion to their share of the vote. Many countries use it, from the Netherlands to New Zealand.
- List PR: vote for a party; seats are shared out by vote share.
- Mixed-member proportional (MMP): local winners plus “top-up” seats to make the totals proportional (Germany, New Zealand).
- Single transferable vote (STV): ranked ballots in multi-member districts (Ireland).
Riverbend, 10 seats, by vote share
| Team | Votes | Exact share of 10 | Seats |
|---|---|---|---|
| Lantern | 34.0% | 3.40 | 3 |
| River | 34.8% | 3.48 | 3 + 1 = 4 |
| Oak | 31.2% | 3.12 | 3 |
Each team gets its whole number (3 each = 9). The last seat goes to the largest remainder: River’s .48. Other formulas, like D’Hondt, can give slightly different results.
No perfect system
| Supporters say | Critics say |
|---|---|
| seats closely match votes | often leads to coalition or minority governments |
| smaller parties get a voice | links between one MP and one area can be weaker |
| fewer “wasted” votes | small parties can hold the balance of power |
Share the seats
A 20-seat council: Team A 45%, Team B 35%, Team C 20%. Use largest remainder. How many seats each?Canadian provinces have held referendums on changing the system (for example B.C. and P.E.I.). Why might people disagree on these choices?
Seat counts versus vote shares
The same votes can produce very different parliaments. Compare carefully.
Compare the shares
| Team | Vote share | FPTP seats (of 5) | Seat share | Gap |
|---|---|---|---|---|
| Lantern | 34.0% | 3 | 60% | +26 |
| River | 34.8% | 1 | 20% | −14.8 |
| Oak | 31.2% | 1 | 20% | −11.2 |
River got the most votes but the fewest seats (tied with Oak). Lantern won narrow pluralities in three ridings and got 60% of the seats with 34% of the vote.
Count them
A vote is sometimes called “wasted” if it went to a loser, or was more than the winner needed. In riding 4, River’s 500 votes were 200 more than needed to beat Oak’s 300. People argue about this term: every vote still counts in the total and sends a message.
Look it up yourself
In real Canadian elections, seat shares and vote shares often differ, sometimes a lot, for parties of every kind. Look up the results of any recent federal or Alberta election from Elections Canada or Elections Alberta and make a table like the one above. Compare several elections and parties, not just one, so you see the pattern rather than a single result.
Elections Canada; Elections Alberta (official results).
Your own table
Pick one real election. For each party, write vote share and seat share. Which party gained most from the system? Which lost most?Why might a system that gives a big seat bonus be described as both a strength and a weakness?
Reading a budget
A budget is a plan written in numbers. Learn to read it like a story.
Riverbend’s $800M operating budget (fictional)
| Service | Spending | Share |
|---|---|---|
| Police & fire | $200M | 25% |
| Transit | $120M | 15% |
| Roads & snow | $96M | 12% |
| Community services | $80M | 10% |
| Parks & recreation | $64M | 8% |
| Debt payments | $64M | 8% |
| Admin & other | $176M | 22% |
Where the money comes from
| Source | Amount | Share |
|---|---|---|
| Property tax | $440M | 55% |
| User fees (transit, rec) | $160M | 20% |
| Provincial & federal grants | $120M | 15% |
| Other | $80M | 10% |
Balanced: $800M in, $800M out. Alberta’s Municipal Government Act requires municipalities to balance their operating budgets.
Operating vs capital
The operating budget pays for day-to-day services (salaries, fuel, snow clearing). The capital budget pays for long-lasting things (bridges, rec centres, buses), often over many years and sometimes with borrowing. Edmonton and Alberta both publish their budgets online: try finding one category you care about.
Make a pie chart
Draw Riverbend’s spending as a pie chart (360° × each share). Which slice surprised you? What would you change, and what would you cut to pay for it?Transit is 15% of spending. How many degrees is its slice?
Per capita figures & percentages
Big numbers sound impressive. Per-person numbers tell the truth.
Per person = fair comparison
Per capita means “per person.” Divide a total by the population.
Riverbend: $800,000,000 ÷ 250,000 people = $3,200 per resident. Police and fire: $200M ÷ 250,000 = $800 each.
A big city will almost always spend more in total than a town. Per capita tells you whether it spends more per person.
Is the budget really growing?
Read numbers carefully
- Percent vs percentage points: a tax rate going from 8% to 10% is up 2 points, but up 25%.
- Totals vs per capita: “Record spending!” may just mean more people.
- Nominal vs real: compare after inflation when years differ.
- Cherry-picked years: ask what the trend looks like over 10 years.
Compare two places
Town A spends $45M for 15,000 people; City B spends $3.3B for 1,000,000. Which spends more per person?A headline says a program’s budget “doubled”. What two questions would you ask?
How taxes are calculated
Simple maths, big debates. Learn the calculation so you can follow the argument.
Illustrative, simplified
- Sales tax: a percentage of a purchase. Alberta has the 5% federal GST and no provincial sales tax: $40 × 5% = $2.
- Property tax: assessed value ÷ 1,000 × the mill rate. A $400,000 home at a (fictional) 7 mills: 400 × 7 = $2,800.
- Income tax: rates rise in steps called brackets.
Progressive brackets (fictional)
10% on the first $50,000, 20% on income above it. Income $70,000:
Myth-buster: moving into a higher bracket only raises the tax on the dollars above the line, never on all your income.
Canada and Alberta, 2026
For 2026, the lowest federal income tax rate is 14%, and Alberta’s lowest rate is 8% on taxable income up to $61,200, with higher rates on higher brackets. Credits, such as the basic personal amount, mean many people owe much less than the bracket maths alone suggests. People disagree about how high taxes should be and who should pay most; that’s a values question, and maths can only show the trade-offs.
Canada Revenue Agency; Government of Alberta. Checked October 2026. Rates change every year.
Calculate it
Using the fictional brackets, find the tax and average rate on $40,000 and on $100,000. What do you notice?A town’s mill rate rises from 7 to 7.35. By what percent did the property tax on the same home rise?
Polling & margin of error
A poll is a careful guess. Learn how careful.
Polls are estimates
A poll asks a sample to estimate what everyone thinks. For a random sample, a common estimate of the 95% margin of error is:
± 1.96 × √(0.25 ÷ n) × 100
| Sample (n) | Margin of error |
|---|---|
| 400 | ± 4.9 points |
| 1,000 | ± 3.1 points |
| 2,500 | ± 2.0 points |
“Accurate within 3.1 points, 19 times out of 20.”
Too close to call?
Questions to ask
- Who was asked? Random sample, or anyone who clicked? Many online panels report a “credibility interval” instead.
- How many? And how many in the group you care about (e.g. Albertans aged 18–24)?
- What was the exact question? Wording changes answers.
- When? Opinions shift. And a poll is a snapshot, not a forecast.
Poll your class
Ask a random half of a group a yes/no question. Compare with the whole group. How close was the sample? What could make it biased?To halve the margin of error, roughly how much bigger must the sample be?
Gerrymandering & riding boundaries
Who draws the map can decide the winner before anyone votes.
Same voters, different maps
Riverbend has 25 voters: 15 Teal, 10 Gold. Split them into 5 districts of 5.
Map A: each district 3 Teal, 2 Gold → Teal 5–0
Map B: three 3–2 Gold districts → Gold 3–2
Gold has 40% of voters but can win 60% of the seats if lines are drawn to pack and crack the other side.
1812
The word comes from Massachusetts governor Elbridge Gerry, whose party approved a district in 1812 that a cartoonist said looked like a salamander: a “Gerry-mander.”
Drawing lines to help one side is still a live issue in some countries, especially the United States, where state legislatures often draw maps.
Independent commissions
Since the Electoral Boundaries Readjustment Act (1964), federal ridings have been drawn by independent commissions in each province after each 10-year census, with public hearings. Ridings are meant to have similar populations, usually within 25% of the provincial average, while respecting communities and geography. Alberta’s provincial boundaries are reviewed by an Electoral Boundaries Commission; the number of seats is set in law and can change (Alberta’s Legislature had 87 seats in 2026, with more planned for the 2027 election).
Elections Canada; Legislative Assembly of Alberta. Checked October 2026.
Draw the lines
On grid paper, make a 6×6 town of 20 Teal and 16 Gold voters. Draw 6 districts of 6 so Teal wins 6–0. Then redraw so Gold wins most. Then draw the fairest map you can.Why might people trust independent commissions more than politicians drawing their own districts?
Quick self-check
Six questions, covering every stop. Show your thinking. Answers are upside down at the bottom; no peeking till you're done.
1 · What’s the difference between a plurality and a majority?
2 · In a ranked ballot count, what happens to the last-place candidate’s ballots?
3 · With 34.8% of the vote, about how many of 10 PR seats should a team get?
4 · Riverbend’s budget is $800M for 250,000 people. What’s the per-capita figure?
5 · What’s the tax on $70,000 with 10% to $50,000 and 20% above?
6 · What’s the margin of error for a random sample of 1,000?
Re-shade your knowledge meter
How much do you know about the math of government now? Flip back to p.3 and shade the meter again in pen (or shade this one). Did any of the four bricks on p.2 move from "Not yet" to "Got it"?
The three-count election
Run a small ranked-ballot vote, then count the very same ballots three ways (FPTP, ranked and proportional) and compare the winners.
1 · My question
If the same group votes with ranked ballots, can I count them three ways and get different winners?
2 · My setup
- Pick 4 options (snacks, books, a class name)
- At least 15 voters rank them 1–4, secretly
- Count FPTP (first choices only)
- Count ranked rounds, then share 10 “seats” by first-choice %
My options:
3 · Results three ways
| Option | 1st choices | Ranked rounds | PR seats /10 |
|---|---|---|---|
| A | |||
| B | |||
| C | |||
| D |
FPTP winner: _______ · Ranked winner: _______
4 · Must-haves
- Secret, fair ballots
- All three counts shown
- Vote share vs seat share compared
- Results explained to the voters
5 · What I found
Did the winner change? Which system did your voters think was fairest, and why?
Keep ballots secret and anonymous. Vote on fun topics, not on people’s personal lives. Destroy ballots after counting.
5 Questions
I Still Have
This module didn't answer everything. Good. Write 5 new questions it didn't answer, one of each type below. There are no wrong questions, only ones nobody's asked yet.
★ Star the one question you'd most like answered. You'll need it on the next page.
1What I'll find out
Break your big question into 2 or 3 smaller ones you could actually answer.
2Where I'll look
3How I'll share it
Who I'll share it with:
Done by: