□→△ · Patterns & Correspondence
Turning counting into computing — finding the rule.
Four chairs per table. With 100 tables, how many chairs — must we count them all?
Distinguish a few examples from a general rule.
The power to know the hundredth without counting to it.
Open the immersive version →A reading path containing this lesson: Quantities that change together · 1/3
Why does this question matter?
A restaurant owner doesn't want to count chairs for 100 tables. Instead of counting 100 times, discover once: 'four per table' — the rule.
The leap happens when the rule is written as a formula, not a sentence. Write '□×4 = △' and plugging in 100 yields 400 instantly. Counting became computing.
This structure — fix □ and △ is determined — is correspondence, later renamed x and f(x): the function. The seed of mathematics' most important concept is planted here.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — four chairs per table. With 5 tables, how many chairs?
Add tables
Watch how many chairs each new table adds — and read the chart vertically too.
| tables □ | 1 | 100? |
|---|---|---|
| chairs △ | 4 | ? |
So — 100 tables?
Read the idea from another angle
Read the chart across and you see the next; read it down and you see the rule.
Reading 4, 8, 12 across shows only '+4 each' (the next term). Reading 1↔4, 2↔8, 3↔12 downward reveals '×4' (any term) — vertical reading is where functional thinking begins.
□ and △ aren't toys — they're the ancestors of variables.
Elementary school's □×4=△ is middle school's y=4x. Only the symbols change; 'leave a slot for the unknown and write the relation' is the same idea.
Now read the definitions with their conditions
What tables and chairs revealed, in mathematical language.
Correspondence
Fix the table count □ and the chair count △ is determined — a relational formula that works for any number in the □ slot.
The joined-tables rule
In one row: 2□ seats along the sides plus 2 at the ends. Same tables, different arrangement, different formula — situations make rules.
The bridge to functions
Rename □×4=△ and you get f(x)=4x. Elementary correspondence IS the function.
Make one more distinction
Finding patterns means guessing the next item.
The real power is computing the 100th or 1000th without counting. Owning the formula '□th = □×4' — not knowing that 16 follows 4, 8, 12 — is the point of pattern-finding.
A rule is always 'grows by so many' — addition.
Join the tables in a row and the chairs become 2n+2 — rules multiply and add, or even double each time (powers). Watching only 'by how much' misses many rules — you need to read the chart vertically, between □ and △.
Where else can this idea help?
Fare rules
Taxi: $4.80 base + $0.80/km → fare = 0.8×□+4.8. Electricity, water, phone bills — all 'base + rate × usage' correspondences.
Calendar rules
If the first Monday is the 3rd, Mondays fall on 3, 10, 17, 24, 31 — the 'by 7' rule: □th Monday = 7×(□−1)+3.
Building blocks
Stack a staircase and n levels take 1+2+…+n blocks — playtime pattern-finding leading straight to series.
Programming
A function in code IS a rule: input fixed, output determined. Writing rules as formulas is the root of computational thinking.
Explain it in your own words
Try another question
Q1Write the wheel count of □ tricycles as a formula. Wheels on 25 tricycles?Show answer ▾
wheels = □×3. Put in 25 and get 75 — one formula instead of counting.
Q2Matchstick triangles in a row: 1 triangle takes 3 sticks, 2 take 5, 3 take 7… □ triangles?Show answer ▾
2×□+1 sticks — the first triangle costs 3, each new one only 2 more. Ten triangles: 21 sticks. The trick is owning both the '+2 each' and the formula 2□+1.
Q3What's the rule in 1↔1, 2↔4, 3↔9, 4↔16?Show answer ▾
△ = □×□ (squares). Read across, the +3, +5, +7 confuses; read down (1↔1, 2↔4) and the rule appears — the power of vertical reading.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Quantities that change together · 1/3
Leads to next
Direct & Inverse Proportion
The special correspondence where 'double □ doubles △' — meet the world's most common rule.
Go to the Direct & Inverse Proportion lab →Related lab
Related lab
Sequences
Where rule-following lists like 4, 8, 12, … get the full treatment.
Go to the Sequences lab →