Intuition The one core idea
Agile is a bet that finding out you were wrong quickly is cheaper than planning to never be wrong . Every symbol on the parent page — points, velocity, WIP, L = λW — exists to measure how fast work flows and how fast feedback returns , so you can steer instead of guess.
This page assumes you know nothing . We build every word, arrow, and letter the parent Agile — Scrum & Kanban note leans on, in an order where each idea rests on the one before it.
Before any math, picture the raw material. A software team turns requests ("let users reset their password") into working features . Each request is a little card. The whole game is: cards come in, cards go out, and we care about how long each card takes and how many are stuck in the middle .
Look at the figure. Three zones — waiting , being worked on , done . Everything below is just names for the quantities you can see here: how many cards sit in the middle, how fast they leave, how long one card lingers.
A batch is how much work you bundle together before showing it to anyone. A big batch = build for a year, then reveal. A small batch = build for two weeks, then reveal.
Intuition Why the parent hates big batches
The picture: imagine baking one giant cake vs many cupcakes. If the recipe is wrong, the giant cake wastes a year of ingredients; a cupcake wastes one tray. Agile bakes cupcakes — each a small increment (a slice of working product) — so a mistake costs one tray, not the whole kitchen.
The parent's word "potentially shippable increment" just means: one cupcake that is actually edible — a slice complete enough to hand to a user.
Definition Sprint and the letter
D
A sprint is a fixed-length stretch of time (1–4 weeks) in which the team builds one increment. We write D for the number of working days in that sprint . So if a sprint is 2 weeks, D = 10 working days.
Why do we need a letter for it? Because in a moment we'll ask "how much work should be left on day 3 of 10?" — and to answer that we compare a running day-count d against the total D .
Definition The day counter
d
d is "which day of the sprint are we on", running from 0 (start, nothing done) up to D (end). It's the x-axis of time inside one sprint .
Here is the first real notation you must not fear.
Intuition Why not just use hours?
Ask a person "how many hours is this task?" and they guess badly. Ask "is this task bigger or smaller than that one?" and they're pretty good. Humans compare well and estimate absolutely badly. So we measure relative size , not time.
A story point is a unit of relative effort/complexity. A task of "5" is meant to feel about 5 times a task of "1". Teams often use jumps like 1 , 2 , 3 , 5 , 8 (Fibonacci-ish) because as tasks get bigger, our ability to tell them apart gets coarser. See Project Estimation & Story Points for the estimation ritual itself.
Common mistake "A point equals some fixed number of hours"
Why it feels right: you want to convert points to a schedule.
The fix: a point has no fixed hour value . It only means something relative to the same team's other points . That's why the parent says comparing two teams' velocities is meaningless.
The parent writes ∑ i = 1 n p i . Let's earn every mark.
p i
p i means "the points completed in sprint number i ". The little i is just a label / house-number . If your last three sprints finished 18, 22, 20 points, then p 1 = 18 , p 2 = 22 , p 3 = 20 .
Definition The sigma symbol
∑
i = 1 ∑ n p i is shorthand for "start i at 1, keep adding p i until i reaches n ." It is nothing more than a lazy way to write p 1 + p 2 + ⋯ + p n .
Worked example Reading the symbol out loud
∑ i = 1 3 p i = p 1 + p 2 + p 3 = 18 + 22 + 20 = 60
The i = 1 under the ∑ = "where to start". The 3 (here n ) on top = "where to stop".
Definition The bar accent
v ˉ (read "v-bar ") is the average of a bunch of v values. The little bar on top is a universal math signal for "averaged".
v ˉ = 3 18 + 22 + 20 = 3 60 = 20 points/sprint
Velocity is just this v ˉ : a measured fact about the team, not a target.
The parent forecasts with ⌈ v ˉ B ⌉ . Two new things: B and the bracket.
B = backlog size
B is the total points of everything still to build — the whole to-do pile measured in points.
Definition The ceiling brackets
⌈ x ⌉
⌈ x ⌉ means "round x up to the next whole number". ⌈ 12.5 ⌉ = 13 . The brackets look like the top corners of a box — a picture-hint that you're pushing up to the ceiling.
Intuition Why up and never down?
If you need 12.5 sprints of work, twelve sprints leave a leftover half. That leftover still needs a whole sprint to run — you cannot buy half a sprint. So we always climb to the next integer. The figure shows the number line: any value in ( 12 , 13 ] lands on 13 .
Backlog B = 240 , velocity v ˉ = 20 : ⌈ 240/20 ⌉ = ⌈ 12 ⌉ = 12 sprints (exact, no rounding). If B = 250 : ⌈ 12.5 ⌉ = 13 sprints — the extra 10 points drag in a whole new sprint.
Definition Function notation
R ( d )
R ( d ) reads "R of d " — "the amount of work R emaining when the day is d ." Feed it a day d , it hands back remaining points. It's a machine: input day → output work-left.
Intuition Why a straight line, and why we bother
A straight line assumes you burn the same W / D points every day — a fair, boring baseline. Its whole job is to be a ruler to lay against reality : if your real remaining-work line sits above the straight one (blue in the figure), you're behind; below, ahead.
Now the second cluster of symbols, for the flow world. Go back to the figure in §0 — the same three zones.
L — items in the system (WIP)
L = the average number of cards sitting in the "being worked on" middle zone . Its everyday name is WIP: Work-In-Progress . Picture cars currently on a stretch of highway.
λ — throughput (the Greek letter "lambda")
λ (say "lambda") = how many cards finish per unit time — the rate cards leave through the Done door. Picture cars exiting the highway per minute. We borrow this letter from queueing theory, where λ always names a rate.
W — time in system (lead / cycle time)
W = the average time one card spends from entering to leaving. Picture how long one car is on the highway. (Same letter W the burndown used for "total work" — context tells them apart; here it means W ait/lead time.)
Intuition Why Kanban cares
Rearrange to W = L / λ . To make each card finish sooner (small W ), you can shrink L (fewer cards in progress) or grow λ (finish faster). Speeding up is hard; capping L is a rule you can just enforce — that is the WIP limit. Same worker speed, shorter wait, because each card queues behind fewer others.
Worked example The numbers from the parent
λ = 5 cards/week, L = 15 cards: W = 15/5 = 3 weeks. Cap WIP so L = 10 : W = 10/5 = 2 weeks. Throughput unchanged, wait cut by a third.
SDLC — the Software Development Life Cycle (SDLC) is the overall lifecycle (plan → build → test → release) that Agile is one way of running.
Waterfall — the Waterfall Model is the "big dam" the parent contrasts Agile with: do every phase once, in order.
CI/CD — Continuous Integration & Delivery (CI-CD) is the machinery that makes tiny increments actually shippable often.
Little's Law L equals lambda W
Cover the right side; can you answer each before revealing?
What does p i mean? The story points completed in sprint number i .
What does ∑ i = 1 n p i expand to? p 1 + p 2 + ⋯ + p n — add every sprint's points.
What is the bar in v ˉ telling you? It's the average of the v values.
Why do we use the ceiling ⌈ ⋅ ⌉ when forecasting sprints? You can't run a fractional sprint, so any leftover work needs a whole extra sprint — round up.
In R ( d ) = W ( 1 − d / D ) , what is R ( 0 ) and R ( D ) ? R ( 0 ) = W (all work remains); R ( D ) = 0 (nothing left).
Why is the ideal burndown a straight line? It assumes a constant burn rate W / D per day — a baseline ruler for reality.
What do L , λ , W each stand for? L = items in progress (WIP); λ = throughput (finished per unit time); W = time one item spends in the system.
State Little's Law and its picture. L = λW — items on the road = exit-rate × time each stays.
To deliver faster (smaller W ) which lever does Kanban actually pull? It caps L (WIP limit), since W = L / λ and λ is hard to force up.
Why are story points relative, not hours? Humans compare sizes well but estimate absolute time badly; points capture relative effort only.