A BEGINNER BRIDGE · ABOUT 55 MINUTES
You do not need to know the symbols yet. Learn one idea, watch it work, then solve a small problem yourself. Mistakes show you what to try next.
Begin with numbersFIVE FOUNDATION MOVES
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BEFORE ANALYSIS I · 10 MIN
Analysis I soon uses fractions, equations, and functions together. Here is the smallest useful step for each: make equal-sized fraction pieces, undo one operation, and read an output at a chosen input.
FRACTIONS · WORKED EXAMPLE
What is 1/2 + 1/4? The denominator tells us the size of each piece. Halves and fourths are different sizes, so first rewrite the half as two fourths.
Keep the denominator 4: you now have three pieces, each one fourth. Adding 2 + 1 and 4 + 4 would describe different pieces.
ONE-STEP EQUATION · WORKED EXAMPLE
The equation 4x = 20 says four times an unknown number x equals 20. To leave x by itself, undo multiplication by 4 with division by 4.
The same operation on both sides keeps the equation balanced. For x + 4 = 20, subtraction would undo addition instead.
FUNCTION GRAPH AND TABLE · WORKED EXAMPLE
In y = f(x), x is the input and y is the output. For this example, the allowed inputs are just 0, 1, and 2, and the output rule is f(x) = 2x + 1. To find f(1), locate x = 1, then read its output: 3. The point is (1, 3).
| x | f(x) = y |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
The domain is the allowed input set, here {0, 1, 2}. These three separate dots show exactly those inputs. A rule can allow other inputs too, but that must be stated or shown; do not assume a line connects separate dots. The table lists the same pairs for text and screen readers.
Try a new fraction, a one-step equation, a table, and a graph’s allowed inputs. Correct answers here show practice completed; they do not establish mastery.
Practice graph. Only the three filled points shown belong to this function; there is no line between them.
| x | f(x) = y |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 4 | 9 |
Try the four checks before moving on; the next step is always available.
Next: functionsALGEBRA AND FUNCTIONS · 10 MIN
A variable is a name for a number we can change. A function takes an input and gives one output. In f(x) = 2x + 3, the rule says “double the input, then add three.”
WORKED EXAMPLE
Check: f(6) = 2 × 6 + 3 = 15. Reversing steps in the opposite order avoids a common mistake.
READ A GRAPH
For the same rule, the pairs (0, 3), (1, 5), and (2, 7) lie on one straight line. The first coordinate is the input; the second is the output. When x increases by 1, the output increases by 2.
The domain is the set of allowed inputs. A real world model may restrict it: a number of students cannot be negative.
Aim for three correct answers. Each check gives a clue if you need another try.
Try the three checks before moving on; the next step is always available.
Next: probabilityINTRODUCTORY PROBABILITY · 12 MIN
An outcome is one possible result. An event groups outcomes we care about. Probability is a number from 0 to 1: 0 means impossible, 1 means certain.
WORKED EXAMPLE
A fair six sided die has outcomes 1 through 6, all equally likely. The event “roll an even number” contains 2, 4, and 6.
The rule “favourable ÷ possible” works here because the six outcomes are equally likely. If they are weighted differently, add their probabilities instead.
EXPECTED VALUE
A game pays €4 with probability 1/4 and €0 with probability 3/4. Multiply each value by its chance and add: 4 × 1/4 + 0 × 3/4 = €1.
Expectation describes the average across many plays. A single play still pays either €4 or €0; it never pays €1.
Calculate a chance, distinguish chance from certainty, then find an expected value.
Try the three checks before moving on; the next step is always available.
Next: table reasoningALGORITHMS AND TABLES · 12 MIN
An algorithm is an unambiguous sequence of steps. A table helps when the next answer depends on earlier ones. Fill only a cell whose required earlier cells are known.
WORKED EXAMPLE
Imagine you record points earned on three days: 2, 3, 1. Let T(i) mean the total through day i.
A recurrence names the earlier answer used to calculate the next one. The base case gives the process somewhere to start.
TRACE THE DEPENDENCY
| Day i | New points | T(i) |
|---|---|---|
| 0 | — | 0 |
| 1 | 2 | 2 |
| 2 | 3 | 5 |
| 3 | 1 | 6 |
In an alignment table, a cell can depend on the cells above, left, and diagonally above. The same habit applies: identify what a cell depends on before filling it.
Trace the update, find the starting value, then choose a valid fill order.
Try the three checks before choosing a lab; the route is always available.
Next: proof reasoningPROOF REASONING · 10 MIN
Checking examples helps you find an idea, but a claim about every allowed number needs an argument that works for any such number. One failing example, called a counterexample, disproves a universal claim.
WORKED ARGUMENT
Because the argument starts with an arbitrary whole number n, it covers every allowed n. Trying 2 + 3 = 5 alone would not.
TEST A UNIVERSAL CLAIM
Someone says “every even number is divisible by 4.” The number 6 is even, but 6 ÷ 4 is not a whole number. That one counterexample is enough to reject the claim.
For a claim that is true, examples and pictures can guide you toward a proof. They do not replace a general explanation.
Separate examples from proof, use a definition, and find a counterexample.
Try the three checks before choosing a lab; the route is always available.
See the next labsAFTER THE BRIDGE
These first steps reuse what you just practised. Each lab adds ideas that this short bridge does not teach.
Use the idea of a counterexample to read “for every” and “there exists.” Induction and calculus come later in the route.
Enter Analysis ITest when two equations can be true together, then explore how models and assumptions affect computed results. Later work uses calculus and linear algebra.
Enter Scientific ComputingAssign chances to outcomes before studying randomness over time. Calculus is needed for the later process models.
Enter Random ProcessesCarry the base case and recurrence habit into sequence alignment. Later models add probability and biological interpretation.
Enter Bioinformatics AlgorithmsCount matches and judge which results matter. The lab then turns these ideas into search and evaluation methods.
Enter Multimedia RetrievalTurn a disease question into a model, then separate an observed pattern from an intervention and its possible harms.
Enter Drug Discovery