How the World Works — Numbers, Patterns and Data

PYP Transdisciplinary Theme: HOW THE WORLD WORKS

Central Idea: Mathematics is a LANGUAGE humans created to describe PATTERNS, solve PROBLEMS, and make SENSE of the world around us.

Lines of Inquiry

How do fractions, decimals, and percentages CONNECT?
How do we collect, organise, and INTERPRET data?
How does mathematics help us make DECISIONS?

1. Fractions, Decimals and Percentages — Three Ways to Say the SAME Thing

½ = 0.5 = 50%. 'These are not DIFFERENT things. They are the SAME number — expressed in three different FORMS. Fractions are good for DIVIDING. Decimals are good for MEASURING. Percentages are good for COMPARING.'

Operations With Fractions

Add/Subtract: find COMMON DENOMINATOR. Multiply: multiply numerators, multiply denominators. Divide: multiply by the RECIPROCAL. 'The Golden Rule: whatever you do to the denominator, you must ALSO do to the numerator — or you change the VALUE.'

Converting Between Forms

Fraction → Decimal: Divide numerator by denominator. 3/8 = 0.375.
Decimal → Percentage: Multiply by 100. 0.375 × 100 = 37.5%.
Percentage → Fraction: Write over 100. Simplify. 25% = 25/100 = ¼.

Real-World Use

'A shop has a 25% SALE. A shirt costs ₹800. How much do you PAY? 25% of 800 = ₹200 discount. You pay ₹600. OR: 75% of 800 = ₹600 — same answer, FEWER steps. Understanding the CONNECTION between fractions, decimals, and percentages gives you the POWER to CHOOSE the easiest method.'

2. Ratio and Proportion — The Mathematics of FAIRNESS

A recipe calls for 2 cups of flour and 1 cup of sugar. Ratio = 2:1. If I want to DOUBLE the recipe → 4 cups flour, 2 cups sugar (the RATIO is PRESERVED). 'Ratio is about RELATIONSHIP. Two quantities in a STABLE relation to each other. Proportion is about EQUIVALENCE. Two ratios that are EQUAL.'

'If 5 notebooks cost ₹200, how much do 8 notebooks cost?' (Proportion: 5/200 = 8/x → x = ₹320). 'This is the UNITARY METHOD — find the value of ONE thing first.'

3. Data Handling — From Information to Understanding

Collecting Data

Primary (collected by YOU — survey, experiment). Secondary (collected by SOMEONE ELSE — books, internet).

Organising Data

TALLY MARKS. Frequency tables. Grouped data (class intervals).

Displaying Data — Choosing the RIGHT Graph

Graph	Best For
Bar Chart	Comparing CATEGORIES
Line Graph	Showing change over TIME
Pie Chart	Showing PROPORTIONS of a whole
Pictograph	Making data FUN and accessible

Analysing Data — The Three M's

Mean (average — sum ÷ count). Median (middle value — not affected by extremes). Mode (most common — the 'popular' one). 'The mean is the USUAL choice — but it can be MISLEADING if there are extreme values. "The average income of 10 people in a room is ₹10 lakh. But that's because one person earns ₹91 lakh and nine earn ₹1 lakh each. The MEDIAN tells the TRUTH: ₹1 lakh."'

4. Probability — What Are the CHANCES?

Probability = Number of favourable outcomes / Total possible outcomes. Scale: 0 (IMPOSSIBLE) → 1 (CERTAIN). Tossing a coin: P(Head) = ½. Rolling a die: P(6) = 1/6.

Experimental vs. Theoretical

'Theoretical: what SHOULD happen (the MATH). Experimental: what ACTUALLY happens (the TRIAL). Toss a coin 10 times — maybe 7 heads, 3 tails. Toss it 1,000 times — it will be MUCH closer to 500/500. This is the LAW OF LARGE NUMBERS.'

Your Summative Assessment

Task: 'The Data Investigation'

Ask a QUESTION about your world: 'How many hours do students in our class sleep?' 2. COLLECT data. 3. ORGANISE it (tables, frequencies). 4. DISPLAY it (at least TWO graphs). 5. ANALYSE it (mean, median, mode, range). 6. Draw a CONCLUSION. 'Mathematics is not about getting the "right answer." It's about ASKING QUESTIONS and using DATA to find ANSWERS.'