Chapter 1 demo doc
Ratios and proportions
In this unit you will learn how to use ratios and proportions to compare quantities and to reason about situations of competition and cooperation. These tools are useful in many everyday and academic contexts - from analysing election representation and pay gaps to solving problems about recipes, sports, maps and science.
Contexts and motivation
Politics and government
Different forms of government (monarchies, democracies, republics) and different systems (parliamentary, presidential) raise questions that can be analysed with ratios. For example:
- What is the ratio of representatives to citizens in a legislature?
- Is that ratio the same in urban and rural areas?
- Can ratios show whether some people have less representation than others?
Personal and cultural expression
Everyday stories and cultural examples can be used to practise proportional reasoning. Consider these imaginative questions:
- Hansel and Gretel: if a gingerbread cottage were scaled up to human size, how would recipe quantities change? How much would the ingredients cost?
- Cinderella: could the prince use proportions to narrow the number of possible owners of the glass slipper?
- Fairy-tale scaling: if a pumpkin becomes a carriage and mice become horses, are all parts scaled by the same ratio?
Key ideas
- Logic is used to simplify quantities and to establish equivalence.
- Related concepts: equivalence, quantity, simplification.
- Global context: identities and relationships - how humans cooperate and compete.
Statement of inquiry
Using a logical process to simplify quantities and establish equivalence can help analyse competition and cooperation.
Overview of learning targets
- Defining and simplifying ratios.
- Dividing a quantity in a given ratio.
- Defining proportion and demonstrating proportional relationships.
- Representing proportional relationships with tables, equations and graphs.
- Finding the constant of proportionality for a proportional relationship.
- Applying proportional reasoning to solve problems.
Prior knowledge
- Round numbers correctly.
- Solve problems involving percentages.
- Convert between different units.
- Solve simple equations.
Introducing ratio and proportion
Competition and cooperation appear together in many activities. Mathematically, a ratio is a comparison of two (or more) quantities measured in the same units. Ratios can be simplified like fractions.
Definition and simple examples
If a team has 12 girls and 8 boys, the ratio of girls to boys is written as 12 : 8 (read "12 to 8").
Simplified form: 12 : 8 = 3 : 2 because both numbers can be divided by 4.
The ratio of girls to the total number of students is 12 : 20 = 3 : 5.
A three-term ratio (boys : girls : total) is 8 : 12 : 20 = 2 : 3 : 5 after simplification.
Investigation 1 - Simplifying ratios
Copy the following kinds of activities into your notebook and complete them:
- Simplify these ratios: 8 : 12, 21 : 35, 18 : 9, 4 : 16, 15 : 35, 45 : 10, 36 : 42.
- Explain how you decided which ratios are equivalent, and generalise a procedure for simplifying ratios.
- Verify the procedure with two different examples.
Reflect and discuss
- Can all ratios be simplified? Explain.
- How is simplifying ratios similar to simplifying fractions?
Activity 1 - Ratios around the room
- Work with a partner to find ratios in the classroom (for example, number of windows : number of doors).
- Play "I Spy" with ratios: one player states a ratio and the other guesses the quantities represented.
- Practice simplifying ratios in the final rounds.
Equivalent ratios
Two ratios that simplify to the same ratio are called equivalent ratios. For example, 12 : 16, 3 : 4 and 36 : 48 are all equivalent because each simplifies to 3 : 4.
Investigation 2 - Equivalent ratios
Complete a table of ratios, find simplified forms and list equivalent ratios. Describe the method used to generate equivalent ratios (multiply or divide both terms by the same non-zero whole number).
Reflect and discuss
- How are equivalent ratios like equivalent fractions?
- How would you simplify a ratio that contains a decimal, for example 0.5 : 3?
Example 1
One famous match was the 1973 "Battle of the Sexes" tennis match. The ratio of games won was 18 : 10 in favour of Billie Jean King. Give three ratios equivalent to 18 : 10.
Ans: 18 : 10 simplifies to 9 : 5. Three equivalent ratios are 9 : 5 (simplified), 36 : 20 (multiply both terms of 9 : 5 by 4), and 90 : 50 (multiply both terms of 9 : 5 by 10). Rule used: multiply or divide both terms by the same non-zero whole number.
Example 2 - Dividing a total in a given ratio
A marathon is about 42 km. Three friends decide to run distances in the ratio 1 : 2 : 3. How far does each person run?
Ans: Sum of ratio parts = 1 + 2 + 3 = 6. Each part equals 42 ÷ 6 = 7 km. So the distances are 7 km, 14 km and 21 km respectively.
After the first runner runs 2 km and withdraws, the remaining distance is 40 km. The other two share it in the ratio 3 : 5. Let r be the common multiplier. Then 3r + 5r = 40 ⇒ 8r = 40 ⇒ r = 5. Distances: 3r = 15 km and 5r = 25 km.
Did you know?
Pheidippides is the legend often cited as the origin of the marathon: he is said to have run from Marathon to Athens in 490 B.C. The modern marathon distance, 26.2 miles (42.195 km), was adopted at the 1908 London Olympics.
Practice 1
Simplify the following ratios. If already simplest, write "simplified". Then give equivalent ratios where requested and solve applied problems:
- Simplify: 3 : 5; 4 : 6; 12 : 16; 100 : 200; 700 : 35; 0.4 : 6; 9 : 10; 1 : 6; 950 : 50; 13 : 200; 1006 : 988; 64 : 16.
- Write three equivalent ratios for each: 2 : 5; 2 : 18; 9 : 27; 10 : 25; 6 : 7; 15 : 45; 8 : 1; 12 : 3.
- Comics example: DC to Marvel original ratio 7 : 5. If kept the same while both add characters, give two possible team sizes. If later the ratio is 21 : 10, is this equivalent? If The Justice League has 91 members with the original ratio, how many Avengers would you expect?
- Draw pictures combining multi-term ratios: green : blue : red flowers 2 : 3 : 5, boys : girls 1 : 2, trees : bushes 1 : 1.
- From a list of ratios, find equivalent groups and explain why.
- Pokemon deck: ratio of Pokemon : trainer : energy = 5 : 2 : 3; for a 60-card deck, how many of each card type?
- Gender pay gap table: convert wages given as decimals to simplified ratios and compare which country shows the largest gap.
- Given ratios and totals, divide the total according to the ratio.
- Regional competition: if ratio Swedish : Danish : Finnish = 14 : 22 : 13 and total is 1 029 students, find counts from each country.
- Solve for unknowns where ratios are stated as equivalent (e.g., 1 : 4 = 2 : k and others).
Reflect and discuss
- Write what is going well in this unit; describe a time when positive thinking helped.
Other representations of ratios
Ratios can be written in several equivalent forms: as a:b, as a fraction a/b, as a decimal and as a percentage. Converting between these forms is useful for comparison. For example:
- 3 : 4 = 3/4 = 0.75 = 75%.
Activity 2 - Competition in government (data handling)
Use a given table of elected officials by gender in several countries. Practice writing the ratio women : men, converting to fraction, decimal and percentage, and compare countries. Discuss which country has the highest proportion of women and whether that passes a 30% quota rule.
Reflect and discuss
- How do you convert a ratio to a percentage?
- Can a three-term ratio like 3 : 5 : 11 be represented as a single percentage? Explain how to express parts of a multi-term ratio as percentages of the whole.
Practice 2
Matching tasks and applied questions:
- Match ratio cards with equivalent fraction and percentage cards.
- University admissions example: express given admittance ratios as fraction, decimal and percentage; decide which representation best shows diversity.
- Summit Series (hockey) data: express games won as ratio/fraction/decimal/percentage; compare effectiveness of representations; compute ratio of goals scored.
- International Space Station examples: examine astronaut nationality ratios and water usage ratios, convert to percentage and fraction; convert an "ISS day" of 6.25% of Earth day into a fraction and find how many ISS orbits per Earth day.
Formative assessment (example)
Eating contests, Olympic prize money and other data sets are used to practise forming ratios, simplifying, and expressing as fractions, decimals and percentages. Sample tasks:
- Hotdog-eating contest tables: find ratio prize money : hotdogs eaten for top competitors in 2017, simplify and express in four forms. Decide which representation is most useful.
- Compare male and female winners across years, express hotdogs per minute, and discuss fairness of contests divided by gender.
Proportions
When two ratios are equal they form a proportion. For example, 2 : 15 = 6 : 45 is a proportion because both simplify to the same ratio.
Solving proportions
An equation showing that two ratios are equal is a proportion. A standard method for solving for a missing value is cross-multiplication. The rule: if a : b = c : d then a × d = b × c.
Investigation 3 - Patterns in proportions
Examine many equivalent fractions/proportions and identify patterns that can be expressed as equations. Use these equations to find missing values and to generate new equivalent ratios.
Example 3 - Finding a missing value using cross-multiplication
Find the missing value in each proportion. Round answers to the nearest hundredth where necessary.
Ans:
For a proportion a : b = c : d we use cross-multiplication: a × d = b × c.
If 8 : 12 = x : 22 then
\[ 8 \times 22 = 12 \times x \]
\[ 176 = 12x \]
\[ x = \frac{176}{12} \approx 14.67 \]
If 5 : x = 9 : 17 then
\[ 5 \times 17 = 9 \times x \]
\[ 85 = 9x \]
\[ x = \frac{85}{9} \approx 9.44 \]
Always check solutions by substituting back into the original proportion.
Activity 3 - Solving proportions: alternative methods
Use algebraic methods and reciprocals to solve proportions. Observe relationships between solving by cross-multiplication and by writing proportions as equations and solving the resulting linear equation.
Practice 3
- Solve a large set of missing-value proportion questions (round where required).
- Decide whether given pairs of ratios are equivalent and justify.
- Map scale problems: use the map scale 1 : 15 000 to convert measurements on the map to real distances and vice versa, using proportions.
- Proportional growth problems: e.g. the Great Trail length progress, Sherpa porter requirements scaled to equipment mass, etc.
Recognizing and using proportional reasoning
Not every relationship is proportional. A relationship is proportional if one variable is always a constant multiple of the other. This constant is called the constant of proportionality.
Investigation 4 - Recognising proportional relationships
Use examples such as Olympic rewards or payments by medals to determine whether the data form a proportional relationship. When plotted with the independent variable on the x-axis and the dependent variable on the y-axis, proportional relationships are straight lines passing through the origin. The equation is y = kx, where k is the constant of proportionality.
Tasks:
- Plot data (for example, payment per gold medal) and describe the graph.
- Find the payment for zero medals and for 10 medals using the proportional rule.
- When payments are paid over years (installments) check whether the relationship between years and total paid is proportional.
- Write rules for recognising proportional relationships from tables, graphs and equations.
Reflect and discuss
- Does a proportional relationship always include the point (0, 0)? Explain.
- How do the graph, equation and table represent the same proportional relationship?
Activity 4 - Keeping the pace (applied proportional reasoning)
Data from a pacemaker runner give times at distances. Use a graph and an equation to test whether the time vs distance relationship is proportional, and use it to estimate times for other distances. Compare your predictions to actual results and discuss sources of difference.
Practice 4
- Decide whether given data sets are proportional. If yes, find the equation y = kx.
- Problems: group singing time inversely proportional to number of singers, rideshare pricing, Olympic money per medal graphs, the Baltic Way chain length vs people, grocery unit-pricing comparisons, and more.
Examples and solutions (summary of methods)
- To check proportionality from a table: compute y/x for each pair; if the ratio is constant, the relationship is proportional and k = y/x.
- To check proportionality from a graph: a straight line through the origin indicates proportionality.
- To check from an equation: if the equation can be written as y = kx (no constant term), it is proportional.
Unit summary
- A ratio compares quantities measured in the same units and can be written as a:b, a/b, a decimal or a percentage.
- Ratios can be simplified; ratios that simplify to the same value are equivalent.
- Two equivalent ratios form a proportion.
- A proportional relationship is one in which one variable is a constant multiple of the other; its graph is a straight line through the origin and its equation is y = kx.
- To solve proportions use cross-multiplication or algebraic rearrangement; always check your results.
Unit review (selected problems)
Work through these to consolidate learning:
- Simplify various ratios and write equivalent ratios.
- Apply ratios to real contexts: mullet hairstyle ratios, workforce education percentages converted to ratios, Olympic team data and refugee team origins.
- Use proportional reasoning to answer speed, scaling and allocation problems (for example, compare athletes' speeds, scale lengths on a map, or allocate total quantities according to a stated ratio).
- Solve problems mixing ratios: if x : y = 4 : 5 and x : z = 2 : 3, find y : z.
- Use proportional models to estimate requirements in cooperative efforts (e.g., number of firefighters per area, or Sherpas per equipment mass).
Summative assessment - Project: What is fair competition?
Explore whether differences such as height give unfair advantage in particular Olympic events. Project tasks:
- Choose a short individual track event (≤ 400 m). Record medalists' heights and times from the last Olympics.
- Calculate each athlete's average speed (distance ÷ time). Represent speed as a decimal.
- Scale race distances in proportion to athletes' heights and compute the new race times (assume constant speed). Analyse whether medal positions change and whether men and women could be compared across sexes if distances were proportional to height.
- Repeat analysis for another event of your choice. Discuss whether height creates unfair advantage and whether competition should be organised by height classes for selected events.
Reflect and discuss
- How precise should calculations be; to how many decimal places should you round?
- What information would you include if you repeated this analysis?
- What makes for fair and healthy competition? Is competition or cooperation more about being equal?
Final suggestions for study
- Practice converting between ratio, fraction, decimal and percentage in many contexts.
- When solving proportions, always check by substitution.
- Use tables, graphs and algebra together: they are different representations of the same relationships.
- Create a study plan: practice problems from this unit daily, and make a one-page mind map summarising key formulas and methods.
FAQs on Chapter 1 demo doc
| 1. What are ratios and how are they used in mathematics? |  |
| 2. What is a proportion and how does it differ from a ratio? | |
| 3. How can understanding equivalence help in solving ratio problems? | |
| 4. What role does simplification play in working with ratios and proportions? | |
| 5. How are ratios and proportions relevant in real-world contexts such as politics and government? | |
