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{{T}}
[[Category:English]]
[[Category:Ratios, Rates, and Proportions]]

{{BR}}
= Introduction =

'''Ratios, Rates, and Proportions''' help you compare quantities, describe how quickly or cheaply something happens, and predict values when quantities change together. In Grades 7–8, these ideas connect arithmetic, [[English:Fractions|fractions]], [[English:Decimals|decimals]], [[English:Percentage|percentages]], [[English:Linear function|linear relationships]], geometry, science, maps, recipes, and everyday decision-making.

By the end of this aiMOOC, you should be able to:
# [[English:Ratio|Ratio]]: Write, simplify, and interpret ratios in several forms.
# [[English:Rate (mathematics)|Rate]]: Compare quantities with different units and calculate unit rates.
# [[English:Proportion (mathematics)|Proportion]]: Recognize and solve equations made from equivalent ratios.
# [[English:Proportionality (mathematics)|Proportional relationship]]: Identify proportional relationships in tables, equations, and graphs.
# [[English:Scale factor|Scale factor]]: Use multiplicative reasoning in scale drawings, recipes, and similar figures.
# [[English:Problem solving|Problem solving]]: Explain your reasoning, check whether an answer is reasonable, and use units correctly.

{{#ev:youtube|https://www.youtube.com/watch?v=HpdMJaKaXXc|500|center}}

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= Ratios: Comparing Quantities =

A '''ratio''' compares two quantities by division. If a class has 12 students wearing blue and 18 students wearing another color, the ratio of blue to other colors is 12:18. Dividing both parts by 6 gives the equivalent ratio 2:3.

Ratios can be written in several ways: 2 to 3, 2:3, or 2/3. The order matters. A ratio of cats to dogs is not the same as a ratio of dogs to cats unless the two quantities are equal.

A ratio may compare:
# [[English:Part-to-part ratio|Part-to-part ratio]]: One part of a group with another part, such as 8 red tiles to 12 blue tiles.
# [[English:Part-to-whole ratio|Part-to-whole ratio]]: One part with the total, such as 8 red tiles to 20 tiles altogether.
# [[English:Equivalent ratio|Equivalent ratio]]: Ratios that represent the same multiplicative comparison, such as 2:3 and 8:12.

[[File:Typical.Aspect.Ratios.png|500px|frameless|center]]

The image above shows how ratios can describe the shape of a rectangle. An aspect ratio such as 4:3 or 16:9 compares width with height. This is the same ratio idea used in screens, photographs, and design.

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== Simplifying and Building Equivalent Ratios ==

To simplify a ratio, divide both terms by the same nonzero number. For example, 24:36 simplifies to 2:3 because both numbers are divided by 12.

To build an equivalent ratio, multiply both terms by the same nonzero number. From 2:3, multiplying by 5 gives 10:15. This is useful in [[English:Ratio table|ratio tables]], where each row keeps the same multiplicative relationship.

A ratio table for 3 notebooks costing €6 could include 1 notebook for €2, 5 notebooks for €10, and 8 notebooks for €16. Each pair has the same cost per notebook, so the relationship is proportional.

{{#ev:youtube|https://www.youtube.com/watch?v=RQ2nYUBVvqI|500|center}}

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= Rates and Unit Rates =

A '''rate''' is a ratio that compares quantities measured in different units. Examples include kilometers per hour, euros per kilogram, words per minute, and liters per second.

A '''unit rate''' tells the amount of one quantity for exactly one unit of another quantity. To find a unit rate, divide.

Example: A cyclist travels 150 km in 3 hours.

150 km ÷ 3 h = 50 km/h.

The unit rate is '''50 kilometers per hour'''.

{{#ev:youtube|https://www.youtube.com/watch?v=qGTYSAeLTOE|500|center}}

[[File:Speedometer-in-driving-car.jpg|500px|frameless|center]]

A speedometer reports a rate: distance traveled per unit of time. Rates always need units because the units explain what is being compared.

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== Comparing Unit Prices ==

Unit rates help you compare offers fairly.

Store A sells 6 notebooks for €9. The unit price is €9 ÷ 6 = €1.50 per notebook.

Store B sells 10 notebooks for €14. The unit price is €14 ÷ 10 = €1.40 per notebook.

If the notebooks are otherwise comparable, Store B has the lower unit price. A good comparison also considers quality, size, waste, and whether you actually need the larger package.

{{BR}}
= Proportions: Equal Ratios =

A '''proportion''' is an equation stating that two ratios are equal. For example:

3/5 = 12/20

Both ratios have the same value, so the equation is true.

You can solve a proportion by using scaling, equivalent fractions, or cross multiplication. Choose a method that makes the structure easy to see.

Example:

3/5 = x/20

Because 5 is multiplied by 4 to get 20, multiply 3 by 4 as well. Therefore x = 12.

You can also use cross products:

3 × 20 = 5 × x

60 = 5x

x = 12

[[File:Equal products in a proportion.svg|500px|frameless|center]]

The cross-product method works because multiplying both sides of an equation such as a/b = c/d by bd gives ad = bc, as long as the denominators are not zero.

{{#ev:youtube|https://www.youtube.com/watch?v=GO5ajwbFqVQ|500|center}}

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== Checking a Proportion ==

A proportion should make sense both numerically and in context. If 4 movie tickets cost €36, then one ticket costs €9. So 7 tickets should cost 7 × €9 = €63. If an algebraic method gave €6.30 or €630, the unit rate helps you notice that the result is unreasonable.

Always check:
# Are corresponding quantities written in the same order?
# Are the units compatible?
# Are the ratios equivalent?
# Is the answer reasonable in the situation?

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= Proportional Relationships =

Two variables have a '''proportional relationship''' when one variable is always a constant multiple of the other. The equation is:

y = kx

The number k is the '''constant of proportionality'''. It is also a unit rate and the slope of the graph.

Example: If each kilogram of apples costs €2.50, then the total cost y for x kilograms is:

y = 2.5x

The constant of proportionality is 2.5 euros per kilogram.

[[File:Proportionality.svg|500px|frameless|center]]

A graph of a proportional relationship is a straight line through the origin. The point (0,0) matters because buying zero kilograms costs zero euros. A straight line that does not pass through the origin may be linear, but it is not proportional.

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== Tables, Equations, and Graphs ==

You can test a table for proportionality by dividing y by x for each pair where x is not zero. If the value y/x stays constant, the relationship is proportional.

For example, the pairs (2, 6), (4, 12), and (7, 21) all have y/x = 3. Their equation is y = 3x.

If the ratios change, the relationship is not proportional. For example, a taxi fare with a fixed starting fee plus a cost per kilometer is linear but not proportional because the cost is not zero when the distance is zero.

{{#ev:youtube|https://www.youtube.com/watch?v=USmit5zUGas|500|center}}

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= Scale Factors and Scale Drawings =

A '''scale factor''' tells how much a figure, map, model, or drawing is enlarged or reduced. If every length in a drawing is twice the matching length in the original, the scale factor is 2.

In a scale of 1:100, one unit on the drawing represents 100 of the same units in reality. For example, 1 cm on the drawing represents 100 cm, or 1 m, in reality.

[[File:Map scale - 8km, 5mi.png|500px|frameless|center]]

Map scales are ratios between distances on a map and real-world distances. The units must be handled carefully before comparing or converting.

Example: On a map with scale 1:100,000, 1 cm represents 100,000 cm in reality. Since 100,000 cm = 1 km, a map distance of 6.4 cm represents 6.4 km.

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= Ratios in Recipes and Measurement =

Recipes are a useful example of proportional reasoning. Suppose a soup recipe for 4 servings uses 600 mL of broth. To make 10 servings, the scale factor is 10/4 = 2.5. Multiply the broth amount by 2.5:

600 mL × 2.5 = 1500 mL

So you need 1500 mL, or 1.5 L, of broth.

[[File:Measuring cups (¼, ½, 1 cup).jpg|500px|frameless|center]]

When scaling a recipe, multiply every ingredient by the same factor. Changing only one ingredient would change the recipe's ratios.

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= Ratios in Paper, Screens, and Design =

Aspect ratios keep shapes similar when they are resized. If a rectangle has width-to-height ratio 3:2, a 15 cm width corresponds to a 10 cm height because 15:10 simplifies to 3:2.

[[File:Paper ratio sizes-plain.svg|500px|frameless|center]]

Proportional resizing appears in printing, photographs, video, digital design, architecture, and technical drawings.

{{BR}}
= Percent as a Rate per Hundred =

A [[English:Percentage|percent]] is a special ratio with a denominator of 100. For example, 35% means 35 per 100, or 35/100 = 0.35.

This connects percent problems to proportions. If 35% of a number is 84, you can write:

35/100 = 84/x

You can also use the decimal equation 0.35x = 84. Both methods express the same proportional relationship.

{{BR}}
= Choosing a Strategy =

Different problems may be solved efficiently in different ways.

# [[English:Equivalent fractions|Equivalent fractions]]: Useful when one denominator or numerator is an obvious multiple of the other.
# [[English:Unit rate|Unit rate]]: Useful for speed, price, density, productivity, and fair comparisons.
# [[English:Cross multiplication|Cross multiplication]]: Useful for a proportion with one unknown when simple scaling is not obvious.
# [[English:Equation|Equation]]: Useful when a proportional relationship is described by y = kx.
# [[English:Graph|Graph]]: Useful for seeing whether the relationship is proportional and comparing rates visually.

Strong mathematical reasoning includes more than getting a number. Explain what each number represents, include units, show why the relationship is proportional, and check the result.

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= Common Mistakes and How to Avoid Them =

'''Mistake 1: Reversing the order of a ratio.''' If a problem asks for boys to girls, keep that order in every equivalent ratio.

'''Mistake 2: Adding instead of multiplying.''' Proportional reasoning is multiplicative. Going from 3 to 9 uses a factor of 3, so the matching quantity must also be multiplied by 3.

'''Mistake 3: Ignoring units.''' A rate such as 60 is incomplete if you do not know whether it means 60 km/h, 60 words/minute, or another unit.

'''Mistake 4: Assuming every straight-line pattern is proportional.''' A proportional graph must pass through the origin.

'''Mistake 5: Cross multiplying without understanding the quantities.''' First make sure corresponding quantities are placed in matching positions.

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= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''What is the simplified form of the ratio 8:12?'''
(2:3)
(!3:2)
(!4:5)
(!8:3)

{{E}}
<br>

{{MC}}

'''A car travels 180 km in 3 hours. What is its unit rate?'''
(60 km per hour)
(!90 km per hour)
(!177 km per hour)
(!540 km per hour)

{{E}}
<br>

{{MC}}

'''Which equation is a true proportion?'''
(3/4 = 9/12)
(!3/4 = 8/12)
(!2/5 = 6/10)
(!5/8 = 15/16)

{{E}}
<br>

{{MC}}

'''In the equation y = 5x, what is the constant of proportionality?'''
(5)
(!0)
(!x)
(!y)

{{E}}
<br>

{{MC}}

'''Six notebooks cost €9. What is the unit price?'''
(€1.50 per notebook)
(!€3 per notebook)
(!€9 per notebook)
(!€54 per notebook)

{{E}}
<br>

{{MC}}

'''Which graph feature is required for a proportional relationship?'''
(It passes through the origin)
(!It is always horizontal)
(!It has a negative slope)
(!It crosses the y-axis above zero)

{{E}}
<br>

{{MC}}

'''A length changes from 6 cm to 15 cm. What scale factor was used?'''
(2.5)
(!1.5)
(!9)
(!21)

{{E}}
<br>

{{MC}}

'''Solve the proportion 5/8 = x/24.'''
(15)
(!10)
(!19)
(!40)

{{E}}
<br>

{{MC}}

'''What does 30 percent mean as a ratio?'''
(30 per 100)
(!30 per 10)
(!3 per 1000)
(!100 per 30 only)

{{E}}
<br>

{{MC}}

'''A recipe uses 2 cups of rice for 6 servings. How many cups are needed for 15 servings?'''
(5 cups)
(!3 cups)
(!7 cups)
(!12 cups)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Ratio || A comparison of two quantities by division
|-
| Rate || A ratio comparing quantities with different units
|-
| Unit rate || A rate for one unit of a quantity
|-
| Proportion || An equation stating that two ratios are equal
|-
| Scale factor || A multiplier used to enlarge or reduce corresponding lengths
|-
| Constant of proportionality || The fixed value k in a relationship of the form y equals kx
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Ratio'''
| Comparison by division
|-
| '''Unit rate'''
| Amount for one unit
|-
| '''Proportion'''
| Equality of two ratios
|-
| '''Constant of proportionality'''
| Fixed multiplier in a proportional relationship
|-
| '''Scale factor'''
| Multiplier connecting corresponding lengths
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Ratio || What compares two quantities by division?
|-
| Rate || What compares quantities measured in different units?
|-
| Proportion || What equation says that two ratios are equal?
|-
| Equivalent || What word describes ratios that have the same value?
|-
| Scaling || What process multiplies related quantities by the same factor?
|-
| Constant || What fixed value appears as k in the equation y equals kx?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Ratios+Rates+and+Proportions </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
A { ratio } compares two quantities by division.
A rate that describes an amount for one unit is called a { unit rate }.
An equation showing that two ratios are equal is a { proportion }.
Equivalent ratios are created by multiplying or dividing both terms by the same { nonzero number }.
In the equation y equals kx, k is the { constant of proportionality }.
A proportional graph passes through the { origin }.
A scale drawing uses a fixed { scale factor } between corresponding lengths.
A percent is a ratio measured per { hundred }.
Before comparing rates, you should check the { units }.
A final answer should be checked for mathematical and contextual { reasonableness }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Ratio Hunt|Ratio Hunt]]: Find five ratios in your classroom, home, or school environment, photograph or sketch each example, and write one sentence explaining what the two quantities compare.
# [[English:Unit Rate Poster|Unit Rate Poster]]: Create a one-page poster that explains one real-life unit rate such as speed, price per item, or words per minute, including a worked example with units.
# [[English:Recipe Scaling|Recipe Scaling]]: Choose a simple recipe and calculate the ingredient amounts for half as many servings and twice as many servings; show the scale factors you used.
# [[English:Ratio Interview|Ratio Interview]]: Interview a classmate or family member about a situation in which they compare quantities, then rewrite the situation as a ratio or rate problem.

{{BR}}
=== Standard ===
# [[English:Best Buy Investigation|Best Buy Investigation]]: Compare at least four package sizes of the same product using unit prices, record your calculations, and explain which option is the best value under different assumptions.
# [[English:Proportional Graph Project|Proportional Graph Project]]: Collect or invent data for a proportional relationship, make a table and graph, write its equation, and explain how the slope relates to the unit rate.
# [[English:Map Scale Challenge|Map Scale Challenge]]: Use a printed or digital map with a scale to estimate three real distances, then explain any sources of measurement error.
# [[English:Video Explanation of Proportions|Video Explanation of Proportions]]: Produce a two- to three-minute video teaching how to solve one proportion in two different ways and how to check the answer.

{{BR}}
=== Advanced ===
# [[English:Nonproportional Comparison|Nonproportional Comparison]]: Find one proportional and one nonproportional real-world relationship, model both with tables, equations, and graphs, and explain the difference.
# [[English:Scale Model Design|Scale Model Design]]: Design a scale model of a room, object, or structure, choose a scale, calculate at least six corresponding measurements, and justify why the model stays proportional.
# [[English:Rate Experiment|Rate Experiment]]: Conduct a safe experiment such as walking a measured distance, filling a container, or typing a passage, calculate rates from repeated trials, and analyze variation in the results.
# [[English:Community Data Investigation|Community Data Investigation]]: Visit or research a relevant local place such as a store, sports facility, transit stop, or museum, collect ratio or rate data, and present a reasoned conclusion supported by calculations.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Model Selection|Model Selection]]: Given a table, graph, equation, and verbal description, decide which representations show proportional relationships and justify each decision using a constant ratio or the origin test.
# [[English:Best Value Reasoning|Best Value Reasoning]]: Compare two pricing plans that include different package sizes or fees, calculate useful unit rates, and explain which plan is better for two different users.
# [[English:Error Analysis|Error Analysis]]: Analyze an incorrect proportion solution, identify the exact reasoning error, correct it, and explain how a unit-rate or estimation check reveals the mistake.
# [[English:Scale Transfer|Scale Transfer]]: Use a scale drawing to calculate an unknown real length, then create a new drawing at a different scale and explain how all corresponding lengths change.
# [[English:Multi-Step Rate Problem|Multi-Step Rate Problem]]: Solve a problem that combines a rate with a unit conversion, show each step with units, and explain why the final unit is appropriate.
# [[English:Proportionality Argument|Proportionality Argument]]: Build a mathematical argument from a real or simulated data set showing whether two variables are proportional, using at least two representations as evidence.

<br>
<br>

{{BR}}
= Evidence of Learning =

Important evidence that you understand the topic includes:

'''Knowledge:''' You can define and distinguish ratios, rates, unit rates, proportions, scale factors, and constants of proportionality.

'''Skills:''' You can simplify ratios, calculate unit rates, solve proportions, convert units when needed, and recognize proportional relationships in tables, equations, and graphs.

'''Reasoning:''' You can explain why two ratios are equivalent, justify whether a relationship is proportional, keep corresponding quantities in the correct order, and check whether an answer is reasonable.

'''Products:''' Your ratio tables, graphs, scale drawings, posters, videos, experiments, or comparison reports show accurate calculations and clearly labeled units.

'''Transfer:''' You can apply proportional reasoning to unfamiliar situations such as shopping, travel, recipes, maps, science measurements, design, sports statistics, and percentages.

<br>
<br>

{{BR}}
= OERs on the Topic =

The English Wikipedia article on [[English:Ratio|Ratio]] provides a broader mathematical reference for ratios and their relationship to proportions.

<iframe> https://en.m.wikipedia.org/wiki/Ratio </iframe>

<br>

{{BR}}
= Linked Learning Areas =

Ratios, rates, and proportions connect multiplicative thinking with algebra, geometry, measurement, data, and real-world problem solving. They prepare you for slope, similarity, percent change, probability, dimensional analysis, and many science applications.

{| align=center
{{:D-Tab}}
'''[[English:Ratios, Rates, and Proportions|Ratios, Rates, and Proportions]]'''
# [[English:Ratio|Ratio]]
# [[English:Rate (mathematics)|Rate (mathematics)]]
# [[English:Unit rate|Unit rate]]
# [[English:Proportion (mathematics)|Proportion (mathematics)]]
# [[English:Proportionality (mathematics)|Proportionality (mathematics)]]
# [[English:Scale factor|Scale factor]]
# [[English:Percentage|Percentage]]
# [[English:Linear function|Linear function]]
# [[English:Similar figures|Similar figures]]
|}

[[Category:English]]
[[Category:Mathematics]]
[[Category:Middle school mathematics]]
[[Category:Grades 7-8]]
[[Category:Ratio]]
[[Category:Proportionality]]
[[Category:Measurement]]
[[Category:Algebra]]

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Ratios, Rates, and Proportions]]
[[Category:Mathematics]]
[[Category:Middle school mathematics]]
[[Category:Grades 7-8]]
[[Category:Ratio]]
[[Category:Proportionality]]
[[Category:Measurement]]
[[Category:Algebra]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Ratios, Rates, and Proportions - Versionsgeschichte

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