Quelltext der Seite English:Ratio proportion and scale drawings - Mathematics Year 6

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= Introduction =

[[File:Cuisenaire-Rods.jpg|500px|center|alt=Coloured mathematical rods of different lengths arranged together.|Rods can model equal unit parts and compared lengths. This photograph supplies no centimetre scale; use stated quantities rather than measuring screen pixels.]]

Does a ratio of two to five mean two fifths of the whole? How does a scale drawing preserve a shape? Connect unit parts, proportional tables and corresponding lengths, then check every quantity against its context.

'''Success criteria.''' Solve relative-size problems; share a quantity unequally using multiplicative reasoning; scale similar shapes; and justify a complete unfamiliar solution with units and an independent check.

Three chapters provide original teaching, worked examples, native mathematics, an original video and unchanged Commons images. All prices, recipes, maps and stock quantities are fictional. The course develops mathematical representations, not observed practical measurement or a complete annual qualification.

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= Read a ratio and share a whole into equal unit parts =

'''A ratio compares named quantities in order.''' Red to blue counters in the ratio 2:5 means two red parts for every five blue parts. Each part contains the same number of counters. There are seven unit parts in the combined whole, not five.

'''Part-to-part and part-to-whole are different.''' The red-to-blue comparison is two to five. The red fraction of all counters is two sevenths; the blue fraction is five sevenths:

<math>\frac{2}{2+5}=\frac{2}{7},\qquad\frac{5}{2+5}=\frac{5}{7}</math>

Do not label the red fraction of the combined whole as two fifths. Two fifths compares red with the blue amount, which is a different denominator.

'''Worked sharing: £42 in the ratio 2:5.''' Seven equal unit parts make £42, so each part is £6. Two parts give £12 and five give £30:

<math>42\div(2+5)=6,\qquad2\times6=12,\qquad5\times6=30</math>

Check both conditions: £12 plus £30 recovers £42, and 12:30 simplifies to 2:5. Checking only the sum would accept many wrong splits.

'''An equivalent fraction route.''' Find two sevenths and five sevenths of the whole. This agrees with the unit-parts method:

<math>\frac{2}{7}\times42=12,\qquad\frac{5}{7}\times42=30</math>

The two methods explain the same structure. One makes a single part explicit; the other makes each share's fraction of the whole explicit.

'''Preserve a ratio multiplicatively.''' Scaling both amounts by the same positive factor preserves the comparison. Thus 2:5 and 4:10 are equivalent. Adding one to each gives 3:6, which is not equivalent: two fifths and three sixths are different ratios.

'''Use a diagram without relying on colour.''' Sketch seven equal boxes labelled R, R, B, B, B, B, B. Give each box £6. The letters and repeated equal parts, not colour or the physical size of a browser box, supply the reasoning.

'''More than two shares.''' Share 96 ml in the ratio 1:2:3. Six equal parts give 16 ml each, so the shares are 16 ml, 32 ml and 48 ml. Their sum is 96 ml; dividing each by 16 recovers 1:2:3.

'''Indivisible objects need interpretation.''' Eleven indivisible counters cannot be split exactly into a 2:3 ratio using every counter, because five equal integer parts would be needed. Fractions of a continuous length can be meaningful; fractions of indivisible counters require an explicit change of context or an allocation rule. Do not silently round and claim the ratio is unchanged.

'''Independent attempt.''' Solve the four sharing tasks in the final chapter. For each, name the ratio order and total, find the unit part, calculate all shares, then check both their sum and their ratio. Explain at least one by a second fraction-of-the-whole method.

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= Use proportional tables, known parts and common units =

'''A multiplicative relationship is an assumption to check.''' In a fictional recipe, two eggs go with 300 g of flour. Scaling every ingredient by the same factor keeps the recipe proportions. This mathematical model ignores practical variations such as egg size.

{| class="wikitable"
! Eggs !! Flour in grams
|-
| 2 || 300
|-
| 4 || 600
|-
| 6 || 900
|-
| 5 || 750
|}

Each egg corresponds to 150 g in this model. To use five eggs, multiply the one-egg quantity by five. Alternatively scale the two-egg row by two and a half. Both routes give 750 g.

'''Adding the same amount does not preserve proportion.''' Changing two eggs and 300 g into four eggs and 302 g adds two to both numerical entries but does not double the flour. The quantities even have different units; a repeated addition of the same bare numeral is not proportional reasoning.

'''When one ratio amount is known.''' Red to blue is 2:5 and there are twelve red counters. The two red unit parts therefore contain six counters each. The five blue parts contain thirty counters, making forty-two altogether. Here twelve is a part, not the combined total. Dividing twelve by seven would answer a different problem.

'''Convert units before forming a same-kind ratio.''' Compare one metre with fifty centimetres. One metre is one hundred centimetres, so:

<math>\frac{100}{50}=2</math>

The length ratio is 100:50, or 2:1. Writing 1:50 without the unit conversion would compare incompatible numerical labels.

'''Unit rates can compare offers.''' Three fictional pens cost £1.50 and five cost £2.75. Assume identical pens, no extra charges and the stated bundle prices. The first rate is £0.50 per pen; the second £0.55 per pen. Compare these rates, not just the bundle totals. A larger bundle is not automatically better value.

<math>1.50\div3=0.50,\qquad2.75\div5=0.55</math>

The difference is five pence per pen. Whether a buyer needs all the pens is a separate contextual question.

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Corbettmaths' original '''Proportion - Primary''' connects ratios with proportional reasoning. Its map example states that one centimetre represents five kilometres. Predict the drawing distance for twenty kilometres: four centimetres. Keep real distance and drawing distance distinct.

'''Read a table critically.''' A table of two, four and six eggs paired with 300, 600 and 900 grams follows the given constant rate. A table beginning two, four and six eggs paired with 300, 500 and 700 grams does not follow that proportional model, even though the flour increases by a regular amount.

'''Justify a method.''' Use a unit rate when one item or unit is useful, a direct scale factor when the multiplier is obvious, or a table to keep several corresponding quantities visible. State the constant relationship and verify a second pair rather than choose a rule from a keyword.

'''Independent transfer.''' Use the recipe table to find flour for eight eggs and eggs for 450 g. Then invent a non-proportional pair of rows and explain which ratio changes. Keep this a numerical model, not advice on food preparation.

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= Scale corresponding lengths and check shape, area and units =

'''Similar shapes preserve corresponding length ratios.''' A rectangle is assigned length 4 cm and width 3 cm. An enlargement with length scale factor three gives length 12 cm and width 9 cm. Every corresponding length is multiplied by the same factor.

<math>4\times3=12,\qquad3\times3=9</math>

Increasing only the length to 12 cm while leaving the width at 3 cm would change the shape's proportions. The result would not be the stated similar enlargement.

[[File:Illustration for the area of a rectangle.svg|500px|center|alt=A rectangle with side variables and its area represented by the product a times b.|Assign hypothetical side values to the variables. Actual display pixels are not centimetres and the source image itself supplies no numerical dimensions.]]

'''Length scale is not automatically area scale.''' The original 4-by-3 rectangle has area 12 square centimetres. The 12-by-9 enlargement has area 108 square centimetres:

<math>4\times3=12,\qquad12\times9=108,\qquad108\div12=9</math>

Both dimensions were multiplied by three, so the area factor is three times three, or nine. The perimeter changes from 14 cm to 42 cm, with factor three. Areas and lengths need different units and different scale reasoning.

'''Find an unknown scale factor.''' Corresponding lengths 5 cm and 10 cm imply factor two from the first shape to the second. A corresponding 2 cm width becomes 4 cm. Reversing direction uses factor one half. Always say which shape is the starting shape.

'''A map with different physical units.''' If one centimetre on a fictional map represents five kilometres, twenty kilometres is represented by four centimetres. A drawing-to-real ratio written using the same units is 1:500000, because five kilometres is five hundred thousand centimetres. The labelled map relationship is the premise; measuring the displayed video thumbnail is not a valid replacement.

'''Four sharing tasks and two scale tasks.''' Attempt these before reading the feedback.

{| class="wikitable"
! Code !! Task
|-
| S1 || Share £42 in the ordered ratio 2:5.
|-
| S2 || Share £72 in the ordered ratio 5:3.
|-
| S3 || Share 84 beads in the ordered ratio 3:4.
|-
| S4 || Share 96 ml in the ordered ratio 1:2:3.
|-
| T1 || Enlarge a 4 cm by 3 cm rectangle by length factor 3. Find both sides, perimeter and area.
|-
| T2 || A 5 cm by 2 cm rectangle has corresponding first side 10 cm in a similar enlargement. Find the factor, second side and area.
|}

'''Check every constraint.''' For a share, test sum and ratio. For a similar shape, test every corresponding side ratio, then calculate perimeter or area in its own unit. For a map, convert units and reverse the stated relationship.

'''Compare representations.''' Seven equal boxes make the 2:5 split easy to explain. A proportional table makes a recipe's constant rate visible. A labelled pair of rectangles makes corresponding lengths explicit. A symbolic calculation is efficient once its quantities are understood; it should not hide which quantity is the whole.

'''An unfamiliar problem.''' A model rectangle has sides 6 cm and 4 cm. Its similar partner has corresponding short side 12 cm. Identify factor three from the 4-to-12 match, then find the long side 18 cm and area 216 square centimetres. Do not match a short side to a long side just because it appears first in a sentence.

'''Complete your explanation.''' Name the ratio order or corresponding sides, justify the multiplicative relationship, retain units and check a different constraint. State limitations such as indivisible stock, a presumed constant recipe rate or an idealised map. A correct numerical model is not proof of an actual measured construction.

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= Interactive Tasks =


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== Quiz: Test Your Knowledge ==

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In a red-to-blue ratio of two to five, what fraction of the combined total is red?
(Two sevenths)
(!Two fifths)
(!Five sevenths)
(!Five halves)
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How many equal unit parts are in a two-to-five split of a whole?
(Seven)
(!Five)
(!Two)
(!Ten)
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What are the ordered shares of forty-two pounds in the ratio two to five?
(Twelve pounds and thirty pounds)
(!Two pounds and five pounds)
(!Twenty-one pounds and twenty-one pounds)
(!Fourteen pounds and twenty-eight pounds)
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Which operation preserves a positive ratio?
(Multiply both quantities by the same positive factor)
(!Add the same number to both quantities in every case)
(!Change only the first quantity)
(!Swap part and whole without relabelling)
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Two eggs correspond to three hundred grams of flour in a proportional model. What corresponds to five eggs?
(Seven hundred and fifty grams)
(!Three hundred and five grams)
(!Six hundred grams)
(!Fifteen hundred grams)
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What is the length ratio of one metre to fifty centimetres?
(Two to one after converting to common units)
(!One to fifty without converting)
(!Fifty to one in the stated order)
(!One to five hundred)
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If the two-part amount in a two-to-five ratio is twelve, what is the five-part amount?
(Thirty)
(!Five)
(!Forty-two)
(!Twelve sevenths)
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A rectangle has both side lengths enlarged by factor three. What happens to its area?
(It is multiplied by nine)
(!It is always multiplied by three)
(!It stays unchanged)
(!It is multiplied by six)
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At one centimetre representing five kilometres, what map distance represents twenty kilometres?
(Four centimetres)
(!One hundred centimetres)
(!Twenty-five centimetres)
(!Four kilometres on the drawing)
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Which checks support a complete solution?
(Check share totals and ratios or all corresponding side factors, retaining units)
(!Check only the first number in the answer)
(!Measure the website picture in pixels and call them centimetres)
(!Assume any regular increase proves direct proportion)
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== Memory Game ==

<div class="memo-quiz">
{|
|-
| Ratio || An ordered comparison of quantities
|-
| Unit part || One equal share underlying a ratio model
|-
| Proportion || A part or relationship considered relative to another quantity
|-
| Scale factor || A multiplier between corresponding quantities
|-
| Corresponding || Matching in role between the compared shapes
|-
| Unit rate || An amount for one item or one unit
|-
| Similar shapes || Shapes with matching angles and corresponding lengths in one ratio
|}
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== Drag and Drop ==

<div class="lueckentext-quiz">
{|
|-
| '''Share a complete total''' || Add ratio parts, find one unit part and calculate every ordered share
|-
| '''Known one-part amount''' || Divide that known amount by its own ratio-part count
|-
| '''Compare same-kind units''' || Convert to a common unit before simplifying the ratio
|-
| '''Enlarge a similar shape''' || Multiply every corresponding length by the same factor
|-
| '''Check an area scale''' || Apply the length factor to both dimensions before comparing areas
|}
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== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| RATIO || Ordered comparison of quantities
|-
| SCALE || A relationship connecting corresponding sizes
|-
| PART || One component of a whole
|-
| WHOLE || The combined total being shared
|-
| SIMILAR || Having matching shape with corresponding sides in one ratio
|-
| RATE || A quantity per unit of another
|}
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== LearningApps ==

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.
<iframe> https://learningapps.org/index.php?s=Ratio+and+proportion </iframe>

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== Cloze Text ==

<quiz display=simple>
{'''Complete the explanation.'''<br>
|type="{}"}
A ratio is an { ordered } comparison. Add its parts to identify the combined { whole }. Each equal building block is a unit { part }. Equivalent ratios preserve a common { multiplier }. A comparison may require common { units }. Matching sides are { corresponding }. A similar enlargement uses one length scale { factor }. Testing both sum and ratio gives a { check }.
</quiz>

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= Open Tasks =


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=== Easy ===

# [[English:Four sharing tasks|Four sharing tasks]]: Solve S1–S4. Name ratio order, total unit parts, the value of one part and every share. Check sum and ratio.
# [[English:Part versus whole|Part versus whole]]: Explain why the red share in a red-to-blue ratio of 2:5 is 2/7 of all counters. Use seven labelled equal boxes.
# [[English:Two scale tasks|Two scale tasks]]: Solve T1–T2. Label corresponding sides and keep length, perimeter and area units separate.
# [[English:Units first|Units first]]: Compare 1 m with 50 cm and explain why a bare ratio 1:50 would be misleading.

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=== Standard ===

# [[English:Recipe transfer|Recipe transfer]]: Use the given constant recipe rate to find flour for eight eggs and eggs for 450 g. Show unit-rate and direct-scaling approaches.
# [[English:Known part versus total|Known part versus total]]: With red:blue = 2:5 and twelve red counters, find blue and total. Contrast this with sharing twelve counters as the total.
# [[English:Compare bundle rates|Compare bundle rates]]: Compare the fictional three-pen £1.50 and five-pen £2.75 bundles. Find unit prices, difference and assumptions.
# [[English:Diagram and fractions|Diagram and fractions]]: Solve the £42 split with equal unit boxes and fractions of the whole. Explain why the two methods agree.

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=== Advanced ===

# [[English:Unknown matching side|Unknown matching side]]: A 6 cm by 4 cm rectangle has a similar partner whose corresponding short side is 12 cm. Find scale factor, other side and area, explaining the matching.
# [[English:Area misconception|Area misconception]]: Contrast length and area scale factors with the 4-by-3 rectangle enlarged by three. Explain why adding three to both dimensions would not give the same enlargement.
# [[English:Map and reversed units|Map and reversed units]]: Use 1 cm representing 5 km to find the drawing distance for 20 km and the same-unit drawing:real ratio. Reverse the result as a check.
# [[English:Model limitation dossier|Model limitation dossier]]: Give an unfamiliar ratio problem and a complete checked solution. Explain an indivisible-object limitation or a case of regular increase that is not proportional.

{{:Open Task - Create a MOOC}}

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= Learning Check =

# [[English:Four ratio shares|Four ratio shares]]: Check S1–S4 in the stated order.
# [[English:Two scale tasks|Two scale tasks]]: Check T1–T2 and dimensional units.
# [[English:Proportional table and known part|Proportional table and known part]]: Check recipe and counter transfer.
# [[English:Rates and common units|Rates and common units]]: Check the conversions and offers.
# [[English:Unfamiliar shape and map|Unfamiliar shape and map]]: Check both transfer problems.
# [[English:Justification and limitations|Justification and limitations]]: Assess a complete contextual method.

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== Worked checks and assessment guidance ==

# S1: £12 and £30. S2: £45 and £27. S3: 36 and 48 beads. S4: 16 ml, 32 ml and 48 ml. Unit parts are respectively £6, £9, 12 beads and 16 ml. Require both the original total and the ordered simplified ratio to be recovered.
# T1: 12 cm by 9 cm, perimeter 42 cm, area 108 cm². Original perimeter is 14 cm and area 12 cm²; length/perimeter factor three and area factor nine differ. T2: length factor two, second side 4 cm, area 40 cm². Original area is 10 cm², so area factor four.
# At 150 g per egg, eight eggs correspond to 1200 g and 450 g to three eggs. For red:blue = 2:5 with twelve red, one part is six, giving thirty blue and forty-two altogether. If twelve were the total, continuous shares would be 24/7 and 60/7; twelve indivisible counters cannot use all stock in that exact ratio. A known ratio part is not automatically the full total.
# 1 m is 100 cm, so 100:50 = 2:1. The first pen rate is £0.50 and the second £0.55 per pen, a 5p difference under the stated identical-product/no-extra-charge assumptions. More items or a lower bundle total alone do not settle value.
# The short-side match 4 cm → 12 cm gives factor three, so the long side 6 cm → 18 cm and area 18 × 12 = 216 cm². For the map, 20 km → 4 cm. Five kilometres is 500000 cm, giving drawing:real 1:500000 in common units. Use stated scales, not on-screen pixel lengths.
# Require named ordered quantities, identified whole or known part, multiplicative relation, suitable diagram/table/fraction method, units and at least two appropriate checks. Eleven indivisible counters cannot use all stock in an exact 2:3 split. A regular additive change is not sufficient to prove proportionality; all ratios or unit rates must agree.

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= Learning evidence =

Keep four checked shares, two similar-shape scale solutions, a proportional-table transfer and a complete unfamiliar context. Include both total and ratio checks, a unit conversion, corresponding-side labels and one stated modelling limitation. No physical measurement or construction competence is certified.

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= OER and Wikipedia =

Checked 11 November 2026. [https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study Department for Education: mathematics programmes of study], Year 6 ratio/proportion and supporting guidance, informs relative sizes, unequal sharing and similar-shape scaling.

Original unit-part diagrams in words, proportional table, fictional stock/price/map contexts, exercises and feedback are newly authored. Length factors and area factors are deliberately distinguished.

Embedded original [https://www.youtube.com/watch?v=bg283xTWtxc Proportion - Primary], corbettmaths. The original is embedded, not copied or described as Commons-licensed. Written models and checks provide an independent route.

Commons assets are unchanged. Rod colours do not supply a numerical scale and no source lengths are inferred from screen pixels. The rectangle's symbolic dimensions are hypothetical variables, not measured numerical labels.

[https://commons.wikimedia.org/wiki/File:Cuisenaire-Rods.jpg Cuisenaire-Rods.jpg], Celcom at en.wikipedia, [https://creativecommons.org/licenses/by/3.0 CC BY 3.0]. Unchanged.

[https://commons.wikimedia.org/wiki/File:Illustration_for_the_area_of_a_rectangle.svg Illustration for the area of a rectangle.svg], Stephan Kulla, [http://creativecommons.org/publicdomain/zero/1.0/deed.en CC0]. Unchanged.

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

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= Linked learning areas =

{| align=center
{{:D-Tab}}
'''[[English:Mathematics|Mathematics]]'''
# [[English:Percentages of amounts and percentage comparison - Mathematics Year 6|Percentages of amounts and percentage comparison - Mathematics Year 6]]
# [[English:Introductory algebra formulae and linear sequences - Mathematics Year 6|Introductory algebra formulae and linear sequences - Mathematics Year 6]]
# [[English:Converting measures and solving time and money problems - Mathematics Year 6|Converting measures and solving time and money problems - Mathematics Year 6]]
|}

[[Category:Mathematics]]
[[Category:Year 6]]
[[Category:Key Stage 2]]
[[Category:Ratio and proportion]]
[[Category:Scale drawings]]
[[Category:Sprache:Englisch]]
[[Category:Locale:en-US]]

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= aiMOOC Projects =

[[Category:AI_MOOC]] [[Category:GPT aiMOOC]]
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