Quelltext der Seite English:Equivalent fractions and comparing fractions - Mathematics Year 5

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= Introduction =

[[File:FractionStrips.PNG|center|500px|Equal-length wholes divided into different numbers of equal parts. Labels as well as colour identify the parts.]]

Three quarters and six eighths can name the same number. Why does changing both written numbers leave the value unchanged? And why can a larger denominator name a smaller piece?

Use equal-sized bars and number lines to explain equivalence, compare fractions and move between mixed numbers and improper fractions. Your final evidence is a checked comparison, an equivalence chain and a short practical decision. This course uses positive denominators and non-negative fractions. You can draw models or describe them clearly; food or special equipment is not required.

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= Keep the whole fixed and explain equivalence =

In a fraction such as three quarters, the '''denominator''' four says how many equal parts make one whole. The '''numerator''' three counts how many of those parts are selected. Equal parts are essential: dividing a bar into four unequal pieces does not make every piece one quarter.

A fraction is also a number. On a number line from zero to one, divide the unit interval into four equal spaces. Three quarters is at the third division after zero. Count spaces, not all the marks including zero. A line model and an area model represent the same value in different ways.

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== Split the parts without changing the selected amount ==

Shade three of four equal parts in a bar. Now split '''every''' quarter into two equal pieces. The whole has eight parts and the shaded region has six. No area was added or removed, so three quarters equals six eighths.

<math>\frac{3}{4}=\frac{6}{8}=\frac{9}{12}</math>

Multiplying both numerator and denominator by the same positive whole number produces an equivalent fraction. The parts become smaller while the number counted increases consistently. Multiplying only the numerator would select a different amount. Adding the same number to top and bottom is not this rule: one half does not equal two thirds.

The reverse operation groups equal parts. Fifteen twentieths can be regrouped into three groups of five selected pieces out of four groups of five in the whole. Dividing top and bottom by five gives three quarters. Use a common factor that divides both exactly.

{{#ev:youtube|Bt60JVZRVCI|640|center}}

Watch the original Khan Academy model. Pause to identify the whole, the equal parts and the selected region before naming the fraction. Its other curriculum label does not define this England Year 5 course.

'''Check:''' Two thirds, four sixths and eight twelfths lie at the same point on a number line with the same unit interval. Three quarters and six eighths also coincide. If your drawings give different endpoints for an allegedly equivalent pair, first check that you used the same-sized whole.

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= Compare fractions by reasoning about the parts =

With the '''same denominator''', the parts have the same size, so the larger numerator gives the larger number: five eighths exceeds three eighths. With the '''same positive numerator''', dividing the same whole into more parts makes each part smaller, so three eighths is less than three quarters. Do not apply either shortcut when both numerator and denominator differ.

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== Use a common denominator ==

Compare five sixths and three quarters. Twelve is a common multiple of six and four. Rewrite both values in twelfths, changing numerator and denominator together.

<math>\frac{5}{6}=\frac{10}{12}</math>

<math>\frac{3}{4}=\frac{9}{12}</math>

Ten twelfths is larger than nine twelfths, so five sixths is larger. Draw two equal-length bars split into twelve spaces to check the result. You have changed the names of the fractions, not the numbers they represent.

Another comparison can use a '''benchmark''' such as one half. Three eighths is below one half because four eighths equals one half. Two thirds is above one half because half of three equal parts would be one and a half parts, while two parts are selected. Therefore three eighths is less than two thirds. A benchmark is useful only when it actually separates the values.

For five eighths and two thirds, both exceed one half. That observation alone does not settle their order. Use twenty-fourths: five eighths equals fifteen twenty-fourths and two thirds equals sixteen twenty-fourths, so two thirds is larger.

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== Order a set and retain equal values ==

To order one third, one half, three quarters, five sixths, seven twelfths and two thirds, use twelfths: the numerators become four, six, nine, ten, seven and eight. In increasing order the original fractions are '''one third, one half, seven twelfths, two thirds, three quarters, five sixths'''. Label a zero-to-one number line to check their positions.

If a list also contains six eighths, keep it beside three quarters with an equality sign. Equal numbers can have different fraction names; ordering is not a reason to discard one of the requested entries.

'''Error clinic:''' 'Five sixths is less than three quarters because six is greater than four.' The denominator-only rule ignores the different numerators. Converting to a shared unit corrects the comparison and shows exactly where that reasoning failed.

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= Convert mixed numbers and improper fractions =

A '''proper fraction''' has numerator smaller than its positive denominator, so its value is less than one. An '''improper fraction''' has numerator at least as large as the denominator, so it represents one or more wholes. The name does not mean it is an incorrect way to write a number.

A '''mixed number''' combines a whole-number part and a proper fractional part. The mixed number written 2 1/3 means '''two plus one third''', not two multiplied by one third.

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== Count the unit fractions ==

Two wholes contain six thirds. One more third gives seven thirds.

<math>2+\frac{1}{3}=\frac{6}{3}+\frac{1}{3}=\frac{7}{3}</math>

The denominator remains three because you are still counting thirds. You have not added the denominators. The same method gives three and two fifths as seventeen fifths: three whole groups of five fifths give fifteen fifths, then add the two fifths.

To go back, divide the numerator by the denominator. Seventeen fifths contains three whole groups of five fifths, with two fifths left. It therefore equals three and two fifths. Check by rebuilding seventeen as three times five plus two.

{| class="wikitable"
! Mixed number !! Unit-fraction count !! Improper form
|-
| 2 1/3 || 2 × 3 + 1 = 7 thirds || 7/3
|-
| 3 2/5 || 3 × 5 + 2 = 17 fifths || 17/5
|-
| 1 3/4 || 1 × 4 + 3 = 7 quarters || 7/4
|-
| 4 5/6 || 4 × 6 + 5 = 29 sixths || 29/6
|}

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== Keep the unit and simplify if useful ==

Fifteen sixths contains two wholes and three sixths. Three sixths equals one half, so the mixed number can be written two and one half. Twelve quarters equals three whole units exactly; no leftover fraction is needed.

A number line for seven thirds must extend beyond one: the point lies one third beyond two. Squeezing every fraction between zero and one hides improper fractions. Use the same unit length for each whole interval.

'''Check:''' Nine quarters equals two and one quarter; eleven fifths equals two and one fifth; thirteen sixths equals two and one sixth; nineteen eighths equals two and three eighths. For each, multiply the whole-number part by the denominator and add the remaining numerator to recover the original numerator.

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= Make a fair comparison and defend your method =

[[File:Cake-quarters.jpg|center|400px|A cake model divided into four labelled quarters. Two equal quarters represent half of this same whole.]]

A fraction comparison as numbers is different from comparing physical amounts from '''different wholes'''. Half of a two-metre ribbon is one metre. Three quarters of a one-metre ribbon is 0.75 metres. One half is the smaller fraction number, but it gives the larger length in this example because the starting wholes differ. Always identify the whole and units in a practical question.

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== Worked decision: choose the longer equal-whole piece ==

Two display ribbons are cut from lengths that each represent one metre. Ribbon A is three quarters of a metre; ribbon B is five sixths of a metre. The task is to choose the longer piece.

Use the common unit of twelfths of a metre. A is nine twelfths; B is ten twelfths. B is longer. The difference is one twelfth of a metre, which can be seen as one extra equal segment on a twelve-part bar. The two ribbon labels must refer to the same one-metre unit for this reasoning to compare the actual lengths directly.

A second check uses distance from one whole: three quarters is one quarter short of one, while five sixths is one sixth short. A sixth is smaller than a quarter of the same whole, so the second ribbon is closer to one metre. Both methods support the same decision.

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== Practise, then inspect the reasoning ==

# Complete and explain the chains: three quarters = ?/8 = ?/12; two thirds = ?/6 = ?/12; three fifths = ?/10 = ?/20.
# Compare five sixths with three quarters, and five eighths with two thirds. Use equivalent fractions or a justified model.
# Order the six fractions from the earlier chapter and mark them on a consistent number line.
# Convert 2 1/3, 3 2/5, 1 3/4 and 4 5/6 to improper fractions, then convert the four independent-check fractions from the previous chapter back to mixed numbers.
# Explain why adding the same number to numerator and denominator is not the equivalence rule. Use one half and two thirds as a counterexample.

'''Worked checks:''' The missing numerators are six and nine; four and eight; six and twelve. Five sixths exceeds three quarters; five eighths is less than two thirds. The improper forms are 7/3, 17/5, 7/4 and 29/6. The reverse checks are 2 1/4, 2 1/5, 2 1/6 and 2 3/8. One half and two thirds lie at different points on the same number line, so adding one to both parts changed the value.

Keep one model and one symbolic check with your final decision. A correct answer with no explanation is weaker evidence than a solution showing the shared unit, a valid conversion and a conclusion with units.

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= Interactive Tasks =


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== Quiz: Test Your Knowledge ==

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What must be true when four pieces of a bar are each called one quarter?
(The pieces are equal parts of the same whole)
(!The pieces may have any different sizes)
(!The bar must be made of a particular material)
(!The numerator must always be four)
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Why can three quarters be renamed six eighths?
(Each quarter is split into two and every selected part is counted consistently)
(!Only the selected region is doubled in size)
(!Adding two to the denominator always preserves a fraction)
(!The whole bar is changed into one twice as long without changing the labels)
{{E}}

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Which fraction is equivalent to three quarters?
(6/8)
(!3/8)
(!6/4)
(!4/5)
{{E}}

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Which comparison correctly uses twelfths?
(Five sixths is ten twelfths and exceeds three quarters, which is nine twelfths)
(!Five sixths is five twelfths and is less than three quarters)
(!Three quarters is twelve twelfths and exceeds five sixths)
(!The denominator alone proves sixths are always the larger amount)
{{E}}

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How do five eighths and two thirds compare?
(Five eighths is less because fifteen twenty-fourths is less than sixteen twenty-fourths)
(!Five eighths is greater because five is greater than two)
(!They are equal because both exceed one half)
(!They cannot be compared as numbers)
{{E}}

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What is the improper fraction for the mixed number two and one third?
(7/3)
(!3/7)
(!2/3)
(!7/6)
{{E}}

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What is fifteen sixths written as a mixed number in simplified form?
(Two and one half)
(!Two and one sixth)
(!Three and one half)
(!One and five sixths)
{{E}}

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Which list is in increasing order?
(One third, one half, seven twelfths, two thirds, three quarters, five sixths)
(!One third, seven twelfths, one half, two thirds, three quarters, five sixths)
(!Five sixths, three quarters, two thirds, seven twelfths, one half, one third)
(!One half, one third, two thirds, seven twelfths, three quarters, five sixths)
{{E}}

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Half of a two-metre ribbon is compared with three quarters of a one-metre ribbon. Which length is greater?
(Half of the two-metre ribbon, because it is one metre)
(!Three quarters of the one-metre ribbon, because three is greater than one)
(!They are equal because both are parts of ribbons)
(!Neither amount can be calculated even when the whole lengths are given)
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Which action always preserves a fraction when the chosen positive whole-number multiplier is used?
(Multiply numerator and denominator by that same number)
(!Multiply only the numerator)
(!Add that number to numerator and denominator)
(!Change only the denominator)
{{E}}


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== Memory Game ==

<div class="memo-quiz">
{|
|-
| Numerator || Counts the selected equal parts
|-
| Denominator || Names how many equal parts make one whole
|-
| Equivalent fractions || Different fraction names for the same number
|-
| Common denominator || A shared unit used to compare converted fractions
|-
| Mixed number || A whole-number part plus a proper fractional part
|-
| Improper fraction || A fraction with numerator at least as large as its positive denominator
|}
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== Drag and Drop ==

<div class="lueckentext-quiz">
{|
|-
| '''Three quarters''' || The same value as six eighths
|-
| '''Five sixths''' || The same value as ten twelfths
|-
| '''Two and one third''' || The same value as seven thirds
|-
| '''Three and two fifths''' || The same value as seventeen fifths
|-
| '''Fifteen sixths''' || The same value as two and one half
|}
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== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| FRACTION || A number written by counting parts of a unit
|-
| NUMERATOR || The top number counting selected equal parts
|-
| DENOMINATOR || The lower number naming the partition of one whole
|-
| EQUIVALENT || Describes fraction names with the same value
|-
| IMPROPER || Describes a fraction representing one or more wholes
|-
| BENCHMARK || A useful reference value such as one half
|}
{{E}}


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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=equivalent+fractions </iframe>

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== Cloze Text ==

<quiz display=simple>
{'''Complete the explanation.'''<br>
|type="{}"}
The top number in a fraction is the { numerator }. The lower number is the { denominator }. Different fraction names with the same value are { equivalent }. A shared lower number used for comparison is a common { denominator }. A whole-number part plus a proper fractional part forms a { mixed } number. A fraction representing one or more wholes can be called { improper }. One half can serve as a comparison { benchmark }.
</quiz>

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= Open Tasks =


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=== Easy ===

# [[English:Identify equal parts and the whole|Identify equal parts and the whole]]: Draw or describe one whole partitioned into four equal parts with three selected. Explain numerator and denominator and why unequal pieces would not support the same labels.
# [[English:Build an equivalence chain|Build an equivalence chain]]: Use the same whole to show three quarters, six eighths and nine twelfths. Explain what happens to every part when the partition becomes finer.
# [[English:Place fractions on a number line|Place fractions on a number line]]: Mark one half and three quarters on a zero-to-one line with four equal spaces. Explain why counting spaces rather than all endpoint marks matters.
# [[English:Convert one mixed number|Convert one mixed number]]: Show why two and one third equals seven thirds. Count the thirds in each whole and explain why the denominator stays three.

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=== Standard ===

# [[English:Complete three equivalence tables|Complete three equivalence tables]]: Complete three quarters in eighths and twelfths, two thirds in sixths and twelfths, and three fifths in tenths and twentieths. Show the multiplier or grouping for each.
# [[English:Compare with a shared unit|Compare with a shared unit]]: Compare five sixths with three quarters and five eighths with two thirds. Give each pair a common denominator and verify one comparison with a model.
# [[English:Order six values with evidence|Order six values with evidence]]: Order one third, one half, three quarters, five sixths, seven twelfths and two thirds. Use common twelfths or a consistent number line and retain the original names in your final order.
# [[English:Convert four mixed and four improper fractions|Convert four mixed and four improper fractions]]: Convert 2 1/3, 3 2/5, 1 3/4 and 4 5/6 to improper form. Convert 9/4, 11/5, 13/6 and 19/8 back to mixed numbers, then reconstruct each numerator to check.

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=== Advanced ===

# [[English:Compare two valid methods|Compare two valid methods]]: Solve the three-quarter-metre versus five-sixths-metre ribbon decision using common twelfths and using the distance each is short of one metre. Explain why the methods agree.
# [[English:Expose a changed-whole error|Expose a changed-whole error]]: Compare half of two metres with three quarters of one metre. Explain why the ordering of the fraction numbers alone does not decide the physical lengths when the starting wholes differ.
# [[English:Correct two invalid fraction rules|Correct two invalid fraction rules]]: Use a diagram or number line to refute adding the same number to numerator and denominator and comparing all fractions only by denominator. Supply one numerical counterexample for each.
# [[English:Create a checked comparison investigation|Create a checked comparison investigation]]: Design a new equal-whole practical comparison with different compatible denominators. Include a model, equivalent forms, units, a decision and a second check. Add a contrasting example where the wholes differ and explain the additional information needed.

{{:Open Task - Create a MOOC}}

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= Learning Check =

# [[English:Explain equivalence through a model|Explain equivalence through a model]]: Explain three quarters equals six eighths and simplify fifteen twentieths without changing the whole.
# [[English:Compare two pairs accurately|Compare two pairs accurately]]: Compare five sixths with three quarters and five eighths with two thirds, showing the shared units.
# [[English:Order a six-fraction set|Order a six-fraction set]]: Put one third, one half, three quarters, five sixths, seven twelfths and two thirds in increasing order.
# [[English:Convert between mixed and improper forms|Convert between mixed and improper forms]]: Convert the four mixed examples and the four independent-check improper fractions from the teaching chapter.
# [[English:Distinguish a number comparison from physical amounts|Distinguish a number comparison from physical amounts]]: Compare three quarters of one metre with five sixths of one metre, then compare half of two metres with three quarters of one metre.
# [[English:Evaluate a proposed rule|Evaluate a proposed rule]]: Explain why adding one to numerator and denominator is not a general equivalence rule and why a larger denominator does not always determine the comparison.

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== Worked checks and assessment guidance ==

# Splitting every quarter into two equal pieces gives eight parts in the whole and six selected parts without changing the shaded region. Multiplying both terms by two preserves the ratio. Fifteen twentieths can be grouped by five to give three quarters. Multiplying or dividing only one term changes the value.
# Five sixths equals ten twelfths and three quarters equals nine twelfths, so five sixths is larger. Five eighths equals fifteen twenty-fourths and two thirds equals sixteen twenty-fourths, so five eighths is smaller. A benchmark of one half alone does not distinguish the second pair because both exceed it.
# The order is one third, one half, seven twelfths, two thirds, three quarters, five sixths. Their equivalent twelfths have numerators four, six, seven, eight, nine and ten. A number line must use the same unit interval for every point.
# The mixed examples become 7/3, 17/5, 7/4 and 29/6. The independent improper examples become 2 1/4, 2 1/5, 2 1/6 and 2 3/8. Reconstruct each original numerator as whole part times denominator plus remaining numerator. If there is no remainder, a whole number is sufficient.
# Five sixths of a metre is longer than three quarters of a metre: ten twelfths exceeds nine twelfths. In the changed-whole example, half of two metres is one metre, which exceeds 0.75 metres. One half is still the smaller fraction number; the larger starting whole changes the resulting amount.
# One half would become two thirds, which is a different point on the same number line. Equivalent fractions instead multiply or divide both terms consistently by the same nonzero factor. A denominator-only comparison works in a restricted same-positive-numerator situation, not for arbitrary fractions; five sixths versus three quarters is a counterexample to a careless general rule.

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= Learning evidence =

'''Knowledge:''' Define numerator, denominator, equivalent fraction, common denominator, proper fraction, improper fraction, mixed number and benchmark.

'''Skills:''' Use equal-whole models and number lines, generate and simplify equivalent fractions, order compatible fractions and convert mixed and improper forms with checks.

'''Products:''' Keep three equivalence chains, two explained pair comparisons, an ordered six-fraction number line, eight conversions and a practical decision with units and a second method. Include a corrected changed-whole error.

'''Assessment:''' Check arithmetic and explanation separately. A drawing must identify equal parts and a consistent whole. A symbolic conversion must change numerator and denominator together. A practical conclusion must specify units and the starting whole. Passing the recall quiz alone does not replace the independent comparison evidence.

'''Transfer:''' Choose a common unit, benchmark or mixed-number form according to the problem; compare methods and explain when differing wholes require extra calculation before physical amounts can be compared.

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= OER and Wikipedia =

[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, Year 5 fractions]. The course follows the supplied editorial route's objectives for equivalence, ordering and mixed/improper forms. The worked ribbon decisions, numerical practice, counterexamples and assessments are original teaching material.

[https://www.khanacademy.org/math/arithmetic-home/arith-review-fractions/visualizing-equiv-frac/v/visualizing-equivalent-fractions Khan Academy: visualizing equivalent fractions] supports the visual equivalence approach. The original [https://www.youtube.com/watch?v=Bt60JVZRVCI video by Khan Academy] is embedded rather than copied or reuploaded.

'''Images, unmodified:''' [https://commons.wikimedia.org/wiki/File:FractionStrips.PNG FractionStrips], Zapotz, [https://creativecommons.org/publicdomain/zero/1.0/ CC0]; equal-length bars support comparison of the same whole. [https://commons.wikimedia.org/wiki/File:Cake-quarters.jpg Cake-quarters], Ian Tresman, [https://creativecommons.org/licenses/by/2.5/ CC BY 2.5]; a labelled cake model supports the equal-parts example. The food image is an optional visual model, not a requirement to prepare or eat food.

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

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= Linked learning areas =

{| align=center
{{:D-Tab}}
'''[[English:Fraction equivalence and comparison|Fraction equivalence and comparison]]'''
# [[English:Factors multiples primes squares and cubes - Mathematics Year 5|Factors multiples primes squares and cubes - Mathematics Year 5]]
# [[English:Adding and subtracting fractions - Mathematics Year 5|Adding and subtracting fractions - Mathematics Year 5]]
# [[English:Fractions as operators and multiplicative scaling - Mathematics Year 5|Fractions as operators and multiplicative scaling - Mathematics Year 5]]
# [[English:Number lines|Number lines]]
# [[English:Mixed numbers|Mixed numbers]]
# [[English:Common denominators|Common denominators]]
|}

[[Category:Mathematics]]
[[Category:Year 5]]
[[Category:Fractions]]
[[Category:Primary education]]
[[Category:Sprache:Englisch]]
[[Category:Locale:en-US]]

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= aiMOOC Projects =

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