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What is a fraction? A fraction names equal parts of a whole. The top number is the numerator. The bottom number is the denominator. The denominator tells how many equal parts make the whole. The numerator tells how many parts are taken. For example, in 3/4 the whole is divided into 4 equal parts and 3 parts are chosen. A clear way to see this is to use real objects: fold a paper into three equal parts to see 1/3, or cut an apple into four equal slices to see quarters.
Fractions of a set: Fractions also show part of a group. If there are 12 students and 3 wear blue shirts, the fraction wearing blue is 3/12. This shows that fractions can describe portions of objects or portions of items in a set. Always check that the parts are equal in size; fractions only work when parts are equal.
Visual models: Drawings help students understand. A circle divided into five equal parts with two shaded shows 2/5. On a number line between 0 and 1, mark equal jumps to show fractions like 1/5, 2/5, 3/5. Unit fractions have numerator 1, for example 1/6. Learning these basic ideas gives a strong foundation for adding, subtracting, multiplying and dividing fractions.
Proper fractions have numerators smaller than denominators and represent less than one whole. Examples are 2/5 and 3/8. Proper fractions are easy to picture as a part of one object or one set.
Improper fractions have numerators equal to or larger than denominators, for example 5/4 or 7/7. They represent one or more whole units. An improper fraction like 9/4 means there are nine parts each of size 1/4 — this is more than two whole units, so we convert it into a mixed number for easier reading.
Mixed numbers show a whole number together with a proper fraction, such as 2 1/3. This means two whole units and one third of another unit. Converting between mixed numbers and improper fractions is an important skill. To change a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator; place this sum over the same denominator. To change an improper fraction to a mixed number, divide the numerator by the denominator: the quotient is the whole part and the remainder becomes the new numerator.
Practise with pictures: draw two full rectangles and a third divided into parts to show a mixed number. Converting back and forth helps when adding or multiplying because some operations are easier with improper fractions and some with mixed numbers.
Equivalent fractions
Simplifying
Use drawings to see equivalence and simplification: draw one rectangle divided into 2 parts and shade one; draw another identical rectangle divided into 4 parts and shade two—the shaded area is the same. Practice making equivalent fractions by multiplying numerator and denominator by 2, 3 or 4, and practise simplifying by checking small common divisors. Always simplify final answers after operations to keep numbers small and clear.
To compare fractions we decide which is larger or smaller. If fractions have the same denominator, the one with larger numerator is bigger. For example 3/7 > 2/7. If denominators are different, change them to a common denominator or make equivalent fractions with the same denominator. Use the least common multiple (LCM) of denominators for an efficient method.
Another method is to compare by converting to decimals or using visual models like pie charts or number lines. To order several fractions, convert all to a common denominator and then put numerators in increasing or decreasing order. When mixed numbers are compared, first compare whole parts, then the fractional parts.
Practice with proper and improper fractions and mixed numbers. Try quick checks like comparing to 1/2: if a fraction with the same denominator has numerator more than half the denominator, it is greater than 1/2. Teach pupils to choose the easiest method depending on numbers given: sometimes converting to a decimal using simple division is fastest; other times LCM is simpler. Regular practice builds skill and speed.
When fractions have the same denominator we add the numerators and keep the denominator the same. This rule works because each fraction is made of equal parts. For example, 2/7 + 3/7 = (2+3)/7 = 5/7. Always check that denominators match before adding. If they do not match, you must change to a common denominator first.
After adding, simplify the fraction to lowest terms if possible. If the numerator is equal to or greater than the denominator, convert the improper fraction to a mixed number by division. For example, 7/4 after addition can be written as 1 3/4. Use diagrams to show why numerators add: draw two identical bars divided into the same number of parts, shade the given parts in each bar, then combine the shaded parts into one bar.
Work on word problems such as combining portions eaten, lengths added or money fractions. Teach checking by making equivalent fractions or by converting to decimals for a quick approximate check. Repetition with different denominators that are the same strengthens procedural fluency and confidence.
Subtraction with the same denominator is like addition: subtract the numerators and keep the denominator. For example, 5/6 − 2/6 = (5−2)/6 = 3/6 = 1/2 after simplifying. This works because both fractions use the same sized parts; taking away is simply removing some of those parts. If the numerator becomes zero, the answer is 0. If the result is negative, interpret it as owing or needing that many parts in context.
With mixed numbers, subtract whole parts and fractional parts carefully. If the fractional part of the minuend (the first number) is smaller than the fractional part of the subtrahend (the number being taken away), borrow 1 whole: convert 1 whole into denominator many parts and add to the fractional part before subtracting. For example, to compute 3 1/4 − 1 3/4, borrow 1 whole from 3 to get 2 5/4, then subtract 1 3/4 to get 1 2/4 = 1 1/2.
Use pictures to show removal of shaded parts, and practice word problems such as measuring left-over lengths, remaining money or portions after sharing. Practising borrowing with diagrams helps prevent mistakes and builds understanding.
When denominators differ, convert fractions to equivalent fractions with a common denominator before adding or subtracting. The least common multiple (LCM) of the denominators gives the smallest useful common denominator and keeps numbers small. For example add 1/4 + 1/6: LCM of 4 and 6 is 12. Convert: 1/4 = 3/12 and 1/6 = 2/12, then add: 3/12 + 2/12 = 5/12.
To subtract, use the same process: find the LCM, write each fraction with that denominator, then subtract numerators and simplify. Always simplify the final answer and convert improper fractions to mixed numbers if needed. Teach a clear step-by-step method: (1) find LCM, (2) make equivalent fractions, (3) add or subtract numerators, (4) simplify, (5) convert to mixed number if necessary.
Visual methods help: redraw shapes so both have the same number of equal parts and then combine or remove shaded parts. Practise with word problems such as adding different fractional lengths, combining pieces of ribbon or subtracting time fractions. Regular practice builds confidence and accuracy when denominators are different.
Multiplying a fraction by a whole number means adding that fraction many times. For example 4 × 1/3 is 1/3 + 1/3 + 1/3 + 1/3 = 4/3. The short rule is to multiply the numerator by the whole number and keep the denominator the same: k × a/b = (k×a)/b. After multiplication simplify or convert to a mixed number if needed.
To multiply two fractions, multiply numerator by numerator and denominator by denominator: a/b × c/d = (a×c)/(b×d). Before multiplying, look for common factors between any numerator and any denominator and cancel them to make the arithmetic easier. Cancelling reduces the product to lowest terms quickly. For example, to compute 2/5 × 3/4, you can multiply directly to obtain 6/20 and then simplify to 3/10, or cancel 2 with 4 first to get 1/2 × 3/2 = 3/4, but in this case cancelling should be chosen correctly.
Use models to understand: to find 2/3 of 12, divide 12 into 3 equal groups (each 4) and take 2 groups: 2×4=8. Practise word problems like portions of quantities, areas and repeated addition to see multiplication of fractions in real contexts.
Dividing a fraction by a whole number means sharing the fraction into equal groups. Use the reciprocal method to make division easy. To divide a/b by a whole number k, compute (a/b) ÷ k = a/b × 1/k = a/(b×k). This is because dividing by k is the same as multiplying by 1/k. For example, (3/4) ÷ 3 becomes 3/4 × 1/3 = 3/12 = 1/4.
To divide by a fraction, multiply by its reciprocal. The reciprocal of c/d is d/c. So (a/b) ÷ (c/d) = (a/b) × (d/c). This switches the second fraction and changes division into multiplication, which we already understand. Always simplify by cancelling before multiplying if possible, then reduce the final answer to lowest terms and convert to mixed numbers when needed.
Use sharing examples: if 3/4 of a cake is to be shared by 3 children, each gets (3/4) ÷ 3 = 1/4. Use visual division by slicing shaded regions into equal shares. Practise with real-life problems like sharing sweets, splitting time or dividing lengths to make the rule meaningful and easy to remember.