Arithmetic procedures with fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Explain why equal-sized parts are needed
Use equivalent fractions to add and subtract

Vocabulary: numerator · denominator · equivalent fraction · common denominator

Retrieve previous learning

Write 1/4 in eighths.

Check: 1/4 = 2/8. Both numerator and denominator are doubled, so the value is unchanged.

Learn the concept

The denominator names the size of a part. To combine 3/8 and 1/4, first rename 1/4 as 2/8. Now 3/8 + 2/8 = 5/8: five parts, each one eighth of the whole.

Subtraction uses the same equal-part principle. For 5/6 − 1/4, use twelfths: 5/6 = 10/12 and 1/4 = 3/12. Removing three twelfths leaves 7/12. The denominator stays 12 because the answer still counts twelfth-sized parts.

Choose a common multiple of both denominators; the lowest common multiple usually makes the calculation shorter. Check size: adding positive fractions makes a larger quantity, while subtracting a smaller positive fraction makes a smaller one.

Worked model

Find 2/3 + 1/6 and choose sixths as the common-sized parts.
Rename 2/3 as 4/6 by multiplying its numerator and denominator by two.
Add equal-sized parts: 4/6 + 1/6 = 5/6.
Check that the total is larger than 2/3 but less than one.

Guided practice

Calculate 3/4 − 1/8.

Hint: Rename quarters as eighths.

Check the guided answer

3/4 = 6/8, so 6/8 − 1/8 = 5/8.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“3/8 + 1/4 = 4/12 because both the tops and bottoms should be added.”

Correction: Adding denominators changes part size. Rename 1/4 as 2/8, giving 3/8 + 2/8 = 5/8.

Extend and reflect

Find two different pairs of fractions with unequal denominators summing to 5/6.

Guidance: One is 1/2 + 1/3. Build another pair by working in twelfths or eighteenths and prove it.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Explain why equal-sized parts are needed. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Securing understanding of addition and subtraction with fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Calculate with mixed numbers
Interpret negative answers and check by the inverse

Vocabulary: mixed number · improper fraction · lowest common multiple · inverse

Retrieve previous learning

Convert 1 1/3 to an improper fraction.

Check: One whole is 3/3, so 1 1/3 = 4/3.

Learn the concept

Converting a mixed number to an improper fraction avoids errors in subtraction. For 1 1/3 − 5/6, write 1 1/3 as 4/3 = 8/6. Then 8/6 − 5/6 = 3/6 = 1/2.

Signed fractions still need common-sized parts. In −2/3 + 5/12, rename −2/3 as −8/12. Moving five twelfths to the right on a number line ends at −3/12 = −1/4.

Check a subtraction by adding the difference to the quantity removed. Here 1/2 + 5/6 = 3/6 + 5/6 = 8/6 = 1 1/3. This is an independent check of both the arithmetic and the mixed-number conversion.

A negative answer is possible when the amount removed exceeds the starting amount. For example 1/3 − 5/6 = 2/6 − 5/6 = −3/6 = −1/2; adding 5/6 to −1/2 returns 1/3.

Worked model

Convert 2 1/4 − 5/6 to 9/4 − 5/6.
Rename both in twelfths: 27/12 − 10/12.
Subtract to get 17/12 = 1 5/12.
Add 5/6 back: 17/12 + 10/12 = 27/12 = 2 1/4.

Guided practice

Calculate −3/4 + 1/6.

Hint: Use twelfths and compare magnitudes.

Check the guided answer

−9/12 + 2/12 = −7/12.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“For 1 1/3 − 5/6, subtract the whole number and fractional parts independently and force both results to be positive.”

Correction: Convert the full mixed number: 1 1/3 = 8/6, then 8/6 − 5/6 = 1/2.

Extend and reflect

Solve x − 7/8 = −1/4 and check your answer.

Guidance: Add 7/8 to both sides: x = 5/8. Substitute to verify 5/8 − 7/8 = −2/8 = −1/4.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Calculate with mixed numbers. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Checking understanding of multiplication with fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Explain a fraction of a quantity as multiplication
Represent a fraction of a fraction by equal subdivisions

Vocabulary: fraction operator · product · area model · unit fraction

Retrieve previous learning

What is 3/4 of 20?

Check: One quarter of 20 is 5, so three quarters is 15.

Learn the concept

The word 'of' often means multiply. For 2/3 of 9/10, split 9/10 into three equal groups. One group is 3/10; two groups are 6/10 = 3/5.

An area model can show the same operation. Split a rectangle into thirds in one direction and tenths in the other. Two of three rows overlap with nine of ten columns in 18 of 30 small cells: 18/30 = 3/5. This explains the rule (a/b) × (c/d) = ac/bd.

Predict size before calculating. Multiplying a positive number by a proper fraction gives only part of the number, so the product must be smaller than the original. This check catches answers such as 18/13 for 2/3 × 9/10.

Worked model

Read 'of' as multiplication and write 3/4 of 2/3 as (3/4) × (2/3).
Multiply numerators to get 3 × 2 = 6.
Multiply denominators to get 4 × 3 = 12, so the product is 6/12.
Simplify to 1/2 and check that it is less than 2/3.

Guided practice

Find 2/5 of 15/8.

Hint: Multiply, then simplify.

Check the guided answer

(2/5) × (15/8) = 30/40 = 3/4.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Multiplication always makes a number bigger.”

Correction: A proper fraction is less than one, so multiplying a positive number by it gives part of the number and makes it smaller.

Extend and reflect

Draw an area model for 2/3 × 3/4.

Guidance: Split a rectangle into 12 cells. Six overlap, so the product is 6/12 = 1/2.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Explain a fraction of a quantity as multiplication. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Securing understanding of multiplication with fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Multiply unit, non-unit and improper fractions
Cancel common factors across a product

Vocabulary: improper fraction · factor · cancellation · simplest form

Retrieve previous learning

Simplify 12/18.

Check: Divide numerator and denominator by 6 to get 2/3.

Learn the concept

To multiply a/b by c/d, multiply the numerators and multiply the denominators. This also works when a numerator exceeds its denominator: 5/4 × 6/5 = 30/20 = 3/2. Unlike addition, multiplication needs no common denominator.

Cancel common factors first to keep the numbers small. In (4/5) × (15/16), divide 4 and 16 by 4, and 15 and 5 by 5. The remaining product is 1 × 3/4 = 3/4. This is valid because the entire numerator and denominator lose the same nonzero factors.

Cancellation works between factors joined by multiplication, not between terms joined by addition. For example 2/(2 + 3) is 2/5, not 1/3. Check by calculating the original expression before trying to simplify.

Worked model

Calculate 7/6 × 9/14.
Cancel 7 with 14 to get 1 and 2; reduce 9 with 6 to get 3 and 2.
Multiply the remaining factors: 3/4.
Check directly: 63/84 simplifies to 3/4.

Guided practice

Calculate 3/8 × 4/9 by cancelling first.

Hint: Reduce 4 with 8 and 3 with 9.

Check the guided answer

(1 × 1)/(2 × 3) = 1/6.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Only numbers within the same fraction can be cancelled.”

Correction: In a product, all numerators and all denominators are factors of one overall fraction. A common factor can be cancelled across the two fractions.

Extend and reflect

Explain why (a/b) × (b/a) = 1 when a and b are nonzero.

Guidance: The combined numerator and denominator are both ab; a nonzero quantity divided by itself is one.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Multiply unit, non-unit and improper fractions. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Deepening understanding of multiplication with fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Multiply mixed numbers using improper fractions
Use sign and size checks

Vocabulary: mixed number · improper fraction · negative factor · estimate

Retrieve previous learning

Write 2 1/2 as an improper fraction.

Check: 2 1/2 = 5/2 because two wholes are four halves.

Learn the concept

A mixed number is a sum, so convert it before using the fraction multiplication rule. For 1 1/2 × 2/3, write 1 1/2 = 3/2. Then 3/2 × 2/3 = 1. Two thirds of one and a half is exactly one.

When both positive factors exceed one, the product exceeds each one. For 1 1/4 × 2 2/5, write 5/4 × 12/5 = 3. Estimating 1.25 × 2.4 gives about three, an independent check. Multiplying just the fractional parts would lose both wholes.

Signs follow the integer rules: one negative factor gives a negative product. For example −1 1/2 × 2/3 = −1. Work out the magnitude, then attach the sign and check whether the size is plausible.

Worked model

Calculate 2 1/4 × 1 1/3 by converting to 9/4 × 4/3.
Cancel the 4s and reduce 9/3 to 3.
The product is 3.
Check: 2.25 multiplied by about 1.33 is about 3.

Guided practice

Calculate 1 2/3 × 3/5.

Hint: Convert 1 2/3 to an improper fraction.

Check the guided answer

5/3 × 3/5 = 1.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“1 1/2 × 2/3 means multiply 1 by 2 and 1/2 by 1/3 separately.”

Correction: The mixed number is one quantity. Convert it to 3/2; then 3/2 × 2/3 = 1.

Extend and reflect

Find the number that multiplies 1 1/2 to make 3.

Guidance: The factor is 2 because 3/2 × 2 = 3. Explain with two groups of one and a half.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Multiply mixed numbers using improper fractions. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Checking and securing dividing a fraction by a whole number

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Interpret division of a fraction as equal sharing
Divide by multiplying by the reciprocal of a whole number

Vocabulary: dividend · divisor · quotient · reciprocal

Retrieve previous learning

What is half of 3/4?

Check: 3/4 × 1/2 = 3/8.

Learn the concept

Dividing 3/4 by 3 means sharing three quarters into three equal amounts. Each share is 1/4, because three groups of 1/4 total 3/4. A bar split into four parts can be distributed one part per group.

For a fraction a/b divided by a nonzero whole number n, calculate (a/b) × (1/n) = a/(bn), then simplify. Thus 2/3 ÷ 4 = 2/12 = 1/6. A quarter of two thirds is one sixth.

Sometimes the numerator can be divided directly, as with 6/7 ÷ 3 = 2/7. This is the same calculation as 6/21 = 2/7. Check any division by multiplying the quotient by the divisor: 2/7 × 3 = 6/7.

Representing 4/9 with four ninth-sized pieces gives another visual check. Sharing them into two equal groups leaves two ninths in each group, so 4/9 ÷ 2 = 2/9.

Worked model

Calculate 5/6 ÷ 5 as five equal shares of five sixths.
Replace division by five with multiplication by one fifth.
Calculate 5/6 × 1/5 = 5/30 = 1/6.
Check 1/6 × 5 = 5/6 and note that the share is smaller than the starting quantity.

Guided practice

Calculate 2/5 ÷ 3.

Hint: Multiply by 1/3.

Check the guided answer

2/5 × 1/3 = 2/15.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Dividing a fraction by a whole number means dividing its denominator by that number.”

Correction: The denominator usually grows: 2/3 ÷ 4 = 2/12 = 1/6. Dividing the denominator by four would make the quantity larger.

Extend and reflect

Find all positive whole numbers n for which (6/7) ÷ n has numerator 1 in simplest form.

Guidance: The numerator becomes one when all factor 6 cancels. Test divisors such as 2, 3, 6, 12 and explain the condition on n.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Interpret division of a fraction as equal sharing. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Dividing a whole number by a fraction

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Interpret division as counting fractional groups
Calculate by multiplying by the reciprocal

Vocabulary: quotient · divisor · reciprocal · grouping

Retrieve previous learning

How many halves fit in 3 wholes?

Check: There are two halves in each whole, so six halves fit in three wholes.

Learn the concept

A whole number divided by a fraction asks how many fractional groups fit. Three metres cut into pieces of length 3/4 metre produces 3 ÷ 3/4 = 4 pieces. In quarter-metres, three metres is 12 quarters and each piece uses three quarters, so 12 ÷ 3 = 4.

Multiplying by the reciprocal records this grouping efficiently: 3 ÷ 3/4 = 3 × 4/3 = 4. For 2 ÷ 2/5, two wholes are ten fifths; each group uses two fifths, giving five groups. The formula 2 × 5/2 = 5 agrees.

When the divisor is between zero and one, the positive quotient is greater than the starting whole number because more than one small group fits into each whole. Check by multiplication: quotient × divisor should return the starting amount.

Worked model

Find 4 ÷ 2/3 by counting thirds.
Four wholes contain 12 third-sized parts.
Each group uses two thirds, so 12 ÷ 2 = 6 groups.
Check with the reciprocal: 4 × 3/2 = 6, then 6 × 2/3 = 4.

Guided practice

Calculate 2 ÷ 1/4 by grouping.

Hint: How many quarters are in each whole?

Check the guided answer

Four quarters per whole make eight quarters in two wholes, so 2 ÷ 1/4 = 8.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Dividing by a fraction always makes a positive number smaller.”

Correction: Dividing by a proper fraction counts small groups and makes the quotient larger: 2 ÷ 1/4 = 8.

Extend and reflect

Explain why 1 ÷ (1/n) = n for every positive whole number n.

Guidance: A whole contains n equal nths, so n groups of size 1/n fit inside one.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Interpret division as counting fractional groups. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Dividing a fraction by a fraction

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Interpret a fraction quotient as the number of groups
Use a nonzero reciprocal and an inverse check

Vocabulary: reciprocal · quotient · inverse multiplication · nonzero divisor

Retrieve previous learning

How many quarter-sized groups fit in 3/4?

Check: Three groups, because 3/4 contains three quarters.

Learn the concept

Dividing 3/4 by 1/2 asks how many half-sized pieces fit in three quarters. One full half fits and a further half of a half fits, giving 1 1/2. Algebraically, 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2.

Multiplying by the reciprocal works because dividing by c/d is undoing multiplication by c/d. For nonzero c and d, (a/b) ÷ (c/d) = (a/b) × (d/c). The divisor cannot be zero because zero has no reciprocal and no quotient can be checked by multiplying by zero.

Check size according to the divisor. Dividing by a fraction below one makes a positive quotient larger; dividing by a fraction above one makes it smaller. Always check with the inverse multiplication: answer × original divisor = original dividend.

Worked model

Calculate 5/6 ÷ 2/3 and identify 2/3 as the nonzero divisor.
Keep 5/6 and invert only the divisor to 3/2.
Multiply 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4.
Check 5/4 × 2/3 = 10/12 = 5/6.

Guided practice

Calculate 3/5 ÷ 9/10.

Hint: Multiply 3/5 by 10/9, then simplify.

Check the guided answer

30/45 = 2/3. Check: 2/3 × 9/10 = 3/5.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“When dividing fractions, invert both fractions.”

Correction: Keep the dividend and multiply by the reciprocal of the nonzero divisor only. Check by multiplying the quotient by the original divisor.

Extend and reflect

Find a divisor that makes 3/4 ÷ divisor = 2.

Guidance: Solve divisor = (3/4) ÷ 2 = 3/8. Check that two groups of 3/8 make 3/4.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Interpret a fraction quotient as the number of groups. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Priority of operations with positive and negative integers, decimals and fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Apply brackets, powers, multiplication and division before addition and subtraction
Convert terminating decimals to fractions when useful

Vocabulary: brackets · exponent · priority of operations · terminating decimal

Retrieve previous learning

Calculate 2 + 3 × 4 and explain the order.

Check: Multiply before adding: 2 + 12 = 14.

Learn the concept

An expression has one value even when it contains different number forms. Evaluate brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication and division have equal priority; the same is true of addition and subtraction.

For −2 + (3/4) × 8, calculate the product first: 3/4 × 8 = 6, then −2 + 6 = 4. The negative sign stays attached to the 2; it is not an instruction to subtract 2 from 3/4 before multiplying.

Convert a terminating decimal to a fraction when that makes the structure easier. For 1.5 ÷ (3/4) − 2, write 1.5 = 3/2; then (3/2) × (4/3) − 2 = 2 − 2 = 0. Use brackets when a sum must be calculated before multiplication, for example (−2 + 3/4) × 8 = −10.

Worked model

Evaluate (1/2 + 1/4)² − 1/8.
Bracket first: 1/2 + 1/4 = 3/4.
Square: (3/4)² = 9/16.
Subtract 1/8 = 2/16 to get 7/16.

Guided practice

Calculate −3 + 2 × 5/4.

Hint: Calculate the product before adding the negative number.

Check the guided answer

2 × 5/4 = 5/2; −3 + 5/2 = −6/2 + 5/2 = −1/2.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“In 2 − 3/4 ÷ 1/2, work left to right from the start.”

Correction: Division takes priority over subtraction. Calculate 3/4 ÷ 1/2 = 3/2 first; then 2 − 3/2 = 1/2.

Extend and reflect

Insert brackets in −2 + 3/4 × 8 so that the answer is −10, then explain why.

Guidance: Use (−2 + 3/4) × 8 = (−8/4 + 3/4) × 8 = −5/4 × 8 = −10.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Apply brackets, powers, multiplication and division before addition and subtraction. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Number · original lesson pack

DASKEREL beta · internal AI review pending

Problem solving with arithmetic procedures involving fractions

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Choose an operation from the meaning of a context
Solve a multi-step fraction problem with units and an inverse check

Vocabulary: model · unit · estimate · inverse check

Retrieve previous learning

What operation finds the number of 3/4-litre bottles that can be filled from 3 litres?

Check: Division: 3 ÷ 3/4, which equals four bottles.

Learn the concept

Read a multi-step question before calculating. Identify the whole quantity, what has already been used, the size of each share, and the unit requested. Draw a bar or write a short equation so that the operation follows the story rather than a keyword alone.

Suppose a 4-metre ribbon loses 1 1/2 metres and the remainder is cut into 1/2-metre pieces. Subtract first: 4 − 1 1/2 = 2 1/2 metres. Then divide: (5/2) ÷ (1/2) = 5 pieces. Five half-metre pieces use 2 1/2 metres, so the result fits the remainder.

Keep units through the calculation. Metres divided by metres per piece gives a number of pieces; the quotient is a count, not another length. Estimate before accepting an answer: from about two and a half metres, half-metre pieces should number about five, not one or twenty.

Worked model

A 3-litre jug loses 3/4 litre. Each cup holds 3/8 litre. Find the number of full cups from the remainder.
Subtract: 3 − 3/4 = 12/4 − 3/4 = 9/4 litres.
Divide by cup size: 9/4 ÷ 3/8 = 9/4 × 8/3 = 6 cups.
Check six cups use 6 × 3/8 = 18/8 = 9/4 litres.

Guided practice

A 2-metre strip has 1/2 metre removed. How many 1/4-metre pieces can be cut from the rest?

Hint: Subtract the removed length before dividing by a piece length.

Check the guided answer

2 − 1/2 = 3/2 metres. Then 3/2 ÷ 1/4 = 3/2 × 4 = 6 pieces.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“To find the number of cups, multiply the remaining liquid by the cup capacity.”

Correction: Number of cups is total volume divided by volume per cup. Multiplication would have the wrong unit and give an implausibly small result.

Extend and reflect

The jug problem gives six cups. If the cup capacity increases to 1/2 litre, how many full cups and how much is left?

Guidance: The remainder is 9/4 litres. Four full half-litre cups use 2 litres, leaving 1/4 litre; a fifth full cup is impossible.