Multiplication and Division with Negative Numbers Worksheets
What are the rules for multiplying and dividing negative numbers?
The multiplication and division of negative numbers follows consistent sign rules: when multiplying or dividing two numbers with the same sign (both positive or both negative), the answer is positive; when the signs differ, the answer is negative. For multiplication: positive × positive = positive, negative × negative = positive, positive × negative = negative. The same rules apply for division: positive ÷ positive = positive, negative ÷ negative = positive, positive ÷ negative (or negative ÷ positive) = negative.
A common error occurs when students apply these rules to addition or subtraction, or when they forget that the rules work identically for both multiplication and division. Exam mark schemes regularly penalise students who write -12 ÷ -3 = -4, showing they've remembered division gives a different operation result but haven't recognised that dividing two negatives follows the same 'same signs give positive' pattern. Teachers often use the phrase 'same signs, positive answer; different signs, negative answer' as a memory aid, though understanding why these rules work helps retention more than rote learning.
Which year groups study multiplication and division with negative numbers?
This topic appears in the Key Stage 3 curriculum, typically introduced in Year 7 and consolidated in Year 8. The National Curriculum requires students to 'use the four operations, including formal written methods, applied to integers, decimals and simple fractions', with negative numbers forming part of the integers strand. Year 7 students encounter these operations after establishing confidence with negative numbers on number lines and in addition and subtraction contexts.
Progression across these year groups involves increasing complexity. Year 7 worksheets generally focus on straightforward calculations with single-digit or simple two-digit negative numbers, ensuring students grasp the sign rules securely. By Year 8, questions incorporate larger numbers, multi-step problems requiring multiple operations, and contexts where students must decide which operation to apply. This builds towards algebraic manipulation at GCSE, where multiplying and dividing terms with negative coefficients becomes routine in simplifying expressions and solving equations.
Why do two negative numbers multiplied together give a positive answer?
The rule that negative × negative = positive stems from the mathematical requirement for consistency across number systems. Teachers can demonstrate this through patterns: starting with 3 × 3 = 9, then 3 × 2 = 6, then 3 × 1 = 3, then 3 × 0 = 0. Continuing this pattern, 3 × -1 must equal -3, and 3 × -2 equals -6. Now reversing the first number: -3 × 2 = -6, -3 × 1 = -3, -3 × 0 = 0, so -3 × -1 must equal 3, following the same increment pattern. This logical consistency underpins all operations with directed numbers.
This concept connects directly to real-world contexts involving reversals. In physics, velocity and acceleration use directed numbers: if an object moving backwards (negative velocity) experiences backwards acceleration (negative acceleration), it moves further backwards, but reversing a backwards motion (multiplying two negatives) produces forward movement (positive direction). Financial models use similar logic with debt and credit reversals, showing students how abstract mathematical rules govern practical applications in STEM fields and economics.
How do these worksheets help students practise negative number operations?
The worksheets provide structured practice moving from straightforward calculations to more complex problems requiring students to apply sign rules accurately under varying conditions. Questions typically begin with isolated multiplication or division problems where students focus solely on applying the correct rule, then progress to mixed operations where they must identify which rule applies to each calculation. This scaffolding helps students build automaticity with sign rules before tackling algebraic contexts where these skills become embedded within larger procedures.
Teachers use these resources for targeted intervention with students who make persistent sign errors, as homework to reinforce classroom teaching, or as starter activities to maintain fluency. The answer sheets allow students to self-mark and identify specific error patterns, whether consistently reversing signs in division, or struggling only when the negative number appears first. Paired work where students explain their reasoning to each other often reveals where conceptual understanding breaks down versus where errors are simply procedural slips.