Integer Operations
Lessons in this chapter
About Integer Operations
Integer Operations takes the positive and negative numbers kids have already met and turns them into a full arithmetic system — one where every operation has to account for sign, not just size.
It starts with add integers and subtract integers — building the number-line model for combining positive and negative values, so a problem like −4 + (−7) is understood as moving further into negative territory, and −4 − (−7) is understood as subtracting a negative, which moves the answer up rather than down.
Multiply and divide integers introduces the sign rules that make integer multiplication and division predictable — two negatives or two positives always give a positive answer, one negative and one positive always give a negative one — so kids stop guessing the sign and start applying a rule.
Kids then use properties of addition and subtraction of integers and use properties of multiplication and division of integers, confirming that the commutative, associative, and identity properties they already know from whole numbers still hold once negative numbers are in the mix — for example, that −3 + 8 gives the same answer as 8 + (−3).
The chapter closes with use the distributive property of integers, applying it to expressions like −2(x − 5), and solve problems on order of operations on integers, working through multi-step expressions that mix all four integer operations and require kids to track sign changes correctly at every step, not just at the end.
Frequently asked questions
It builds on the basic addition and subtraction of positive and negative numbers kids already know, adding multiplication and division, checking the commutative/associative/identity properties still hold with signed numbers, and combining all four operations into multi-step expressions using order of operations.
Same signs give a positive answer, different signs give a negative answer — two negatives or two positives multiplied or divided always come out positive, one negative and one positive always come out negative.
Subtracting a negative number undoes moving in the negative direction, which is the same as moving in the positive direction — so −4 − (−7) becomes −4 + 7, which equals 3. Subtract Integers has kids practice rewriting a subtraction of a negative as an addition before calculating.
Treating a problem like −8 + 5 the same as 8 + 5 and losing track of the sign. The reliable rule is to subtract the smaller absolute value from the larger one, then keep the sign of whichever number has the larger absolute value — for −8 + 5, that's 8 − 5 = 3, kept negative, giving −3.
Distributing a negative number flips the sign of every single term inside the parentheses, not just the first one — so −2(x − 5) becomes −2x + 10, because a negative times a negative gives a positive. Use Distributive Property of Integers is where this sign-flipping gets practiced deliberately.
A sign error made early in a multi-step expression carries through every operation after it, so getting the order of operations right — and re-checking the sign after each step — matters more here than in a single-operation problem. Solve Problems on Order of Operations on Integers is built entirely around this kind of multi-step, sign-sensitive problem.