Course Guide - USAID Teacher Education Project

It is also important for students to understand that when multiplying a negativenumber by a positive number, the result is never “greater” than the factors. Havethem look at the results of their multiplication and see if they notice this patternbefore mentioning it.• However, when two negative numbers are multiplied, the result is positive. In thissituation, the number line and repeated addition are less useful models to helpstudents understand why this is so.One model that is helpful is to look at patterns with the understanding that ournumber system is logical and consistent.(-3) x 4 = - 12(-3) x 3 = - 9(-3) x 2 = - 6(-3) x 1 = - 3(-3) x 0 = 0(-3) x (-1) = ?The answer cannot be -3; that was the answer to (-3) x 1. So it must be somethingelse. To maintain the pattern of the answer becoming 3 greater each time, (-3) x (-1)must be +3.Recall that in Session 1 this week we discussed the concept of opposites. Since (-1)is the opposite of 1, then the answer to (-3) x (-1) must be the opposite of (-3) x 1.However logical this is, students may not be convinced. Here is where a real lifeexample can make this concept easier to understand. One useful negative context istime past. For example, I have a $200 car payment taken out of my savings accounteach month. This can be thought of as -$200. How much more was in my savingsaccount 3 months ago (-3 for time past). The answer is that 3 months ago I had +$600 more in my account.• Models for division of integers: Since division is the inverse of multiplication, thesame models described above apply. Although repeated addition is useful inexplaining integer multiplication, students find it more difficult to apply that idea tothe division of integers. Thus, beginning by building on what they learned aboutpatterns and the idea of the inverse may be more helpful. Note that by creating thechart in this format, the idea of dividing a negative by a negative immediately resultsin four examples of a positive product.(-3) x (+ 4) = - 12 so - 12 ÷ (-3) = + 4(-3) x (+ 3) = - 9 so - 9 ÷ (-3) = + 3(-3) x (+ 2) = - 6 so - 6 ÷ (-3) = + 2(-3) x (+ 1) = - 3 so - 3 ÷ (-3) = + 1(-3) x 0 = 0 so 0 ÷ (-3) = 0(-3) x (-1) = + 3 so + 3 ÷ (-3) = - 1• Fact Families: Just as we could build a fact family for the inverse operations ofmultiplication and division of whole numbers:3 x 4 = 124 x 3 = 1212 ÷ 4 = 3