Question

fluency and skills practice | name: lesson 11 understanding multiplication with negative integers continued 3 use the distributive property to show why the product (-6)(-3) is positive. the first step is done for you. (-6)(-3) + (-6)(3) = (-6)(-3) + 3 4 mark’s work to simplify (-3)(-5)(-2) is shown. explain his error and show how to find the correct product. (-3)(-5)(-2) = (-15)(-2) = 30

Explanation:

Problem 3: Use the distributive property to show why the product \((-6)(-3)\) is positive.

Step 1: Simplify the right - hand side

First, simplify the expression inside the brackets on the right - hand side of the equation \((-6)(-3)+(-6)(3)=(-6)[(-3)+3]\). We know that \((-3)+3 = 0\). So the right - hand side becomes \((-6)\times0=0\).
So we have the equation \((-6)(-3)+(-6)(3)=0\).

Step 2: Analyze \((-6)(3)\)

Calculate \((-6)(3)\). The product of a negative number and a positive number is negative, so \((-6)(3)=- 18\). Let \(x = (-6)(-3)\). Then our equation from Step 1 is \(x+(-18)=0\).

Step 3: Solve for \(x\)

To solve for \(x\) in the equation \(x - 18=0\) (since \(x+(-18)=x - 18\)), we add 18 to both sides of the equation. Using the addition property of equality, \(x-18 + 18=0 + 18\). So \(x=18\), and since \(x = (-6)(-3)\), we have shown that \((-6)(-3)=18\), which is positive.

Problem 4: Mark's work to simplify \((-3)(-5)(-2)\)

Step 1: Identify Mark's error

Mark first calculated \((-3)(-5)=-15\), but the product of two negative numbers should be positive. The correct calculation of \((-3)(-5)\) is \(15\) (because the product of two negative integers is positive: \((-a)(-b)=ab\) for positive integers \(a\) and \(b\)).

Step 2: Calculate the correct product

First, calculate \((-3)(-5)\) correctly. \((-3)(-5)=15\) (using the rule that the product of two negative numbers is positive).
Then multiply this result by \(-2\): \(15\times(-2)\). The product of a positive number and a negative number is negative, so \(15\times(-2)=-30\).

Answer:

s:

• For problem 3, \((-6)(-3)=18\) (positive).
• For problem 4, Mark's error was in calculating \((-3)(-5)\) as \(-15\) (it should be \(15\)), and the correct product of \((-3)(-5)(-2)\) is \(-30\).