Materials & Prep
Usual classroom setup only. Display or write the equations used in the lesson. Prepare the independent problem set and exit ticket below.
Opening
DisplayWrite 4 + (-4) = 0. Students verify that the expression equals zero using their existing understanding of opposites and integer addition.
AskIf both sides are multiplied by 3, what equation results? Students write 3(4 + (-4)) = 3(0), then use the distributive property to obtain 12 + 3(-4) = 0. Ask what 3(-4) must equal. Students should conclude -12 because 12 + (-12) = 0.
ConnectPoint out that students just used a familiar fact, multiplying by zero, and the distributive property to determine a product they may not have memorized. Tell them that the same reasoning will determine the sign of a negative times a negative.
Direct instruction
ModelStart with the true equation 4 + (-4) = 0, but multiply both sides by -3: -3(4 + (-4)) = -3(0). The right side is 0. Distribute on the left: (-3)(4) + (-3)(-4) = 0.
Think aloudI already know (-3)(4) = -12 because a negative times a positive is negative. Substituting gives -12 + (-3)(-4) = 0. The second term must be 12, because -12 + 12 = 0. Therefore, (-3)(-4) = 12. The positive result is not an isolated rule. It is required if multiplication is to keep working with the distributive property.
ConfrontAsk students to commit with a thumb signal: should (-3)(-4) be negative or positive before reviewing the result. Address the error “two negatives make a negative.” That phrase confuses adding two negative numbers with multiplying two negative factors. If the product were negative, then -12 plus that product could not equal zero.
DefineState the general conclusion: for nonzero numbers, the product of two negative numbers is positive because the distributive property and the fact that multiplying by zero gives zero force that result. Then record the sign procedure: same signs produce a positive product; different signs produce a negative product. The procedure summarizes the reasoning rather than replacing it.
Guided practice
Have students work with a partner and complete each chain, writing a reason beside the final step.
- (-2)(5 + (-5)) = 0, so (-2)(5) + (-2)(-5) = 0. Since -10 + (-2)(-5) = 0, (-2)(-5) = __.
- (-4)(3 + (-3)) = 0, so (-4)(3) + (-4)(-3) = 0. Since -12 + (-4)(-3) = 0, (-4)(-3) = __.
- Use the same structure to explain why (-6)(-2) is positive, not negative.
Check for understandingAsk a selected pair to explain item 2 without saying only “the signs are the same.” Listen for the sequence: distribute, use (-4)(3) = -12, then identify the additive opposite needed to make zero. If a student claims the answer is -12, have the student substitute it into -12 + product = 0 and evaluate the contradiction.
ScaffoldProvide the sentence frame: “Because (a + (-a)) = 0, distributing gives + = 0. Since the first term is , the second term must be .” Students may first fill in the numerical values before writing the explanation.
Independent work
Students complete the following individually. Require one or two sentences of justification for items 2 and 4.
- Calculate: (-7)(-3), (-8)(2), and 5(-6).
- Use the distributive property and zero to explain why (-7)(-3) is positive.
- A student writes, “(-5)(-4) = -20 because there are two negative signs.” Identify the error and correct the reasoning.
- Find the missing value and explain: (-9)(6) + (-9)(-6) = 0, so (-9)(-6) = __.
CirculateCheck that students distinguish the known product (-9)(6) = -54 from the unknown product and that they use the additive opposite of -54, not simply the same number with a guessed sign.
ExtensionGeneralize the argument with variables: explain why (-a)(-b) must equal ab when a and b are positive. Students should begin with (-a)(b + (-b)) = 0 and identify the two terms after distributing.
Closing
Exit ticketUse the distributive property to explain why (-7)(-2) = 14. Your response must begin with (-7)(2 + (-2)) = 0, show the distributed equation, and state why the unknown product must be positive. Collect responses to check whether students can explain the reason rather than only state the sign rule.