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High School Algebra
Domain Reasoning with Equations and Inequalities
Cluster Understand solving equations as a process of reasoning and explain the reasoning.
Standard Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Task Zero Product Property 2

Zero Product Property 2


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Alignments to Content Standards: A-REI.A.1
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Task

The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers is zero. In symbols, where $a$ and $b$ represent numbers, if $ab=0$, then $a = 0$ or $b=0$. This steps below provide a proof of this property starting with the equation $ab=0$.

  1. If $a=0$, then the property is true. Explain.
  2. Assume that $a \ne 0$. Then $a$ has a reciprocal. Explain.
  3. Since $a$ has a reciprocal, we can multiply both sides of the equation by $\frac{1}{a}$. What effect does this move have on the left side of the equation? On the right side of the equation?
  4. Explain why these steps prove the Zero Product Property.

IM Commentary

This task is part of a series of tasks that lead students to understand and apply the zero product property to solving quadratic equations. The emphasis is on using the structure of a factorable expression to help find its solutions (rather than memorizing steps without understanding).  

In this particular task, we are trying to get students to prove the zero product property, which is the lynchpin in understanding how to solve quadratic equations by factoring. In tasks that follow in this series, students will apply this property to solving quadratic equations and justifying their solutions.

Getting students to understand what a proof looks like at this level of mathematics is a challenge, and students may very well struggle with the abstract nature of this task, which attempts to scaffold the required reasoning with sturdy enough footholds. It is recommended that the task Zero Product Property 1 immediately precede this task. It sets students up to express regularity in repeated reasoning (MP.8) and state the ZPP, and then this task generalizes that observation. Another recommendation would be for the class to summarize important findings like this one in students notebooks or on a wall in the classroom to supports students in making sense of what is important and what they are expected to know.

Other tasks in this series:

Solution

  1. If $a=0$ is true, then at least one of the numbers $a$ and $b$ is $0$ so the Zero Product Property is true.
  2. Every nonzero number has a reciprocal: the reciprocal of $a$ is $\frac{1}{a}$.
  3. The left side becomes $b$ and the right side becomes $0$ so the new equation is $b = 0$.
  4. It shows that if $a \ne 0$, then $b=0$. This means that at least one of $a$ and $b$ must be $0$ so the Zero Product Property is true.

Zero Product Property 2

The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers is zero. In symbols, where $a$ and $b$ represent numbers, if $ab=0$, then $a = 0$ or $b=0$. This steps below provide a proof of this property starting with the equation $ab=0$.

  1. If $a=0$, then the property is true. Explain.
  2. Assume that $a \ne 0$. Then $a$ has a reciprocal. Explain.
  3. Since $a$ has a reciprocal, we can multiply both sides of the equation by $\frac{1}{a}$. What effect does this move have on the left side of the equation? On the right side of the equation?
  4. Explain why these steps prove the Zero Product Property.
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