Integer Multiplication

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Continuing patterns

(a): Continue the following patterns, and explain why it makes sense to continue them in that way.

\begin{align*} 4\times 3 &= 12 \\ 4\times 2 &= \\ 4\times 1 &= \\ 4\times 0 &= \\ 4\times (-1) &= \\ 4\times (-2) &= \\ 4\times (-3) &= \\ \end{align*}

\begin{align*} 3\times 6 &= 18 \\ 2\times 6 &= \\ 1\times 6 &= \\ 0\times 6 &= \\ (-1)\times 6 &= \\ (-2)\times 6 &= \\ (-3)\times 6 &= \\ \end{align*}

\begin{align*} (-7)\times 3 &= -21 \\ (-7)\times 2 &= \\ (-7)\times 1 &= \\ (-7)\times 0 &= \\ (-7)\times (-1) &= \\ (-7)\times (-2) &= \\ (-7)\times (-3) &= \\ \end{align*}
(b): What rule of multiplication might a student infer from the first pattern?
(c): What rule of multiplication might a student infer from the second pattern?
(d): What rule of multiplication might a student infer from the third pattern?

Using properties of operations

Suppose we do not know how to multiply negative numbers but we do know that $4\times 6=24$ . We will use this fact and the properties of operations to reason about products involving negative numbers.

(a): What do we know about $A$ and $B$ if $A+B=0$ ?
(b): Use the distributive property to show that the expression $4\times 6 + 4\times (-6)$ is equal to $0$ . Then use that fact to reason about what $4\times (-6)$ should be.
(c): Use the distributive property to show that the expression $4\times (-6) + (-4)\times (-6)$ is equal to $0$ . Then use that fact to reason about what $(-4)\times (-6)$ should be.

Walking on a number line

Matt is a member of the Ohio State University Marching Band. Being rather capable, Matt can take $x$ steps of size $y$ inches for all integer values of $x$ and $y$ . If $x$ is positive it means face North and take $x$ steps. If $x$ is negative it means face South and take $|x|$ steps. If $y$ is positive it means your step is a forward step of $y$ inches. If $y$ is negative it means your step is a backward step of $|y|$ inches.

(a): Discuss what the expressions $x \cdot y$ means in this context. In particular, what happens if $x = 1$ ? What if $y=1$ ?
(b): If $x$ and $y$ are both positive, how does this fit with the “repeated addition” model of multiplication?
(c): Using the context above and specific numbers, demonstrate the general rule: \[ \text{negative}\cdot \text{positive} = \text{negative} \] Clearly explain how your problem shows this.
(d): Using the context above and specific numbers, demonstrate the general rule: \[ \text{positive}\cdot \text{negative} = \text{negative} \] Clearly explain how your problem shows this.
(e): Using the context above and specific numbers, demonstrate the general rule: \[ \text{negative}\cdot \text{negative} = \text{positive} \] Clearly explain how your problem shows this.

2024-10-10 13:37:45