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Multiplication as Repeated Addition
Problem 3
Jenny was asked to compute $243192 \times 4$. She wrote:
\[ 243192 \times 4 = 8\, |\, 16\, |\, 12\, |\, 4\, |\, 36\, |\, 8.\]
- What was Jenny thinking about? Is her answer correct?
- Translate Jenny’s answer into a number that the rest of the world can understand.
- Use Jenny’s method to find the answers to these multiplication exercises. Be sure to translate your answers into familiar base 10 numbers.
\[156 \times 3 =\qquad 2873 \times 2 = \qquad 71181 \times 5 = \qquad 3726510392 \times 2 =\]
Problem 4
Can you adapt Jenny’s method to solve these problems? Write your answers in base eight. Try to work directly in base eight rather than converting to base 10 and back again!
\begin{align*}
&156_{\text{eight}} \times 3_{\text{eight}} = \\ &2673_{\text{eight}} \times 4_{\text{eight}} = \\ &36255772_{\text{eight}} \times 2_{\text{eight}} =
\end{align*}
Jenny might have been thinking about multiplication as repeated addition. If we have some number $N$ and we multiply that number by 4, what we mean is:
\[ 4 \cdot N = N + N + N + N.\]
If we take the number 243192 and add it to itself four times using the “combining method,” we get
- 2 + 2 + 2 + 2 = 8 ones,
- 9 + 9 + 9 + 9 = 36 tens,
- 1 + 1 + 1 + 1 = 4 hundreds,
- and so on.
Notation
Notice that we have used both × and · to represent multiplication. It’s a bit awkward to use × when you’re also using variables. Is it the letter x? Or the multiplication symbol ×? It can be hard to tell! In this case, the symbol · is more clear.
We can even simplify the notation further, writing 4N instead of 4 · N. But of course we only do that when we are multiplying variables by some quantity. (We wouldn’t want 34 to mean 3 · 4, would we?)
Problem 5
Here is a strange addition table. Use it to solve the following problems. Important: Don’t try to assign numbers to A, B, and C. Solve the problems just using what you know about the operations!
A + B B + C 2A 5C 3A + 4B
Think / Pair / Share
How does an addition table help you solve multiplication problems like 5C?
