Exploding Dots

2.5 ADDENDUM: Some Fun Extra \(1 \leftarrow 10\) Questions

Lesson materials located below the video overview.

 

Question 50:  MULTIPLYING BY TEN

 

Quickly compute each of the following:

a) \(263207 \times 3\)

b) \(563872 \times 9\)

c) \(673600023 \times 2\)

 

d) Use the \(1 \leftarrow 10\) machine to explain why multiplying a number in base \(10\) by \(10\) results in simply placing a zero at the end of the number.

 

e) Comment on the effect of multiplying a number written in base \(b\) by \(b\).

 

Comment: One can go much further with this thinking. What is the effect of dividing a number written in decimal notation by ten? By one-hundred?

 

 

Question 51: MULTIPLICATION BY ELEVEN

 

Here’s a trick for multiplying two-digit numbers by \(11\):

To compute \(14 \times 11\), say, split the \(1\) and the \(4\) and write their sum, \(5\), in between:

\(14 \times 11=154\).

To compute \(71 \times 11\) split the \(7\) and the \(1\) and write their sum, \(8\), in between:

\(71 \times 11 = 781\).

In the same way:

\(20 \times 11 = 220\)

\(13 \times 11 = 143\)

\(44 \times 11 = 484\)

 

It also works if we do not carry digits:

 

\(67 \times 11 = 6|13|7 = 737\)

\(48 \times 11 = 4|12|8 = 528\)

 

 

a)    Why does this trick work?

b)    Quickly, what’s \(693 \div 11\) ?

c)     Quickly work out \(133331 \times 11\).

 

A number is a palindrome if it reads the same way forwards as it does backwards. For example, 124454421 is a palindrome.

 

d)    TRUE OR FALSE and WHY: Multiplying a palindrome by \(11\) produces another palindrome.

 

Question 52: AN UNUSUAL WAY TO DIVIDE BY NINE 

 

Here’s an unusual way to divide by nine.To compute \(21203 \div 9\), say, read “\(21203\)” from left to right computing the partial sums of the digits along the way:

ex11001

and then read off the answer:

\(21203 \div 9  = 2355 R 8\).

In the same way,

\(1033 \div 9 = 1 | 1+0 | 1 + 0 + 3 | R  1 + 0 + 3 + 3  = 114  R  7\)

and

\(222 \div 9 = 246  R  8\).

 

Why does this trick work? (Is the act of “carrying” digits a problem?)

 

HINT: Does it help to think of division by nine as division by “ten plus negative one”?

 

Comment: Alternatively … What is \(\frac{1}{9}\) as a decimal? What do you obtain if you multiply \(21203\) by this decimal?

 

 

Question 53:  IS IT PRIME? 

 

Is \(18^{1000} – 1\)  prime?

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