Exploding Dots
2.1 Introducing Decimals
Lesson materials located below the video overview.
Up to now our machines have consisted of a row of boxes extending infinitely far to the left. Why not have boxes extending to the right as well? (Mathematicians like symmetry!)
Let’s work specifically with a \(1 \leftarrow 10\) machine and see what boxes to the right could mean.
Comment: It has become convention to separate boxes to the left from the ones to the right with a decimal point. (At least, this is what the point is called in the base ten world!)
What is the value of the first box to the right of the decimal point? If we denote its value as \(x\) , we have that ten \(x\)s is equivalent to \(1\). (It is a \(1 \leftarrow 10\) machine after all.)
From \(10x = 1\) we get that \(x = \frac{1}{10}\).
Call the value of the next box to the right of the decimal point \(y\).
From \(10y = \frac{1}{10}\) we get \(y=\frac{1}{100}\).
If we keep doing this, we see that the boxes to the right of the decimal point represent the reciprocals of the powers of ten.
EXAMPLE: The decimal \(0.3\) is represented by the picture:
It represents three groups of \(\frac{1}{10}\), that is:
\(0.3 = \frac{3}{10}\).
EXAMPLE: The decimal \(0.007\) is represented by the picture:
It represents the fraction \(\frac{7}{1000}\).
Question 54:
a) What fractions do the following decimals represent:
\(0.09\) \(0.003\) \(0.7\) \(0.0000003\)
b) Write the following fractions as decimals:
\(\frac{1}{1000}\) \(\frac{7}{100}\) \(\frac{9}{10}\)

Of course, some decimals represent fractions that can simplify (reduce) further. For example:
\(0.5 = \dfrac{5}{10} = \dfrac{1}{2}\).
Conversely, if a fraction can be rewritten to have a denominator that is a power of ten, then it is easy to convert it to a decimal. For example, \(\frac{3}{5}\) can be written as \(\frac{6}{10}\) and so we have:
\(\dfrac{3}{5} = 0.6\).
Question 55:
a) What fractions (in simplest terms) do the following decimals represent? \(0.05\) \(0.2\) \(0.8\) \(0.004\) b) Write the following fractions as decimals: \(\dfrac{2}{5}\) \(\dfrac{1}{25}\) \(\dfrac{1}{20}\) \(\dfrac{1}{200}\) \(\dfrac{2}{2500}\) 
Question 56: Some people read \(0.6\), for example, out loud as “point six.” Others read it out loud as “six tenths.” Which is more helpful for understanding what the number really is? Why do you think so? 
Here is a more interesting question:
What fraction is represented by the decimal \(0.31\)?
There are two ways to think about this.
Approach 1: From the picture of the \(1 \leftarrow 10\) machine we see: \(0.31 = \frac{3}{10} + \frac{1}{100}\).
We can add these fractions by find a common denominator: \(\frac{3}{10} + \frac{1}{100} = \frac{30}{100} + \frac{1}{100} = \frac{31}{100}\).
Thus \(0.31\) is the fraction \(\frac{31}{100}\).
Approach 2: Let’s unexploded the three dots in the\(\frac{1}{10}\) position to produce an additional \(30\) dots in the \(\frac{1}{100}\) position.
Thus we can see right away that \(0.31 = \frac{31}{100}\).
Question 57: Brian is having difficulty seeing that \(0.47\) represents the fraction \(\frac{47}{100}\). Describe the two approaches you could use to help explain this to him. 
Question 58: A teacher asked his students to each draw a \(1 \leftarrow 10\) machine picture of the fraction \(\frac{319}{1000}\). JinJin drew:
Subra drew: The teacher marked both students as correct. Are each of these solutions indeed valid? Explain your thinking. JinJin said all that she would need to do in order to get Subra’s solution is to perform some explosions. What did she mean by this? Is she right? 
Question 59: MULTIPLE CHOICE!
1: The decimal \(0.23\) equals: (A) \(\dfrac{23}{10}\) (B) \(\dfrac{23}{100}\) (C) \(\dfrac{23}{1000}\) (D) \(\dfrac{23}{1000}\)
2: The decimal \(0.0409\) equals:
(A) \(\dfrac{409}{100}\) (B) \(\dfrac{409}{1000}\) (C) \(\dfrac{409}{10000}\) (D) \(\dfrac{409}{100000}\)
3: The decimal \(0.050\) equals:
(A) \(\dfrac{50}{100}\) (B) \(\dfrac{1}{20}\) (C) \(\dfrac{1}{200}\) (D) None of these
4: The decimal \(0.000208\) equals
(A) \(\dfrac{51}{250}\) (B) \(\dfrac{51}{2500}\) (C) \(\dfrac{51}{25000}\) (D) \(\dfrac{51}{250000}\)

Question 60:
a) What fraction is represented by each of the following decimals? \(0.567\) \(0.031\) \(0.4077\) \(0.101\)
b) Write each of the following fractions as decimals:
\(\dfrac{73}{100}\) \(\dfrac{519}{1000}\) \(\dfrac{71}{1000}\) \(\dfrac{7001}{10000}\)
c) Write each of the following fractions as decimals:
\(\dfrac{7}{20}\) \(\dfrac{16}{25}\) \(\dfrac{301}{500}\) \(\dfrac{17}{50}\) \(\dfrac{3}{4}\)

Question 61: (CHALLENGE)
a) What fraction does the decimal \(2.3\) represent?
b) What fraction does \(17.04\) represent?
c) What fraction does \(1003.1003\) represent?

Question 62: Let’s explore the question:
Do \(0.19\) and \(0.190\) represent the same number or different numbers?
Here are two dots and boxes pictures for the decimal 0.19: Here are two dots and boxes picture for the decimal 0.190 a) Explain how one “unexplosion” establishes that the first picture of \(0.19\) equivalent to the second picture of \(0.19\). b) Explain how several unexplosions establishes that the first picture of \(0.190\) equivalent to the second picture of \(0.190\). c) Explain how explosions and unexplosions in fact establish that all four pictures are equivalent to each other. d) So … does \(0.190\) represent the same number as \(0.19\)?

EXAMPLE: How does \(12 \frac{3}{4}\) appear as a decimal?
Answer: Now \(12 \frac{3}{4} = 12 + \frac{3}{4}\). Let’s write the denominator as a power of ten by multiplying the numerator and denominator of fractional part each by \(25\):
\(12\dfrac{3}{4} = 12 + \dfrac{75}{100}\).
Thus we can now see that:
\(12\dfrac{3}{4} = 12.75\).
Question 63: Write each of the following numbers in decimal notation.
\(5 \dfrac{3}{10}\) \(7 \dfrac{1}{5}\) \(13 \dfrac{1}{2}\) \(106 \dfrac{3}{20}\) \(\dfrac{78}{25}\) \(\dfrac{9}{4}\) \(\dfrac{131}{40}\)

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