A penny is about \(\frac{1}{16}\) of an inch thick.
Here we need to perform the division \(239,000 \,\,miles \div \frac{1}{16} inches\). Because the units, miles and inches, are different we need to find out how many inches are in each mile (or equivalently, convert the \(239,000\) miles to inches). We have
Since it takes \(16\) pennies to make a stack one inch high, with \(5,000,000,000\) pennies we could make a stack $$ \frac{5,000,000,000}{16} \text{inches} $$ high. The question asks how many miles this is so we need to convert inches to miles. For this, we use the facts that there are \(12\) inches in a foot and \(5280\) feet in a mile:
If we write \(x\) for the number of pennies in a stack which reaches the moon then we have the equation $$ x \,\times \frac{1}{16} \text{inch} = 239,000 \,\, \text{miles}. $$ We can solve this equation as follows: $$ x \,\,\, \text{inches} = 16 \times 239,000 \,\, \text{miles} \times \frac{5280 \,\,\,\text{feet}}{\text{mile}} \times \frac{12 \,\,\,\text{inches}}{\text{foot}} $$ Simplifying we find
Since the pennies being stacked all have the same thickness, the ratio of the number of pennies in a stack to the height of the stack does not depend on how many pennies are in the stack. In other words if \(a\) denotes the number of pennies in a stack and \(h\) is the function so that \(h(a)\) denotes the height of that stack then the ratios \((a:h(a))\) are equivalent for any positive number \(a\) of pennies.
Here we have \(a = 5,000,000,000\). According to the previous paragraph we need to find \(h(5,000,000)\) we need to solve the equation $$ \left(1:\frac{1}{16} \text{inch}\right) = \left( 5,000,000,000: x \,\, \text{miles} \right). $$ We have $$ \left(1:\frac{1}{16} \text{inch}\right) = \left( 5,000,000,000: 5,000,000,000 \times \frac{1}{16}\,\, \text{inches} \right). $$ So to solve for \(x\) we need to convert \(\frac{5,000,000,000}{16} \text{inches}\) to miles which can be done as in the previous solution:
So \(x\), the height of the stack of \(5,000,000,000\) pennies, is about \(5000\) miles.
Scientific notation is appropriate for this problem as the number of pennies involved is very large and nicely represented using exponential notation. Either of the above solutions can be adapted for the use of exponential notation and we have chosen here to use the first method.
One billion is one with \(9\) zeroes or \(1 \times 10^9\). So \(5\) billion pennies is \(5 \times 10^9\) pennies. Since it takes \(16\) pennies to make a stack one inch high, with \(5 \times 10^9\) pennies we could make a stack $$ \frac{5 \times 10^9}{16} \text{inches} $$ high. The question asks how many miles this is so we need to convert inches to miles. For this, we use the facts that there are \(12\) inches in a foot and \(5280\) feet in a mile:
If we write \(x\) for the number of pennies in a stack which reaches the moon then we have the equation $$ x \,\times \frac{1}{16} \text{inch} = 239,000 \,\, \text{miles}. $$ We can solve this equation as follows: $$ x \,\,\, \text{inches} = 16 \times 239,000 \,\, \text{miles} \times \frac{5280 \,\,\,\text{feet}}{\text{mile}} \times \frac{12 \,\,\,\text{inches}}{\text{foot}} $$ Simplifying we find
Commentary
This task can be made more hands-on by asking the students to determine about how many pennies are needed to make a stack one inch high. In addition, for part (b) students could be invited to research this question. Useful information which would need to be compiled to get total mintage figures can be found at Annual mintage of Lincoln pennies. From 1909 through 2009 the total number of pennies minted is 455,627,740,918 according to Total mintage of Lincoln pennies.
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise. In particular, before they start working on the problem, the teacher may wish to ask the students whether or not they think it would be possible to reach the moon with this giant stack of pennies.
Four solutions are offered to the problem stressing different mathematical ideas:
The first of these solutions uses multiplication as well as division because there is conversion of units involved. Note that the second of these solutions leads naturally to the introduction of a function (representing the height of a stack of \(x\) pennies) and so, although it also strongly stresses ratio language, it would need to be adapted in order to be appropriate at the sixth grade level. As written, the answer to part (c) meets the 6-RP.3 standard. Students should eventually be comfortable with all three approaches. Note too that throughout the teacher needs to provide some guidance in terms of the level of accuracy with which results should be recorded: because the number of pennies minted is an estimate and the thickness of the pennies is an estimate, no more than one or two significant digits should be recorded in the answers; this is important because it makes each successive calculation easier if numbers are rounded.
The number of pennies made in the last 100 years is about 100 times the number of pennies made in 2011. The vast majority of these pennies, however, have been made in the last 40 years. In 1922, for example, only a little over 7 million pennies were made while in 1982 the total exceeded a staggering 17 billion.