9.3.4 Math in Archaeology

Calculator SymbolThe calculator can be accessed on the left-hand side bar under Toolkit.

In parts of the world, footprints from prehistoric human civilizations have been found preserved in either sand or volcanic ash. From these tracks, it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running. This can be done by using three formulas.

height equation left hand side equals right hand side seven postfix multiplication length of foot
relative stride length equation left hand side equals right hand side stride length divided by hip height
hip height equation left hand side equals right hand side four postfix multiplication length of foot

Note that no units have been included in these formulas, so it is important to make sure that the same units, for example centimeters, are used throughout the calculation. If the value of the relative stride length is less than 2, the person was probably walking, and if the value is greater than 2.9, the person was probably running.

Activity Symbol Activity: Footprints in the Sand

From one set of footprints, the length of the foot is measured as 21.8 cm and the stride length as 104.6 cm. What does the data suggest about the height and the motion of the person who made these footprints?

Hint symbol

Comment

Break the problem down into simpler parts. Start by asking yourself what you need to know to tell if the person was running or walking.

Answer Symbol

Answer

height equation left hand side equals right hand side seven postfix multiplication length of foot

So, the estimated height of the person = seven multiplication 21.8 cm = 153 cm (to the nearest cm). [ How do you think these formulas have been derived? How reliable do you think these estimates are? ]

To work out if the person was running or walking we need to know the associated relative stride length. This is given by:

relative stride length equals stride length divided by hip height

So now we need to calculate the hip height and then substitute it into the relative stride length formula.

hip height equation left hand side equals right hand side four postfix multiplication length of foot

The hip height = four multiplication 21.8 cm = 87.2 cm.

So, the relative stride length equation sequence equals 104.6 divided by 87.2 equals 1.2 (rounded to 1 decimal place).

As the relative stride length is less than 2, the person was probably walking.

Notice that the relative stride length has no units. The stride length and the hip height are both measured in cm, and when we divide one by the other, these units cancel each other out, just like canceling numbers. If you go on to study physics or other sciences, you will find many examples like this.

Notice that to find the relative stride length here, we tackled the problem by using the two formulas step by step, but you could also have combined these two formulas into one:

relative stride length equals stride length divided by four multiplication foot length

This new formula could be used directly.

Activity Symbol Activity: What's Wrong?

A student used the formula: relative stride length equals stride length divided by four multiplication foot length to calculate the relative stride length in the footprints activity.

Use this formula to work out the relative stride length and check that your answer agrees with the result in the last activity. The student entered the key sequence for the calculation as 104.6 division four multiplication 21.8. This gave an answer of approximately 570, so the student concluded that the person was probably running very fast. Can you explain where the student made the mistake?

Answer Symbol

Answer

The calculator will perform this calculation from left to right, using the PEMDAS rules (order of operations), which treat multiplication and division as equally important. So, it will first divide 104.6 by 4 to get 26.15, and then multiply by 21.8 to get approximately 570. However, this is not the correct calculation from the formula.

[ Can you explain why these two calculations are equivalent? ] The stride length (104.6) should be divided by left parenthesis four postfix multiplication foot length right parenthesis, so the calculation should be 104.6 division open four multiplication 21.8 close or alternatively 104.6 division four division 21.8. Enter both these expressions into your calculator and check that these both give the correct answer.

Note that the student should have been expecting an answer between about 0 and 5, so an answer of 570 is suspicious! Also if the student had worked out an estimate for the answer first, the mistake might have been noticed.

Now, look over the formulas you have used in this unit and also any others that you might be familiar with from your home, work, or hobbies. What are the common steps in using a formula? If you were asked to write down a list of tips for using a formula, what would you say? Here are a few suggestions—feel free to add your own!

Tips for Using a Formula

  • First, check that the formula can be applied to your particular problem.
  • Check what values you need to substitute and also what units these should be measured in. Convert the measurements if necessary.
  • Substitute the values into the formula carefully.
  • Make a rough estimate for the answer.
  • Use PEMDAS to work out the resulting calculation, step by step.
  • Explain your steps carefully using words like “substituting” and “converting.”
  • Check that your answer seems reasonable, both practically and from the estimate.
  • Round your answer appropriately.
  • Remember your concluding sentence and the units.

9.3.3 Math in Business

9.4 Your Formulas—Using a Spreadsheet