Interpreting Scientific Notation

Interpreting Scientific Notation exercise

• What is scientific notation?

  Hints

  Our knowledge of powers of ten helps us compare numbers in scientific notation by understanding that multiplying by 10 raises the number by one power, and dividing by 10 lowers the number by one power.

  For instance, when comparing $4.5 \times 10^6$ and $7.2 \times 10^5$, we know that $4.5 \times 10^6$ is larger because it has a higher power of ten ($6$ compared to $5$).

  Solution

  Scientific notation (or standard form) is a way of writing numbers that are very large or very small using powers of ten.

  For example, $6.02 \times 10^{23}$ represents a very large number, and $3.2 \times 10^{-5}$ represents a very small number.

• Which unit of measurement is more appropriate for expressing the length?

  Hints

  Reflect on the standard units used for measuring distances and their relevance in everyday contexts.

  Which unit is commonly used for longer distances?

  When dealing with longer distances, using a unit like kilometres allows for easier comparisons and calculations.

  It avoids the need for extensive conversions and simplifies data analysis.

  Solution

  Sarah's measurement is $3.2 \times 10^{5}$ centimetres.

  Jack's measurement is $3.2$ kilometres.

  In this case, measuring in kilometres would be more appropriate for determining the length of the river.

• Determine which unit of measurement is more practical for describing the pool's volume.

  Hints

  Consider the scale of the measurement.

  Think about which unit (cubic centimetres or cubic metres) would result in a more manageable and easy-to-understand number for describing the size of a swimming pool.

  Reflect on which unit (cubic centimetres or cubic metres) is commonly used for larger volumes like swimming pools in everyday conversations or real-world applications.

  Solution

  Using cubic metres is more practical for describing a pool's volume because it results in a smaller, more manageable number without the need for scientific notation (or standard form).

  Additionally, cubic metres are a commonly used unit for measuring larger volumes like swimming pools, making it more familiar and understandable in everyday contexts.

• Convert a number in scientific notation.

  Hints

  Consider the power of $10$.

  The index $2$ indicates how many places to move the decimal point.

  Recall that $10^2$ = $100$.

  Solution

  $5 \times 10^2$ is equivalent to $500$.

  The company needs $500$ flyers.

• Which is the more appropriate unit?

  Hints

  Metres are a standard unit of length used in scientific contexts, especially for large distances like the distance from Earth to the Moon.

  Using metres provides a more appropriate and universally understood measurement.

  Using grams provides a more precise and detailed measurement for a scientific experiment analysis.

  Feet and inches are imperial measurements. Remember, feet are bigger than inches.

  Solution

  The distance from Earth to the Moon - $3.84\times 10^{5}$ metres

  The mass of a tennis ball - $7.2\times 10^{5}$ grams

  The volume of a swimming pool - $9.2\times 10^{4}$ cubic feet

• How long did each journey take?

  Hints

  Convert $1.26 \times 10^4$ to an ordinary number.

  $1.26 \times 10^4$ is equivalent to 12,600.

  Divide 12,600 by 60 to find the number of minutes it is equal to.

  12,600 $\div$ 60 = 210

  How many hours is 210 minutes equal to?

  Repeat the steps above for Pippa's journey making sure to check the units used.

  Solution

  Carla's journey took 210 minutes or 3 $\frac{1}{2}$ hours.

  Pippa's journey took 225 minutes or 3 $\frac{3}{4}$ hours.

  For this length of journey, minutes or hours would be more appropriate units to use than days or seconds.