Lesson 6

Find That Factor

  • Let’s calculate factors that scale numbers.

6.1: Multiplication and Division

Here are some multiplication and division equations. Write the missing pieces. The first one is completed, as an example.

  1. \(6 \div 2 = 3\) and \(2 \boldcdot 3 = 6\)
  2. \(20 \div 4 = 5\) and \(\underline{\hspace{.5in}}\)
  3. \(\underline{\hspace{.5in}}\) and \(1.5 \boldcdot 12 = 18\)
  4. \(9 \div \frac14 = 36\) and \(\underline{\hspace{.5in}}\)
  5. \(12 \div 15 = \underline{\hspace{.5in}}\) and \(\underline{\hspace{.5in}}\)
  6. \(a \div b = c\) and \(\underline{\hspace{.5in}}\)

6.2: Scaling Segments

For each question, the length of the second segment (on the right) is some fraction of the length of the first segment (on the left). Complete the division and multiplication equations that relate the lengths of the segments. 

figure 1

Left, line segment from origin to 0 comma 14. Right, line segment from 1 comma 0 to 1 comma 7.

figure 2

Left, line segment from origin to 0 comma 4. Right, line segment from 1 comma 0 to 1 comma 12.

figure 3

Left, line segment from origin to 0 comma 12. Right, line segment from 1 comma 0 to 1 comma 8.

\(7 \div 14 = \frac12\)

\(14 \boldcdot \frac12 = 7\)

\(\boxed{\phantom{33}} \div \boxed{\phantom{33}} = 3\)

\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}} = 12\)

\(8 \div 12 = \boxed{\phantom{33}}\)

\(12 \boldcdot \boxed{\phantom{33}}=8\)

figure 4

Left, line segment from origin to 0 comma 8. Right, line segment from 1 comma 0 to 1 comma 12.

figure 5

Left, line segment from origin to 0 comma 14. Right, line segment from 1 comma 0 to 1 comma 3.5.

figure 6

Left, line segment from origin to 0 comma 15. Right, line segment from 1 comma 0 to 1 comma 6.

\(\boxed{\phantom{33}} \div \boxed{\phantom{33}}=\boxed{\phantom{33}}\)

\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}}=\boxed{\phantom{33}}\)

\(\boxed{\phantom{33}} \div \boxed{\phantom{33}}=\boxed{\phantom{33}}\)

\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}}=\boxed{\phantom{33}}\)

\(\boxed{\phantom{33}} \div \boxed{\phantom{33}}=\boxed{\phantom{33}}\)

\(\boxed{\phantom{33}} \boldcdot \boxed{\phantom{33}}=\boxed{\phantom{33}}\)

6.3: Medicine Wears Off

Some different medications were given to patients in a clinical trial, and the amount of medication remaining in the patient’s bloodstream was measured every hour for the first three hours after the medicine was given. Here are graphs representing these measurements.

Medicine A

scatterplot, horizontal axis, time in hours. Vertical axis, medicine remaining in milligrams. 

Medicine B

Scatterplot, horizontal, time in hours, 0 to 6 by 1, vertical, medicine remaining in milligrams, 0 to 60 by 10. Points at 0 comma 50, 1 comma 10, 2 comma 2, 3 comma 0 point 4.

Medicine C

Scatter plot with 4 points plotted. 

Medicine D

Scatter plot with 4 points plotted.

Medicine E

Scatter plot with 4 points plotted.

 

  1. For one of these medicines, the relationship between medicine remaining and time is not exponential. Which one? Explain how you know.
  2. For the other four medicines:
    1. How much was given to the patient?
    2. By what factor does the medicine remaining change with each passing hour?
    3. How much medicine will remain at 4 hours?
  3. Which medicine leaves the bloodstream the quickest? The slowest? Explain how you know.

Summary