Lesson 6

Find That Factor

These materials, when encountered before Algebra 1, Unit 5, Lesson 6 support success in that lesson.

6.1: Multiplication and Division (10 minutes)

Warm-up

In this warm-up, students recall what they know about related multiplication and division equations.

Launch

Display this image for all to see, and ask students how they would express the relationship between the quantities pictured:

Diagram. Two circles contain 3 dots.
Language students might use is “2 groups of 3” or “6”. Next to the image, write two corresponding equations: \(2 \boldcdot 3 = 6\) and \(6 \div 2 = 3\). Before students begin working, tell them that it’s not necessary to draw diagrams unless they find the diagrams helpful. The task is to come up with the missing equation, so that each relationship is expressed using both multiplication and division.

Student Facing

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 \(\underline{\hspace{.5in}}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Either display the corresponding equations, or ask students to share their responses. It is possible to write equations that are correct but look different. For example, a response of \(5 \boldcdot 4 = 20\) or \(18 \div 12 = 1.5\). All of these equivalent forms should be validated. The important point is that a relationship between three numbers involving division can also be expressed as a relationship involving multiplication.

6.2: Scaling Segments (20 minutes)

Activity

This representation creates a bridge between multiplication-as-scaling and the distance of a point from the \(x\)-axis given its \(y\)-coordinate. In the associated Algebra 1 lesson, students will need to interpret a graph representing an exponential relationship and determine the growth factor. This activity pares that down to focusing on only two points on the graph, and provides students a tool (multiplication and division equations) to extract relevant information.

Launch

It’s recommended that students try this task without using a calculator, but provide access to calculators if the calculations present too great a barrier.

Student Facing

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.

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

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

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

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

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

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

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to take a few minutes to look at the equations associated with segments that grow in length, and those associated with segments that shrink in length. Ask, “How can you tell from an equation if a second segment will be longer or shorter?” If students struggle to answer this question, use the examples to point out that some factors are greater than 1 and some are less than 1.

6.3: Medicine Wears Off (15 minutes)

Activity

In this practice activity, students can continue to write corresponding division and multiplication equations (as in the previous activity) in order to determine a decay factor.

Launch

Provide access to calculators so that students can focus on looking for a common decay factor rather than on doing computations.

Student Facing

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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share their responses and their reasoning process. If students write division and multiplication equations to make sense of the given information and extract the growth factor, write these equations next to the graph. In order to find the amount of medicine at 4 hours, highlight the approach of multiplying the amount at 3 hours by the decay factor that was found earlier in the previous question.