Lesson 9

This warm-up encourages students to connect and reason algebraically about various computational relationships and patterns from previous exercises. While students may evaluate each side of the equation to determine if it is true or false, encourage students to think about the following ideas in each:

1. The multiplicative relationships between the factors. Multiplying one factor by 2 and dividing the other by 2 on the left side of the equation results in the two factors on the right hand side.
2. In this case, the factors on the left hand side of the equation are adjusted in the same manner as the first equation, however since the operation is division, this strategy results in one side being 4 times the value of the other.
3. This equation applies the same reasoning as the first equation except the factors are adjusted by multiplying and dividing by 4.

Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.

Students may think the same strategies that work for multiplication can be applied to division. For these students ask them for a context to demonstrate what happens we we double a dividend and halve the divisor.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students if or how the factors in the problem impacted the strategy choice. To involve more students in the conversation, consider asking:

• “Do you agree or disagree? Why?”
• “Who can restate ___’s reasoning in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Will that strategy always work? How do you know?”

After each true equation, ask students if they could rely on the same reasoning to determine if other similar problems are equivalent. After each false equation, ask students how the problem could be changed to make the equation true.

Note: if it is possible in your local environment, we recommend using the digital version of this activity.

This activity is intended to help students identify correspondences between parts of a table, graph, and an equation of a line through the origin in the first quadrant. Students are guided to notice:

• Any pair of positive values \((a, b)\) determine a proportional relationship.
• Given a point \((a,b)\) other than the origin on the graph of a line through the origin, the constant of proportionality is always \(\frac{b}{a}\).
• In an equation \(y=\frac{b}{a} x\) that represents the relationship, the constant of proportionality appears as the coefficient of \(x\).
• The constant of proportionality is the \(y\)-coordinate when \(x\) is 1, that is, \((1,\frac{b}{a})\) is a point on the graph.

In the print version, students plot one point and draw a ray that starts at the origin and passes through their point. They also create a table and an equation to represent the relationship. They respond to a series of questions about their representations. Then, they compare their representations of their relationship with the representations of other relationships created by members of their group.

In the digital version (recommended), students interact with a dynamic sketch while recording observations and responding to prompts. In the sketch, students are able to manipulate the graph of a proportional relationship, while changes to an associated table and equation automatically update.

When all students have completed the activity, ask them to share their responses with a partner or small group. After a minute, go around the room asking each group to share one thing from the last question, and display these for all to see. When each group has shared, ask if there were any important observations that were missed.

Ensure that all the important connections are highlighted:

• A graph of a line through the origin and passing through the first quadrant represents a proportional relationship.
• The value of \(\frac ba\) computed from any point \((a,b)\) on that line (other than the origin) is the constant of proportionality.
• An equation of the relationship is given by \(y = kx\) where \(k\) is \(\frac ba\) for any point \((a,b)\) on the graph other than the origin.

The purpose of this activity is to help students understand derived units and rates. It is intended to help students see that a proportional relationship between two quantities is associated with two rates. The first rate indicates how many hot dogs someone eats in one minute (number of hot dogs per minute), and the second indicates how many minutes it takes to eat one hot dog (number of minutes per hot dog). The need for care with units (MP6) will become very clear should any students express rates in seconds per hot dog. If any students make this choice, it can be an opportunity to make connections between different ways of representing the same situation.

Monitor for students who take each approach.

Select students to share their reasoning. An important point to bring out in the discussion is that we can describe the rate of hot dog eating in two different ways as hot dogs per minute or minutes per hot dog.

Consider asking the following sequence of questions:

• “At what rate did Andre eat hot dogs?” (Some students might say \(\frac{10}{3}\) or three and a third, while some say \(\frac{3}{10}\) or 0.3.)
• “Well, you’re both right, but we need more information to know what you’re talking about. Can you be more precise?” (This will prompt students to modify their response and say “\(\frac{10}{3}\) or three and a third hot dogs per minute” and “\(\frac{3}{10}\) or 0.3 minutes per hot dog.”)

Reassure students that either response is correct, as long as units are included. The important thing is that we communicate the meaning of the number clearly. Highlight the fact that \(\frac{3}{10}\) and \(\frac{10}{3}\) are reciprocals of each other.