Lesson 3

The activities in this lesson require students to observe tables of values, look for patterns, and generalize their observations into equations. This warm-up prompts students to think about how they could go about analyzing the values in the table and to articulate their reasoning.

Arrange students in groups of 2. Display the table for all to see. Explain that the quantities in each column are related.

Ask groups to try to find a relationship and pay attention to how they go about doing so. Emphasize that the goal is not to successfully find a relationship. It is to notice the strategies they use when attempting to figure out what the relationship might be.

Invite groups to share their strategies and record them for all to see. If not already described by students, apply each strategy using the values in the table, or ask students to give an example of how it could be applied.

Some students may notice that each time \(x\) increases by 2, \(y\) increases by 8 more than the previous time. Others may notice that the \(y\)-values are 1 less than the square numbers 1, 4, 9, 25, and 49, and that these numbers are the squares of the listed \(x\)-values, and from there concluded that the relationship is along the lines of: "square \(x\) and subtract 1 to get \(y\)." Neither observations are essential, but consider asking if they see any special patterns in either column that could help determine the relationship.

If no one mentions plotting the pairs of values as a way to understand the relationship between \(x\) and \(y\), and if time permits, consider displaying a graph such as shown (or displaying a blank coordinate plane and plotting the points together).

​​​​​​

Ask students to keep in mind the different strategies as they work on the activities in the lesson.

Previously, students wrote equations to model relationships presented via verbal descriptions. In this partner activity, students are presented with pairs of values that represent quantities and take turns describing the relationship between the quantities—first using words, and then using equations. 

To do so, they need to interpret the values in context, look for structure or patterns, and generalize them (MP7). As they do so, students also practice reasoning quantitatively and abstractly (MP2). As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

As students discuss their thinking, listen for the different ways they describe the same relationship. For example, here are some ways students might describe the relationship in the second table ("meters from home" and "meters from school"):

• The distance from home and the distance from school always add up to 400.
• The distance from school is always 400 minus the distance from home.
• As the distance from home, \(x\), increases by a number, the distance from school, \(y\), decreases by the same number. \(x\) starts at 0 and \(y\) starts at 400. 
• The distance between home and school is 400 meters. The table seems to be telling us about a person traveling from home to school and how their distance to home and distance to school change along the way. 

Select students with different analyses and descriptions to share later.

Keep students in groups of 2. Give students a few minutes of quiet time to study the tables of values in the first question. Then, ask partners to take turns describing the relationships in the tables before moving on to the second question. As one person explains, the partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they reach an agreement.

Consider asking partners to also take turns matching the tables to the equations in the second question.

Some students may not know what is meant by “amount deposited” in Table D. Clarify this term to students if needed.

If time is limited, consider asking each group to analyze only two of the tables.

The relationship in Table D may not be obvious to students. Encourage students who get stuck to look at the equations in the last question and try to figure out which equation desribes the relationship between the numbers in the table.

Invite previously identified students to share how they thought about one of the relationships in the first question. Start with students who reasoned only in terms of numerical operations, and move toward those who interpreted the quantities in context (as shown in the Activity Narrative). If possible, record and display their descriptions for all to see, and highlight the connections between the different responses. 

Discuss with students whether or how their ways of thinking about each relationship affected the work of matching the tables and equations. If not brought up in students' comments, point out that some ways of describing a relationship could make it easier to identify or write a corresponding equation. To really understand what's happening in the situation, however, often requires carefully interpreting the operations that relate the two quantities.

In an earlier activity, students discerned the relationship between two quantities by analyzing and looking for patterns in tables of values. They then described each relationship in words and identified a corresponding equation. In that activity, all relationships were linear.

This activity offers new opportunities to identify and represent relationships between pairs of quantities. In each of the first two situations, the relationship is an inverse variation. (Students do not need to know the term or the concept to be able to reason about the relationships.) The last situation involves a proportional relationship. Because the given information involves three quantities (volume in gallons, in cups, and in fluid ounces), students need to reason carefully about how two of them (volume in gallons and in fluid ounces) are related.

Monitor for different ways students use to figure out and articulate how the quantities are related. For instance, students may:

• Use broad and qualitative descriptions ("As the base length increases, the height decreases").
• Give specific and quantitative descriptions ("The product of the base length and the height is always 48").
• Use diagrams, tables, graphs, or equations to make sense of the relationships and to illustrate them.

Identify students using varying strategies and ask them to share during discussion later.

Tell students that they will now describe the relationship between two quantities in some new situations. 

If time is limited, ask students to focus on the first two questions.

If students assume that the relationships must be expressed as equations and they get stuck, clarify that verbal descriptions, tables, or other representations are just as welcome.

When answering the first question, students may look only at the relationship between the first few rows of the table and say that the \(y\)-values are decreasing by 8 each time, not noticing that this is not always the case. Encourage them to look farther down the table and to also look at the relationships between the values in the columns.

Some students may have trouble getting started on the question about the volume of milk or setting up a table. Ask students which units are given in the problem, or suggest the headings “gallons,” “cups,” and “fluid ounces.” Then, ask them to use the given information to complete a row in the table. This might involve trying a different unit to start with. (For example, if they start with \(\frac12\) gallon and struggle to find the equivalent amount in cups and fluid ounces, try starting with "4 cups" or "8 cups.") A more direct hint is to suggest finding the number of fluid ounces in 8 cups.

Select previously identified students to share their responses and thinking. Sequence the presentation in the order of precision, starting with the broader descriptions or illustrations and ending with equations. (Remind students who use equations to specify what the variables represent.) If students write equations in different forms to describe the same relationship, record and display the equivalent equations for all to see.

If no students considered using tables to make sense of the pairs of quantities, ask how these would help and show an example. A table, for instance, can be particularly helpful for reasoning about the last two relationships.

Display the equations that can represent the three situations. Highlight that writing equations is an efficient way to capture the constraints in a situation.

To help students connect the equations to prior work, ask students which quantities vary and which remain constant in each equation. Point out that these equations are also equations in two variables, but unlike the equations we saw in the previous activity, not all of these represent linear relationships.