Lesson 3

This prompt gives students opportunities to see and make use of structure (MP7). The specific structures they might notice is the table and how it relates to a linear relationship between \(x\) and \(y\) (specifically, that \(y = 3x + 6\)).

Monitor for students who:

• describe patterns only vertically or only in terms of \(x\) or \(y\) (e.g., “it” or “\(y\)” goes up by 3, “it” starts at 6)
• describe \(y\) in terms of \(x\)
• describe a relationship between \(x\) and \(y\), such as \(y\) goes up 3 every time \(x\) goes up 1
• notice the linear relationship
• attempt to identify the missing value in the table

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the relationship between the table and an equation does not come up during the conversation, ask students to discuss how they can come up with the missing value. It is not necessary to generate an equation like \(y=3x+6\) at this time.

The purpose of this activity is for students to recall the connection between tables and equations. In the associated Algebra 1 lesson, students are given tables and are asked to come up with an equation to represent a possible relationship describing all the sets of two associated numbers in the table. Here, they do the reverse. Students practice using an equation to complete a table by substituting one value and determining the other. In the process, students also practice solving equations, for example, solving \(12=2n+1\). 

Arrange students in groups of 2.

The purpose of this discussion is to ensure that students know how tables and equations connect to represent the same information. Here are sample questions to promote class discussion:

• "How does the equation help you to complete the table?" (The equation gives the rule for the table. You can substitute one value into the equation and then find the other value.)

• "Think about the equation \(m=2n+1\). Sometimes you know \(n\) and need to find \(m\). Sometimes you know \(m\) and need to find \(n\). How are these different from each other?" (When given the value of any variable, you can substitute the value for the variable, then solve. In some cases, it can be easier if you rearrange the equation to isolate a variable before substituting in the value you know. For example, \(m = 2n+1\) can be rearranged into \(n = \frac{m-1}{2}\) which may be an easier form to use when given a value for \(m\) and seeking a value for \(n\).)

In this activity, students get a chance to practice applying their skills at connecting multiple representations of the same linear relationship.

They can use the structure of the card-matching task to check their thinking and make sense of the concepts they are practicing, as wrong matches will make some piles uneven or a representation that seems to match one representation in a pile may not match another.

Arrange students in groups of 2.

Demonstrate how to set up and find matches. Choose a student to be your partner. The graphs included in the blackline master will not be used until a later lesson.  Mix up the cards and place them face up. Point out that the cards contain either a table, an equation, or a situation. Select two styles of cards and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree, for example, by explaining your mathematical thinking or asking clarifying questions.

Give each group a set of cut-up slips for matching. 

Once all groups have completed the matching, discuss:

• “Which matches were tricky? Explain why.”

• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Be sure to collect the sets of cards for use in a future activity.