Tables, Words, and Equations

The purpose of today's lesson is to get students to move from expressing a rule in words (like they did in the previous lesson, Introduction to Functions) to expressing a rule symbolically.  You can start by posting a table that uses a two operation rule on the board.  Ask students if they can find the pattern and ask them to add two more values to the table.

Draw their attention to the top of the table where it says In and Out.  Have them notice that next to the In, it now says (x) and next to the Out, it now says (y).  Tell students that today, they will be doing some translating.  That is, they will be translating some rules in words (like those they worked on yesterday) into algebra.  

They can start with the opening example. After students have found the pattern and added two more values to the table, ask them to write the rule in words using the In/Out language from yesterday.  They might say, for example, "the Out is 2 times the In, plus 1."  Now tell them they will begin translating to algebra.  We are going to call the In x (let students know they could really call it any letter) and the Out, y.  Ask them, "how do we say y is 2 times x, plus 1, in algebra?  Students may write something like x * 2 + 1.  Now is a good time to explain some standard algebraic notation and terminology if students don't have it already.  I like to say that yes x * 2 means x times 2, but that in algebra, the standard way to write that is 2x.  You can talk about x and y as being variables here, and introduce variable as a vocabulary word.  You can also tell students that the 2 in front of the x is called a coefficient and the 1 at the end of the equation is a constant term.

Let students know that in today's activity, they will be following this same process with some new In/Out tables so they have a chance to practice this kind of translation into algebra.

Let students get working in small groups or in pairs on some In/Out tables that they will represent as a rule in words and as an equation.  

Things to watch for as you circulate:

• Many students will want to jump right to writing equations and will be tempted to skip the written rule. At this point, I insist on students writing both representations of the pattern.  I think this is especially helpful when they are trying to translate something to algebra that they don't know how to notate.
• Students may come up with all kinds of neat ways to represent Questions 2 and 3, where the In value is squared.  I love to see how students translate "multiply x by itself."  Watch out for the 2x representation and spend some time eliciting the difference between x * x and 2x.  Ask if anyone knows another way to write x * x.
• Students often struggle with the last table and there are different ways you can approach it.  Above all, encourage students to notice and look for a pattern. Sometimes asking the whole class what they notice might come up with the idea that the rows all add across to equal 20.  If you get this idea from the class, you can ask them how they would represent that idea with x and y.  From there, students can work toward x + y = 20.  You can spend lots of time here talking about how this equation is slightly different than the ones they have been working with and ask students how they might get the equation to look like the others (in other words, solve for y, though you don't need to use this language).  Once students write an equation based on the Out, or y value, have them check to make sure it works.  They may now be seeing the table entries in a different way.

I like to end class with a matching activity that really drives the idea of the multiple representations in this lesson home.  Ask students to come up with a rule of their own. Once they have their rule, ask them to represent it in a table and as an equation.  

Give each student three strips of newsprint (or you can use pieces of construction paper).  Tell students to write their rule in words on one piece of paper, their table on another, and their equation on the third.  You might tell students to use different marker colors or to disguise their writing so it is not too easy to see which rule matches with which table and equation.  While they are working, you can designate three sections on a board or a wall in your room.  Label the three sections "Problem in Words," "Equation," and "Table."  

As students finish, collect their strips of paper, mix them up, and them tape them up on the wall under the appropriate sections, but in a random order.  Ask students to match up the tables, rules, and equations.

It may be interesting to note where students start and which representations they are most comfortable working between. If they need guidance, you could choose one of the equations and ask which rule in words matches with that equation.  As students work, you can ask them what details make matching easy or hard.  You can also ask students how they know their matches are correct.

Since you have been working as a whole group at the end of class, you can close class by asking students to report out verbally on the following question: What do you ask yourself when you are looking for rules in a table?

You might want to record these student ideas as Tips that others can refer to when they are stumped!