Solving Equations with Tables and Graphs

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Objective

SWBAT use tables and graphs to solve equations

Big Idea

This lesson is the beginning of formal equation-solving methods. Tables and Graphs lead to exact or approximate solutions.

Launch

20 minutes

To begin today's lesson I project Launch -- Tables and Graphs on the board as students enter the classroom. I ask the class to read the two questions and answer them in their notebooks. I will use a Think, Pair, Share strategy here, so I am really asking the students to prepare by working in their notebooks on their own. In a few minutes I will ask students to discuss their thinking with an elbow partner, and then write a collective answer in their notebooks. We will go over these answers, so I let the students know that they should be ready to share their responses.

The main idea that I want my students to be thinking about in this activity is that the solutions to an equation can be represented on a graph, as well as in a chart. By this I mean that the line the students are exploring in this launch section is a set of solutions of the form y = 4x + 3.

We will also talk about the difference between an exact and an approximate solution. The content here provides various ideas that can be discussed as a whole group. I have some of my own that I make sure I lead the class into (Questions 2).

Question 1: Tables A and D are both representations of the graph.

Question 2:

• Solutions to a linear equation can be found in both a table and a graph
• Solutions to a linear equation can involve decimals (because lines are continuous)
• A table or graph does not show the complete set of possible solutions
• The scale or window of a graph can be changed, if desired, to see the solution to a particular equation
• The format for a table can be changed, if desired, to see the solution to a particular equation

Here are some reasons for making sure the previous ideas are discussed as a group:

• It is common for my students to not realize that a line is a continuous set of points and that there are always points between points. Many think that the solutions are integers.
• Some of my students do not realize that a line runs infinitely in opposite directions, is a representation of a complete set of points following a pattern like y = 4x + 3.  They sometimes see the representation on paper as the whole picture. This occurs with tables as well. Students may think that the only solutions to an equation are those visible on the table. This may not always be the case.
• Too many of my students do not realize that the scale on the x or y axis can vary and that the "window" on a graph can change. This is a cause of many mistakes that my students make on exams. For example, it is common for my students to assume that the scale on a graph reflects x- and y- increasing by 1 unit per mark.

To conclude the discussion of today's Launch, I will ask students to provide the missing equation in each table.

New Info/Application

25 minutes

As we begin this section of the lesson, I will leave the Launch image on the board. At this point, my students should recognize that both Table A and Table D represent solutions to equations that are represented on the graph.

Next, I will write the following equations on the board and ask everyone to solve each for x:

1. 4x + 3 = 19
2. 4x + 3 = 9
3. 4x + 3 = 10

I am interested in seeing if my students will refer immediately to the graph or the tables on the board. I'll observe for a second, then remind them of this possibility.

Once all students are done solving the three equations, I call on students to not only give me answers, but to indicate how they arrived at their answer. The solution for question 3 is often more difficult for my students. I expect students to give a good approximation and to explain how they got it. We will discuss what makes a good answer.

Next, I hand each student the Application Graphs and Tables handout. I will have students continue to work with their same partner. On this worksheet I expect some students to report that they are able to figure out the answer mentally. I also expect that they will dismiss making tables and graphs as time spent uselessly. When a students raises this issue with me class, I emphasize that I am looking for more than the numerical answer to the equation; I'm also interested in how well he/she can relate solutions to equations, tables and graphs.

I expect that the word problems in the second half of the worksheet will be more challenging to my students than those in the first section. I want them to see the relationship between the given information and the representations (equations in particular) that they use to model the situation. A major goal for this unit is for students to be able to translate problem situations into algebraic equations.  This will not be easy for some students at the beginning, but with guidance, such translations should become automatic by the end of the unit.  See reflection

Closure

5 minutes

Before ending today's lesson I ask each student to complete this Exit Slip. Students should realize that the indicated point, despite being on the line, is NOT the solution. Question 1 is really checking if students are interpreting equations and graphs in a connected way, or are they working superficially (looking for matching numbers)?

Question 2 asks students to validate a solution.  They can solve or use substitution. I expect that many will substitute the given value for x in the equation to determine if it is a solution or not. I include Question 2b to make sure that students realize that the value must be substituted for ALL of the variables, x, in the equation. I am commonly asked this question by my students. So, here I am asking the question anticipating their possible confusion. Most will not currently combine like terms before solving. We will address this idea in a later lesson.

Exit Tickets are collected as students leave.