Tables and Equations.....They're related!
Lesson 8 of 14
Objective: SWBAT write equations from tables by identifying the independent and dependent variables.
Students will be completing a problem from illustrative mathematics where they will need to write an algebraic expression from the information given and be able to extend and use their expression to find another solution (SMP 7: looking for the rule, SMP 8: applying the rule to find another solution), I liked this problem because the students could use the visual to help them find a pattern. Once the students find the pattern, they can use a table to record their information (SMP 5: using tools strategically). This task gives the students a good opportunity to work together to generate and compare equivalent expressions. While students are finding their expression to represent the situation, have them identify the terms of their expression by asking them to say what quantities they represent from the diagram. Students can make a table of values and look for patterns. As you walk around, students should be able to explain to you how the diagram, table and expression are connected. For students that struggle, encourage them to record their data in a table and extend the table to find the information needed.
Solution: n + 2 is the expression. N stands for the number of tables and 2 represents how many can fit around a table.
As you walk around, notice the way the students are finding the solution. If the expression ends up being n + 2, then they are working it correctly. I say this because you may see some different expressions, but when simplified, represent n + 2.
Once students have the expression, they can use it to extend the pattern to find how many tables will be needed for 26 children. N + 2 = 26, answer 24 tables.
Tools: illustrative math problem.
The vocabulary is part of CCSS 6.EE.C.9. Students need to know what an independent and dependent variable represent. I will give the students the definitions and have them write this down in their notes. As a little trick, and to get them thinking in terms of x and y, I tell the students: The independent variable goes “in” which represents our x value and the dependent variable comes “out” which represents our y value. I also tell them that no matter what variable is being used, it still means x and y. I will also be showing them a word problem example to help make sense of the two vocabulary words.
Tools: vocabulary words
I’ve chosen 4 different tables to show how equations are formed. This will be guided practice. Students will be responding to questioning and taking notes under my guided supervision.
The first table, the equation (rule) is already given. I will be discussing with the students the independent and dependent variable to make sense of the vocabulary. I like to refer to the table as the input/out box. The value on the right is substituted in for the variable. The rule or function changes the value of x to the output or dependent variable. Once we have our x and y values, we can take this information and place it in the coordinate grid to graph it. Graph the results and connect the lines. Ask, “what do the connected values result in?” Students should see that the result of the values placed on in the grid form a straight line.
Next, I will be showing them an equation and a table with the values already filled in. The purpose of this example is to show students that the y value depends on what the x is. Ask the students, “how do we know these solutions come from this equation”. Students should be able to say that if we plug in the x values, the output would be the y values. We can substitute and check. We will again talk about independent and dependent variables and graph the line in the grid. Watch to make sure students are using the correct values on the grid. Remind them that the input represents the x and the output represents the y.
Next, the students will be looking at a real life situation. They will be asked to identify the independent and dependent variable. Once they have done that, they will need to complete the table. Ask the students if there is an algebraic expression (rule/function), that will represent this situation? Students should see that x + 15 is the expression. The marked up price is the dependent variable and the price is the independent variable. Now, ask the students “do we know how to find the price of any item in this store?” Students should be able to say, yes, because we know the rule we can substitute in any number for the variable and find the marked up price. (SMP 7 and 8: students will look for patterns and make a generalization about what they notice)
Then, the students will be writing the equation, filling in the table, and graphing the information on the grid. If students use x and y instead of the designated variable, ask them if they think it represents the same information? They should know that no matter what variable is being used, it still can represent the information. However, students should be precise when dealing with equations and write the equation using the variable suggested.
Finally, students will be looking at a line on a graph and they will need to explain the change in y as x increases by 1. This will require students to use a table to keep track of their information. Ask students how they could represent the ordered pairs in order to look for a pattern? (SMP 5). Then students will set up a table using x and y. On the top, x will increase by 1, on the bottom, students will need to look for the pattern (SMP 7) in order to write the equation. In this case the equation would be y = x + 1. Walk students through the information in the table to help them see the pattern. If the students try to write x + 1, ask them what equals x + 1? They should say “y”.
Tools: Tables and Equations examples
The students will be working independently to show what they know about writing equations from tables. We have been working with writing equations, finding key words, and graphing. Students will be given a problem from illustrative math to help support their learning. I will be collecting this as evidence of student learning. Students will begin by completing the table. I’m not anticipating any issues with this part. Then students are asked to write an expression, using specified variables, to represent the table. They are also asked to identify the independent and dependent variables. If students are struggling with this part, have them use their notes for a resource. Next, they are going to graph the information from the table. This may get students stumbling because the numbers are bigger. Students can be asked what they can do to represent all numbers on the number line? They should be able to tell you they can use intervals or skip count. Again, watch for correct values being plotted in the grid.
Finally, students are going to be asked to use the information found to answer questions. Students can use the table, expression, or grid to answer the questions.
This activity supports the following practices:
SMP1: students will need to make sense of the problem
SMP2: Students will need to know what the numbers tell them
SMP4: Students will need to model the math with an expression.
Tools: Summative Assessment
I’m going to use the summative assessment for my wrap up. I’m going to give the students time to work with another student to describe, explain, and justify their solutions. In order to get the students in groups, I’m going to use HUSUPU. Once students are working with a partner, give them time to share.(SMP 3: students will need to prove their solutions are correct) End the class with a whole class discussion about strategies used.
Final question: how did you determine your solutions were correct? I would like students to say the following: it followed a pattern in the table, I plugged it in the expression and found the result, or when I graphed it, it formed a line.