“Today we are going to look at expressions that contain variables. A variable is a letter representing an unknown number in an expression. Remember that an expression is different than an equation. An expression does not contain an equal sign. When solving for a variable, you determine what number value the variable represents. Symbolic expressions that contain variables are used in accounting, building, and even in science. The goal for this lesson is to be able to look at a written statement and translate it into a symbolic expression that contains a variable.”
Project the ABC Variables template (M-6-6-1_ABC Variables.doc) for students to see. “Notice the alphabet letters in the top row. In the bottom row, you’ll see that each letter has been given a value. What would be the value of the expression h + a + p + p + y?” Give students time to work with each other to determine the value of this expression. Ask students to share how they approached solving this expression. Record responses on the board. (Possible answer: 11 + 1 + 2(17) + 25 = 71) Say, “Variables can be substituted for values. Once we know the values of the variables, we can evaluate the expression. Let’s try another example: m + a + t + h.” Give students time to work with each other to determine the value of this expression. Ask students to share how they evaluate this expression. Record responses on the board. (Possible answer: 15 + 1 + 21 + 11 = 48)
“Now you are going to look at some statements and practice translating each of them into a symbolic equation that contains a variable.”
Write this statement on the board: Twelve more than a number is fifty-two. As you work through each part of the statement, write the equation on the board. Say, “To be able to write this as a symbolic equation, we have to break the statement apart and translate it into numbers and symbols. Twelve more than tells me that we are adding, so the equation will start with 12 + . The next part of the statement is a number. That would be the variable. We could use any letter for the variable. The equation so far is 12 + n. The last part of the statement is fifty-two. This tells me where the equal sign should go, so the equation would be 12 + n = 52.”
“Let’s look at another example where we read a statement and then turn it into a symbolic equation that contains numbers, symbols, and variables.” Write the following on the board: Six is fifteen less than three times a number. As you explain each step, write it out for students to see. “I can begin by dividing the statement into parts:
“Six is | fifteen less than | three times | a number.
“I can translate Six is into 6 =
“This is subtraction, but fifteen less than tells me I have to subtract 15 from a first number.
“The phrase three times can be translated into 3 ×
“The phrase a number can be translated into the variable m.
“So, the equation would be 6 = 3 × m − 15.”
Divide students into small groups and hand each student a copy of the Signal Words Template (M-6-6-1_Signal Words Template.doc). Have students brainstorm signal words that could help them identify what operation is being referred to in a written expression. Once groups have had a chance to brainstorm, create a class list on chart paper. As an alternative, first create a class list on chart paper and have students transfer the information to their template. This will be an important resource that students can refer back to during this lesson/unit.
At this point, it would be beneficial to show students the different ways to represent multiplication. Distribute the Ways to Show Multiplication and Division sheet (M-6-6-1_Ways to Show Multiplication and Division.doc) and encourage students to use it as a resource during this unit.
To increase fluency in writing equations with variables, divide students into small groups and hand each student a copy of the Silent Pass with Variables worksheet (M-6-6-1_Silent Pass with Variables and KEY.doc). Put a copy of the worksheet in the center of each group. “The best way to improve how you write equations with variables is practice. We will do an activity called Silent Pass with Variables. Each person in your group will need to use a different color so it is easy to distinguish which questions each person has answered. You will answer one question on your worksheet using your designated color. When you are done, trade papers with someone from your group and correct any errors, and then answer a new question. Once you are finished, trade papers again making sure you get a different sheet and repeat the process until time is up.” (Set time limit at 10 to 15 minutes.) “Try to finish as many questions as possible in the time given. Before you begin, let me remind you of the title…SILENT Pass with Variables.”
While students are completing this activity, monitor their performance. By noticing which color distinguishes each student, you can determine which students are having difficulty. When the time is up, bring the class together and discuss any of the questions in which students need clarification. Remind students that there may be more than one right answer when writing an expression or equation because of the multiple ways to express the operations of addition, subtraction, multiplication, and division. Refer to the Signal Words template the class created on chart paper.
Use this opportunity to find out what may be confusing to students. Review items on the Silent Pass worksheet. Allow students time to form questions. They will likely want to ask about different wording for some of the written expressions.
In the following activity, students will work independently on writing symbolic equations that contain numbers, variables, and symbols. The expressions will be embedded in word problems.
- Hand each student a copy of the Equations & Expressions Involving Variables worksheet (M-6-6-1_Equations & Expressions Involving Variables and KEY.doc).
- Remind students that they must first identify the relationship that is being expressed in each problem by breaking the expression/equation into smaller parts.
- Ask students to identify which part of the expression or equation needs to be represented by a variable.
- Tell students to write a symbolic expression to answer the question posed by the problem.
- Remind students that they need not solve the equation. Students are practicing writing symbolic equations.
During this activity, students will have the opportunity to revise and refine their thinking related to this concept. Be available to provide on-the-spot support for those students who may need some prompting to get through the process of writing expressions or equations that contain variables.
After students have completed the worksheet, bring them together and discuss what generalizations they can make about when to use addition, subtraction, multiplication, and division. (Possible answers: Adding is when you are putting items together; “less than” means you are subtracting, but you have to be careful which number goes first; when you “times” a number, it’s multiplication; and when you need equal parts you divide.) Allow students to pair up and compare their answers from the Equations & Expressions Involving Variables worksheet. If their answers do not agree, have students go to another pair of students to determine why their answers differ. Monitor student interaction and performance. Set up a location where you will be available to help students who are having difficulty understanding the concepts, even with the support of their peers. Through observation, call small groups together to remediate misconceptions.
Extension:
This lesson is designed to introduce students to the concept of translating written expressions into symbolic expressions/equations. At this point, students are not expected to solve these types of problems. There might be times in this lesson when an explanation of the use of parentheses in mathematical expressions/equations may be necessary. In Lesson 3, you will find a more in-depth look at order of operations and properties.
Following are some ways to tailor this activity to meet the needs of your class.
- Routine: Make a set of flashcards that can be written as expressions or equations using variables, similar to the ones used in this lesson. (For a sample, see M-6-6-1_Routine Flash Card Example.doc.)
- Small Group, Tiered Problems: Based on student proficiencies, the difficulty of the problems can vary. For those students who are challenged by the process of translating from written statements to expressions/equations that contain a variable, break the statements apart so students can look at smaller pieces at a time (for example, four times | a number | is | forty-eight).
- Expansion: Students who are proficient in translating statements into symbolic expressions/equations can work on solving for the unknown variable. At this point, students can solve using any method that makes sense to them. For example, students can use guess and check to see if they can identify numbers that can be substituted for the variables to make the equation true.
