Substituting Numbers for Letters
Students express the lengths of trains as algebraic expressions and then substitute numbers for letters to find the actual lengths of the trains.
- An algebraic expression can be written to represent a problem situation. More than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.
- To evaluate an algebraic expression, a specific value for each variable is substituted in the expression, and then all the calculations are completed using the order of operations to get a single value.
Goals and Learning Objectives
- Evaluate expressions for the given values of the variables.
Introduce the lesson by discussing what the class learned in the last lesson. Tell students that today they will use expressions to find the actual lengths of the trains. The goal of the lesson is to evaluate the expressions for the given values of the variables.
- The expression that describes the length of 3 locomotives in a row is 3a.
- The length of the 3 locomotives is 3 ⋅ 70, or 210, feet.
This Burlington Northern and Santa Fe (BNSF) Railway locomotive is about 70 feet long.
- Write an expression to describe the length of 3 locomotives in a row, where a represents the length of 1 locomotive.
- Evaluate your expression for a = 70.
To evaluate your expression for a = 70, you need to replace any occurrence ofa in your expression with the value 70
Discuss the Math Mission. Students will write algebraic expressions to represent the lengths of trains and then find the actual train lengths by substituting values for the variables.
Express the lengths of the trains as algebraic expressions and then substitute to find the actual lengths.
Make a Train
Students will use the interactive to build trains and then write expressions representing the length of their trains.
SWD: Provide all students who have accommodations access to a calculator when performing the calculations necessary in this activity.
Mathematical Practice 7: Look for and make use of structure.
As students write two different expressions for their train, identify students who use the properties of operations to shorten an expression.
Mathematical Practice 8: Look for and express regularity in repeated reasoning.
Students may begin to see general methods for creating the shortest expression to represent their train lengths, and they may reason that the shortest expression is easiest to evaluate.
Student has difficulty getting started.
- Describe the task in your own words to your partner.
- What cars can you use to make your train? How many cars do you need to use?
- How will you represent a car’s length?
- How many expressions will you write for the length of your train?
- How can you find the actual length of your train?
Student has a solution.
- Describe how you wrote the two expressions for the length of your train.
- Explain how you know that the two expressions represent the same train length.
- How did you find the actual length of your train?
- Which of your two expressions was easier to evaluate? Explain why.
- Answers will vary. Possible answer: My train has 1 locomotive, 1 flatcar, 3 boxcars, 2 tank cars, and 1 caboose.
- The two expressions that represent the length of my train are:
a + c + b + c + d + c + d + e
a + b + 3c + 2d + e
- The length of my train is 575 feet.
70 + 55 + 75 + 55 + 50 + 55 + 50 + 80 = 70 + 75 + 3(55) + 2(50) + 80 = 490
Make a Train
- Use the Creating Trains interactive to make a train that has at least 8 cars. The table shows the types of cars that you can use to make your train.
- Write two different expressions that represent the length of your train. Use the letters specified in the table to represent the lengths of the different types of cars.
- Find the length of your train by replacing the letters in each of your expressions with the values given for each letter in the table. INTERACTIVE: Creating Trains
- Did you make a train and write an expression for the train?
- Did you replace each variable in the expression with the appropriate car length from the table?
- Once you complete these steps, you can calculate the length of the train.
Prepare a Presentation
Preparing for Ways of Thinking
Look for student work that substitutes the actual values into each of the expressions for the train length.
- 5a + 0d = 5a + 0 = 5a. Any number times 0 is 0, so 5a + 0d is equivalent to 5a.
- 5a + d is not equivalent to 5a. The variable d can represent any number, so if d = 1, say, then
5a + d = 5a + 1, which is not equivalent to 5a.
Prepare a Presentation
- Explain how you wrote your two different expressions for one of your trains.
- Describe your methods for finding the simplest form of the expression for the train you chose to present.
- Compare the expressions 5a + 0d and 5a. Are they equivalent? Justify your thinking.
- Compare the expressions 5a + d and 5a. Are they equivalent? Justify your thinking.
Have students give their presentations. Then facilitate the discussion to help students see that substituting the given values for the variables into either of the two expressions for the length of a given train will result in the same total length for that train. Compare and contrast the steps used in evaluating the different expressions. Students should also see that some ways of writing an expression are much easier to evaluate than others.
- What two expressions did you write for the length of your train? How do you know both expressions represent the length of your train?
- What is the value of each expression? Why are the values the same?
- Compare the steps used in evaluating each expression. Which expression was easier to evaluate? Explain your thinking.
- In general, what expressions do you think are easiest to evaluate? Explain why.
Students might notice that once they substitute values into the expression, evaluating an expression is just like doing the arithmetic they already know how to do. You might want to point this out, if students do not mention it.
ELL: Be cognizant that ELLs may encounter difficulties when they have to express themselves in a language other than their primary language. If you hear that they say the right things but use the wrong grammar structure, show signs of agreement and softly rephrase using the correct grammar. Use the student’s words as much as possible.
Ways of Thinking: Make Connections
Take notes about your classmates’ approaches to evaluating the expressions they wrote.
As your classmates present, ask questions such as:
- What cars did you use in your train?
- How do your two expressions represent your train?
- What properties did you use to rewrite your expression to combine like terms?
- Did you get the same answer when you evaluated each of your expressions? If not, what did you do?
- Which expression was easier to evaluate? Why?
- What is the length of your train? How do you know?
Evaluate Expressions Using Substitution
As students are working, ask questions such as the following:
- What is the variable in this expression?
- For what value of the variable are you evaluating the expression?
- How did you evaluate the expression? Explain the steps you took.
- What does s represent in the problem about the area of the square?
- What expression would you evaluate to find the perimeter of the square?
SWD: Consider the prerequisite skills for this task/skill. Students may need direct instruction, review, and guided practice when working with decimals, fractions, and exponents.
- 2x + 3 for x = 4
2(4) + 3 = 8 + 3 = 11
- 5(x − 3) for x = 6
5(6 − 3) = 5(3) = 15
- 2x + 5 + 3y for x = 2.5 and y = 5.25
2(2.5) + 5 + 3(5.25) = 5 + 5 + 15.75 = 25.75
- (a + b) for a = 5 and b = 4
(5 + 4) = (9) = 6
- for s = 8 inches
The area of the square is 64 square inches.
Evaluate Expressions Using Substitution
Evaluate each of these expressions:
- 2x + 3 if x = 4
- 5(x – 3) if x = 6
- 2x + 5 + 3y if x = 2.5 and y = 5.25
- (a + b) if a = 5 and b = 4
- If the area of a square is , and a side is equal to 8 inches, what is the area?
How to Evaluate an Expression
A Possible Summary
To evaluate an expression, you plug the values of the variables into the expression in place of the variables and then do all of the arithmetic, using the appropriate order of operations and the properties of operations.
Additional Discussion Points
- An algebraic expression can be written to represent a problem situation.
- More than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.
Summary of the Math: How to Evaluate an Expression
Write a summary about how to evaluate an expression.
Check your summary.
- Do you explain what to do with the variables when you evaluate an algebraic expression?
- Do you mention order of operations?
- Did you discuss the properties of operations?
Reflect On Your Work
Have each student write a brief reflection before the end of class. Review the reflections to find out how students thought the problems in this lesson were different from the problems in the last lesson.
Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.
The problems in this lesson were different from the problems in the last lesson because …