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Comparing Tape Diagram Solutions to Algebraic Solutions




 


Video solutions to help Grade 7 students learn how to compare tape diagram solutions to algebraic solutions.

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New York State Common Core Math Module 2, Grade 7, Lesson 17


Lesson 17 Student Outcomes

Students use tape diagrams to solve equations of the form px + q = r and p(x + q) = r, (where p, q, and r, are small positive integers), and identify the sequence of operations used to find the solution.

Students translate word problems to write and solve algebraic equations using tape diagrams to model the steps they record algebraically.

Lesson 17 Summary

Tape Diagrams can be used to model and identify the sequence of operations to find a solution algebraically.

The goal in solving equations algebraically is to isolate the variable.

The process of doing this requires “undoing” addition or subtraction to obtain a 0 and “undoing” multiplication or division to obtain a 1. The additive inverse and multiplicative inverse properties are applied, to get the 0 (the additive identity) and 1 (the multiplicative identity).

The addition and multiplication properties of equality are applied because in an equation, A = B, when a number is added or multiplied to both sides, the resulting sum or product remains equal.


Example 1
Expenses on Your Family Vacation
John and Ag are summarizing some of the expenses of their family vacation for themselves and their three children, Louie, Missy, and Bonnie. Create a model to determine how much each item will cost, using all of the given information. Then, answer the questions that follow.

Scenario 1

During one rainy day on the vacation, the entire family decided to go watch a matinee movie in the morning and a drive- in movie in the evening. The price for a matinee movie in the morning is different than the cost of a drive-in movie in the evening. The tickets for the matinee morning movie cost each. How much did each person spend that day on movie tickets if the ticket cost for each family member was the same? What was the cost for a ticket for the drive-in movie in the evening?







Scenario 2

For dinner one night, the family went to the local pizza parlor. The cost of a soda was If each member of the family had a soda and one slice of pizza, how much did one slice of pizza cost?



 

Discussion / Lesson Questions for Algebraic Approach

1. When solving an equation with parentheses, order of operations must be followed. What property can be used to eliminate parentheses; for example, 3(a + b) = 3a + 3b ?

2. Another approach to solving the problem is to eliminate the coefficient first. How would one go about eliminating the coefficient?

3. How do we “undo” multiplication?

4. What is the result when “undoing” multiplication in any problem?

5. What mathematical property is being applied when “undoing” multiplication?

6. What approach must be taken when solving for a variable in an equation and “undoing” addition is required?

7. How can this approach be shown with a tape diagram?

8. What is the result when “undoing” addition in any problem?

9. What mathematical property is being applied when “undoing” addition?

10. What mathematical property allows us to perform an operation (or, “do the same thing”) on both sides of the equation?

11. How are the addition and multiplication properties of equality applied?

Exercise 1

The cost of a babysitting service on a cruise is $10 for the first hour, and $12 for each additional hour. If the total cost of babysitting baby Aaron was $58, how many hours was Aaron at the sitter?

How can a tape diagram be used to model this problem?

How is the tape diagram for this problem similar to the tape diagrams used in the previous activity?

How is the tape diagram for this problem different than the tape diagrams used in the previous activity?



Lesson 17 Problem Set

3. Jillian exercises 5 times a week. She runs 3 mi. each morning and bikes in the evening. If she exercises a total of 30 miles for the week, how many miles does she bike each evening?




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