What you should learn
To solve equations involving more than one operation To solve problems by working backward NCTM Curriculm Standards 2, 6 - 10
To solve equations involving more than one operation
To solve problems by working backward
NCTM Curriculm Standards 2, 6 - 10
In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:
Multi-step equations Consecutive Integers Number Theory
Multi-step equations
Consecutive Integers
Number Theory
Introduction: On June 9, 1993, Japan's Crown Prince Naruhito wed former diplomat Masako Owada. It took Ms. Owada about 2 1/2 hours to put on her wedding kimono, a 12-layered silk garment that weighed aobut 30 pounds. The prince wore a kimono of bright orange - the "sunrise color" - that can only be worn by the heir to the throne.
After the wedding, the crown prince changed into a tuxedo and the princess into a bridal gown to formally announce their marriage to the Emperor and Empress. To change into the bridal gown, Princess Masako and her attendants had to remove the wedding kimono, a process that was similar to putting on the garment, only in reverse order. You will solve certain kinds of problems by working in reverse order, or working backward. Working backward is one of the many problem-solving strategies that you can use to solve problems. Here are some other problem-solving strategies.
draw a diagram
make a table or chart
make a model
guess and check
check for hidden assumptions
use a graph
solve a simple (or a similar) problem
elminate the possibilities
look for a pattern
act it out
list the possibilities
identify subgoals
Exercise 1: Due to melting, an ice sculpture loses one-half its weight every hour. After 8 hours, it weights 5/16 of a pound. How much did it weigh in the beginning?
Make a table to show weight as a function of time. Work backward to find the original weight
8
7
6
5
4
3
2
1
0
5/16
2(5/16) = 10/16 = 5/8
2(5/8) = 10/8 = 5/4
2(5/4) = 10/4 = 5/2
2(5/2) = 10/2 = 5
2(5) = 10
2(10) = 20
2(20) = 40
2(40) = 80
Recall that the sculpture loses one-half its weight every hour. Multiply the current weight of the sculpture by 2 to find its weight an hour before. Continue multiplying until you reach the original weight. The original weight of the sculpture was 80 pounds.
Recall that the sculpture loses one-half its weight every hour. Multiply the current weight of the sculpture by 2 to find its weight an hour before. Continue multiplying until you reach the original weight.
The original weight of the sculpture was 80 pounds.
To solve equations with more than one operation, often called multi-step equations, you will undo the operations by working backward.
Exercise 2: Mrs. Guzman's washing machine needs fixed. Since her machine is pretty old, she doesn't want to spend more than $100 for repairs. A service call will cost $35, and the labor will be an additonal $20 per hour. What is the maximum number of hours that the repairperson can work and keep the total cost at $100?
Explore Read the problem and define the variable. Let h represent the maximum number of hours the repair can take. Plan Write an equation 35 + 20h = 100 Solve Work backwards to solve the equation 35 + 20h = 100 35 - 35 + 20h = 100 - 35 20h = 65 (20h)/20 = 65/20 h = 65/20 = 13/4 = 3 1/4 The repairperson has up to 3 1/4 hours to fix the washing machine. Examine Check to see if the answer makes sense. 3 1/4 * 20 = 65 65 + 35 = 100
Explore
Read the problem and define the variable. Let h represent the maximum number of hours the repair can take.
Read the problem and define the variable.
Let h represent the maximum number of hours the repair can take.
Plan
Write an equation 35 + 20h = 100
Write an equation
35 + 20h = 100
Solve
Work backwards to solve the equation 35 + 20h = 100 35 - 35 + 20h = 100 - 35 20h = 65 (20h)/20 = 65/20 h = 65/20 = 13/4 = 3 1/4 The repairperson has up to 3 1/4 hours to fix the washing machine.
Work backwards to solve the equation
35 + 20h = 100 35 - 35 + 20h = 100 - 35 20h = 65 (20h)/20 = 65/20 h = 65/20 = 13/4 = 3 1/4
35 - 35 + 20h = 100 - 35
20h = 65
(20h)/20 = 65/20
h = 65/20 = 13/4 = 3 1/4
The repairperson has up to 3 1/4 hours to fix the washing machine.
Examine
Check to see if the answer makes sense. 3 1/4 * 20 = 65 65 + 35 = 100
Check to see if the answer makes sense.
3 1/4 * 20 = 65 65 + 35 = 100
3 1/4 * 20 = 65
65 + 35 = 100
You have seen a multi-step equation in which the first, or leading, coefficient is an integer. You can use the same steps if the leading coefficient is a fraction.
Exercise 3: Solve each equation.
a. (y/5) + 9 = 6 (y/5) + 9 = 6 (y/5) + 9 - 9 = 6 - 9 y/5 = -3 5(y/5) = -3(5) y = -15 Check: (y/5) + 9 = 6 (-15/5) + 9 = 6 -3 + 9 = 6 6 = 6 The solution is -15. b. [(d - 2)/3] = 7
a. (y/5) + 9 = 6
(y/5) + 9 = 6 (y/5) + 9 - 9 = 6 - 9 y/5 = -3 5(y/5) = -3(5) y = -15 Check: (y/5) + 9 = 6 (-15/5) + 9 = 6 -3 + 9 = 6 6 = 6 The solution is -15.
(y/5) + 9 = 6
(y/5) + 9 - 9 = 6 - 9
y/5 = -3
5(y/5) = -3(5)
y = -15
Check: (y/5) + 9 = 6 (-15/5) + 9 = 6 -3 + 9 = 6 6 = 6
Check:
(y/5) + 9 = 6 (-15/5) + 9 = 6 -3 + 9 = 6 6 = 6
(-15/5) + 9 = 6
-3 + 9 = 6
6 = 6
The solution is -15.
b. [(d - 2)/3] = 7
Consecutive integers are integers in counting order, such as 3, 4, 5. Beginning with an even integer and counting by two will result in consecutive even integers. For example, -6, -4, -2, 0, and 2 and 2 are consecutive even integers. Beginning with an odd integer and counting by two will result in consecutive odd integers. For example, -1, 1, 3, and 5 are consecutive odd integers.
The study of numbers and the relationships between them is called number theory. Number theory involves the study of odd and even numbers.
Exercise 4: Find three consecutive odd integers whose sum is -15.
Let n = the least odd integer Then n + 2 = the next greater odd integer and n + 4 = the greatest of the three odd integers. n + (n + 2) + (n + 4) = -15 3n + 6 = -15 3n + 6 - 6 = -15 - 6 3n = -21 (3n)/3 = -21/3 n = -7 n + 2 = -7 + 2 = -5 n + 4 = -7 + 4 = -3 The consecutive odd integers are -7, -5, and -3.
Let n = the least odd integer
Then n + 2 = the next greater odd integer
and n + 4 = the greatest of the three odd integers.
n + (n + 2) + (n + 4) = -15 3n + 6 = -15 3n + 6 - 6 = -15 - 6 3n = -21 (3n)/3 = -21/3 n = -7 n + 2 = -7 + 2 = -5 n + 4 = -7 + 4 = -3
n + (n + 2) + (n + 4) = -15
3n + 6 = -15 3n + 6 - 6 = -15 - 6 3n = -21 (3n)/3 = -21/3 n = -7
3n + 6 = -15
3n + 6 - 6 = -15 - 6
3n = -21
(3n)/3 = -21/3
n = -7
n + 2 = -7 + 2 = -5
n + 4 = -7 + 4 = -3
The consecutive odd integers are -7, -5, and -3.
Further Application: Exploration: Graphing Calculators
You can use a graphing calculatro to solve multi-step equations. The solve( function will solve an equation if it is rewritten as an expression that equals zero, or most commonly, in the form ax + b - c = 0. This function also requires that you include a guess about the solution to the equation. The format is as follows. solve(expression, variable, guess) To solve the equation 3x - 4 = 2, access the math function by pressing the key. Then choose 0, for solve. Enter the expression 3x - 4 - 2, the variable x, and a guess about its solution. If we guess -1, then this appears on the calculator screen: solve(3x-4-2,X,-1). When you press , the solution, 2, appears on the screen. Your Turn a. Work through the examples in this lesson by using a graphing calculator. b. Describe the process of using a graphing calculator to solve equations in your own words. c. Why did we input the equation 3x - 4 = 2 as 3x - 4 - 2? d. How would you use a graphing calculator to solve the equation 3x - 4 = -2x + 6?
You can use a graphing calculatro to solve multi-step equations. The solve( function will solve an equation if it is rewritten as an expression that equals zero, or most commonly, in the form ax + b - c = 0. This function also requires that you include a guess about the solution to the equation. The format is as follows.
solve(expression, variable, guess)
To solve the equation 3x - 4 = 2, access the math function by pressing the key. Then choose 0, for solve. Enter the expression 3x - 4 - 2, the variable x, and a guess about its solution. If we guess -1, then this appears on the calculator screen: solve(3x-4-2,X,-1). When you press , the solution, 2, appears on the screen.
Your Turn
a. Work through the examples in this lesson by using a graphing calculator. b. Describe the process of using a graphing calculator to solve equations in your own words. c. Why did we input the equation 3x - 4 = 2 as 3x - 4 - 2? d. How would you use a graphing calculator to solve the equation 3x - 4 = -2x + 6?
a. Work through the examples in this lesson by using a graphing calculator.
b. Describe the process of using a graphing calculator to solve equations in your own words.
c. Why did we input the equation 3x - 4 = 2 as 3x - 4 - 2?
d. How would you use a graphing calculator to solve the equation 3x - 4 = -2x + 6?
Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.
Homework: The homework to be assigned for tonight would be: 17 - 47 odd, 49 - 56
Alternative Homework: Enriched: 18 - 44 even, 45 - 56
Extra Practice: Students book page 762 Lesson 3-3
Extra Practice Worksheet: Click Here.
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