WRITE AND EVALUTE NUMERICAL EXPRESSIONS WITH ORDER OF OPERATIONS
INTRODUCTION
The objective for this lesson is, the student will use the order of operations to write and evaluate numerical expressions, including expressions with exponents.
The skill students should have in order to help them in this lesson include, evaluate numerical expressions with the four operations.
We will have three essential questions that will be guiding our lesson. Number One, explain the meaning of the term “verbal expression.” “Give an example of a verbal expression. Number Two, explain the meaning of the term “numerical expression.” Give an example of a numerical expression. And Number Three, in a numerical expression with several operations, in what order do you complete the operations when evaluating the expression?
Begin by completing the warm-up for this lesson on matching words with their operations, to prepare for the lesson on writing and evaluating numerical expressions with order of operations.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Marvin’s grandmother is going to give him some money. Marvin can choose to get one hundred dollars from his grandmother, or he can choose to take two dollars that his grandmother will double every day for ten days. If he takes the second offer, he will have to pay her back ten dollars at the end of ten days. Which offer will give Marvin more money?
In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, which offer will give Marvin more money?
Now that we have identified the question we need to put this question in our own words in the form of a statement. This problem is asking to me to find the offer that gives Marvin more money.
During this lesson we will learn how to write and evaluate numerical expressions with order of operations in order to complete the SOLVE problem at the end of the lesson.
VERBAL AND NUMERICAL EXPRESSIONS
Read the following expression: four plus five minus two
What do you notice about the expression? It has words; has addition and subtraction.
What do you call an expression that has just words? A verbal expression
Why do we call this a verbal expression? It uses only words.
Read the following expression: four plus five minus two
What do you notice about the expression in Question Four? Has no words, has addition and subtraction
What do you can an expression that has only numbers? A numerical expression
Why do we call this a numerical expression? It has only numbers and operation symbols
If we look at the two expressions, they have the same meaning, but are written in different languages. In math we translate between verbal and numerical expressions.
Write a definition for Verbal Expression. The definition is an expression with words only.
Write a definition for Numerical Expression. Definition: An expression with numbers and symbols only.
The quotient of twenty-two and eleven, increased by five, decreased by seven
What does the word “quotient” mean? Division
Why is the division problem of twenty-two and eleven written as a fraction? The fraction bar is one way to represent division.
Complete the rest of the table.
REVIEW WITH ORDER OF OPERATIONS
In the past, you have learned that there is an order for solving numerical expressions with more than one operation. What is this order?
What is the first operation that should be done to evaluate the expression? Operations in parentheses.
What comes next in the order of operations? Multiplication and division left to right
What comes last in the order of operations? Addition and subtraction left to right
The first step is parentheses. ( )
What does the parentheses step really mean? Complete any operations that are inside the grouping symbols whether they are parentheses, brackets, or braces.
You may have heard of the term PEMDAS (Please Excuse My Dear Aunt Sally.) The P is for Parentheses but we have to remember that we can use grouping symbols such as braces or brackets as well.
We will be using GEMDAS, where the G stands for grouping symbols, not just parentheses.
What is the next step? We complete all multiplication and division from left to right.
What are the two most common ways we may see division represented in order of operations? With a division sign or with a fraction bar.
If I have a problem that has twenty-five divided by five times two, which operation is performed first? Why? Division, it is on the left.
If I have a problem that has eight times two divided by four, which operation is performed first? Why? Multiplication, it is on the left.
What is the last step? We complete all addition and subtraction from left to right.
If I have a problem that has six plus two minus three, which operation is performed first? Why? Addition, it is on the left.
If I have a problem that has fifteen minus two plus nine, which operation is performed first? Why? Subtraction, it is on the left.
Five times (six) minus the sum [eight plus three]
In Problem Five, which operation is performed first? Eight plus three
Why is eight plus three performed first? It is grouped with brackets.
Now we have the problem five times (six) minus {eleven}
What operation is next? Multiplication
Five times six is equal to thirty. And now our problem is thirty minus {eleven}
What operation is next? Subtraction
What is the value of the expression? Nineteen
Let’s look at Problem Six. Eight divided by two minus one plus five times three
In Problem Six, which operation is performed first? Eight divided by two
Why is eight divided by two performed first? Multiplication and division go left to right.
Now our problem is four minus one plus five times three.
Which operation is next? Multiplication
Now our problem is four minus one plus fifteen
What operation is next? Subtraction
Three plus fifteen
What operation is last? Addition
What is the value of the expression? Eighteen
The difference of [twenty-nine minus the sum of (four plus seven)] times two
How is Problem Seven different from Problems Five and Six? Multiple grouping symbols.
Which operation is performed first? Four plus seven
Now our problem is [twenty-nine minus (eleven)] times two
Why is four plus seven performed first? It is the innermost grouping.
Which operation is next? Subtraction. Twenty-nine minus eleven
What operation is next? Multiplication
What is the value of the expression? Thirty-six
EXPONENTS
These will be the words that will help to answer the questions in the graphic organizer.
Let’s look at the following expression?
How would you use the term “base?” Baseball? Bottom of a platform?
In the expression two cubed or two to the third, what term might we use to describe the two? Base
What do you notice about the three in the expression above? It is to the right of the base. It is smaller in size and is written near the top of the base.
When we use the word “smaller” we are referring to the physical size, not the magnitude of, the number.
What term can we use to identify the three? Exponent
How might we read this expression? Two raised to the third power or two cubed.
What connection is there between the use of the word “cubed” for the exponent of three to the shape of a cube? A cube has three dimensions – length, width, and height.
How can we write the expression of two cubed using two as the factor? Two times two times two
What is the value of two times two times two? Eight
What is the value of two cubed? Eight
How can we write the expression of four to the fifty power using four as a factor? Four times four times four times four times four
What is the value of four times four times four times four times four? One thousand twenty-four
What is the value of four to the fifth power? One thousand twenty-four
Conclusion:
A numerical expression with a base and an exponent can be written as a multiplication problem where the base is the factor and the exponent tells us how many of the factors to multiply. When we write an expression with an exponent as the product of the factors we expand the expression and call this expanded form.
ORDER OF OPERATIONS WITH EXPONENTS
Let’s look back at the order of operations. Up to this point, we have only worked with grouping symbols and the four basic operations, but sometimes we have exponents in our expression that we need to simplify. Parentheses or other grouping symbols, Exponents, Multiplication and Division from Left to Right, Addition and Subtraction from Left to Right.
What does the E in GEMDAS stand for? Exponents
If there are exponents in a numerical expression, when should we find their value? After we compute within the grouping symbols, and before multiplication and division.
Sixty-four divided by two to the fourth power plus one
What should be done first to simplify this numerical expression? Simplify the number with the exponent. Explain. There are no grouping symbols so we compute the value with the exponent before the rest of the computation.
What is the value of the number with the exponent? Sixteen
Now we have the problems sixty-four divided by sixteen plus one
What is the next step in finding the value? Division
Sixty-four divided by sixteen is equal to four.
Our problem is now four plus one.
What is the last step in finding the value? Addition
Our expression is five.
Twenty-five divided by (one half) squared plus the difference of (eight minus three)
What should be done first to evaluate this numerical expression and why? Subtraction, it is in parentheses.
What is the value when we subtract? Five
Our problem is now twenty-five divided by (one half) squared plus (five)
What is the next step in finding the value? The exponent
Find the value of the exponent after the step. One half squared is one fourth.
What is the next step in finding the value? Division
Twenty-five divided by one fourth is one hundred.
Our problem is now one hundred plus (five)
What is the last step in finding the value? Addition
Our expression simplifies to one hundred five.
When simplifying numerical expressions we follow these guidelines:
First we complete any computation within grouping symbols.
Second, we complete any computation of values with exponents.
Third, we complete all multiplication and division left to right.
Last, we complete all addition and subtract left to right.
FOLDABLE ON WRITING AND EVALUATING EXPRESSIONS
Place a sheet of paper on your desk. The paper should be horizontal (wider across).
Fold the top of the paper down to almost the bottom of the page. Leave about half of an inch at the bottom. Crease the fold at the top.
Fold the paper in half from left to right and crease the fold.
Fold the paper in half again from left to right and crease the fold.
Now open the paper back up to see that you have created four sections.
Left the top flap up and cut to the fold on the three creases to create four flaps.
Label the first flap with “Write Expressions.” On the inside complete the section with the given information.
Label the second flap with “Order of Operations.” On the inside complete the section with the given information. On the inside complete the section with the given information.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson.
The question was, Marvin’s grandmother is going to give him some money. Marvin can choose to get one hundred dollars from his grandmother, or he can choose to take two dollars that his grandmother will double every day for ten days. If he takes the second offer, he will have to pay her back ten dollars at the end of the ten days. Which offer will give Marvin more money?
In Step S, we Study the Problem. Underline the question and put the question in our own words in the form of a statement. This problem is asking me to find the offer that gives Marvin more money.
In Step O, we Organize the Facts. We begin by identifying the facts. We go back to the original problem and read it again. We place a strike mark or a vertical line at the end of each fact. Marvin’s grandmother is going to give him some money./ Marvin can choose to get one hundred dollars from his grandmother,/ or he can choose to take two dollars that his grandmother will double every day for ten days./ If he takes the second offer, he will have to pay her back ten dollars at the end of ten days./
After we identify the facts, we eliminate any unnecessary facts. We can cross out the first sentence because that fact does not contain any information that will help us determine which offer will give Marvin more money.
After we eliminate the unnecessary facts we list the necessary facts. There are two offers that Marvin has. The first is one hundred dollars or two dollars doubled every day for ten days, minus ten dollars.
In the L Step, Line Up a Plan. Write in words what your plan of action will be. Set up and evaluate a numerical expression to find out how much money the second offer will be so that I can compare it to the first offer. I will need to use exponents.
Choose an operation or operations. Multiplication and subtraction
V, Verify Your Plan with Action. Estimate your answer. I think the second offer will be better, because I know an exponent will increase the amount of money pretty fast.
Carry out your plan. The first offer gives Marvin a hundred dollars. The second offer we have two to the tenth power minus ten which is equal to one thousand twenty-four minus ten which gives Marvin one thousand fourteen dollars.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, I found which of the offers is more money.
Is your answer reasonable? Compare your answer to the estimate. Yes, because the second offer is more money.
Is your answer accurate? Check your work. Yes.
Write your answer in a complete sentence. Marvin should take his grandmother’s second offer.
CLOSURE
Now let’s go back and discuss the essential questions from the beginning of the lesson.
Our first question was, explain the meaning of the term “verbal expression.” Give an example of a verbal expression. A verbal expression has only words. An example is – the product of ten and twelve added to seven.
Number Two, explain the meaning of the term “numerical expression.” Give an example of a numerical expression. A numerical expression has only numbers and symbols. An example is, six plus eighteen divided by two cubed.
Number Three, in a numerical expression with several operations, what order do you complete the operations in when evaluating the expression? First, complete any operations in parentheses or other grouping symbols, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.