In this lesson, we will use the order of operations to evaluate and simplify expressions that contain numbers and variables.

Transcription

1 Show Me: Expressions M8081 Could we sit in a classroom on the other side of the world and still make sense of the mathematics? The answer is yes! Of course, we might not understand exactly what the teacher is saying, but we would recognize some of the math conventions being used. Mathematicians around the world have adopted certain mathematical conventions. For example, it s important that we all use the same order of operations when evaluating expressions. In this lesson, we will use the order of operations to evaluate and simplify expressions that contain numbers and variables. A variable expression is a combination of math symbols that includes numbers, variables, and operation signs. Here is an example of a mathematical expression. This special example is called a numerical expression, because it contains only operation signs and numbers. Let s recall the order of operations. There are four steps. First, simplify expressions inside parentheses. Second, evaluate exponents or radicals. Third, multiply and divide, in order, from left to right. Finally, add and subtract, in order, from left to right. We can simplify this numerical expression by following the order of operations.

2 First, we will simplify the expression inside the parentheses. Thirteen minus three equals ten. Next, we will multiply and divide, in order, from left to right. Six times ten equals sixty. Sixty divided by twelve is five, our answer. Now, try to simplify this expression. What is the value of the expression two times the quantity ten minus four plus five? Expressions are not always numerical only. Some expressions may contain variables. In the expression five X minus Y, the variables are X and Y. To evaluate this expression, we must substitute the given values of X and Y into the expression. Let s see how this works. First, let s substitute negative four for X in the expression. Then, let s substitute two for Y. Now that the expression is only numerical, we can evaluate it. We can evaluate the expression, as we did before, using the order of operations. We will start by multiplying from left to right. Five times negative four equals negative twenty. Then, we subtract to get negative twenty-two. Thus, the value of the expression is negative twenty-two. Try this one. Evaluate the expression five V minus six W when V equals four and W equals two. So far, all of the expressions we ve evaluated used positive and negative integers numbers such as six and negative three. But, expressions can also contain decimals or fractions. Fortunately, we can evaluate these expressions in a similar way. Look at this expression. It s slightly different from the previous expressions because it involves a decimal number five-tenths.

3 Let s evaluate the expression. First, we will substitute the values of the variables P and Q into the expression. Then, we will follow the order of operations. We simplify the expression in the parentheses first. Twenty-six minus ten equals sixteen. Next, we multiply the terms in the numerator. Five-tenths times sixteen equals eight. Finally, we divide eight by four and get the answer, two. Try this one. What is the value of the expression one point six times the quantity X minus Y divided by two when X equals ten and Y equals five? Expressions can contain fractions, too. Let s find the value of this expression: M divided by twelve plus the quantity three-fourths minus N when M equals two and N equals fiveeighths. Let s start by substituting two for M and five-eighths for N in the expression. Now we can simplify using the order of operations. First, we will subtract the terms in the parentheses. To subtract, we rewrite the fractions so they share a common denominator. Here, we can use eight as the common denominator. Six-eighths minus five-eighths equals one-eighth. Next, we add two-twelfths and oneeighth. Again, we rewrite the fractions so they share a common denominator. This time, we can use twenty-four as the common denominator. The expression becomes four twenty-fourths plus three twenty-fourths. The sum is seven twenty-fourths. So, this expression equals seven twenty-fourths when M equals two and N equals five-eighths.

4 Here s another example that contains a fraction. What is the value of the expression three-fourths times the quantity X minus twelve plus Y when X equals two-thirds and Y equals one-sixth? Click Solution to see one way to simplify the expression. We start by substituting two-thirds for X and one-sixth for Y in the expression. Now, we can simplify using the order of operations. First, we will subtract the terms in parentheses. To subtract, we rewrite the fractions so they share a common denominator. Here, we can use three as the common denominator. Two-thirds minus thirty-six thirds equals negative thirty-four thirds. Next, we will multiply three-fourths and negative thirty-four thirds. To make this multiplication easier, we can divide three out of the numerator and three out of the denominator. Also, we can factor two from negative thirty-four in the numerator and two from four in the denominator. The result is the fraction negative seventeen-halves. To add negative seventeen-halves and one-sixth, we, once again, rewrite the fractions so they share a common denominator. This time we can use six as the common denominator. The expression becomes negative fifty-one sixths plus one-sixth. The sum is negative fifty-sixths, which simplifies to negative twenty-five thirds. So, this expression equals negative twenty-five thirds when X equals two-thirds and Y equals one-sixth. Now, try evaluating an expression with fractions. Enter the value of the expression P times the quantity one-half plus Q minus three when P equals four and Q equals threefourths. Here is a summary of the important ideas in this lesson.

5 A variable expression is a combination of math symbols that includes numbers, variables, and operation signs. Expressions that contain only operation signs and numbers are called numerical expressions. A numerical expression does not have variables. We can evaluate and simplify numerical expressions using the order of operations. To evaluate a variable expression, change it to a numerical expression by substituting the given values for the variables. Then, simplify using the order of operations. If you d like to review this activity again, click Review. If you re ready to exit, click Done.