Can you divide before multiplying

Ask : What is the value of this expression? Walk students through evaluating the expression. Ask : What happens if I switch the addition and multiplication symbols? What value would I get? Ask : Did we get different values when we changed the operations? This result will probably not surprise your students. They most likely know that performing different operations on the same numbers will give different values.

If time permits and students are ready, challenge them to find an expression where switching the addition and multiplication symbols like you did results in the same value. If any students succeed, have them show how they derived the expressions. Note that it is only possible when the middle number is 1 e. How do you think I could do that? Draw attention to the parentheses. Say : We call these symbols parentheses.

If there are parentheses in an expression, do whatever is inside the parentheses first. Say : Now, let's finish calculating the value. Is that the same value we got before? Help students notice that the value isn't the same as either the original expression or the expression with the operation symbols switched.

Developing the Concept: Order of Operations Materials: Whiteboard or way to write for the class publicly Prerequisite Skills and Concepts: Students should be familiar with order of operations and feel prepared to practice it. Make sure students understand clearly that the order of operations requires them to perform multiplication before addition. Ask : What happens if I want to add 3 and 5 before I multiply by 8? Allow students to discuss ideas of how to override the order of operations.

Do not tell students that they are right and wrong. Instead, encourage mathematical discourse and compare differing opinions in order to correct misconceptions. Note that there are many possible answers! For example, the problem could explicitly say "add 3 and 5 first," or historically, there have been other ways of grouping, such as using horizontal bars over the expression. If they don't mention parentheses, remind them of what you did in the first lesson.

Today we're going to practice finding the value of expressions with and without parentheses and see what difference the parentheses make. Write the following three expressions publicly for all students to see. Allow time for students to finish calculating. Then have student volunteers report what they found. Ask : Did you get the same value for all three expressions?

Why or why not? Students should notice that expressions 1 and 3 yield the same value while expression 2 is different. Discuss that expression 2 requires that we add before multiplying while expressions 1 and 3 have us multiply before adding. The goal is for students to see that the use of parentheses sometimes changes the value of an expression and sometimes doesn't.

Write the following two expressions publicly for all students to see. In the example that follows, the parentheses are not a grouping symbol; they are a multiplication symbol. In this case, since the problem only has multiplication and division, we compute from left to right. Be careful to determine what parentheses mean in any given problem. Are they a grouping symbol or a multiplication sign? This expression has multiplication and division only. The multiplication operation can be shown with a dot.

Since this expression has only division and multiplication, compute from left to right. The parentheses still mean multiplication; the additional braces are a grouping symbol. According to the order of operations, compute what is inside the braces first. Notice that the braces caused the answer to change from 1 to 4. Compute the addition in parentheses first. Then, perform division. Finally, add and subtract from left to right. The correct answer is Finally, with only subtraction and addition left, add and subtract from left to right.

The Order of Operations. Grouping symbols include. So far, our rules allow us to simplify expressions that have multiplication, division, addition, subtraction or grouping symbols in them. What happens if a problem has exponents or square roots in it?

We need to expand our order of operation rules to include exponents and square roots. If the expression has exponents or square roots, they are to be performed a fter parentheses and other grouping symbols have been simplified and before any multiplication, division, subtraction and addition that are outside the parentheses or other grouping symbols. Note that you compute from more complex operations to more basic operations. Addition and subtraction are the most basic of the operations.

You probably learned these first. Multiplication and division, often thought of as repeated addition and subtraction, are more complex and come before addition and subtraction in the order of operations.

Some examples that show the order of operations involving exponents and square roots are shown below. This problem has addition, division, and exponents in it. Use the order of operations. Perform division before addition. This problem has exponents and multiplication in it. Simplify 3 2 and 2 3. This problem has parentheses, exponents, and multiplication in it. To multiply terms, multiply the coefficients and add the exponents on each variable.

The number of terms in the product will be equal to the product of the number of terms. Of course, there may well be like terms which you will need to combine. Long Division of Polynomials. The polynomial you are dividing by is called the divisor. The polynomial you are dividing it into is called the dividend.

The answer is called the quotient. The difference between the quotient times the divisor and the dividend is called the remainder. Divide the first term in the dividend into the first term in the divisor. This gives you the first term in the quotient.

Multiply this term in the quotient by the divisor and subtract this product from the dividend. Repeat the process with the remainder until you have a remainder whose degree is smaller than the degree of the divisor. Definition: A fraction or rational expression is the answer to a division problem of polynomials.

The Fundamental Fact of Fractions. If you multiply or divide the top and bottom of a fraction by the same thing, you get a different name for the same number. Reducing or Simplifying Fractions. Factor the top and bottom until you get factors that cannot be factored further. If you find the same factor on both the top and bottom, you can cancel them. If after factoring the top and bottom as much as possible, if there are no common factors in the top and bottom, the fraction is reduced to lowest terms.

Adding Subtracting Fractions. If you have common denominators, add subtract the numerators. If not, find common denominators. To find common denominators, factor all the denominators and fill in the missing factors. You can multiply the bottom by whatever you want so long as you multiply the top by the same thing.

Multiply the tops and multiply the bottoms. You can cancel either before or after you multiply. Invert the divisor and multiply. You can make any change you want on one side of an equation so long as you make the same change on the other side.

One of the most common techniques is to get rid of a term on one side by subtracting it from both sides. When you get rid of a term on one side, it pops up on the other side with its sign changed. Moving a term from one side to the other and changing its sign is called transposing the term. If you move a factor from one side to the other, move it across the fraction bar. Steps in solving first degree equations. Clear Denominators: Multiply both sides by a common denominator.

Simplify: Remove parentheses and combine like terms. Transpose known terms to one side and unknown terms to the other. Divide both sides by the coefficient of the unknown. Steps in solving quadratic equations by factoring. Steps in solving quadratic equations by completing the square. Steps in solving quadratic equations using the quadratic formula.