Order of Operations. Unit Description. Susan Mercer

Transcription

1 Order of Operations Unit Description Susan Mercer

2 The NCTM standards contend that... learning mathematics without understanding has long been a common outcome of school mathematics instruction. (page 20 NCTM standards) In particular this is true for order of operations. Traditionally, textbooks have presented order of operations as a convention, a cluster of rules to be followed and memorized in order to get the right answer. For example, Mc Dougal Littell states, in the California Middle School Mathematics: Concepts and Skills Course 1 textbook, To make sure everyone gets the same result when an expression is evaluated, mathematicians have established order of operations. These rules do not provide understanding; nor do they build on students prior knowledge of numbers and operations. Students are expected to apply the rules without comprehending why they are doing what they are doing. Furthermore, the NCTM standards argue that Students who memorize facts of procedures without understanding often are not sure when or how to use what they know, and such learning is often quite fragile. (page 20 NCTM standards) Mathematical expressions are presented to students out of context, without any meaning attached to the numbers or operations. This leads to mistakes primarily because students tend to solve mathematical expression from left to right rather than looking at the problem as a whole. In my experience, students view mathematical expressions as abstract and meaningless and tend to decode them in the same way as they decode a sentence or paragraph, from left to right. Also, many teachers encourage students to use the acronym Please Excuse My Dear Aunt Sally (PEMDAS) to remember the order in which to evaluate mathematical expressions. Unfortunately, PEMDAS does not provide an explanation and students tend to solve the multiplication before the division rather than multiplication and division from left to right and addition before subtraction rather than addition and subtraction from left to right. The order of operation model described in this article requires students to represent mathematical expressions in three different ways: 1) verbally; 2) using a drawing; and 3) symbolically using addition and/or subtraction only. These representations help students see the mathematical expressions and help them evaluate them by finding the number of objects in the representation. By using this model students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically. (page 37 NCTM standards) Modeling order of operations using different representations helps students comprehend how to evaluate a mathematical expression by making sense of the parts of the expression rather than applying memorized rules. In addition, scaffolding methodologies are used throughout. The unit starts by reviewing what students already know and build on that knowledge. Each new concept or idea is introduced one at a time and layered upon the concepts already presented. S. Mercer - Order of Operations Description page - 2

3 This model was developed on addition and multiplication seeing that students learned these two concepts in the primary grades. Addition is the operation of combining two or more elements or groups of elements and multiplication denoting successive additions. For example: three times seven is the same as three groups of seven and this is the same as Further, exponents can be expressed as successive multiplications and multiplications can be expressed as successive additions. For example: three to the power of two is the same as three times three, which is the same as three groups of three which can be written as three plus three plus three. Mathematically: 32 = 3 3 = A graphic organizer is used to help students appreciate the different representations of a mathematical expression. The above example presents the graphic organizer. In the middle oval students write the mathematical expression. In the top left box, students record the expression in words as they read it from left to right. In the top right box, students represent the expression using individual objects and/or groups of objects. In the bottom left box, students use the pictorial representation to rewrite the mathematical expression using only addition and/or subtraction. In the final box, students write the answer after evaluating the mathematical expression. When using this model, it is very important not to provide students with the order of operation rules as they are stated in the textbook. The hierarchies between the different operations need to be discovered and generalized by the students as they move through the unit. The teacher is a key element in guiding and facilitating student discussion in order for them to make the needed connections and generalizations. It is crucial for the teacher to model how to complete the graphic organizer. The teacher should model each example by asking probing questions and recording the students answers on the graphic organizer. S. Mercer - Order of Operations Description page - 3

4 The teacher introduces the graphic organizer by reviewing multiplication with a simple problem such as three times seven. The above example shows the completed graphic organizer for the problem three times seven. In the top left box, students represent the problem using words three groups of seven ; in the top right box students draw three circles to represent three groups and inside each group they draw seven objects; in the bottom left box the students represent the multiplication problem as successive additions, in this case seven plus seven plus seven. Students might not remember or know the concept of multiplication as successive additions therefore it is important for them to complete the bottom left box by observing their picture. Students need to know that to evaluate an expression means, in this context, to find out how many objects they have in total. This is easily accomplished by looking at the picture. The bottom right box shows the total number of objects. If students are familiar with the representation of multiplication as groups of and successive additions, then they should evaluate one or two more multiplication problems. If this is a new concept for them then more problems are necessary. Next, an addition operation is added to the multiplication problem. It is crucial that the first example shows the addition first when reading the problem from left to right because in this way students do not make the erroneous assumption that expressions are evaluated as they are read from left to right. It is very important for teachers to model on the board or overhead how to complete the graphic organizer and model the different expressions, especially when a new component or level of difficulty is added to the previous problems. The above example displays a complete graphic organizer for the expression five plus two times three. It is essential for the teacher to help students make connections between the different representations and the idea that even though the five is first when the expression is read from left to right, mathematically the number of objects in the groups is evaluated first. S. Mercer - Order of Operations Description page - 4

5 three plus two groups of eight plus two Emphasis should be placed on the number of objects. to solve this problems? with addition only two groups of six plus four groups of four Emphasis should be placed on the number of objects with addition only 28 to solve this problems? The above examples demonstrate the completed graphic organizer for different expressions that include addition and multiplication. At this point it is important to challenge the students to find shortcuts. For example, some students may ask can t we just write the number, instead of drawing all the pictures. Or they may suggest, writing directly twelve instead of or writing 16 instead of writing It is crucial for the teacher to challenge students to explain why they are able to write their shortcuts and where in the different boxes or representations they can find the short cut. For example: is two groups of six in the verbal representation, it is 2 6 in the expression box, it is the two groups of six in the drawing and it is in the expression with addition only box. It is key to have students note that they are not really solving the problem from left to right when using their shortcut but rather looking at the problem as a whole and finding the groups to find out how many objects are in each group. It is critical for the teacher to constantly ask probing questions such as: How many groups do you have? How many objects are in each group? What part of the expression represents the number of groups? What part of the expression represents the number of objects in each group? What is outside of the groups? Are all the groups the same? Why? Why not? S. Mercer - Order of Operations Description page - 5

6 Once students are able to represent and evaluate expressions that involve addition and multiplication, parenthesis is incorporated into the expression. This is a new type of problem and the teacher needs to model how to solve the first expression with the students, while asking probing questions such as: How is this problem the same as the ones we did before? How is it different? It is very important for students to make connections with what they learned previously and realize that the logic that they applied before is still valid with the new concept. Different shapes or colors can be used to represent what is inside each group but it is important that students focus on the total number of objects in each group and not their shape or color. Also, the teacher needs to ask students: What is inside the group? How many groups are there? 4 (2 + 3) The above examples display the graphic organizer completed for two problems that include parenthesis. Next, students solve different problems that include addition, multiplication and parenthesis. S. Mercer - Order of Operations Description page - 6

7 Emphasis should be placed on the number of objects. three groups of two groups of four to solve this problems? with addition only 24 Subsequently, students apply what they have learned to problems involving successive multiplications. The above example presents the expression three times two times four. The picture represents three groups, each one containing two groups of four elements. During the discussion time students are able to describe shortcuts such as six groups of four or three groups of eight. This is an appropriate time to discuss the commutative and associative properties. Next, students are presented with expressions including exponents. Students are introduced to this concept with the expression three to the power of two. three to the power of two is three times three, which is represented as three groups of three. Using an expression with addition only, three to the power of two is three plus three plus three. The answer is nine. After modeling one or two examples with exponents only, exponents and addition is presented to students. The graphic organizer below shows a completed graphic organizer for the expressions using addition and exponents Once students are familiar with the different types of expressions that involve addition, multiplication, parenthesis and power of two, subtraction is introduced to the model. S. Mercer - Order of Operations Description page - 7

8 ten minus nine 10-9 Problem 10 9 with addition and/or subtraction only 1 First students need to review the concept of subtraction as taking away. A simple expression such as 10 minus 9 can be used as review. Students complete the graphic organizer for subtraction. It is critical for the teacher to show students what notation will be used to represent take away in order for all students to use the same visual aid. Next, the teacher presents an expression such as students write ten minus three groups of two. When representing this expression students need to draw ten objects and then take away three groups of two. Mathematically using addition and/or subtraction students need to record Ten represents the starting number of objects and two is the number of objects the students need to take away three times in order to represent the expression accurately. take away four groups of four from six groups of three to solve this problems? with addition and/or subtraction only. S. Mercer - Order of Operations Description page - 8

9 four groups of nine minus five 4 ( 9-5) with addition and/or subtraction only. The above example shows a completed graphic organizer that combines subtraction and multiplication and the above example llustrates an expression that combines parenthesis and subtraction. It is imperative to introduce each type of situation individually and model it for the students. In this way students are able to see how to represent each expression and can conceptually understand what is being done to solve each expression. Due to space constrains not all the different examples can be presented. Sabrina has six boxes of crayons, with eight crayons in each box. She uses ten crayons. How many crayons does she have left? crayons with addition and/or subtraction only. The graphic organizer can also be used to solve word problems. Using this model helps students write expressions given a word problem. The above example illustrates a word problem where students are asked not only to solve the problem but also to write a mathematical expression to represent the problem. Representing problems visually helps students see the groups and the objects that are added that are not in groups. S. Mercer - Order of Operations Description page - 9

10 Once students are familiar with the different types of problems and their representations, it is important for students to work in groups to discuss, draw conclusions and summarize in what order expression are evaluated. To help this process students are asked to complete a summary table that has three columns. Evaluating s with the Evaluating s without the s ten plus three groups of four = ( 2 + 1) five plus three groups of two plus one = ( 2 + 1) Order the expression was evaluated 1) multiplication 2) addition 1) parenthesis 2) multiplication 3) addition The above table illustrates this process. In the first column students evaluate an expression using a picture. Next, they are asked: How would you evaluate this expression without the representation? It has been my experience that many students when asked this question revert to looking at the problem from left to right. But as soon as they realize that the answer they get from evaluating the expression from left to right does not coincide with the answer they arrived at by drawing the picture they self-correct. The graphic organizer provides students with immediate feedback; they can check their answers by looking at the picture. Lastly students are asked: While evaluating the expression without the picture, what operation did you do first? Why? What did you do next? Students record in the last column the order they evaluated the expression. After several examples, a pattern emerges that enables students to generalize the order of operation rules. These generalizations can then be extended to expressions that are difficult to represent with a picture such as expressions with fractions, decimals, and division. At this point it is necessary to review with the students the importance of evaluating an expression from left to right if it includes only addition and subtraction. This can be done by writing the following expression on the board: and asking students to represent it using words and a picture. Students will write ten minus four plus three and will represent the expression by drawing ten objects, taking away four and finally adding three leaving nine objects. Next, ask students: Do you get the same answer if you add four and three and then subtract the answer from ten? Why not? Students quickly realize that if they add first they are subtracting seven from ten instead of subtracting four then adding three. Using several of these examples can be S. Mercer - Order of Operations Description page - 10

11 used to reinforce and emphasize the importance of evaluating an expression from left to right if it only includes addition and subtraction. After students are familiar with these types of problems, the same type of activity can be used with multiplication and division. Probing questions such as: How can you explain to a friend how to solve a problem if you cannot draw a picture? Do you always solve problems from left to right? Is there an order in which problems are solved? Is there a hierarchy? Students should be able to articulate the conclusion they have found regarding order of operations. In my experience each student can arrive at two or three conclusions, such as all multiplications can be expressed as additions but not vice versa. Consequently a class discussion where all the conclusions are presented, recorded and summarized is critical. This dialogue allows students to thoroughly understand how to evaluate mathematical expressions. By the end of this unit students should be able to answer the following key questions: Why are addition and subtraction done last? What are parentheses used for in an Why were the rules of order of operations established? Why were they selected? Are the rules arbitrary or is there a reason for them? Extension Using the same reasoning process, this model can be used with variables to introduce the distributive property. The first expression should be 3a represented as three groups of a which is the same as a + a + a. The graphic organizer below shows how this model can be used to teach distributive property by changing only the last box from to without Parenthesis. It is crucial for the teacher to continually ask students: How many groups are there? What is in each group? What part of the expression is not part of the group? What part of the expression denotes the number of groups? What part of the expression indicates the number of objects in each group? five plus two groups of a plus three b plus four 5 b a b b 4 b a b b a + b + b + b a + b + b + b ( a + 3b + 4 ) 2a + 6b + 13 with addition only without parenthesis S. Mercer - Order of Operations Description page - 11

12 Conclusion This model is not magical. It requires the teacher to plan and prepare the examples carefully and to continually ask probing questions that help students make the connections between the mathematical expression and its representations. Furthermore, the teacher needs to assess the needs of his/her students and determine the pace and number of examples each type of problem requires before introducing the next level difficulty. Is the time and effort worthwhile? Absolutely! By using this unit students deepen their understanding of number and operation and improve their number sense S. Mercer - Order of Operations Description page - 12

13 Order of Operations Key Susan Mercer

14 Multiplication Emphasis should be placed on the number of objects. three groups of seven with addition only Emphasis should be placed on the number of objects. four groups of five with addition only S. Mercer - Order of Operations Key page - 1

15 Emphasis should be placed on the number of objects. three groups of three with addition only Emphasis should be placed on the number of objects. one group of nine with addition only S. Mercer - Order of Operations Key page - 2

16 Multiplication and Addition five plus two groups of three Emphasis should be placed on the number of objects with addition only nine plus three groups of four Emphasis should be placed on the number of objects with addition only S. Mercer - Order of Operations Key page - 3

17 Emphasis should be placed on the number of objects. seven groups of four plus two with addition only 30 Emphasis should be placed on the number of objects. ten plus six groups of two with addition only 22 S. Mercer - Order of Operations Key page - 4

18 three plus two groups of eight plus two Emphasis should be placed on the number of objects. with addition only Emphasis should be placed on the number of objects. two groups of six plus four groups of four with addition only 28 S. Mercer - Order of Operations Key page - 5

19 Emphasis should be placed on the number of objects. five plus two groups of four plus one with addition only four groups of one plus five groups of two Emphasis should be placed on the number of objects with addition only S. Mercer - Order of Operations Key page - 6

20 Emphasis should be placed on the number of objects. ten groups of four plus nine with addition only 49 Emphasis should be placed on the number of objects. five groups of two plus seven groups of two with addition only S. Mercer - Order of Operations Key page - 7

21 Multiplication, Addition and Parenthesis Emphasis should be placed on the number of objects. two groups of three plus one 2 ( 3 + 1) with addition only 8 Emphasis should be placed on the number of objects. four groups of two plus three 4 ( 2 + 3) with addition only S. Mercer - Order of Operations Key page - 8

22 Emphasis should be placed on the number of objects. three groups of three plus two plus six with addition only 3 ( 3 + 2) problems? six plus four groups of one plus one Emphasis should be placed on the number of objects (1 + 1) with addition only 14 S. Mercer - Order of Operations Key page - 9

23 four groups of two plus two groups of six plus two Emphasis should be placed on the number of objects ( 6 + 2) with addition only 24 problems? seven groups of four plus one plus four groups of three with addition only 7 (4 +1) problems? S. Mercer - Order of Operations Key page - 10

24 Emphasis should be placed on the number of objects. three groups of two groups of four problems? with addition only 24 six groups of two groups of three Emphasis should be placed on the number of objects with addition only problems? S. Mercer - Order of Operations Key page - 11

25 three groups of three groups of three problems? with addition only 27 four plus three groups of two groups of five with addition only 34 problems? S. Mercer - Order of Operations Key page - 12

26 five times five therefore five groups of five with addition only 25 three times three therefore three groups of three with addition only 9 S. Mercer - Order of Operations Key page - 13

27 four groups of four plus ten problems? with addition only 26 three groups of three plus three groups of five with addition only 24 problems? S. Mercer - Order of Operations Key page - 14

28 one plus six groups of six plus two groups of three with addition only 43 problems? two groups of two plus three groups of three plus one ( 3 + 1) with addition only 16 problems? S. Mercer - Order of Operations Key page - 15

29 three plus six groups of six plus three groups of three with addition only problems? nine groups of nine plus five groups of four plus seven with addition only ( 4 + 7) problems? S. Mercer - Order of Operations Key page - 16

30 Sabrina has five nickels and ten dimes. How much money does Sabrina have? with addition only 125 cents or $1.25 problems? Sabrina has ten nickels, five quarters and three pennies. How much money does Sabrina have? with addition only problems? S. Mercer - Order of Operations Key page - 17

31 Sabrina has seven eggs in her refrigerator. She buys three more dozens eggs. How many eggs does she have? problems? with addition only 43 eggs Sabrina has three boxes of crayons with eight crayons in each box. She buys six more boxes with ten crayons in each box. How many crayons does she have? 8 crayons 8 crayons 8 crayons problems? with addition only 64 crayons S. Mercer - Order of Operations Key page - 18

32 Subtraction take away five from twelve twelve minus five 12-5 Expressi on with addition and/or subtraction only take away nine from ten ten minus nine 10-9 Expressi on 10 9 with addition and/or subtraction only 1 S. Mercer - Order of Operations Key page - 19

33 take away five from two groups of three with addition and/or subtraction only. 1 take away two groups of four from ten with addition and/or subtraction only. 2 S. Mercer - Order of Operations Key page - 20

34 take away five groups of two from seven groups of two problems? with addition and/or subtraction only. take away four groups of four from six groups of three problems? with addition and/or subtraction only. S. Mercer - Order of Operations Key page - 21

35 two groups of seven minus three 2 ( 7-3) with addition and/or subtraction only. four groups of nine minus five 4 ( 9-5) with addition and/or subtraction only. S. Mercer - Order of Operations Key page - 22

36 Sabrina has six dimes and ten quarters. She spends fifty cents. How much money does Sabrina have left? with addition and/or subtraction only. 260 cents or $2.60 Sabrina has ten nickels, one quarter and ten dimes. She spends 75 cents. How much money does Sabrina have left? cents or $1.00 with addition and/or subtraction only. S. Mercer - Order of Operations Key page - 23

37 Sabrina has six boxes of crayons, with eight crayons in each box. She uses ten crayons. How many crayons does she have left? crayons with addition and/or subtraction only. Sabrina bought two dozen eggs and uses eight to do an omelette. How many eggs does she have left? eggs with addition and/or subtraction only. S. Mercer - Order of Operations Key page - 24

38 Sabrina has 10 cans of soda. She buys seven 6-packs of soda. She drinks two sodas. How many sodas does she have left? 6 cans 6 cans 6 cans 6 cans 6 cans 6 cans cans cans with addition and/or subtraction only. Sabrina has five dozen pens. She gives away two dozen to her friends. How many pens does she have left? 12 pens 12 pens 12 pens 12 pens 12 pens pens with addition and/or subtraction only. S. Mercer - Order of Operations Key page - 25

39 Order of Operations Name: Period: Date:

40 Multiplication 3 7 with addition only 4 5 with addition only S. Mercer - Order of Operations page - 1

41 3 3 with addition only 1 9 with addition only S. Mercer - Order of Operations page - 2

42 Multiplication and Addition with addition only with addition only S. Mercer - Order of Operations page - 3

43 with addition only with addition only S. Mercer - Order of Operations page - 4

44 with addition only with addition only S. Mercer - Order of Operations page - 5

45 with addition only with addition only S. Mercer - Order of Operations page - 6

46 with addition only with addition only S. Mercer - Order of Operations page - 7

47 Multiplication, Addition and Parenthesis 2 ( 3 + 1) with addition only 4 ( 2 + 3) with addition only S. Mercer - Order of Operations page - 8

48 3 ( 3 + 2) + 6 with addition only (1 + 1) with addition only S. Mercer - Order of Operations page - 9

49 ( 6 + 2) with addition only 7 (4 + 1) with addition only S. Mercer - Order of Operations page - 10

50 3 2 4 with addition only with addition only S. Mercer - Order of Operations page - 11

51 3 3 3 with addition only with addition only S. Mercer - Order of Operations page - 12

52 5 2 with addition only 3 2 with addition only S. Mercer - Order of Operations page - 13

53 with addition only with addition only S. Mercer - Order of Operations page - 14

54 with addition only ( 3 + 1) with addition only S. Mercer - Order of Operations page - 15

55 with addition only ( 4 + 7) with addition only S. Mercer - Order of Operations page - 16

56 Sabrina has five nickels and ten dimes. How much money does Sabrina have? with addition only Sabrina has ten nickels, five quarters and three pennies. How much money does Sabrina have? with addition only S. Mercer - Order of Operations page - 17

57 Sabrina has seven eggs in her refrigerator. She buys three more dozens eggs. How many eggs does she have? with addition only Sabrina has three boxes of crayons with eight crayons in each box. She buys six more boxes with ten crayons in each box. How many crayons does she have? with addition only S. Mercer - Order of Operations page - 18

58 Subtraction 12-5 with addition and/or subtraction only 10-9 with addition and/or subtraction only S. Mercer - Order of Operations page - 19

59 2 3-5 with addition and/or subtraction only with addition and/or subtraction only. S. Mercer - Order of Operations page - 20

60 with addition and/or subtraction only with addition and/or subtraction only. S. Mercer - Order of Operations page - 21

61 2 ( 7-3) with addition and/or subtraction only. 4 ( 9-5) with addition and/or subtraction only. S. Mercer - Order of Operations page - 22

62 Sabrina has six dimes and ten quarters. She spends fifty cents. How much money does Sabrina have left? with addition and/or subtraction only. Sabrina has ten nickels, one quarter and ten dimes. She spends 75 cents. How much money does Sabrina have left? with addition and/or subtraction only. S. Mercer - Order of Operations page - 23

63 Sabrina has six boxes of crayons, with eight crayons in each box. She uses ten crayons. How many crayons does she have left? with addition and/or subtraction only. Sabrina bought two dozen eggs and uses eight to do an omelette. How many eggs does she have left? with addition and/or subtraction only. S. Mercer - Order of Operations page - 24

64 Sabrina has 10 cans of soda. She buys seven 6-packs of soda. She drinks two sodas. How many sodas does she have left? with addition and/or subtraction only. Sabrina has five dozen pens. She gives away two dozen to her friends. How many pens does she have left? with addition and/or subtraction only. S. Mercer - Order of Operations page - 25