Lesson 2 Homework Practice Numerical Expressions Worksheets

To help students learn about the need for the order of operations, introduce the A Lesson from Aunt Sally practice worksheet (M-5-6-1_A Lesson from Aunt Sally-Worksheet and KEY.docx). Distribute a copy of the A Lesson from Aunt Sally practice worksheet to all students.

“Jeremiah and his Aunt Sally are working on his math homework. Jeremiah has to find the value of these expressions:

  • 18 – 4 + 2
  • 24 + 8 ¸ 4
  • 15 – 2 × 3
  • 36 ÷ 2 × 9

“But, Jeremiah and his Aunt Sally always get different answers. ‘Why does Aunt Sally always get a different answer than I do?’ Jeremiah wondered. In today’s lesson, we will learn about the order of operations. This will help us understand why Jeremiah and Aunt Sally get different answers. First, Jeremiah needs your help. Work together in pairs to find the value of these expressions. Show your work in the ‘Try It’column. The second column will be used later.”

Observe students as they work in pairs. Do all students evaluate these expressions performing the operations strictly from left to right? This is what we would expect. Students read left to right, so they tend to also want to perform the operations from left to right. Yet, some students may first perform the operation they perceive to be the easiest. The table shows Jeremiah’s and Aunt Sally’s solutions. While you are observing, make note of any students who get these same answers. You may want to call on them to ask about the process they used when the class discussion of these solutions occurs later in the lesson.

After the pairs of students are finished evaluating these expressions, write a table on the board of Jeremiah’s and Aunt Sally’s solutions. Note, only write the values of the expressions Jeremiah and Aunt Sally computed, as noted in boldface. Do not write the processes shown in the table. They are provided for use in class discussion.

 

Jeremiah

Aunt Sally: Order of Operations

18 – 4 + 2 = 18 – 6 = 12

18 – 4 + 2 = 14 + 2 = 16

24 + 8 ÷ 4 = 32 ÷ 4 = 8

24 + 8 ÷ 4 = 24 + 2 = 26

15 – 2 × 3 = 13 × 3 = 39

15 – 2 × 3 = 15 – 6 = 9

36 ÷ 2 × 9 = 36 ÷ 18 = 2

36 ÷ 2 × 9 = 18 × 9 = 162

 

“Let’s continue to help Jeremiah. For each expression, work together in pairs again to understand how both Jeremiah and Aunt Sally got their solutions. To do so, follow these steps:

  • For each expression, decide if your solution is the same as Jeremiah’s, Aunt Sally’s, or neither.
  • Now, let’s try again. Try to understand the solution you did not have. For example, if you got Jeremiah’s solution, try to determine how Aunt Sally got her solution. If you didn’t get either of the solutions, try to determine how both Jeremiah and Aunt Sally got their solutions.
  • Record your work in the ‘Try It Again’ column. Be prepared to share your thinking with the class.”

Observe students as they work in pairs. The purpose is to help students understand that performing the computations in different orders results in different solutions. This activity will motivate the need for an order of operations that all agree on.

Write each expression on the board. Ask students to write the processes Jeremiah and Aunt Sally used for each expression on the board. Then ask students to explain the processes used by both Jeremiah and Aunt Sally.

Now ask students, “Why did Jeremiah and Aunt Sally get different solutions for each expression?” Students will likely mention that Jeremiah and Aunt Sally performed the operations in different orders.

Next, introduce students to the order of operations.

“Mathematicians agree on an order of operations. This is a specific order used so everyone is sure to get the same value. Today we will use the order of operations to decide whether Jeremiah or Aunt Sally got the correct solutions for each expression. Mathematicians say Parentheses (P)or grouping symbols first, Exponents (E) next, then Multiplication and Division (MD) from left to right, and finally Addition and Subtraction (AS) from left to right. Let’s use the order of operations to determine the correct answers for each of these expressions.”

Help students evaluate the expressions using the order of operations as shown here. [Note: As shown, it may be helpful to have students circle the operation they should perform at each step. Not all students will need to use circles, but it helps some students to focus on a single task during each step.]

Expression:      18 – 4 + 2 =    

Notice: There are no Parentheses or Exponents in this expression.There is also no Multiplication/ Division in this expression.

               [Perform Addition/Subtraction left to right]


     

 

Expression:      24 + 8 ÷ 4 =

Notice: There are no Parentheses or Exponents in this expression.

            [Perform Multiplication/Division left to right]

                       [Perform Addition/Subtraction left to right]

                          

 

Expression:      15 – 2 × 3 =

Notice: There are no Parentheses or Exponents in this expression.

                    [Perform Multiplication/Division left to right]

          [Perform Addition/Subtraction left to right]

                            

 

Expression:   36 ÷ 2 × 9

Notice: There are no Parentheses or Exponents in this expression.

                [Perform Multiplication/Division left to right]

                   

            

“Using the order of operations that mathematicians agree upon, we now know that Aunt Sally had the correct values for the expressions. But, how will we remember the order of operations? Notice, the acronym PEMDAS represents the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Many students remember PEMDAS using the sentence ‘Please Excuse My Dear Aunt Sally.’” Encourage students to create their own sentences to remember PEMDAS, if they choose. Also, it is very helpful to display the order of operations in the classroom while students are first learning it.

Write the following expressions on the board:

  • 4 + 5 × 8 – 3²
  • 2 + 18 ÷ 3 × 4 – 1
  • 5 – 2³ ÷ 4 × 2

Ask students to work in pairs to determine the value of these expressions. Encourage them to follow the order of operations carefully so they get the correct value for each expression.

Observe students as they work. Find pairs of students who found the correct values (35, 25, and 1). When students are finished working, ask one pair of students to write and explain the processes they used for each expression on the board. (Working as a pair, students tend to feel less threatened when sharing their work, and students can support each other in both recording and explaining the process.) Be sure to have students ask questions about processes they don’t yet understand. It is important that all questions are answered before students begin the practice activity.

For practice using the order of operations, introduce the What Happened to Aunt Sally? worksheet (M-5-6-1_What Happened to Aunt Sally Practice Worksheet and KEY.docx). Distribute the What Happened to Aunt Sally? worksheet to all students.

“There are 16 expressions on the worksheet. Each expression has a corresponding letter. Evaluate the expression. On page 2, find all boxes with the value of the expression and write its corresponding letter. For example, if the value of expression E is 128, you would write E in the boxes above every 128 in the puzzle.”

Monitor students’ progress as they work. Provide necessary interventions and support as needed. Students may need to be reminded of the order of operations. Also, remember to suggest that struggling students circle the operation to be performed in each step. Also you will likely need to remind students that multiplication and division are done from left to right in the same step, as are addition and subtraction. In the acronym PEMDAS, students often think multiplication is done before division, but these operations are performed at the same time. Similarly, this also holds true for addition and subtraction.

With 5 to 8 minutes left in the class period, distribute an index card to each student. Present the expression 4 + 5² – 8 × 3 + 7, and ask students to work individually to find the value of this expression. Remind students that it is important they write down each step in the process, as this is an opportunity for you to assess both what they know and what concepts they have yet to master.

Collect all of these “exit slips” before students exit the classroom. Review the exit slips before the next class period to identify common errors students are making and specific students who may benefit from extra support. (The value of the expression is 12.)

Extension:

Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year.

  • Routine: Throughout the year, use this Web site to keep students fresh on the concept of order of operations. Students canpractice evaluating numerical expressions without grouping symbols at this site.The site is interactive and records how many expressions students evaluate correctly in 2 minutes.http://cemc2.math.uwaterloo.ca/mathfrog/english/kidz/order.shtml
  • Small Group: Students who need additional practice may find this online game, Calculator Chaos, to be beneficial.Students practice using operations to generate a target number given a broken calculator, with select keys and numbers. http://www.mathplayground.com/calculator_chaos.html
  • Expansion: Students who are ready for a greater challenge may find it in this online game, One to Ten.Students use the order of operations and select numbers to generate the numbers 1 through 10. Generating an expression is usually more difficult for students than evaluating given expressions. So, the One to Ten Game can be used to challenge students:http://www.theproblemsite.com/games/onetoten2.asp

“In mathematics, we often need to rewrite words and phrases using numbers and symbols. Today we will practice writing and interpreting mathematical phrases—more often called mathematical expressions.”

Present these situations:

  • “Lija has four cousins. Amelia has three less than five times as many cousins as Lija. How many cousins does Amelia have?
  • Hans has six pencils and Russell has twelve pencils. The teacher bought three times as many pencils as the number Hans and Russell have together. How many pencils did the teacher buy?”

Now help students learn how to write expressions to represent these situations. Many students will probably want to find the answers and that is okay. However, it is important to help them write the numerical expression that represents the situation. A student’s ability to translate words into mathematical symbols and numerals is very important when engaging in problem solving.

Ask students if they have ideas about how to write an expression for the number of cousins Amelia has. If there are multiple ideas, record them on the white board. Work as a class to determine which of the expressions is correct. This will require students to use their working knowledge of the order of operations from a previous lesson. (Both 4 × 5 – 3 and 5 × 4 – 3 are correct expressions.)

Ask students, “What clues did you find in the situation to help write the expression? What specific words were clues for you?” Students will likely say “times as many” indicated multiplication. Students will also likely say “less than” indicated subtraction.

Mathematical expressionsare like phrases in English class—they do not have any “punctuation” (e.g., an equal sign in math). Mathematical equationsare like sentencesin English class—they have “punctuation” (e.g., an equal sign in math). So, 4 × 5 is an expression and 4 × 5 = 20 is an equation.

Again, ask for ideas for writing an expression for the number of pencils the teacher bought in the second situation. Work as a class to determine which of the expressions students created are correct. (Both 3 × (6 + 12) and (6 + 12) × 3 are correct expressions.) It is important that students use parentheses in this situation to indicate the sum of 6 and 12 must be found first, and the sum should then be multiplied by 3. For this reason, students’ understanding of the order of operations is very important to this lesson.

Ask students, “What clues did you find in the situation to help write the expression? What specific words were clues for you?” Students will likely say “times as many” indicated multiplication and “together” indicated addition.

Help students review some key terms before continuing. “There are four basic operations: addition, subtraction, multiplication, and division. The result of each operation has its own term. The result of adding two numbers is a sum. The result of subtraction is a difference. The result of multiplying two numbers is a product. The result of dividing two numbers is a quotient.” Write the following table on the board, and encourage students to record this in their math notebooks. The last column of the table will be completed after students finish the Writing Numerical Expressions practice worksheet.

 

Operation

Result

Other “Clue” Words or Phrases

Addition

Sum

 

Subtraction

Difference

 

Multiplication

Product

 

Division

Quotient

 

 

Now distribute the Writing Numerical Expressions practice worksheet (M-5-6-3_Writing Numerical Expressions Practice Worksheet and KEY.docx). Ask students to work in pairs to write expressions for each of the six situations. Monitor students’ progress as they work.

Provide necessary interventions and support as needed. English language learners and other students struggling with literacy skills may need extra support translating between words and symbols. Focus on the key words and phrases compiled in the table above. You may also choose to put these words and phrases on a word wall in your classroom.

Ask six students to volunteer to write the expressions on the board for each of the situations. Ask other students to verify that they have the same expressions. If they have a different expression, ask them to write that on the board as well. Notice that two or three expressions can be written for each situation (the KEY is also provided in M-5-6-3_Writing Numerical Expressions Practice Worksheet and KEY.docx). Ask the class to use the order of operations to verify that the different expressions students have written for a particular situation all have the same value.

After students have completed the Writing Numerical Expressions Practice Worksheet, ask them to identify any other “clue” words or phrases to complete the last column of the table. “Review the situations. Did you identify any other clue words or phrases in these situations that indicated a specific operation?” Students will likely identify some of the words and phrases listed below in the third column. If not, suggest these and help them identify the situation from the Writing Numerical Expressions Practice Worksheet in which the word or phrase was used.

 

Operation

Result

Other “Clue” Words or Phrases

Addition

Sum

Total, In all, More than

Subtraction

Difference

Less than

Multiplication

Product

Times as many, Times as large

Division

Quotient

Divided by

For more practice interpreting numerical expressions, introduce the Expression Matching Games 1, 2, and 3 (M-5-6-3_Expression Matching Games 1, 2, and 3.docx).

Each pair of students will need a copy of each of the three versions of the game. It is important to prepare these in advance, as the 10 cards in each game set need to be cut out prior to play. If possible, copy each game set onto a different color of paper. This will help keep the different versions of the game separate from each other.

Ask students to work in pairs. Distribute Expression Matching Game 1 and introduce Game 1 as follows. “We will now use a matching game to practice interpreting numerical expressions. The matching game consists of 10 cards. Five of the cards have a word phrase on them, and five have a mathematical phrase or expression on them. The goal is to match each word phrase with the corresponding mathematical expression. To begin, put all 10 cards on your table with the writing showing. Work together to find all of the matching sets of cards.”

When all groups are finished with Expression Matching Game 1, ask students which expressions were the most challenging to match with their phrases. If students do not identify it as difficult, be sure to discuss the phrase “four less than the sum of 7 and 3” as it is often the most challenging for students. Students often think this should be written as 4 – (7 + 3). Remind students that the order of a subtraction problem is important. Finding “four less than” 20, for example, actually means you are subtracting 4 from 20, written as 20 – 4.

Now, distribute Expression Matching Games 2 and 3 to each pair of students. For Game 2, you may want to suggest students place the 10 cards so the writing cannot be seen. Students may then take turns flipping over two cards. If the two cards they uncover are a match, they keep that pair of cards. The student with the most matches when the game is over is considered the winner. A third Expression Matching Game is also provided. The difficulty increases slightly from Game 2 to Game 3.

With 5 to 8 minutes left in the class period, distribute an index card to each student. Present the expression 7 + 8 × 3, and ask students to write a word phrase describing this expression. Also, present the word phrase “4 less than the product of 9 and 2”. Ask students to write a mathematical expression to represent this phrase.

Collect all of these “exit slips” before students exit the classroom. Review the exit slips before the next class period to identify common errors students are making and specific students that need extra support. (7 + 8 × 3could be described as 7 more than the product of 8 and 3. The expression 9 × 2 – 4 or (9 × 2) – 4 can represent “4 less than the product of 9 and 2”.)

Extension:

Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year.

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