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Writing and Interpreting Numerical Expressions

Lesson Plan

Writing and Interpreting Numerical Expressions

Objectives

Students learn to write and interpret numerical expressions. Students will:

  • match a numerical expression to a word phrase, and match a word phrase to a numerical expression.
  • create a word phrase representing a numerical expression, and create a numerical expression representing a word phrase.

Essential Questions

  • How is mathematics used to quantify, compare, represent, and model numbers?
  • How can mathematics support effective communication?
  • How can expressions, equations and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?

Vocabulary

  • Braces: Symbols used to group certain parts of a mathematical expression, { }.
  • Brackets: Symbols used to group certain parts of a mathematical expression, [ ].
  • Numerical Expressions: A mathematical combination of numbers, operations, and grouping symbols.
  • Order of Operations: The steps used to evaluate a numerical expression: 1) Simplify the expressions inside grouping symbols. 2) Evaluate all powers. 3) Do all multiplications and/or divisions from left to right. 4) Do all additions and/or subtractions from left to right.
  • Parentheses: Symbols used to group certain parts of a mathematical expression, ( ).

Duration

60–90 minutes

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

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Related Materials & Resources

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Formative Assessment

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    • Use the expression matching games to determine whether students need additional instruction.
    • The exit slip can be used to determine if students understand how to both write and interpret numerical expressions.

Suggested Instructional Supports

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    Scaffolding, Active Engagement, Modeling, Explicit Instruction
    W: The lesson deals with writing and interpreting numerical expressions. The goal is to help support students in translating between word phrases and mathematical phrases (called expressions). 
    H: Introduce the situations about Amelia’s cousins and the teacher’s pencils, and engage students in a discussion about them. Lead students to understand that it is important to be able to translate between word phrases and the corresponding number phrases or expressions. 
    E: Encourage students to write expressions for many different situations. Ask students to present to the class the expressions they wrote. Peer teaching is a very powerful tool. The more exposure students get to different explanations of the same concept, the more likely they are to understand and retain the concept. Reinforce student thinking about what words or phrases are clues for each of the four basic operations: addition, subtraction, multiplication, and division. 
    R: Use the three Matching Games to help students refine their ability to translate between word phrases and numerical expressions. Students working in pairs can also verify each other’s thinking. 
    E: The Matching Games can be used to assess students’ understanding, as there is only one correct way to create the five pairs of cards. Monitor student responses and clarify any misconceptions. Use the exit slip to evaluate individual understanding of both writing and interpreting numerical expressions. 
    T: The lesson may be adapted to meet the needs of your students by using the suggestions in the Extension section. 
    O: The lesson gets students familiar with translating between word phrases and numerical expressions. The lesson starts with situations that need to be translated into numerical expressions, but ends with matching games that require students to translate between word phrases and numerical expressions. A focus on clue words and phrases for each operation helps students to be more successful in translating word problems into numerical expressions. 

Instructional Procedures

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    “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 expressions are like phrases in English class—they do not have any “punctuation” (e.g., an equal sign in math). Mathematical equations are like sentences in 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 × 3 could 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.

Related Instructional Videos

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Final 05/10/2013
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