Lesson Question:How can identifying key words help students solve mathematical word problems?
Lesson Overview:In this lesson, students will take turns acting as "math coaches" who will assist other students in solving word problems by identifying key words that usually indicate specific mathematical operations.
Length of Lesson:One hour to one hour and a half
Instructional Objectives:Students will:
- brainstorm key words that usually indicate specific mathematical operations
- create flash cards to review the relationships between key words and operations
- coach one another in collectively solving mathematical word problems
- synthesize their knowledge of word problems by writing some of their own
- student notebooks
- white board
- computers with Internet access
- index cards (four per student)
- "Solving Word Problems through Translation" sheets (one per small group) [click here to download]
Solving an authentic word problem:
- Open the class by having the students solve an authentic word problem in pairs: "There are _________ (fill in the number) students in this classroom. I need to distribute four index cards per student. How many index cards do I need?"
- Circulate around the room as students work, ensuring that students are multiplying the number of students in the classroom by four in order to determine how many index cards are needed.
- Have a student volunteer come to the front board to write the mathematical equation that he or she used to determine the answer to the problem. Then, above that equation, write the original word problem and ask students which specific word in the problem let them know that they needed to multiply the two numbers in order to determine the number of index cards needed for the class (i.e., per).
Brainstorming key words that indicate mathematical equations:
- Explain to the class that whether they notice it or not, they are constantly interpreting key words in word problems in order to determine which mathematical operations to use in solving the problems.
- On the white board, display the Visual Thesaurus word map for "arithmetic operation" and then click on the meaning "a mathematical operation involving numbers" in order to reveal the four mathematical operations: addition, subtraction, multiplication, and division.
- Draw a four-quadrant table on the board and write a different mathematical operation title in each quadrant: addition, subtraction, multiplication, and division. Write the word "per" under the title "multiplication" and have students brainstorm additional key words that belong under each of the four mathematical operation categories.
- If students get stuck in this brainstorming process, you could suggest different key words (within the context of simple word problems) and have students direct you where to write the words in the table. At the end of this brainstorming session, make sure you have at least the following words and phrases listed in your table:
Mathematical Operations and Key Words
Creating key word flash cards:
- Have a student count out the number of index cards that the class determined in the warm up problem and distribute four cards to each student.
- Direct students to create four flash cards — one for each of the four mathematical operations. On the blank side of each card, they should boldly write an operation and its symbol (i.e., +, -, x, where is the division symbol?), and on the reverse, lined sign they should list the key words associated with that operation. (Students should base these flash cards on the table you created on the front board.)
Playing the role of "math coach":
- Organize the class into small groups of no more than three to four students in each group, and explain that they will be using their new flash cards as visual aids in math coaching!
- Distribute a "Solving Word Problems through Key Words" sheet to a student in each group and explain that the student with the sheet will act as the reader and recorder during the first round. The reader and recorder's job is to read a word problem aloud and to allow his fellow "math coaches" to advise him on which mathematical operation to follow in solving the problem.
- Advise the math coaches in the class to listen to the word problem closely, to advise the reader and recorder to underline any key words in the problem that they detect, and to follow the flash card mathematical operation that they decide to "flash."
- Direct groups to complete the "Solving Word Problems through Key Words" sheet, alternating the role of reader and recorder so that each student has at least one or two turns in that role.
Sharing word problem answers and strategies:
- Invite students to the front of the classroom to explain their group's word problem strategies and how key words led to determining which mathematical operations to use in each problem.
- For homework, assign students the task of writing some of their own word problems containing some of the key words discussed in class but not previously used on the "Solving Word Problems through Key Words" sheet.
Extending the Lesson:
- To further challenge students, you could give them additional word problems that challenge them to interpret the same key words in somewhat confusing contexts (e.g., "I have eight jelly beans, which is three fewer than my brother has. How many jelly beans does my brother have?") Or, you could also introduce word problems involving multiple mathematical operations (e.g., "A 6000 seat stadium is divided into 3 sections. There are 2000 seats in Section 1, and there are 1500 more seats in Section 2 than in Section 3. How many seats are in Section 2?")
- Check whether or not groups accurately solved each of the ten word problems and underlined appropriate key words in the "Solving Word Problems through Key Words" sheet.
- Assess students' original word problems to see if they appropriately incorporated key words to indicate specific mathematical operations.
Benchmarks for Mathematics
Standard 1. Uses a variety of strategies in the problem-solving process
Level II (Grades 3-5)
1. Uses a variety of strategies to understand problem situations (e.g., discussing with peers, stating problems in own words, modeling problem with diagrams or physical objects, identifying a pattern)
2. Represents problems situations in a variety of forms (e.g., translates from a diagram to a number or symbolic expression)
3. Understands that some ways of representing a problem are more helpful than others
4. Uses trial and error and the process of elimination to solve problems
5. Knows the difference between pertinent and irrelevant information when solving problems
6. Understands the basic language of logic in mathematical situations (e.g., "and," "or," "not")
7. Uses explanations of the methods and reasoning behind the problem solution to determine reasonableness of and to verify results with respect to the original problem
Level III (Grades 6-8)
1. Understands how to break a complex problem into simpler parts or use a similar problem type to solve a problem
2. Uses a variety of strategies to understand problem-solving situations and processes (e.g., considers different strategies and approaches to a problem, restates problem from various perspectives)
3. Understands that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages
4. Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions
5. Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms
6. Generalizes from a pattern of observations made in particular cases, makes conjectures, and provides supporting arguments for these conjectures (i.e., uses inductive reasoning)
7. Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems (i.e., uses informal deductive methods)
8. Understands the role of written symbols in representing mathematical ideas and the use of precise language in conjunction with the special symbols of mathematics
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