• The Zero and Eight worksheet (M-3-5-1_Zero and Eight and KEY.docx) may be used to determine how well students are able to translate and solve real-world problems involving multiplication and division.
• Use the Multiply or Divide worksheet (M-3-5-1_Multiply or Divide and KEY.docx) for a quick assessment of student ability to choose multiplication or division in real-world problems.
• Use the Lesson 1 Exit Ticket (M-3-5-1_Lesson 1 Exit Ticket and KEY.docx) to quickly evaluate students’ understanding of translating and solving real-world multiplication and division problems.

Using Manipulatives to Model Multiplication and Division

“Today we are going to discuss how to take real-world problem situations and write them as number sentences. Then we will solve the number sentences. Let’s start with some manipulatives. I’m going to distribute some color tiles. Find a partner. Each pair of students will get about 40 colored tiles.”

Students should work in pairs to complete this activity. Distribute at least 40 color tiles to each pair of students.

Distribute a copy of the Problem Solving worksheet (M-3-5-1_Problem Solving and KEY.docx) to each student.

Ask for a volunteer, “Will someone please read the first example aloud?” After a student has read the problem, ask students to explain the situation. “Can someone please explain the problem aloud?” Students will likely be able to explain at least some aspects of the problem. Ask students probing questions until the important information (who and what) has been stated, and write the information on the board using short phrases as follows.

            There are 3 kids.          Each kid has 6 cars.              How many cars in all?

Ask students to model this problem. “Work together with your partner to model this problem using color tiles.”

When students are finished, ask one pair of students to show how they modeled the problem using color tiles. Then demonstrate how to draw a picture of the model. Ask students to draw the model on their practice worksheet. (You may want to draw a person next to each group to represent the 3 kids.)

Now, ask students “What number sentence can we write for this problem?” Likely some students will suggest  or . It is important to focus on the multiplication number sentence and remind students that it is an alternative way to represent repeated addition.

Also, remind students that the number sentence  is appropriate because the sentence can be read as “3 groups of 6 are equal to 18”. Although  by the commutative property, the number sentence  is not appropriate because there are not six groups of three cars. If we help students read the multiplication as 3 groups of 6, they can use this meaning of multiplication to help differentiate between multiplication and division word problems.

“Now let’s change the problem just a bit. Instead, say we know there are 18 toy cars to be split equally among 3 kids. What number sentence can we write for this problem?” Guide students towards realizing that this situation uses division (18 ÷ 3 = 6) because we are splitting a certain number into equal groups. Also emphasize the opposite relationship between the original multiplication problem () and the new division problem (18 ÷ 3 = 6).

Ask students to work in pairs to complete the Problem Solving worksheet (M-3-5-1_Problem Solving and KEY.docx).

When students have completed the worksheet, ask a pair of students to model each problem and write the corresponding number sentence. When students solve real-world problems, they often struggle to determine what operation to use to solve the problem. For each problem be sure to ask the pair of students or the entire class, “How did you know what operation to use? How do you know if you should multiply or divide to solve the problem?” (I used multiplication when there was repeated addition; I used division when I needed to split a number into equal groups.) Language to describe these differences includes the fact that multiplication is used when there are a number of groups that are equal in size and you want to find the total amount, and division is used when there is one large group and you want to divide it into a number of groups that are equal in size.

Translating Words to Multiplication or Division Problems

When solving real-world problems, students often struggle the most to translate the sentences into mathematical symbols. This activity focuses on helping students learn to identify key terms that indicate multiplication or division.

Distribute a copy of the Multiply or Divide? worksheet (M-3-5-1_Multiply or Divide and KEY.docx) to all students.

Introduce the worksheet. “Notice on the worksheet there are two columns. In the column on the left are problems written in words. Number sentences are in the right column. Please work together in pairs again. Match the word problem with the correct number sentence. Be careful. Notice many number sentences look alike, such as . The goal is to decide if the problem is a multiplication or a division problem. Let’s look at the first one together.” Begin as follows, “What number sentence do you think represents number 1?” (B, ) “Why do you think number 1 is a multiplication problem?” (There are 15 pies or groups, each pie or group has 3 apples, and the goal is to determine the total number of apples in all of the pies.) Help students focus on multiplication as determining the total when there are a number of equal groups. Be sure students still have access to the color tiles from the previous activity. Some students may want to use the tiles to model the problem to identify if the problem requires multiplication or division. “Now work with your partners to complete the worksheet.”

After students are finished with the practice worksheet, it is important to have pairs of students explain each problem. The focus should be on how they determined whether it required multiplication or division.

Continue asking students to explain the correct number sentence for each problem. The following is a summary of how you may want to discuss number 3, as this is the first division problem. “What number sentence do you think represents number 3?” Many groups will likely say G, which is . “Why do you think number 3 is a division problem?” Ask a student or pair of students to explain. Students will likely say, “The total is 14 bananas, and the total number of bananas must be divided into 2 equal groups for the 2 monkeys; the goal is to determine how many bananas each monkey gets to eat.” Help students focus on division as starting with the total amount and dividing the total into equal groups.

[Note: Division can occur two different ways. One way division presents itself is that the total is divided into a specific number of groups, and the goal is to find how many are in each group. This is what happened in the monkey problem. The total was 14 bananas, it was divided into 2 equal groups, and the answer was the number of bananas in each group. A second way division presents itself is that the total is divided into groups of a specific size, and the goal is to decide how many groups can be created. Number 8 is this type of problem. There are a total of 8 pencils, the pencils are put into groups of 2, and the goal is to determine how many groups there will be. If students focus on division as being given the total amount and dividing it into equal groups, they will be ready for both types of division, even without understanding the subtle difference in how division can be presented.]

Distribute copies of the Zero and Eight worksheet (M-3-5-1_Zero and Eight and KEY.docx) to all students. Ask students to complete this worksheet either in class or at home. Use the worksheet to assess students’ ability to translate and solve real-world multiplication and division problems.

Extension:

Use the suggestions below to modify the lesson as needed. The Routine section provides ideas to review lesson concepts throughout the year. The Small Group section offers additional practice opportunities for students who could benefit from them. The Expansion section gives opportunities for students who are ready for a challenge beyond the requirements of the standard.

• Routine: During the school year, ask students to create real-world problems that involve multiplication and division. For example, if there are 24 students and 48 cookies, how many cookies can each student have?
• Small Group: Students who need additional practice may be pulled into small groups to work on using manipulatives to model the action in word problems. Focus on helping students identify the action of “combining equal groups” as multiplication and the “separating into equal groups” as division. This Web site includes additional multiplication and division word problems that can be used with the small groups.

http://www.beaconlearningcenter.com/WebLessons/CameronsTrip/default.htm

• Expansion: The three Web sites listed below are suggested for students who are looking for a greater challenge. They all include multistep real-world problems for students to solve. Many also require students to use two operations. http://www.prongo.com/farm/game.html

http://www.studyzone.org/testprep/math4/d/twostep4p.cfm http://www.mathplayground.com/WordProblemsWithKatie2.html