## Multiplying Integers Using Videotape

- Lesson

In this lesson, students experience beginning-algebra concepts through discussion, exploration, and videotaping. The concept of multiplication of integers is presented in a format which encourages understanding, not simply rote memorization of facts. This lesson plan is adapted from the article, "A Videotaping Project to Explore the Multiplication of Integers", by Marcia B. Cooke, which appeared in Arithmetic Teacher, Vol. 41, No. 3 (November 1993) pp. 170-171.

Begin the class discussion by saying, "Today we are going to explore both positive and negative rates. Imagine a rate that is positive when a person moves from left to right in front of the number line and negative when he or she moves right to left." Allow time for student comments to indicate that they understand the idea. Continue by saying, "Next, imagine continuing the rate for one minute, two minutes, and so forth."

Do some examples at the chalkboard of rates, such as:

+10 yards/minute × 1 minute

+10 yards/minute × 2 minutes

+10 yards/minute × 3 minutes

...

Repeat the same examples for:

-10 yards per minute × 1 minute

-10 yards per minute × 2 minutes

-10 yards per minute × 3 minutes

...

Left-to-right movement in front of the number line is the model for forward rates. Right-to-left movement denotes negative rates.

Initially, the following situations are demonstrated:

**positive time x positive rate = positive distance**

and

**positive time x negative rate = negative distance.**

Present the other two situations as opposites of the former two:

**negative time x negative rate = positive distance**

and

**negative time x positive rate = negative distance.**

Students may ask, "How could you have a negative time? Time moves only forward." Clearly, the students may be discomfited by the use of a model that accounted for only half the multiplication products. Allow students the chance to discuss this problem in small groups of two or three. Students may suggest (and if they don't, you may need to guide them into suggesting), "Maybe it's like a film that is rewinding."

Next, select two student volunteers. One student may operate the video camera while another student is walking approximately fifteen yards forward carrying a green "I am walking forward" sign. After that, the student will backward carrying a pink "backward" sign.

Before watching the videotape, discussed the motions the students think they will be seeing on the video monitor. Distribute copies of the student activity sheet and ask students to predict the apparent motion.

Students should have little difficulty correctly and confidently
predicting the outcomes of the first three situations. When the tape of
the students walking forward is rewound, she *appears* to be walking backward, just as the tape of a student walking backward operated in the play-forward mode *appears*
to be walking backward. Despite the logical symmetry, the class may not
be able to commit to an outcome if the tape of a student walking
backward is played in "rewind" mode. Eventually, students should see
that a negative rate times a negative time yields a positive distance!
You may need to play the tape over and over to reassure students,
double-checking that you see the pink backward sign in the rewind mode.

### Reference

Cooke, Marcia B. "A Videotaping Project to Explore the Multiplication of Integers." Arithmetic Teacher, 41 (November 1993): 170-171.

- Video camera
- Videotaping Activity Sheet

### Learning Objectives

Students will:

- Develop an experimental model for multiplication of integers.
- Predict outcomes, conduct experiments, and reflect on the results.

### NCTM Standards and Expectations

- Develop meaning for integers and represent and compare quantities with them.

- Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

- Develop and analyze algorithms for computing with fractions, decimals, and integers and develop flue

### Common Core State Standards – Mathematics

Grade 7, The Number System

- CCSS.Math.Content.7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

Grade 7, Expression/Equation

- CCSS.Math.Content.7.EE.B.3

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP6

Attend to precision.