## Too Big or Too Small?

• Lesson
6-8
3

In this lesson, students develop number sense through a series of three hands-on activities. Students explore the following concepts: the magnitude of a million, fractions between 0 and 1, and the effect of decimal operations.

Included here is a selection of problems and activities, appropriate for the middle grades classroom, for which the underlying theme is the development of number sense. These activities can be used in varied ways to generate discussion and to extend student thinking about number-related concepts. The discussion that arises as students describe their thinking will certainly give insight into their thinking and will help in evaluating students' development of number sense.

Activity 1: Exploring The Size of a Million Dollars

This activity explores whether one million dollars will fit into a standard suitcase. If so, how large would the suitcase need to be? How heavy would it be? You may have students work in small groups (2 or 3 students per group) to explore these questions.

Begin the investigation by telling the following story:

Just as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2 million in a contest. The money was sent to him in two suitcases, each containing$1 million in one-dollar bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Can he make you a millionaire?

Involve students in formulating and exploring questions to investigate the truth of this claim. For example:

• Can $1,000,000 in one-dollar bills fit in a standard-sized suitcase? If not, what is the smallest denomination of bills you could use to fit the money in a suitcase? • Could you lift the suitcase if it contained$1,000,000 in one-dollar bills? Estimate its weight.

Calculators should be available to facilitate and expedite the computation for analysis.

Note: The dimensions of a one-dollar bill are approximately 6 inches by 2.5 inches. Twenty one‑dollar bills weigh approximately 0.7 ounces.

You may wish for students to locate these facts about dollar bills on their own, using internet or other appropriate resources. The students will also need to determine the dimensions of a "standard" suitcase.

Activity 2: Estimating Fractions Between 0 and 1

The model suggested here is easy to make and will help you evaluate your students' understanding of fractions between 0 and 1. Encourage students to make estimates using familiar benchmarks (e.g., ½, ¼, ¾).

Copy the Circle Template onto light-colored cardstock.

 Circle Template

Give each student a copy and ask them to cut out the circles and make a cut in the radius of each.

Have students put the circles together so that they can see the fractions printed on one side of one circle. Ask questions such as these:

• Show a small part of the shaded circle (less than ¼). Can you name the part represented?
• Show a large part of the shaded circle (greater than ¾). Can you name the part represented?

Ask students to reverse the circle with the printed fractions so that they cannot see the fractions. Ask students if they can:

• Show a fraction that is a little bigger than ½. What name can you give it?
• Show a fraction that is between ½ and ¾. What name can you give it?

Continue asking questions that allow students to show their understanding of the fractions represented.

Other fraction models should also be used to evaluate students' understanding of fractions.

Activity 3: Exploring The Effect of Operations on Decimals

This activity provides an opportunity for students to explore the effect of addition, subtraction, multiplication, and division on decimal numbers.

Write the problem (as described next) on the chalkboard or overhead. Ask students to discuss what they notice. Lead a discussion that focuses on these key points:

In computing the product of 4.5 and 1.2, a student carefully lined up the decimals and then multiplied, bringing the decimal point straight down and reporting a product of 54.0.

Reflection on the answer should have caused the student to realize the product was too big. Multiplying 4.5 by a number slightly greater than 1 produces an answer a little more than 4.5. Instead, this student applied an incorrect procedure (line up the decimals in the factors and bring the decimal point straight down) and did not reflect on whether the resulting answer was reasonable.

Tell students that they will be playing a game to practice decimal operations and their effects. Encourage students to trace several paths through the maze while always looking for the path that will yield the greatest increase in the calculator's display. Note: Students often shy away form dividing by decimals less than 1, so you may want to discuss the general effect of dividing by a number less than 1 or multiplying by a number greater than 1.

Give each student a calculator and a copy of the Maze Playing Board activity sheet.

 Maze Playing Board Activity Sheet

Students are to choose a path through the maze. To begin, have the students enter 100 on their calculator. For each segment chosen on the maze, the students should key in the assigned operation and number. The goal is to choose a path that results in the largest value at the finish of the maze. Students may not retrace a path or move upward in the maze.

In pairs or in groups of three, students should discuss their strategies (after playing the game) and what worked best for them.

Students should be able to achieve a score in the thousands. The path highlighted below gives a result of roughly 6332.

Possible follow-up activities include finding the path that leads to the smallest finish number or finding a path that leads to a finish number as near the start number (100) as possible.

References:

• Reys, Barbara J., et al. Developing Number Sense in the Middle Grades, 5, 8, 9, 22, 29, 41, 55, 56. Reston, VA: NCTM, 1991.
• Hoffer, Alan R., ed. Mathematics Resource Project: Number Sense and Arithmetic Skills. Palo Alto, CA: Creative Publications, 1978. Used with permission of the University of Oregon.
• Morris, Janet. How to Develop Problem Solving Using a Calculator. Reston, VA: National Council of Teachers of Mathematics, 1981.
• One thousand or more fake dollar bills (play money or rectangular sheets of paper the approximate size of a dollar bill)
• Scissors
• One copy of Circle Template (on colored cardstock) for each student
• Calculators
• Decimal Maze Activity Sheet
Extensions
1. The Decimal Maze can be modified depending on the level of your students and the topics covered in your classroom. For instance, the maze could be limited to positive whole numbers using only the operations of addition and subtraction for young students, whereas the maze could include scientific notation and exponents for older students. The Blank Maze can be modified to fit your needs.
 Blank Maze

none

### Learning Objectives

Students will:
• Develop intuition about number relationships
• Estimate computational results
• Develop skills in using appropriate technology

### NCTM Standards and Expectations

• Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

### Common Core State Standards – Mathematics

Grade 6, The Number System

• CCSS.Math.Content.6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

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.

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.