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Fraction × and ÷ Diagrams

# Fraction × and ÷ Diagrams

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How To Do It:

1. In this activity, students will use graph paper and the shading of

rectangles to picture multiplication and division of fractions. Often

students just memorize the rules for these operations, but never

know why these rules work.

thus

=

×

1

5

of a 2 × 5 grid

1

2

of a 2 × 5 grid

1 × 1 = 1

2

5

10

1

2

of

1

5

=

1

10

Use the large graph paper or an overhead projector to work

through several problems with the class. For example, with the

(illustrated above), begin by helping

problem 1/2 × 1/5 =

students understand that because the denominators are 2 and 5,

they will need to utilize a 2-unit by 5-unit grid. Then shade the

first column and mark it 1/5, as it is 1/5 of the whole rectangle.

Next shade the bottom row and mark it 1/2, because it is 1/2 of the

whole rectangle. Finally, note where these shaded areas overlap

and find the fraction that represents this overlapped region. In this

problem, the fractional answer is 1/10, and therefore 1/2 of the 1/5

portion equals 1/10 of the entire rectangle. It has thus been shown

that 1/2 × 1/5 = 1/10. (Note: Some students find it helpful to think

of this situation as 1/2 of 1/5 = 1/10.)

2. A similar process is followed when dividing a fraction by a fraction,

as demonstrated in the Example below. Finally, the slightly more

complex issue of multiplying (or dividing) a mixed number by

a mixed number is illustrated in the Extensions section. After

demonstrating as many examples as necessary, have students try

some problems on their own.

Example:

The students depicted here are working cooperatively on drawing a

. The area of the first shaded region is

diagram for 1/4 ÷ 1/3 =

3 square units, and the area of the second shaded region is 4 square units,

so the answer is 3/4. Notice the students’ thoughts about the division

process, including the application the last student came up with.

Fraction ì and ữ Diagrams

155

Extensions:

When they are ready, students might be asked to visualize and diagram

mixed number situations, such as those that follow.

1. The diagram below shows a means of representing the problem

.

1-1/3 × 2-1/4 =

1

1

1

1

4

(Note: 1 whole area is 3 × 4 or 12 spaces)

1

3

1

1

13 × 24 =

4

3

156

×

9

4

=

36

12

or 3 (whole areas)

Computation Connections

2. Applying the numerals used in Extension 1 to the process of

division yields quite a different result. In this situation, 1-1/3 ÷

might be illustrated as shown below.

2-1/4 =

1

1

3

2

(16 squares) ữ (27 squares) =

Fraction ì and ÷ Diagrams

1

4

16

27

157

Chapter 43

Decimal Squares

×

×

×

×

×

Total group activity

Cooperative activity

Independent activity

Concrete/manipulative activity

Visual/pictorial activity

Abstract procedure

Why Do It:

Students will compare decimals and compute problems with

decimals using a visual model.

You Will Need:

A supply of Decimal Squares (reproducible is provided), highlighter pens, and colored pencils or crayons are required.

How To Do It:

1. This activity will make use of Decimals Squares to

compare, add, subtract, and multiply decimal numbers.

Each Decimal Square is a 10-unit by 10-unit square; in

other words, it has an area of 100 square units. Each

small square represents one hundredth of the Decimal

Square, or .01. Each column (of 10 squares) represents

one-tenth, or 0.1, of the Decimal Square.

Start by shading 0.2 (two columns of 10 square units)

on a Decimal Square (using an overhead projection

system, if possible) and explain that it represents twotenths of the whole square. Then shade 0.32 (three

158

columns and 2 square units) on a Decimal Square and explain that

it represents thirty-two hundredths.

2. Next use a Decimal Square or squares and colored writing utensils

to shade areas that show the problem, and discuss how to find the

answer to the problem using the shaded squares. Below each Decimal Square, show the problem and the solution to the problem.

The solution should be an inequality sign or a decimal number.

See the Examples below to use as demonstrations. (Note: When

designing problems for this activity, be careful that the answers

do not include the thousandth place, or the problems will not be

possible to solve using a Decimal Square.)

Examples:

The following problems can be used to explain the use of Decimal

Squares to students.

1. Shade 0.6 in the first Decimal Square (starting on the left) and 0.21

in the second (starting on the left). Then compare the shaded areas

to discover the answer to the problem, ‘‘To solve 0.6

0.21, fill

in the blank with >, <, =.’’

0.6 > 0.21

2. Shade 0.25 using one color and 0.3 using

another color. Add up the entire area to

discover the sum of 0.25 and 0.3.

0.25 + 0.3 = 0.55

Decimal Squares

159

another color. The part of the figure

from the total area. Ask students

what decimal represents the area

remaining.

– 0.2

0.46 – 0.2 = 0.26

4. Shade 0.5 vertically and 0.4 horizontally. The number of squares

in the rectangular area of overlap represent the solution to the

problem 0.5 × 0.4.

0.4

0.5

0.5 × 0.4 = 0.20

Extensions:

Have students try the following problems using Decimal Squares.

1. Insert <, >, or = to make the statement 0.34

0.3 true.

2. Insert <, >, or = to make the statement 0.5

0.50 true.

3. Find 0.4 + 0.27.

4. Find 0.72 + 0.15.

5. Find 0.54 − 0.3.

6. Find 0.68 − 0.4.

7. Find 0.5 × 0.6.

8. Find 0.3 − 0.8.

160

Computation Connections

9. Using the Decimal Square below, state the decimal multiplication

problem represented by the rectangle shaded.

Rectangle A →

0.4 × 0.5 = .20

Rectangle B →

Rectangle C →

D

A

E

Rectangle D →

Rectangle E →

Rectangle F →

Rectangle G →

Rectangle H →

Rectangle I →

Decimal Squares

B C

F

H

I

G

161

Decimal Squares

162

Computation Connections

Chapter 44

Square Scores

×

×

×

Total group activity

Cooperative activity

Independent activity

Concrete/manipulative activity

Visual/pictorial activity

Abstract procedure

Why Do It:

Students will practice with addition, subtraction, multiplication, or division facts, using logical-thinking strategies in a

game setting.

You Will Need:

Square Score Grids (provided at the end of this activity) are

required. Usually one per pair of students is enough to start

with. Once they are familiar with the activity, players might

also devise grids for each other (see Extensions). Pencils and

pens of different colors are also needed.

How To Do It:

Square Scores is usually played by two students on one grid.

The grid contains 5 rows and 7 columns of dots. In the

middle of a group of four adjacent dots is a math problem.

Each student uses a pencil or pen of a different color, and at

her or his turn draws a vertical or horizontal line between any

two adjacent dots. Play continues in this manner until a line

is drawn that closes a square. The student who draws that

line must attempt to answer the problem contained within

163

that box. If the problem is answered correctly, that student is allowed

to claim the square and to shade or mark it. If the student gives an

incorrect answer, the square is marked with an X and no credit is

allowed. (Students might check their answers with a calculator or an

answer sheet.) When all squares are closed, the students count the boxes

claimed to see how many facts they knew.

Example:

The players pictured below are practicing their multiplication facts for

6s, while also attempting to capture as many squares as possible. Thus

far Juanita has captured and marked the three squares marked \\\\, and

Jose has claimed the two facts marked ////.

Extensions:

1. If students need practice with a certain operation, such as subtraction, then the grid should utilize only those types of problems.

However, if mixed practice is desirable, a different grid might

include a combination of addition, subtraction, multiplication, or

division or even fractions or decimals.

164

Computation Connections

2. Square Scores also works well as a team game when it is played on

the overhead projector. In such a setting, the team members are

allowed a strategy conference (for two minutes), and then the team

leader draws the line for that turn. Play continues in this manner

until all squares on the overhead transparency are surrounded and

marked. The winning team is the one that has captured the most

squares.

3. Players can easily devise their own grids by writing equations

designated for practice on blank grids (see model provided) or

by using one-inch or larger graph paper. (Note: The grid designer

should also create an answer key.) The designed grid can be

photocopied and tried by several other players.

4. Advanced levels of the game might include having three, four, or

more players competing on the same grid, and could include bonus

squares (enclosing problems more difficult than those typical for

Square Scores

165

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