Jossey-Bass Teacher - Math Wise Phần 9 pdf

How To Do It:

1. Tell the students that for this activity they will need to stack

oranges, as grocery stores sometimes do. Ask them how they

think orange stacks stay piled up without falling down. Discuss

how the stacks are usually in the shape of either square- or

triangular-based pyramids. Then allow the students to begin help￾ing with the orange-stacking experiment.

2. The players might begin by analyzing patterns for square-based

pyramids of stacked oranges, because these are sometimes easier

to conceptualize than pyramids with triangular bases. Have them

predict and then build the succeeding levels. The top (Level 1) will

have only 1 orange. Challenge students to determine how many

oranges will be required for the next level down (Level 2). After

discussing the possibilities for Levels 3 and 4, build the structure

as a class. Ask students how they might determine the number of

oranges that would be needed to build an even larger base (Level 5),

given that there are not enough additional oranges to build one.

3. It may be sufficient for young students to predict and build

the structures for Levels 1 through 4. As they build, students

in grades 2 through 5 will develop their logical-thinking skills.

Older students (grades 6 through 8), however, will often logi￾cally analyze the orange-stacking progression and be able to

discover a pattern and eventually a formula for determining

the number of oranges at each level. Students will find that

from the top down, Level 1 = 1 orange; Level 2 = 4 oranges;

Level 3 = 9 oranges; Level 4 = 16 oranges; and Level 5 will

require 25 oranges. Have students determine how many oranges

will be needed for Levels 6, 8, 10, or even 20, instructing them to

write a statement or a formula that they can use to tell how many

oranges will be needed at any designated level (see Solutions).

4. When they are ready, students can be challenged with stacking

oranges as triangular-based pyramids. With 35 oranges, partici￾pants will be able to predict, build, and analyze Levels 1 through 5

of the pyramid. Ask them further to determine how many oranges

will be needed for Level 6, Level 10, and so on. As before,

instruct them to write a statement or a formula that will find

how many oranges will be needed at any designated level (see

Solutions).

342 Logical Thinking

Example:

The students below have diagrammed the oranges needed at each level of

a square-based pyramid stack. Their comments help reveal their logical

thinking.

Extensions:

1. When they are finished with the orange-stacking experiments,

allow participants to eat the oranges (after they wash their hands).

Also, see how the oranges might be used in the same manner

as the watermelons in Watermelon Math (p. 232), prior to their

being eaten.

2. Students can represent the findings from both the square- and

triangular-based orange-stacking experiments as bar graphs,

and then analyze, compare, and contrast them.

3. Challenge advanced students to create orange stacks that have

bases of other shapes, such as a rectangle using 8 oranges as the

length and 5 oranges as the width. Learners might also be asked

to find, in the case of a 7-orange hexagon base, how many oranges

would be needed in the level above it, how many they would need

to form a new base under it, and so on.

Stacking Oranges 343

Solutions:

1. Solutions for the square-based orange-stacking experiment: Initially,

participants will often notice that Level 2 has 3 more oranges

than Level 1, Level 3 has 5 more than Level 2, and Level 4 has

7 more than Level 3. This realization will allow them to figure

out the number of oranges needed at any level, but the required

computation will be cumbersome! A more efficient method would

be for the participants to recognize that all of the levels are

square numbers. That is, Level 1 = 12 = 1 orange; Level 2 = 22 =

4 oranges; Level 3 = 32 = 9 oranges, and so on.

2. Solutions for the triangular-based orange-stacking experiment: The

hands-on stacking of oranges in triangular-based pyramids is quite

easy to comprehend; however, as the following explanation notes,

the abstract-level logical thinking is a bit more complex. The

participants will notice that Level 2 has 2 more oranges than

Level 1, Level 3 has 3 more than Level 2, and so on. Thus it can

be seen that the total number of oranges at any level is equal to

the number at the prior level, plus the additional oranges needed

at the new level (which, for the orange stacks, is the same as the

level number). For instance, the total number of oranges required

at Level 4 will be 6 oranges (the total for Level 3) plus 4 oranges

(which is the level number), or 6 + 4 = 10 oranges. The following

table may help clarify matters:

Level (from the Top Down) Number of Oranges

1 1

2 3 = 1 + 2

3 6 = 3 + 3

4 10 = 6 + 4

5 15 = 10 + 5

6 21 = 15 + 6

344 Logical Thinking

Chapter 88

Tell Everything

You Can

Grades 2–8

× Total group activity

× Cooperative activity

× Independent activity

× Concrete/manipulative activity

× Visual/pictorial activity

× Abstract procedure

Why Do It:

Students will investigate, compare, and contrast the logical

similarities and differences of varied objects using mathemat￾ical ideas.

You Will Need:

A variety of objects (see Examples) that have at least one

attribute in common are required.

12 1

2

3

4

5 6 7

8

9

10

11

How To Do It:

1. Display two mathematical items that at first glance

appear to have few, if any, similarities. For instance,

the square design and the clock face shown above seem

345