STACKINGSQUARES

MONICA & ELENA 10.2

[1] [2] [3]

[4] [5] [6]

72 72 7218

18

182

2

2

2

22

22

2

FIND STACKS OF SQUARES WITH THE EXACT SAME HEIGHT AS A SQUARE WITH AREA 72 SQUARE CENTIMETERS

728

8

8 2

2

8

872 728

2

2

2

2

ARE THERE ANY SQUARES THAT WOULD HAVE NO STACKS THAT ARE THE SAME HEIGHT?

Yes, there are squares that have no stacks. There can only be a same-height stack if the factor(s) of the area is a perfect square. Perfect squares are numbers that are the product of two identical numbers.

E.g. 2 x 2 = 4 4 is the perfect square 3 x 3 = 9 4 x 4 = 16

If you continue, perfect squares include the numbers 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on.

The number 69 is an example of a square that cannot make any stacks (or at least any same-height stack). The factors of 69 are only 1, 3, 23, 69 and none of the factors are perfect squares, so you can’t make a natural stacks with a square of 69. The same goes for other numbers like 1, 3, 17, 23, 34 and other prime numbers. They cannot make stacks that are the same height.

We can make same-height square stacks, hahaha!72 69

We can’t...45

34

EXPLAIN HOW TO FIND THE STACKS THAT WOULD MATCH A GIVEN SQUARE IN HEIGHT

*We are using a square with an area of 72 square centimeters as our example.

72

Factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

1. FIND THE FACTORS OF THE AREA

2. LOOK AT THE FACTORS OF THE AREA AND FIND IF ANY OF THE FACTORS ARE ALSO PERFECT SQUARES

4, 9, 16, 25, 36, 49, 64, 81, 100

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72↑

Factors of 72

Perfect Squares↓

16

25

49

64

81

100

Perfect Squares↓

Factors of 72↓

4936

1 2

3 68

12 18

2473

↑Factors of 72 &Perfect Squares

3. FIND THE SQUARE ROOT(S) OF THE PERFECT SQUARE(S) AND WITHIN THE SQUARE, DIVIDE IT INTO ROWS AND COLUMNS ACCORDING THE

SQUARE ROOT (E.G. √4=2 SO DIVIDE THE SQUARE INTO 2X2 GRID)

√36=6√4=2 √9=3

2 COLUMNS2 ROWS

3 COLUMNS3 ROWS

6 COLUMNS6 ROWS

5. DIVIDE THE AREA BY THE PERFECT SQUARE

(OR NUMBER OF SQUARES WITHIN THE SQUARE)

72 ÷ 4 = 18 72 ÷ 9 = 8 72 ÷ 36 = 2

18 8 2

HOW WOULD ALL THIS WORK FOR CUBES INSTEAD OF SQUARES?

125cm³27cm³

8cm³

5x5x5

2x2x2

3x3x3

27cm³

27cm³54cm³ 3x3x3

Stacking cubes is essentially the same as stacking squares but instead of finding the square root, we need to find the cube root.

2x2x2= 83x3x3=274x4x4=645x5x5=1256x6x6=1927x7x7=3438x8x8=5129x9x9=819

3+2=5