1. Analyze your set of squares and describe the side lengths you found.
Draw squares of various sizes by connecting dots. Draw squares with as many different areas as possible. Label each square with its area. Include at least one square whose sides are not horizontal or vertical.
Analyze your set of squares and describe the side lengths you found.

2. Can you draw a square that has an area of 1 square unit?
Can you draw a square that has an area of 2 square units?

3. Draw squares with different areas by connecting dots.

4. Draw squares with different areas by connecting dots.
4 unit2
1 unit2
9 unit2
16 unit2
2 unit2
5 unit2
8 unit2
10 unit2
What is the area of each square?
What is the side length of each square?

5. UPRIGHT squares & TILTED SQUARES
4 unit2
1 unit2
9 unit2
16 unit2
2 unit2
5 unit2
8 unit2
10 unit2
For which kind of square - upright or tilted- is it easier to find the length of a side? Why?
Some of the squares above are called “perfect squares”. Which ones do you think are “perfect squares?’ Why?

6. How do you find the length of each side?
4 unit2
1 unit2
9 unit2
16 unit2
2 unit2
5 unit2
8 unit2
10 unit2
Find the length of each side of each square.

7. What is the length of each side of each square above?
Below are all of the squares you could draw on the 5 dot by 5 dot grid, if each vertex of your square had to be on a dot.
What is the length of each side of each square above?
A very common mistake is for students to look at the picture, and count dots diagonally to try and find the length of the line. On a graph, you can only look at vertical and horizontal lines & count the space between dots to try and find the length, but you cannot do that for diagonal lines.

8. Some people try to visualize turning the figure so that it is placed upright so that the lines go vertically and horizontally. However, it is hard to get an accurate measurement for how long each line is because you cannot tell exactly where it landed between dots.

9. 1 unit2
2 unit2
4 unit2
5 unit2
8 unit2
9 unit2
10 unit2
16 unit2
Think about the answers that would seem reasonable for the side lengths of the squares above. What makes the answers reasonable?

10. 1 unit2
2 unit2
4 unit2
5 unit2
8 unit2
9 unit2
10 unit2
16 unit2
Think about the answers that would seem reasonable for the side lengths of the squares above. What makes the answers reasonable?

11. If you know the length of the side of the square, how would you find the area of the square?
 If you know the area of the square, how would you find the length of the side?
 Some of the squares you drew above are called “perfect squares”. Which ones do you think are “perfect squares”? What do you think makes a “perfect square”?   
Make a list of the first 20 perfect squares.
 A square that has an area of 60 square units is not a perfect square. Estimate what you think the length of its sides would be. Explain.

12. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
#^2 25 36 49 64 81
100
121
144
169
196
225
256
289
324
361
400

13. 60 Make a list of the first 20 perfect squares.
 A square that has an area of 60 square units is not a perfect square. Estimate what you think the length of its sides would be. Explain. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
#^2 25 36 49 64 81
100
121
144
169
196
225
256
289
324
361
400 7 8 49 64
The side lengths for a square with an area of 60 would have to be between 7 and 8. I know this because 7x7=49 and 8x8=64, and I would need a number times itself that is between 49 & 64. Since 60 is closer to 64 than it is to 49, the square root of 60 would be closer to 8 than it is to is about two-thirds of the way between 49 & 64. It would be about 7.6 or 7.7 60
Square Area 49 64
Side length 7 ? 8

14. The area of a square is the length of a side multiplied by itself
The area of a square is the length of a side multiplied by itself. This can be expressed by the formula A= s x s, or A = s2. If you know the area of a square, you can work backward to find the length of a side. For example, suppose a square has an area of 4 square units. To find the length of a side, you need to figure out what positive number multiplied by itself equals 4. Since 2 x 2 = 4, the side length is 2 units. We call 2 a square root of 4.
In general, if A = s2, then s is called a square root of A. Since x 2 = 4 and -2 x -2 = and -2 are both square roots of 4. Every positive number has two square roots. The symbol for the positive square root is  . We write 4 = 2 to give the positive root. If you are asked for the negative root, the text will put a negative sign in front of the radical such as -4 = -2.
If they want both positive and negative roots, they will write it a “plus or minus” sign like +4 = +2

15. What is the value of the 1 ? What is the value of the -9 ?
1 = 1
-9 = - 3
16=  4
25 = 5
-50  -7.1
There is no number that you can square and get a negative number

16. Many people define a rational number as one that you can write like a fraction. They reason that an irrational number is one that you CANNOT write like a fraction. A more technical way of saying that is to say it cannot be written as a ratio of two integers.
When you take the square root of any whole number that is not a perfect square, your answer comes out to a decimal that does not terminate nor does it repeat. It is an irrational number. The cube root of any integer that is not a perfect cube is an irrational number.
The decimal form of an irrational number does not terminate nor does it repeat.
You will need to categorize numbers as rational or irrational. You will need to be able to estimate the decimal value of irrational numbers.

17. Activity 2: Investigating Cubes
Position the unit cubes so that they make as many larger cubes as you can. Record the number of unit cubes used and the length of each edge in a table.

18. Position the unit cubes so that they make as many larger cubes as you can. Record the number of unit cubes used and the length of each edge in the table below.
Length of each side of the face (edge)  1  2  3  4  5  6  7
Number of unit cubes in larger cube
You don’t have enough cubes to build these ones, how can you figure out how many cubes you’d need if you were able to build it?

19. Position the unit cubes so that they make as many larger cubes as you can. Record the number of unit cubes used and the length of each edge in the table below.
Length of each side of the face (edge)  1  2  3  4  5  6  7
Number of unit cubes in larger cube  8
27 
 64
 125
 216
343

20. The larger cubes you constructed using unit cubes are “perfect cubes”
 The larger cubes you constructed using unit cubes are “perfect cubes”. What makes them “perfect cubes”?
Make a list of the first 10 perfect cubes.
To the right is an example that is not a perfect cube.
Why is it not a perfect cube?
What is the volume of the cube to the right? Explain.

21. Each of those cubes has whole number side lengths.
 The larger cubes you constructed using unit cubes are “perfect cubes”. What makes them “perfect cubes”?
Each of those cubes has whole number side lengths.
Make a list of the first 10 perfect cubes. 1 2 3 4 5 6 7 8 9 10 27 64
125
216
343
512
729
1000

22. If you had a larger cube with a volume of 30 cubic units, how long would the length of the edge be? Explain. 1 2 3 4 5 6 7 8 9 10 27 64
125
216
343
512
729
1000 30
It would be a little more than 3 units long. Since 3 cubed is 27 and 4 cubed is 64, it would be a lot closer to 3 than it is to 4.
Side length
Volume