How many rectangles?

2 x 2 squares: 5

3 x 3 squares: 2

4 x 4 squares: 2

16 squares

1 x 3 rectangles: 4

1 x 4 rectangles: 2

2 x 4 rectangles: 4

3 x 4 rectangles: 2

24 rectangles (no double counting of squares as rectangles)total of 40

How many squares do you see?

How many rectangles?

How many rectangles?

1 x 1 squares: 6

2 x 2 squares: 5

3 x 3 squares: 2

4 x 4 squares: 2

2 x 2 squares: 5

3 x 3 squares: 2

4 x 4 squares: 2

plus the orange square

16 squares

16 squares

1 x 2 rectangles: 8

1 x 3 rectangles: 4

1 x 3 rectangles: 4

1 x 4 rectangles: 2

2 x 3 rectangles: 4

2 x 4 rectangles: 4

3 x 4 rectangles: 2

2 x 4 rectangles: 4

3 x 4 rectangles: 2

24 rectangles (no double counting of squares as rectangles)total of 40

Q: A Math quiz has multiple-choice questions, each with choices. A student passes the quiz with or more correct questions. If the student randomly picks their answers, what is the probability of passing the quiz?

A: 7/27 = 1/27 + 6/27

Scenario 1: The student gets all 3 correct (1/3) x (1/3) x (1/3) = 1/27

Scenario 2: The student gets 2 correct. 1/3 x 1/3 x 2/3 but this can happen 3 different ways as the student could get the 1st one incorrect or the 2nd or the 3rd.

so (1/3)x (1/3) x (2/3) x 3 = 6/27

1/27 + 6/27 = 7/27.

Factoring out 2^2016 from both the numerator and the denominator:

results in 5/3.

Look at where the graph crosses the x axis…then check those x values by seeing that the y value is 0 on the table. Students can also prove their answer by substituting the x values into the equation to see if the result is 0.