1. Construct the first 10 rows.
2. Identify different families or sets of numbers in the diagonals.
3. Relate the numbers in the triangle to the row numbers.
4. Examine sums of rows. Relate row sums to the row numbers.
5. For each row, form two sums by adding every other number. Compare sums within and between rows. Describe the patterns that emerge and why they occur.
6. Describe how the triangle is developed recursively.

Refers to standard AII.P.3 (12.P.3) (TIMSS)
Problem: Brighto soap powder is packed in cube-shaped cartons that measure 10 cm on each side. The company decides to increase the length of each side by 10%. How much does the volume increase?

Solution: (10 + 1)³ - 10³ = (10³ x 1⁰ + 3 x 10² x 1¹ + 3 x 10¹ x 1² + 10⁰ x 1³) - 10³ = 331, therefore the volume increases by 331cm³.

Refers to standards AII.P.8, AII.P.11, and AII.P.12 (12.P.8, 12.P.11, and 12.P.12)
A stone is thrown straight up into the air with initial velocity v₀ = 10 feet per second. If one neglects the effects of air resistance, after t seconds the height of the stone is h=v₀t-½gt² (until the stone hits the ground), where g ≈ 32 feet per second squared (the gravitational acceleration at the Earth's surface). What is the greatest height that the stone reaches, and when does it reach that height?

Refers to standards AII.P.8, AII.P.11, and AII.G.2 (12.P.8, 12.P.11, and 12.G.2)
A stabilizing wire (guy wire) runs from the top of a 60 foot tower to a point 15 feet down the hill (measured on the slant) from the base of the tower. If the hill is inclined 11 degrees from the horizontal, how long does the wire need to be?

Refers to standards AII.P.8, AII.P.11, and AII.G.1 (12.P.8, 12.P.11, and 12.G.1)
Students replicate the experiment in which Eratosthenes calculated the circumference of the earth and got a remarkably good answer. They locate some schools roughly due north or south of their school and connect with students in those schools through electronic mail. Students in each school agree that on a given day, at high noon, they will measure the shadow cast by a vertical stick on level ground. After sharing the measurements of the stick and the shadow, students use trigonometric ratios to determine the angle of the sun's rays. Using this information, along with the approximate distance between the schools, students use proportions to find an approximation of the earth's circumference. This example can be extended to sharing data with students from other states and countries.

Refers to standards AII.P.8, AII.P.11, and AII.G.1 (12.P.8, 12.P.11, and 12.G.1)
How far from the horizontal must a sheet of plywood 4 feet wide be rotated to fit through a doorway 30 inches wide?

Refers to standards AII.P.8 and AII.P.11 (12.P.8 and 12.P.11)
A solution's pH depends on the concentration of hydrogen ions per liter of the solution. The formula for determining a pH is where H+ is the number of gram atoms of hydrogen ions per liter. The pH of neutral water is 7. Acidic solutions have a pH that is lower than 7, basic solutions have a pH that is higher than 7.

1. What is the pH of a solution with 4.231 x 10^-5 gram atoms of hydrogen ions per liter?
2. A certain juice has a pH of 3.9. Find the concentration of hydrogen ions of the juice.

Refers to standards AII.P.1, AII.P.11, and PC.P.9 (12.P.1, 12.P.11, and 12.P.12)

1. State the relationship between the position of car A and that of car B at t = 1 hr. Explain.
2. State the relationship between the velocity of car A and that of car B at t = 1 hr. Explain.
3. State the relationship between the acceleration of car A and that of car B at t = 1 hr. Explain.
4. How are the positions of the two cars related during the time interval between t = 0.75 hr. and t = 1 hr.? (That is, is one car pulling away from the other?) Explain.

Refers to standards AII.P.8, AII.P.11, and AII.G.2 (12.P.8, 12.P.11, and 12.G.2)
A stabilizing wire (guy wire) runs from the top of a 60 foot tower to a point 15 feet down the hill (measured on the slant) from the base of the tower. If the hill is inclined 11 degrees from the horizontal, how long does the wire need to be?

Refers to standards AII.P.8, AII.P.11, and AII.G.1 (12.P.8, 12.P.11, and 12.G.1)
Students replicate the experiment in which Eratosthenes calculated the circumference of the earth and got a remarkably good answer. They locate some schools roughly due north or south of their school and connect with students in those schools through electronic mail. Students in each school agree that on a given day, at high noon, they will measure the shadow cast by a vertical stick on level ground. After sharing the measurements of the stick and the shadow, students use trigonometric ratios to determine the angle of the sun's rays. Using this information, along with the approximate distance between the schools, students use proportions to find an approximation of the earth's circumference. This example can be extended to sharing data with students from other states and countries.

Refers to standards AII.P.8, AII.P.11, and AII.G.1 (12.P.8, 12.P.11, and 12.G.1)
How far from the horizontal must a sheet of plywood 4 feet wide be rotated to fit through a doorway 30 inches wide?

Refers to standard AII.D.2 (12.D.6) (EDC, Inc)
There are 9 points on a paper. No three are on the same line. How many different triangles can be drawn with vertices on these points?

Refers to standard AII.D.2 (12.D.6)
There are eight McBride children, three girls and five boys. How many different ways are there of forming groups of McBride children containing at least two of the three girls?

Refers to standard AII.D.2(12.D.6 )
Some services that involve electronic access require clients to choose a six-digit password. In an effort to increase security of the passwords, clients cannot use combinations that correspond to actual dates, nor can they use two identical digits in succession, nor passwords with one digit appearing three or more times. How many "secure" passwords are available?