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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.2
TASK 2.10.2: MORE CHANCES,
MORE NUMBERS
— IS THERE A PATTERN?
Solutions
1. If the numbers 2, 4, 6, and 7 (without repetition) randomly replace the letters a,
b, and c in the formula ax + by = c, what is the probability that the slope will
be negative? Justify your answer
As in the previous problems, only the coefficients of a and b will determine the slope of
the equation. If the signs of a and b are the same, the slope will be negative. In all of
these 24 possible situations (4!), the slope will be negative. Therefore, the probability
that the slope will be negative is 24/24 or 100%.
2. If the numbers −2,− 4, −6, and −7 (without repetition) randomly replace the
letters a, b, and c in the formula ax + by = c, what is the probability that the
slope will be negative? Justify your answer.
Changing the sign of the coefficients to negative does not change the probability of a
negative slope. The signs of the coefficients are the same in each of the 24 possible
combinations, so the slopes will all be negative.
3. If the numbers 2, 4, 6, and −7 (without repetition) randomly replace the letters
a, b, and c in the formula ax + by = c, what is the probability that the slope will
be negative? Justify your answer
If only one of the replacement numbers is negative, the probability of a negative slope
becomes 50%. We know that if the signs of the coefficients for x and y are different the
slope for that situation will be positive. These possibilities exist. The negative number
will replace b twice with each other positive number replacing a. Ex. 2x − 7y = 6 and
2x − 7y = 4. Since there are three other positive numbers that can replace the coefficient
of x, this produces 6 positive slopes. The negative will also replace the coefficient of x six
times. The probability for a positive slope will be 12 out of 24 and the probability for a
negative slope will also be 12 out of 24.
If −7 is not used the slope is negative, because both a and b are (+)
If c = −7, the slope is negative, because both a and b are (+).
If b = −7, the slope is positive, because a and b have different signs.
If a = −7, the slope is positive, because a and b have different signs.
4. If the numbers 2, 4, −6, and −7 (without repetition) randomly replace the letters
a, b, and c in the formula ax + by = c, what is the probability that the slope will
be negative? Justify your answer
There are 24 possible combinations for this situation. A negative number can be the first
coefficient and matched with a positive coefficient for y four times. (4 x 2 =8) Each
negative can be the coefficient of y and matched to a positive coefficient of x four times,
(4 x 2 = 8). This makes 16 possible combinations of different signs for the coefficients
for x and y resulting in a 2 out of 3 chance of a positive slope or a 1 out of 3 chance of a
negative slope. (8 out of 12)
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
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Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.2
5. If the numbers 2, −4, −6, and −7 (with no repetition) randomly replace the
letters a, b, and c, in the equation ax + by = c, what is the probability that the
slope will be negative? Justify your answer.
The probability of obtaining a negative slope with three negative numbers is 50%.
The probability is the same as with one negative number, as in #3.
Teaching notes
Make sure students are aware that the sign of the slope is determined by the values of
the variables a and b. In the summary and discussion of this task, have students
explain the statement “If the signs of the coefficients for x and y are the same, the
slopes will be negative. If the signs are different, the slope will be positive.”
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 2. Linear Functions; Topic 10. Probability; Task 2.10.2
TASK 2.10.2:
TASK 2.10.2: MORE CHANCES, MORE NUMBERS — IS THERE A PATTERN?
1. If the numbers 2, 4, 6, and 7 (without repetition) randomly replace the letters a, b,
and c in the formula ax + by = c, what is the probability that the slope will be
negative? Justify your answer.
2. If the numbers −2,− 4, −6, and −7 (without repetition) randomly replace the letters
a, b, and c in the formula ax + by = c, what is the probability that the slope will
be negative? Justify your answer.
3. If the numbers 2, 4, 6, and −7 (without repetition) randomly replace the letters a,
b, and c in the formula ax + by = c, what is the probability that the slope will be
negative? Justify your answer.
4. If the numbers 2, 4, −6, and −7 (without repetition) randomly replace the letters a,
b, and c in the formula ax + by = c, what is the probability that the slope will be
negative? Justify your answer.
5. If the numbers 2, −4, −6, and −7 (without repetition) randomly replace the letters
a, b, and c in the formula ax + by = c, what is the probability that the slope will
be negative? Justify your answer.
November 22, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.