A. Exploring Integer Operations. Consider the integers -2 and 5.
1. What is the sum of -2 and 5? - The sum of -2 and 5 is 3.
Is the result positive or negative? -Positive.
Can you explain why? -The result is positive because when you add a negative number (-2) to a positive
number (5), the sum will be positive.
How can you combine -2 and 5 to find their sum? Is there a rule or property you can apply? - We
combine -2 and 5 by adding them together: -2 + 5 = 3.
2. What is the product of -2 and 5? - The product of -2 and 5 is -10.
How can you find the product of -2 and 5? What mathematical operation do you need to perform? - We
find the product by multiplying -2 and 5: -2 × 5 = -10.
Is there a rule or property that can help you determine the sign of the product when multiplying a
negative and a positive number? - The rule or property that helps determine the sign of the product
when multiplying a negative and a positive number is that the product of a positive number and a
negative number is always negative.
3. Can you find an integer that, when added to -2, gives a sum of 0? - Yes
If so, what is that integer? - The integer that, when added to -2, gives a sum of 0 is 2.
What can you conclude about the relationship between -2 and this integer? - The relationship between
-2 and 2 is that they are additive inverses or opposites of each other because their sum is zero.
B. Consider the fraction 3/5 and the decimal 0.6.
1. Which one is larger, the fraction 3/5 or the decimal 0.6? How can you determine this? - To determine
which one is larger, we can compare the fraction 3/5 and the decimal 0.6. Since 0.6 is equivalent to
6/10, we can compare 3/5 and 6/10 directly. To do this, we can find a common denominator, which in
this case is 10. We can multiply the numerator and denominator of 3/5 by 2 to get 6/10. Now we can
see that both fractions, 3/5 and 6/10, are equal. Therefore, the fraction 3/5 and the decimal 0.6 are
equal in value and neither one is larger than the other.
How can you compare the size of a fraction and a decimal? Is there a method or strategy you can use? To compare the size of the fraction 3/5 and the decimal 0.6, we can convert them to a common form.
We can convert the fraction to a decimal or the decimal to a fraction.
Can you convert the fraction 3/5 into a decimal or the decimal 0.6 into a fraction to make the
comparison easier? - Converting the fraction 3/5 to a decimal, we divide 3 by 5: 3 ÷ 5 = 0.6. Therefore,
the fraction 3/5 is equal to the decimal 0.6, so they are the same value.
2. How can you express the fraction 3/5 as a decimal? - ? - To express the fraction 3/5 as a decimal, we
can divide the numerator (3) by the denominator (5). Performing this division, we get 0.6.
How can you ensure that it is correct? - To ensure correctness, we can check if the decimal
representation terminates or repeats. In this case, 0.6 terminates, which means it ends after a finite
number of digits. Therefore, expressing the fraction 3/5 as 0.6 is correct.
3. What does the decimal 0.6 mean as a fraction? - The decimal 0.6 means 6/10 as a fraction.
How can you express it in simplest form? - To express it in simplest form, we can divide both the
numerator and the denominator by their greatest common divisor, which is 2. Thus, 0.6 simplifies to 3/5.
C. Exploring Terminating and Repeating Decimals. Consider the fraction 2/3 and its decimal
representation.
1. Express the fraction 2/3 as a decimal. Is the decimal terminating or repeating? Why? - The fraction 2/3
as a decimal is 0.6666... The decimal representation of 2/3 is repeating because the digit 6 repeats
indefinitely.
2. What does it mean for a decimal to terminate? - A decimal is said to terminate if it ends after a finite
number of digits.
Can you think of other examples of terminating decimals? - For example, 0.25 is a terminating decimal
because it ends after two digits. Other examples of terminating decimals include 0.5 and 0.75.
3. What does it mean for a decimal to repeat? - A decimal is said to repeat if it contains a repeating
pattern of one or more digits.
Can you think of other examples of repeating decimals? - For example, 0.3333... is a repeating decimal
because the digit 3 repeats indefinitely. Another example is 0.121212..., where the pattern 12 repeats.
4. Can you determine if the decimal representation of 2/3 is terminating or repeating without actually
calculating it? How? - Yes, we can determine if the decimal representation of 2/3 is repeating without
calculating it. Since the denominator of 2/3 is 3, and 3 is a prime number, the decimal representation will
always be repeating.
5. How can you convert a repeating decimal into a fraction? - To convert a repeating decimal into a
fraction, we can use algebraic manipulation.
Can you demonstrate this process with an example? - Let's take the repeating decimal 0.3333... as an
example. Let x = 0.3333... Multiplying both sides by 10, we get 10x = 3.3333... Subtracting the original
equation from this new equation eliminates the repeating part, giving us 9x = 3. Solving for x, we find x =
1/3. Therefore, 0.3333... is equal to 1/3.
6. Can you convert a terminating decimal into a fraction? - A terminating decimal can always be
converted into a fraction by expressing it as a ratio of the decimal number to a power of 10. For example,
the terminating decimal 0.25 can be written as 25/100, which simplifies to 1/4.
How is it different from converting a repeating decimal? - The process is different from converting a
repeating decimal because there is no repeating pattern to consider.