Problem 1: Comparing Positive and Negative Integers
Replace the box with <, >, or = to make a true sentence:
4 ☐ -4
Remedial
Solution
The correct symbol is **>** (greater than).
4 > -4
- **Step 1: Understand the numbers.** You are comparing a positive number (4) and a negative number (-4).
- **Step 2: Use the number line rule.** On a number line, any positive number is always to the right of any negative number.
- **Step 3: Apply the comparison rule.** Since positive numbers are always greater than negative numbers, 4 is greater than -4.
Problem 2: Finding Absolute Value
Evaluate the expression:
|-6|
Remedial
Solution
The value of |-6| is **6**.
- **Step 1: Understand absolute value.** Absolute value means the distance a number is from zero on the number line.
- **Step 2: Count the distance.** The number -6 is 6 units away from 0.
- **Step 3: State the answer.** Since distance is always positive, the absolute value of -6 is 6.
Problem 3: Comparing Identical Integers
Replace the box with <, >, or = to make a true sentence:
-7 ☐ -7
Remedial
Solution
The correct symbol is **=** (equal to).
-7 = -7
- **Step 1: Analyze the numbers.** You are comparing the integer -7 to itself.
- **Step 2: Apply the rule of equality.** Any number is always equal to itself. Therefore, -7 is equal to -7.
Problem 4: Ordering a Simple Set of Integers
Order the integers in the set from least to greatest:
{4, -5, 0}
Remedial
Solution
The correct order is **{-5, 0, 4}**.
- **Step 1: Identify the negative number.** The only negative number is -5. All negative numbers are less than 0 and positive numbers, so -5 is the smallest.
- **Step 2: Place zero.** Zero comes after all negative numbers but before all positive numbers.
- **Step 3: Identify the positive number.** The only positive number is 4. It is the greatest number in the set.
- **Step 4: Combine them in order.** Following the steps, the order is -5, then 0, then 4.
Problem 5: Integers in Real-World Contexts
During a football game, a quarterback was tackled for a loss of six yards. Use an integer to express this change in field position.
Remedial
Solution
The integer that represents a loss of six yards is **-6**.
- **Step 1: Identify the key phrase.** The phrase is "a loss of six yards."
- **Step 2: Determine the direction.** A "loss" or moving backward is represented by a negative sign. A "gain" would be represented by a positive sign.
- **Step 3: Write the integer.** Since it's a loss of 6, the integer is -6.
Problem 6: Ordering Integers
Order the integers in the set from least to greatest:
{-5, 5, 3, -1}
On-Level
Solution
The correct order is **{-5, -1, 3, 5}**.
- **Step 1: Find all negative numbers.** The negative numbers are -5 and -1.
- **Step 2: Order the negative numbers.** When comparing negative numbers, the one with the larger absolute value is smaller. |-5| is 5 and |-1| is 1. Since 5 is larger than 1, -5 is smaller than -1. So the order starts with -5, -1.
- **Step 3: Find all positive numbers.** The positive numbers are 5 and 3.
- **Step 4: Order the positive numbers.** 3 is less than 5.
- **Step 5: Combine the lists.** The final order is {-5, -1, 3, 5}.
Problem 7: Adding Absolute Values
Evaluate the expression:
|-8| + |-3|
On-Level
Solution
The value of the expression is **11**.
- **Step 1: Evaluate the first absolute value.** The absolute value of -8, or |-8|, is 8 because -8 is 8 units from zero.
- **Step 2: Evaluate the second absolute value.** The absolute value of -3, or |-3|, is 3 because -3 is 3 units from zero.
- **Step 3: Add the results.** Add the two values together: 8 + 3 = 11.
Problem 8: Ordering Temperatures
At 6:15 a.m. the temperature was -8°F. At 12:15 p.m. the temperature was -12°F. At 6:16 p.m. the temperature was -10°F. Order the temperatures from least to greatest (coldest to warmest).
On-Level
Solution
The correct order is **-12°F, -10°F, -8°F**.
- **Step 1: Identify all the temperatures.** The temperatures are -8°F, -12°F, and -10°F.
- **Step 2: Understand negative temperature.** With negative numbers, a "larger" number (like -12) is actually colder, or "less," than a "smaller" number (like -8).
- **Step 3: Order the numbers.** On a number line, -12 is the farthest to the left, making it the least. -10 is next, and -8 is the farthest to the right, making it the greatest (warmest) of the three.
Problem 9: Evaluating an Expression
Evaluate the expression if a = -3 and b = 0:
|a| - b
On-Level
Solution
The value of the expression is **3**.
- **Step 1: Substitute the values.** Replace 'a' with -3 and 'b' with 0 in the expression.
|-3| - 0
- **Step 2: Evaluate the absolute value.** The absolute value of -3, or |-3|, is 3.
- **Step 3: Perform the subtraction.** 3 - 0 = 3.
Problem 10: Comparing on a Number Line
Replace the box with < or > to make a true sentence:
3 ☐ -4
On-Level
Solution
The correct symbol is **>** (greater than).
3 > -4
- **Step 1: Locate the numbers on the number line.** Find the position of 3 and -4.
- **Step 2: Compare their positions.** The number 3 is located to the right of -4.
- **Step 3: Apply the rule.** Any number to the right on the number line is greater than any number to its left. Therefore, 3 is greater than -4.
Problem 11: Ordering a Complex Set of Integers
Order the integers in the set from least to greatest:
{-48, -30, -49, -8, 3, -4}
Extended
Solution
The correct order is **{-49, -48, -30, -8, -4, 3}**.
- **Step 1: Identify the negative numbers.** They are -48, -30, -49, -8, and -4.
- **Step 2: Order the negative numbers.** To find the smallest number, find the negative number that is farthest from zero (has the largest absolute value). -49 is the farthest, so it is the smallest. Then comes -48, then -30, then -8, and finally -4, which is closest to zero. The order so far is -49, -48, -30, -8, -4.
- **Step 3: Identify the positive numbers.** The only positive number is 3.
- **Step 4: Combine the ordered lists.** Place the positive number at the end, as it is the greatest. The final order is {-49, -48, -30, -8, -4, 3}.
Problem 12: Adding Two-Digit Absolute Values
Evaluate the expression:
|-13| + |-7|
Extended
Solution
The value of the expression is **20**.
- **Step 1: Evaluate |-13|.** The distance of -13 from zero is 13. So, |-13| = 13.
- **Step 2: Evaluate |-7|.** The distance of -7 from zero is 7. So, |-7| = 7.
- **Step 3: Add the results.** 13 + 7 = 20.
Problem 13: Evaluating a Complex Expression
Evaluate the expression if a = -1, b = 2, and c = -8:
|a + c| + b
Extended
Solution
The value of the expression is **11**.
- **Step 1: Substitute the values.** Replace the variables with their numbers:
|(-1) + (-8)| + 2
- **Step 2: Perform the operation inside the absolute value bars first.** Adding two negative numbers: -1 + (-8) = -9.
|-9| + 2
- **Step 3: Evaluate the absolute value.** The absolute value of -9 is 9.
9 + 2
- **Step 4: Perform the final addition.** 9 + 2 = 11.
Problem 14: Interpreting Data from a Table
A golfer keeps track of his scores by noting the difference from par. Compare the golfer's scores on the 7th and 13th holes using <, >, or =.
| Hole | 6 | 7 | 13 | 18 |
|---|---|---|---|---|
| Score vs. Par | -2 | +1 | -1 | -1 |
Extended
Solution
The correct comparison is **1 > -1**.
- **Step 1: Find the score for the 7th hole.** Look at the table. The score for hole 7 is +1 (or simply 1).
- **Step 2: Find the score for the 13th hole.** The score for hole 13 is -1.
- **Step 3: Compare the two integers.** You need to compare 1 and -1.
- **Step 4: Apply the comparison rule.** Any positive number is greater than any negative number. Therefore, 1 is greater than -1.
Problem 15: Finding the Pattern
Describe the pattern in the sequence below. Then find the next three terms.
10, 8, 4, -2, ...
Extended
Solution
**Pattern:** You subtract the next even number (2, 4, 6, 8, ...) to get the next term in the sequence.
The next three terms are **-10, -20, and -32**.
- **Step 1: Find the difference between the first two terms.** 10 - 2 = 8.
- **Step 2: Find the difference between the second and third terms.** 8 - 4 = 4.
- **Step 3: Find the difference between the third and fourth terms.** 4 - 6 = -2.
- **Step 4: Identify the pattern.** The amount being subtracted is increasing by 2 each time (you subtract 2, then subtract 4, then subtract 6).
- **Step 5: Apply the pattern to find the next terms.**
- From -2, subtract the next even number, which is 8: -2 - 8 = **-10**.
- From -10, subtract the next even number, which is 10: -10 - 10 = **-20**.
- From -20, subtract the next even number, which is 12: -20 - 12 = **-32**.