Number Line Basics
Where is the origin?
The origin is at 0.
What is the additive inverse of 8?
−8.
Give the definition of a natural number and name three examples
Natural numbers are counting numbers (usually 1, 2, 3, ...). Examples: 1, 7, 12.
What symbol do we use to show a negative number? Give one example
A minus sign before the number, e.g., −6.
Starting at 0, move 3 units to the right. What number do you land on? Then from that number move 4 units to the left. What number do you land on?
A: 3; then 3 − 4 = −1.
Give the multiplicative inverse (reciprocal) of 3/4 and show the product that proves your answer.
Multiplicative inverse is 4/3 because (3/4)·(4/3) = 1
What are integers? Give two examples and two non-examples.
Integers are whole numbers and their negatives: ..., −2, −1, 0, 1, 2, ... Examples: −3, 10. Non-examples: 1/2, 0.75.
Define absolute value and compute |−9|
Absolute value is the distance from zero; |−9| = 9.
Describe how to find the distance between −2 and 5 on the number line. Give the numerical distance.
Find the absolute value of their difference: |−2 − 5| = |−7| = 7. Distance = 7 units.
Define "zero pair" and give one example using integers.
A zero pair is a positive and a negative of the same value together that sum to 0. Example: (+5) + (−5) = 0.
Is 0 considered a natural number? Explain and state how 0 is classified among the sets in this board.
Typically 0 is not a natural number (counting numbers start at 1); 0 is an integer and is the origin on the number line.
Give two examples of zero pairs and show that each sums to zero
(+2) + (−2) = 0; (+10) + (−10) = 0
If you start at −3, move 6 units to the right, then 2 units to the left, what is your final position? Show steps using the number line.
A: −3 → (right 6) to 3 → (left 2) to 1. Final position = 1.
State the additive inverse and multiplicative inverse for −2, and show calculations that verify each.
Additive inverse: 2 because (−2) + 2 = 0. Multiplicative inverse: −1/2 because (−2)·(−1/2) = 1.
Provide a short real-world context where negative integers and natural numbers are used together; then identify which values would be integers and which would be natural numbers.
Example: Elevation relative to sea level: 300 ft above sea level (natural number) and −50 ft below sea level (negative integer). 300 is a natural number; −50 is an integer.
Identify the additive inverse and absolute value of 0. Explain why these answers make sense.
Additive inverse of 0 is 0 because 0 + 0 = 0. Absolute value of 0 is 0 because distance from 0 to itself is 0.
On a horizontal number line, explain why adding a negative number moves you left and subtracting a negative number moves you right. Use an example: start at 4 and add −7, then start at 4 and subtract −7.
A: Adding −7 means move 7 units left: 4 + (−7) = −3. Subtracting −7 means add the additive inverse: 4 − (−7) = 4 + 7 = 11, so move 7 units right.
Use the number line to explain why a number and its additive inverse are the same distance from zero. Use x = −4 as your example and include absolute value notation.
Both x and its inverse −x lie on opposite sides of 0 at equal distance. For x = −4, |−4| = 4 and |−(−4)| = |4| = 4, so distances from 0 are equal.
Explain the difference between rational numbers and integers. Provide an example of a number that is rational but not an integer and explain why
Rational numbers can be written as a fraction p/q with integers p, q ≠ 0. Integers are whole numbers without fractional parts. Example: 3/2 is rational but not an integer because it is not a whole number.
A student claims that the absolute value of a sum equals the sum of absolute values for all integers: |a + b| = |a| + |b|. Give a counterexample using integers and explain why the claim fails in that case.
Counterexample: a = 3, b = −5. |3 + (−5)| = |−2| = 2. |3| + |−5| = 3 + 5 = 8. 2 ≠ 8, so equality fails because signs can cancel when adding.