Integer World Quests
Maya has $15 in her bank account. She deposits $20, withdraws $45 to buy shoes, and then gets paid $30 for babysitting. What is her final account balance?
Answer: $20
Deposits are positive (+) and withdrawals are negative (-).
Start at $15: 15 + 20 = $35.
Spend $45: $35 - $45 = -$10 (she owes the bank $10).
Earn $30: -$10 + $30 = $20.
In the coordinate pair (-3, 5), which number tells you how far to move left or right?
Answer: -3 (the x-coordinate)
Coordinate pairs are written as (x, y). The first number (x) tells you to move left or right. The second number (y) tells you to move up or down.
If a shape is dilated by a scale factor of 2, what happens to its side lengths?
Answer: They double (get multiplied by 2).
Dilation means making a shape bigger or smaller like a camera zoom. A scale factor of 2 means every side length is multiplied by 2, making the shape twice as big.
The Grocery Store Receipt:-
Leo buys 3 bags of apples for $4.25 each and 2 boxes of cereal for $3.80 each. He pays with a $20 bill. How much more money does Leo need to give the cashier to cover the total cost?
Answer:$0.35
Calculate the cost of the apples: 3 × $4.25 = $12.75.
Calculate the cost of the cereal: 2 × $3.80 = $7.60.
Add them together to find the total bill: $12.75 + $7.60 = $20.35.
Compare the total bill to his $20 bill: $20.35 - $20.00 = $0.35. Leo is short by $0.35!
What is 1/2+1/4?
Answer: 3/4
Student Explanation: Convert 1/2 to an equivalent fraction with a denominator of 4: 1/2=2/4. Now add the top numbers: 2/4+1/4=3/4. (Think of two quarters plus one quarter = 3 quarters).
At 8:00 AM, the temperature was -6°F. By noon, it rose by 14°F. By 8:00 PM, a storm dropped the temperature by 20°F. What was the temperature at 8:00 PM?
Answer: -12°F
Imagine a vertical thermometer. Start at -6°F.
Going up 14 degrees: -6 + 14 = +8°F (it's above zero now).
Dropping 20 degrees: 8 - 20 = -12°F (12 degrees below zero).
Which quadrant is the point (-4, 7) located in?
Answer: Quadrant II (Quadrant 2)
Starting at the origin (0, 0), go left 4 units (negative x) and up 7 units (positive y). That lands in the top-left section, which is Quadrant II
What algebraic rule represents dilating a point (x,y) by a scale factor of 3?
Answer: (x,y) → (3x,3y)
To scale a point on a grid by 3, multiply both the x-coordinate and y-coordinate by 3.
The Road Trip Fuel Trap
A delivery truck uses 1/8 gallon of fuel for every mile driven. The driver starts with a full tank of 24 gallons. After driving 120 miles, how many gallons of fuel are left in the tank?
Answer: 9 gallons
Find how much gas was used in 120 miles: 120 × (1/8) = 15 gallons.
Subtract the used gas from the full 24-gallon tank: 24 - 15 = 9 gallons remaining.
Find an equivalent fraction for 2/5 with a denominator of 10.
Answer: 4/10
To change the bottom number from 5 to 10, multiply by 2 (5 X 2=10). Do the exact same to the top number: 2 X 2=4. So, 2/5=4/10
A submarine starts at 200 feet below sea level (-200 feet). It dives down another 150 feet, then rises 80 feet, and finally dives 100 feet lower. What is the submarine's final depth?
Answer: -370 feet (or 370 feet below sea level)
Diving means going deeper into negative numbers (-). Rising means going up (+).
Start at -200.
Dive 150 feet: -200 - 150 = -350 feet.
Rise 80 feet: -350 + 80 = -270 feet.
Dive 100 feet: -270 - 100 = -370 feet.
If you reflect the point (6, -2) over the x-axis, what are the new coordinates?
Answer: (6, 2)
Reflecting over the horizontal line (x-axis) is like flipping across a mirror. The x-coordinate stays the same (6), but the y-coordinate flips from -2 to 2
Point A is at (2,4). If it is dilated by a scale factor of 12, what are the new coordinates A'?
Answer: (1,2)
Multiplying by 12 means taking half of each number:
2 X 1/2 = 1 and 4 X 1/2 = 2
So A' is at (1,2).
The Phone Plan Split
Four friends share a monthly cell phone bill of $142.60. The plan includes a flat maintenance fee of $10.60, and the rest of the bill is split equally among all 4 friends. How much does each friend pay?
Answer:
$33.00
Take away the flat maintenance fee first: $142.60 - $10.60 = $132.00.
Divide the remaining total among the 4 friends: $132.00 ÷ 4 = $33.00 per person.
What is 3/4 written as a decimal?
Answer: 0.75 Student Explanation: Think of money: 3 quarters out of 4 quarters equals 75 cents ($0.75). Or divide 3/4=0.75.
In a trivia game, you get +50 points for a correct answer and -25 points for an incorrect answer. Team A answered 8 questions correctly and 3 questions incorrectly. What was Team A's final score?
Answer: 325 points
Points gained: 8 × 50 = 400 points.
Points lost: 3 × (-25) = -75 points.
Combine them together: 400 + (-75) = 325 points.
What is the distance between the points (3, 5) and (3, -4) on a coordinate grid
Answer: 9 units
Both points have the same x-coordinate (3), so count vertically on a number line from 5 down to -4: 5 steps down to 0, plus 4 more steps down to -4 gives 9 total units.
Point P was at (3,6) and moved to P'(9,18) after a dilation. What was the scale factor?
Answer: 3
Check how many times larger the numbers became: 33=9 and 63=18. Since both coordinates were multiplied by 3, the scale factor is 3.
The Target Range Challenge
You are creating a decimal number using the digits 2, 5, 8, and 9 (using each digit only once).
Your number must be greater than 52 but less than 53.
What is the only decimal number you can make that fits this rule?
Answer:
52.89
To make a number between 52 and 53, the tens and ones digits MUST be 5 and 2 (making 52).
That leaves the digits 8 and 9 for the decimal places. If you place 9 first (.98), it gives 52.98. Placing 8 first gives 52.89, which fits perfectly between 52 and 53.
Which fraction is bigger: 2/3 or 3/5?
Answer: 2/3
Student Explanation: Convert both to decimals or find a common denominator (15):
2/3=10150.67 and 3/5=915=0.60
Since 10 out of 15 is larger than 9 out of 15, 23 is larger.
An elevator starts in an underground parking garage on floor -3. It goes UP 12 floors, goes DOWN 15 floors, and then goes UP 8 floors. What floor is the elevator on now?
Answer: Floor 2 (or 2nd floor)
Going UP is positive (+), going DOWN is negative (-).
Start at -3.
A shape is translated using the rule (x + 2, y - 5). What direction does the shape move?
Answer: 2 units RIGHT and 5 units DOWN
Adding 2 to x moves the shape to the right on the horizontal axis. Subtracting 5 from y moves the shape down on the vertical axis.
Triangle ABC has a vertex A located at the point (-4, 6).
First, Triangle ABC is dilated by a scale factor of ½ with the origin as the center of dilation to create Triangle A'B'C'.
Then, Triangle A'B'C' is translated using the rule (x, y) → (x + 5, y - 7) to create Triangle A''B''C''.
What are the coordinates of point A''?
(3,−4)
Break this problem into two simple steps:
Step 1: Do the Dilation (Scale down by ½):
A scale factor of ½ means you multiply both coordinates of point A(-4, 6) by ½ (or divide them by 2):
x-coordinate: -4 × ½ = -2
y-coordinate: 6 × ½ = 3
So, point A' is at (-2, 3). Step 2: Do the Translation (Slide the point):
Apply the rule (x + 5, y - 7) to your new point A'(-2, 3):
Move x to the right by 5: -2 + 5 = 3
Move y down by 7: 3 - 7 = -4
So, final point A'' is at (3, -4).
The Running Track Relay
A track runner trains by running 6 laps on Monday, each taking 1 1/2 minutes.
On Tuesday, she runs 8 laps, but cuts her time down to 1 1/4 minutes per lap.
Did she spend more total time running on Monday or Tuesday, and by how many minutes?
Tuesday, by 1 minute
Calculate Monday's total running time: 6 × 1 1/2 = 9 minutes.
Calculate Tuesday's total running time: 8 × 1 1/4 = 10 minutes.
Compare the times: Tuesday (10 minutes) was longer than Monday (9 minutes).
Find the difference: 10 - 9 = 1 minute.
Solve: 5/6-1/3.
Answer: 3/6 (or 1/2)
Change 1/3 into an equivalent fraction with 6 on the bottom: 1/3 X 2/2=2/6. Subtract the numerators: 5/6-2/6=36, which simplifies to 1/2