Number Sense
Show on a number line: What number is the additive inverse of 6
-6
If 3 shirts cost $21, what is the unit rate price per shirt?
$7 per shirt.
Simplify by factoring out the common factor: 6x - 30
6(x-5)
What is the circumference of a circle with radius 3?
Circumference = 2πr = 2π(3) = 6π.
From the list [2, 5, 7, 7, 10], what is the mean and median?
Mean = (2+5+7+7+10)/5 = 31/5 = 6.2; Median = 7
What is the absolute value of the distance between -3 and 5 on a number line? Show how you found it.
Distance = | -3 - 5 | = | -8 | = 8.
A recipe uses 2 cups of flour for every 5 cups of sugar. For 20 cups of sugar, how many cups of flour are needed?
(2/5)20 = 8 cups of flour.
Solve and check: 3x + 5 = 20
3x = 15 → x = 5.
Find the area of a circle with diameter 10. (Use formula and show steps.)
Radius = 5; Area = πr^2 = π(25) = 25π.
Explain why a random, representative sample matters when making conclusions about a population. Give one example.
(Sample must reflect population diversity; e.g., polling only one neighborhood won't represent the whole city.)
Find the prime factorization of 84 and write it using exponents.
84 = 2^2 * 3 * 7 so written with exponents: 22⋅3⋅7
Sales tax is 6%. If an item costs $40 before tax, what is the total cost after tax?
Tax = 0.0640 = $2.40; total = $42.40
Solve for x in p(x + q) = r form: If 5(x + 2) = 35, what is x?
5(x+2)=35 → x+2=7 → x=5.
A rectangular prism is 4 cm by 3 cm by 10 cm. What is its volume?
Volume = 4310 = 120 cm^3.
Given data set [4, 6, 6, 9, 15], find the range and interquartile range (IQR). (Show steps.)
Range = 15 - 4 = 11. For IQR: order data (4,6,6,9,15), Q1=6, Q3=9, IQR=3.
Evaluate and justify using properties of operations: (–1)(–1) and explain why it equals 1. (Use number properties in your explanation.)
(–1)(–1) = 1 because multiplying two negatives gives a positive; distributive and sign rules justify this.
A map uses a scale of 1 inch : 50 miles. If two towns are 3.4 inches apart on the map, how many miles apart are they in real life? Show proportional reasoning.
3.4 in * 50 miles/in = 170 miles.
Define slope as rise over run. What is the slope of the line through (2, 3) and (5, 12)?
Slope = (12 - 3) / (5 - 2) = 9/3 = 3.
A cylinder has radius 2 cm and height 7 cm. What is its volume? Show formula and work.
Volume cylinder = πr^2h = π(2^2)(7) = 28π28π cm^3.
A bag contains 3 red, 2 blue, and 5 green marbles. If one marble is drawn at random, what is the probability it is blue? Express as a fraction and decimal. Classify this probability as unlikely, equally likely, or likely.
Probability blue = 2 / (3+2+5) = 2/10 = 0.2 = 20% → unlikely
A number squared equals 2, and another squared equals 9. Which is the square root of each, and explain the inverse relationship between squaring and square roots. Also, explain whether each square root is rational or irrational.
Square roots: √2 is about 1.414 (irrational), √9 = 3 (rational). Squaring and square rooting undo each other for nonnegative numbers.
A store marks up an item by 25% and then later gives a 20% discount on the new price. If the original price was $80, what is the final price? Show steps and explain why percent increase then decrease is not the same as a single percent change.
Original $80 → +25% = $100 → -20% = $80. Final price $80 (coincidence here returns to original). Show calculation: 801.25=100; 1000.8=80.
Graphing/Modeling: Write an equation y = mx that represents a proportional relationship with unit rate 4. Then explain how the unit rate (constant of proportionality) relates to slope. Give a real-world context for your equation.
y = 4x; unit rate 4 equals slope 4; example: dollars earned = $4 per hour worked.
A scale drawing has a scale 1:200. On the drawing, a park has area 12 square centimeters. What is the actual area in square meters? (Show reasoning converting scale and units.)
Scale 1:200 means linear scale factor 200, area scale factor 200^2 = 40,000. 12 cm^2 on drawing → actual 12 * 40,000 = 480,000 cm^2 = 48 m^2.
Create a simple probability model for flipping two fair coins. List the sample space, assign probabilities, and predict the relative frequency of getting exactly one head after many trials. Then explain how observed frequencies might differ from the model.
Sample space for two coins: {HH, HT, TH, TT}. Prob(exactly one head) = 2/4 = 1/2 ≈ 0.5. Predicted relative frequency ≈ 50%;