Grade 8 Mathematics — Standards-Aligned Skills Examination
A calculation-focused, standards-walkthrough exam covering the CCSS-M Grade 8 standards (8.NS, 8.EE, 8.F, 8.G, 8.SP). Each problem cites the standard(s). Form B complements Form A.
Instructions
- Time limit: 120 minutes. Calculator permitted unless an item says otherwise.
- Show all work. Bare answers earn at most half credit.
- Each problem header cites CCSS-M codes the items target.
- Classify each number as rational or irrational. Justify each in one phrase:
\( \tfrac{4}{7},\ \sqrt{49},\ \sqrt{50},\ \pi,\ 0.\overline{75},\ 0.121221222\ldots,\ \sqrt[3]{27},\ \sqrt[3]{20} \)
- Express each as a fraction in simplest form: \(0.\overline{6},\ 0.\overline{27},\ 0.583\overline{3}\). For the first two, show the algebra: let \(x = 0.\overline{6}\), multiply, subtract.
- Approximate each irrational to two decimal places without a calculator, by squeezing between consecutive squares: \( \sqrt{30},\ \sqrt{75} \).
- Order from smallest to largest: \( \sqrt{15},\ 3.8,\ \tfrac{15}{4},\ \pi \).
- Simplify, expressing answers with positive exponents only:
(i) \( 3^4 \cdot 3^{-2} \) (ii) \( \dfrac{2^7}{2^3} \) (iii) \( (5^2)^3 \) (iv) \( 4^{-2} \cdot 4^{5} \) (v) \( \dfrac{x^3 \cdot x^{-5}}{x^{-1}} \) (vi) \( (2 a^3)^4 \)
- Compute each \(\sqrt{}\) or \(\sqrt[3]{}\) (perfect powers): \( \sqrt{144},\ \sqrt{0.16},\ \sqrt[3]{64},\ \sqrt[3]{125} \).
- Convert to scientific notation:
(i) 47{,}000 (ii) 0.000{,}059 (iii) 6{,}300{,}000 (iv) 0.0000{,}004
- Operate in scientific notation:
(i) \( (3 \times 10^5)(4 \times 10^{-2}) \) (ii) \( \dfrac{6 \times 10^{8}}{2 \times 10^{3}} \) (iii) \( (5 \times 10^{-3}) + (2 \times 10^{-3}) \)
- Find the slope of the line passing through:
(i) \( (2, 5) \) and \( (8, 14) \) (ii) \( (-3, 4) \) and \( (5, -2) \) (iii) \( (1, 7) \) and \( (4, 7) \) (iv) \( (3, -1) \) and \( (3, 8) \)
- Find the slope and \(y\)-intercept; then write the equation in slope-intercept form:
(i) \(2x + 3y = 12\) (ii) \(x - 4y = 8\) (iii) line through \((-1, 5)\) and \((3, -3)\)
- Solve each linear equation. Show every step:
(i) \( 3(x - 4) = 5x + 2 \) (ii) \( \dfrac{2x + 1}{3} = 5 \) (iii) \( 4x - 2(x + 3) = 8 \) (iv) \( 0.5x + 1.2 = 0.3x + 2.4 \)
- Classify the number of solutions for each equation:
(i) \( 2(x + 1) = 2x + 2 \) (ii) \( 3x + 4 = 3x + 9 \) (iii) \( 5x - 3 = 2x + 9 \)
- Solve each system by substitution or elimination:
(i) \(\begin{cases} y = 3x - 1 \\ 2x + y = 14 \end{cases}\) (ii) \(\begin{cases} 5x + 2y = 16 \\ 3x - 2y = 8 \end{cases}\) (iii) \(\begin{cases} x + 3y = 9 \\ 2x - y = 4 \end{cases}\) (iv) \(\begin{cases} 4x - 3y = 1 \\ 2x + 5y = 19 \end{cases}\)
- For each system, determine the number of solutions without solving. Justify by comparing slopes/intercepts:
(i) \(\begin{cases} 3x + y = 5 \\ 6x + 2y = 11 \end{cases}\) (ii) \(\begin{cases} 2x - y = 4 \\ 4x - 2y = 8 \end{cases}\)
- Determine whether each is a function. Justify in one phrase (one input → one output).
(i) \(\{(1,2), (3,4), (5,6)\}\) (ii) \(\{(1,2), (1,3), (4,5)\}\) (iii) the equation \(y = 2x + 1\) (iv) the equation \(x = y^2\)
- For \( f(x) = -2x + 7 \), compute \(f(0), f(3), f(-2)\), and find \(x\) for which \(f(x) = -1\).
- For each table, decide whether the relationship is linear. If so, find the slope and write the equation in slope-intercept form.
(i) \(x\) 0 1 2 3 \(y\) 5 8 11 14 (ii) \(x\) 0 1 2 3 \(y\) 1 2 4 8 - A water tank has 240 gallons and is being drained at 6 gallons per minute. Write a function \(V(t)\). Find when the tank is empty.
Triangle has vertices \( A(2, 1), B(5, 1), C(5, 4) \).
- Translate by \(\langle -4, 3\rangle\). State the new coordinates.
- Reflect across the \(x\)-axis. State the new coordinates.
- Reflect across the \(y\)-axis. State the new coordinates.
- Rotate 90° counterclockwise about the origin. State the new coordinates.
- Dilate from the origin by scale factor 2. State the new coordinates. State whether the image is congruent or merely similar to the original.
- Two parallel lines are cut by a transversal. Two corresponding angles are \( (3x + 18)° \) and \( (5x - 12)° \). Find \(x\).
- Same setup: alternate-interior angles are \( (4x + 5)° \) and \( (2x + 35)° \). Find \(x\).
- The exterior angle of a triangle equals the sum of the two non-adjacent interior angles. If two interior angles are 47° and 68°, find the exterior angle adjacent to the third interior angle.
- Find the missing side of each right triangle. Express each in simplest radical form when not a perfect square; also give a decimal approximation to two places.
(i) legs 6, 8 → hyp ___ (ii) legs 5, 12 → hyp ___ (iii) hyp 10, leg 6 → leg ___ (iv) legs 7, 24 → hyp ___ (v) legs 1, 2 → hyp ___
- Decide whether each side-length triple forms a right triangle:
(i) 9, 12, 15 (ii) 8, 15, 17 (iii) 5, 6, 8 (iv) 7, 24, 25
- Find the distance between each pair of points using the Pythagorean theorem:
(i) \( (1, 3) \) and \( (5, 6) \) (ii) \( (-2, 4) \) and \( (3, -1) \) (iii) \( (0, 0) \) and \( (8, -6) \)
- Find the diagonal of a rectangle with sides 9 and 12. (The diagonal is the hypotenuse of a right triangle whose legs are the two sides.)
- Compute each volume. Express in terms of \(\pi\) (exact) and to one decimal (use \(\pi \approx 3.14\)):
(i) cylinder, radius 4, height 10 (ii) cone, radius 6, height 8 (iii) sphere, radius 5 (iv) hemisphere, radius 7
- Find the missing dimension:
(i) cylinder, volume \(48\pi\), height 3 → radius ___ (ii) cone, volume \(36\pi\), radius 3 → height ___ (iii) sphere, volume \(\dfrac{32\pi}{3}\) → radius ___
- Find the volume of a hemisphere of radius 4 cm. Express in terms of \(\pi\) and as a decimal to one place.
Eight students recorded study hours \(x\) and quiz scores \(y\):
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| \(y\) | 62 | 68 | 72 | 78 | 82 | 85 | 88 | 92 |
- Plot the data on a labeled coordinate plane.
- Sketch a line of best fit (by eye). Identify two points your line passes through.
- A line of best fit for these data is given by \( \hat y = 4.2x + 58.4 \). Interpret the slope in context.
- Describe the direction (positive or negative) and the strength (strong or weak) of the association by inspecting the scatterplot. Justify in one sentence.
- Use the given line of best fit to predict the score for a student who studies 5.5 hours. Round to the nearest whole point.
One hundred eighth-graders were classified by participation in a sport (Yes / No) and music ensemble (Yes / No):
| Music: Yes | Music: No | Total | |
|---|---|---|---|
| Sport: Yes | 22 | 38 | 60 |
| Sport: No | 18 | 22 | 40 |
| Total | 40 | 60 | 100 |
- Compute relative frequencies (proportions) for each cell. Show the table.
- Compute the conditional relative frequency of music participation, given sport participation: \(P(\text{Music = Yes} \mid \text{Sport = Yes})\).
- Compute \(P(\text{Music = Yes} \mid \text{Sport = No})\). Compare to (b). Comment in one sentence on whether the table suggests an association between sport and music participation.