Grade 8 Mathematics — Testing-Out Examination
A student who passes this examination has demonstrated mastery of the Common Core Grade 8 standards (8.NS, 8.EE, 8.F, 8.G, 8.SP) and is eligible to advance directly to Algebra 1.
Student Name
Proctor
Score
Instructions
- Time limit: 120 minutes total. Part I (no-calculator) about 45 minutes; Part II (calculator-permitted) about 75 minutes.
- Show all work. Final answers without supporting work earn at most half credit.
- For graphs, label both axes and at least one specific point.
- Express irrational numbers exactly (e.g., \(\sqrt{2},\ \pi\)) unless the problem requests rounding.
Part I · No Calculator · Problems 1–4 · 35 pts · ~45 min
1.
Number system, exponent rules, and scientific notation
8 points
- Classify each number as rational or irrational. Justify each in one short phrase: \[ \tfrac{7}{3}, \quad \sqrt{16}, \quad \sqrt{27}, \quad 0.\overline{45}, \quad \pi, \quad 0.1010010001\ldots \]
- Simplify each, expressing your answer with positive exponents only:
(i) \( 2^3 \cdot 2^{-5} \) (ii) \( \dfrac{x^4 \cdot x^{-7}}{x^{-2}} \) (iii) \( (3 m^2)^3 \) (iv) \( (2^{-2})^3 \)
- Compute \( (4 \times 10^7)(2.5 \times 10^{-3}) \) and write your answer in proper scientific notation.
- Estimate \( \sqrt{27} \) between two consecutive integers, and explain how you decided. Then state which is larger, \( \sqrt{27} \) or \( 5.2 \), and justify by squaring.
2.
Linear equations and slope
10 points
- Solve for \(x\): \( 4(x - 3) - 2x = 5x + 7 \). Show every step.
- Solve and classify each equation as having one solution, no solution, or infinitely many solutions:
(i) \( 3(2x + 1) = 6x + 3 \) (ii) \( 5x - 4 = 5x + 6 \) (iii) \( 2(x + 4) = 3x - 1 \)
- The points \((2, -1)\) and \((6, 7)\) lie on a line. Find the slope; then write the equation in slope-intercept form. Show your work.
- Two linear functions are described:
Function A: \(y = -2x + 9\)
Function B is given by the table:
\(x\) 0 2 4 6 \(y\) 5 2 -1 -4 Which function has the larger \(y\)-intercept? Which has the steeper slope (in absolute value)? Justify each.
3.
Simultaneous linear equations
8 points
- Solve graphically by sketching both lines on a coordinate plane: \[ \begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases} \] Identify the point of intersection. Verify your answer algebraically.
- Solve algebraically (substitution or elimination): \[ \begin{cases} 3x + 2y = 16 \\ x - y = 2 \end{cases} \] State which method you used and why it was efficient.
- A class of 30 students sold pies and cakes for a fundraiser. Pies sold for $8; cakes sold for $12. The total raised was $284. Set up and solve a system of equations to find how many pies and how many cakes were sold. Define your variables clearly.
4.
Pythagorean theorem and applications
9 points
- A right triangle has legs of length 9 and 12. Find the hypotenuse exactly.
- A right triangle has hypotenuse 13 and one leg of length 5. Find the other leg exactly.
- A triangle has side lengths 8, 15, and 17. Determine whether it is a right triangle. Justify by checking the Pythagorean relation.
- Find the distance between the points \(P(-3, 2)\) and \(Q(5, 8)\) using the Pythagorean theorem applied to a right triangle on the coordinate plane. Sketch the right triangle and label its legs.
- A rectangular field is 60 m by 80 m. Find the length of the diagonal across the field. Sketch the right triangle and label its legs.
Part II · Calculator Permitted · Problems 5–10 · 65 pts · ~75 min
5.
Functions and rate of change
10 points
Three functions are given in different representations.
Function A: \( y = 4x + 2 \).
Function B: the linear graph passing through \((-1, 7)\) and \((3, -1)\).
Function C: shown by the table.
| \(x\) | 1 | 3 | 5 | 7 |
|---|---|---|---|---|
| \(y\) | 5 | 11 | 17 | 23 |
- Find the slope and \(y\)-intercept of each function. Show your work for each.
- Rank the three functions by rate of change (greatest to least, by absolute value of slope).
- Which function has the greatest \(y\)-intercept?
- Construct a fourth linear function \(D\) verbally: “A taxi service charges a $4 base fare plus $1.50 per mile.” Write the function and identify its slope and intercept in context.
6.
Geometric transformations
10 points
Triangle \(\triangle ABC\) has vertices \(A(1, 1)\), \(B(4, 1)\), \(C(4, 5)\).
- Translate \(\triangle ABC\) by the vector \(\langle -5, 2 \rangle\). State the coordinates of the image \(\triangle A'B'C'\).
- Reflect \(\triangle A'B'C'\) (the image from (a)) across the \(x\)-axis. State the coordinates of \(\triangle A''B''C''\).
- Plot all three triangles (original, after translation, after reflection) on a single coordinate plane and label the vertices.
- Are \(\triangle ABC\) and \(\triangle A''B''C''\) congruent or merely similar? Justify in one sentence — what kinds of transformations preserve distance vs. only ratio?
- Now apply a dilation centered at the origin with scale factor 2 to \(\triangle ABC\), producing \(\triangle DEF\). State the coordinates of \(D, E, F\). Compute the ratio \( \text{area}(\triangle DEF) / \text{area}(\triangle ABC) \). Explain in one sentence why area scales as the square of the linear scale factor.
7.
Volume of curved solids
10 points
- Find the volume of a cylinder with radius 5 cm and height 12 cm. Express exactly (in terms of \(\pi\)) and decimally to the nearest cubic cm.
- Find the volume of a cone with radius 5 cm and height 12 cm. Express both exactly and decimally. Note the relationship between this volume and the cylinder in (a).
- Find the volume of a sphere with radius 5 cm. Express both exactly and decimally.
- A cylindrical can has volume \(144\pi\) cm\(^3\) and height 9 cm. Find its radius. Show your equation.
- A sphere has volume \(36\pi\) cm\(^3\). Find its radius. Show your equation.
8.
Bivariate data — scatterplot & line of best fit
10 points
The hours of TV watched per week and the GPAs of 8 randomly selected high school students:
| TV hours \(x\) | 2 | 5 | 7 | 10 | 12 | 15 | 18 | 22 |
|---|---|---|---|---|---|---|---|---|
| GPA \(y\) | 3.8 | 3.6 | 3.5 | 3.2 | 3.0 | 2.7 | 2.5 | 2.0 |
- Plot the eight data points on a labeled coordinate plane (x-axis: TV hours; y-axis: GPA). Choose appropriate scales.
- Sketch (by eye) a line of best fit. Identify two points your line passes through.
- A line of best fit for these data is given by \(\hat y = -0.08x + 3.85\). Interpret the slope in context. Use a complete sentence.
- Use the given line of best fit to predict the GPA of a student who watches 8 hours of TV per week. Round to one decimal. Then comment in one sentence on whether you should trust this prediction (interpolation vs. extrapolation).
9.
Two-way tables & association
9 points
One hundred fifty middle-school students were surveyed about whether they prefer pizza or burgers, broken out by gender.
| Pizza | Burgers | Total | |
|---|---|---|---|
| Boys | 32 | 43 | 75 |
| Girls | 49 | 26 | 75 |
| Total | 81 | 69 | 150 |
- What proportion of all students prefer pizza? What proportion of all students are girls?
- What proportion of boys prefer pizza? What proportion of girls prefer pizza? Compute each as a relative frequency.
- Based on your answers in (b), is there evidence of an association between gender and food preference? Justify in one or two sentences.
- Construct the table of conditional relative frequencies (each cell divided by its row total). Round to two decimals. State what the conditional row totals must equal.
10.
Multistep modeling — water tank
16 points
A cylindrical water tank has radius 3 ft and height 10 ft. It is initially full. A drain at the bottom releases water at a steady rate of 2 ft\(^3\) per minute.
- Find the initial volume of water in the tank. Express exactly (in terms of \(\pi\)) and decimally to the nearest cubic foot.
- Let \(V(t)\) denote the volume of water (in ft\(^3\)) remaining in the tank \(t\) minutes after draining begins. Write \(V(t)\) as a linear function of \(t\). Identify the slope and the \(V\)-intercept and state the meaning of each in context.
- How many minutes does it take to drain the tank completely? Solve algebraically, then round to the nearest minute. Express your answer in minutes-and-seconds form as well.
- Sketch \(V(t)\) on a labeled coordinate plane for \(0 \le t \le \) (the time you found in (c)). Identify both axis intercepts and state their meanings in context.
- The tank refills from above at a rate of 5 ft\(^3\) per minute, while the drain still releases 2 ft\(^3\) per minute. The tank starts empty when the refill begins; refill and drain run simultaneously. Write the new volume function \( \tilde V(t) \) and find the time at which the tank becomes full. Round to the nearest minute.
- Compare your two functions \(V(t)\) (drain only) and \(\tilde V(t)\) (refill with simultaneous drain). Sketch both on the same axes and identify the time at which they would yield the same volume — if any. Justify your answer.