Troy School District · Mathematics · Grade 6
Form A · 120 minutes · 100 points

Grade 6 Mathematics — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Grade 6 standards (6.RP, 6.NS, 6.EE, 6.G, 6.SP) and is eligible to advance directly to Grade 7 Mathematics.

Student Name
Proctor
Score

Instructions

Part I · No Calculator · Problems 1–4 · 35 pts · ~45 min
1. Ratios, rates, and proportional reasoning 10 points
  1. A trail mix contains 3 cups of nuts for every 2 cups of dried fruit. If a baker uses 12 cups of nuts, how many cups of dried fruit are needed? Set up a ratio table to show your reasoning.
  2. A car travels 180 miles in 3 hours. Find the unit rate (miles per hour). Then find how far the car travels in 7 hours, assuming the same rate.
  3. A 12-oz box of cereal costs $4.20; a 18-oz box of the same cereal costs $5.85. Find the unit price (cents per ounce) for each. Which box is the better value? Show your work.
  4. Express each ratio in simplest form: \( 24 : 18 \), \( 45 : 30 : 15 \), \( \dfrac{2}{3} : \dfrac{4}{9} \).
2. Operations with fractions, decimals, and integers 10 points
  1. Compute exactly. Show your work and give answers in simplest form:

    (i) \( \dfrac{3}{4} + \dfrac{5}{6} \)     (ii) \( 4\dfrac{1}{2} - 1\dfrac{2}{3} \)     (iii) \( \dfrac{2}{5} \cdot \dfrac{15}{8} \)     (iv) \( \dfrac{9}{10} \div \dfrac{3}{5} \)

  2. Compute (decimals): (i) \( 12.4 + 8.07 - 5.39 \)     (ii) \( 4.2 \cdot 0.8 \)     (iii) \( 6.3 \div 0.9 \).
  3. Compute with integers (signed numbers):

    (i) \( -8 + 15 \)     (ii) \( 6 - (-9) \)     (iii) \( -7 \cdot 4 \)     (iv) \( -36 \div (-9) \)

  4. Find the GCF of 36 and 54. Find the LCM of 12 and 18. Briefly explain in one sentence the difference between GCF and LCM.
3. Expressions and one-variable equations 10 points
  1. Evaluate the expression \( 3a^2 - 2b + 5 \) when \(a = 4\) and \(b = -3\). Show every step using the order of operations.
  2. Simplify by combining like terms: \( 5x + 7 - 2x + 3x - 4 \).
  3. Use the distributive property to expand: \( 4(2y - 3) + 6 \).
  4. Solve each one-step equation. Show the inverse operation explicitly.

    (i) \( x + 17 = 25 \)     (ii) \( 6y = 42 \)     (iii) \( \dfrac{m}{4} = 9 \)

  5. Solve a two-step equation: \( 3x + 7 = 22 \). Show every step.
4. Coordinate plane and absolute value 5 points
  1. Plot the points \(A(-3, 4)\), \(B(2, 4)\), \(C(2, -1)\), \(D(-3, -1)\) on a coordinate plane. Connect them in order. What shape is formed? Find its perimeter and area by counting units.
  2. Find \(|{-7}| - |{3}|\) and \(|{-2 + 5}|\). Show your work.
  3. The point \( (-5, 7) \) is reflected over the \(y\)-axis. State the new coordinates.
Part II · Calculator Permitted · Problems 5–10 · 65 pts · ~75 min
5. Percent applications 10 points
  1. Convert each: 0.45 to a percent; 7/20 to a percent; 175% to a decimal; 8% to a fraction in simplest form.
  2. Find 30% of 240. Show your work using either a proportion or multiplication.
  3. What percent of 80 is 28?
  4. A meal costs $48. A 20% tip is added. Find the total cost.
  5. A T-shirt is on sale for $18, marked down 25% from its original price. Find the original price. Show the equation you set up.
6. Area of polygons (composite figures) 10 points
  1. Find the area of a parallelogram with base 9 cm and height 5 cm.
  2. Find the area of a triangle with base 14 in and height 6 in.
  3. Find the area of a trapezoid with parallel sides of length 8 ft and 12 ft, and height 5 ft. Show the formula \( A = \tfrac{1}{2}(b_1 + b_2) h \).
  4. A composite figure is formed by joining a 6-by-8 rectangle and a triangle whose base is 8 (the rectangle's top) and whose height is 4 (extending upward). Sketch the figure on a coordinate grid; then compute its total area.
  5. An irregular polygon has vertices at \((0, 0)\), \((6, 0)\), \((8, 4)\), \((6, 7)\), \((0, 7)\). Find the area by decomposing into a rectangle and a triangle. Show the decomposition.
7. Volume and surface area of rectangular prisms 10 points
  1. A rectangular prism has dimensions 4 cm by 7 cm by 12 cm. Find its volume.
  2. Find the total surface area of the prism in (a). Show the six face areas as a sum (or use the formula \(2(\ell w + \ell h + w h)\) and verify).
  3. A rectangular box has dimensions \(2\dfrac{1}{2}\) ft by \(1\dfrac{1}{2}\) ft by 4 ft. Find its volume in cubic feet. Show the multiplication of mixed numbers.
  4. How many small cubes of edge length \(\dfrac{1}{2}\) ft are needed to fill the box in (c)? Show your reasoning by either dividing the volume or counting cubes per dimension.
  5. A net (unfolded prism) for a rectangular box is laid flat. The net consists of how many rectangles? Sketch one possible net for a 3-by-4-by-5 box and label the dimensions of each rectangular face.
8. Statistics — center, spread, and displays 12 points

A 6th-grade class collected the daily reading minutes of 12 students:

15, 22, 30, 30, 35, 40, 42, 45, 50, 55, 60, 90

  1. Compute the mean. Round to one decimal.
  2. Find the median.
  3. Find the mode.
  4. Find the range and the interquartile range (IQR). Show how you split the data to compute Q1 and Q3.
  5. Sketch a box plot of the data above a number line. Label all five summary values.
  6. The 90 in the data set is much larger than most other values. Identify it as an outlier by inspection and justify in one sentence (e.g., it is far above the rest of the data). If 90 is removed, compute the new mean and compare to the original. Comment in one sentence on which measure of center (mean or median) is more affected by the outlier.
9. Proportional reasoning & mixed-rate problem 10 points

A printer prints 24 pages in 6 minutes. A second printer prints 35 pages in 7 minutes.

  1. Find the unit rate (pages per minute) for each printer.
  2. Which printer is faster? By how many pages per minute?
  3. If both printers run simultaneously, how long will it take to print 100 pages? Set up an equation using the combined rate and solve. Round to the nearest tenth of a minute.
  4. If only the slower printer runs, how long will it take to print 100 pages? Round to the nearest tenth of a minute.
10. Multistep modeling — class trip budget 13 points

A 6th-grade class of 28 students is planning a one-day field trip. The bus rental is a flat $480. Each museum ticket is $14.50 per student plus $18 per chaperone. Two chaperones will accompany the class. Lunch costs $9 per student and $11 per chaperone.

  1. Compute the total cost of the museum tickets (students + chaperones).
  2. Compute the total cost of lunches.
  3. Compute the grand total cost of the trip (bus + tickets + lunches).
  4. If the cost is divided equally among the 28 students, how much should each student pay? Round up to the nearest cent (parents typically pay in whole-cent amounts).
  5. The class has $200 in their fundraiser account. After applying this $200 to the total, how much does each student now pay? Round up to the nearest cent.
  6. Suppose 2 students cannot attend at the last minute, but the bus and chaperones still go. Recompute the per-student cost (using only museum tickets and lunches for the remaining 26 students, plus the full bus and the 2 chaperones). Round up to the nearest cent. Compare to (e) — by how much did each student's share increase?