Troy School District · Mathematics · Grade 5
Form A · 120 minutes · 100 points

Grade 5 Mathematics — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Grade 5 standards (5.OA, 5.NBT, 5.NF, 5.MD, 5.G) and is eligible to advance directly to Grade 6 Mathematics.

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Proctor
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Instructions

Part I · No Calculator · Problems 1–4 · 35 pts · ~50 min
1. Place value, powers of 10, and order of operations 10 points
  1. Write the value of the digit 7 in each number:

    (i) 47.236     (ii) 0.0875     (iii) 7,142.0     (iv) 132.470

  2. Find each answer and explain the place-value pattern in one sentence:

    (i) \( 4.36 \times 10 = ?\)     (ii) \( 4.36 \times 100 = ? \)     (iii) \( 4.36 \div 10 = ? \)     (iv) \( 4.36 \div 100 = ? \)

  3. Round 27.4856 to the nearest:

    (i) tenth     (ii) hundredth     (iii) thousandth     (iv) whole number

  4. Find the value using the order of operations. Show every step:

    (i) \( 5 + 3 \times (8 - 2) \)     (ii) \( 24 \div (4 + 2) - 3 \)     (iii) \( 18 - 2 \times (5 + 3) \div 4 \)

2. Decimal operations to the hundredths 10 points
  1. Find (show your work, vertical alignment with decimal points):

    (i) \( 12.45 + 8.7 + 0.36 \)     (ii) \( 25.6 - 17.84 \)

  2. Multiply (show standard algorithm):

    (i) \( 4.6 \times 1.2 \)     (ii) \( 0.25 \times 8.4 \)     (iii) \( 7.05 \times 0.6 \)

  3. Divide (show your work; you may multiply both divisor and dividend by a power of 10 to get a whole-number divisor):

    (i) \( 14.4 \div 6 \)     (ii) \( 4.8 \div 0.4 \)     (iii) \( 16.5 \div 0.25 \)

  4. Estimate (round each number to a convenient value first; then find the exact answer to check): \( 4.93 \times 6.18 \). State your estimate, your exact product, and the difference between them.
3. Fraction operations with unlike denominators 10 points
  1. Add or subtract; write the answer in simplest form (find a common denominator first):

    (i) \( \dfrac{2}{3} + \dfrac{1}{4} \)     (ii) \( \dfrac{5}{6} - \dfrac{1}{4} \)     (iii) \( 2\dfrac{1}{2} + 1\dfrac{3}{8} \)     (iv) \( 4\dfrac{1}{6} - 1\dfrac{2}{3} \)

  2. Multiply (write in simplest form):

    (i) \( \dfrac{3}{5} \times \dfrac{10}{9} \)     (ii) \( \dfrac{2}{3} \times 12 \)     (iii) \( 1\dfrac{1}{2} \times 2\dfrac{2}{3} \)

  3. Divide (write in simplest form):

    (i) \( \dfrac{3}{4} \div \dfrac{1}{2} \)     (ii) \( 6 \div \dfrac{2}{3} \)     (iii) \( \dfrac{5}{6} \div 4 \)

  4. Place each result on a number line marked from 0 to 4: \( \dfrac{7}{4}, \quad \dfrac{8}{3}, \quad 1\dfrac{1}{8}, \quad \dfrac{15}{4} \).
4. Coordinate plane (Quadrant I) 5 points
  1. Plot \(A(2, 5)\), \(B(7, 5)\), \(C(7, 2)\), \(D(2, 2)\) on a coordinate plane. Connect them in order. Identify the shape and find its area and perimeter.
  2. List the coordinates of two points whose \(x\)-coordinate is twice their \(y\)-coordinate (e.g., \((2, 1)\)). Plot both, and observe the line they lie on.
  3. State the coordinates of the origin. State the coordinates of any point on the \(x\)-axis (other than the origin).
Part II · Calculator Permitted · Problems 5–10 · 65 pts · ~70 min
5. Multistep word problems with fractions and decimals 10 points
  1. A pitcher contains \(2\dfrac{1}{4}\) gallons of juice. Maria pours out \(\dfrac{3}{8}\) gallon for breakfast and \(1\dfrac{1}{2}\) gallons for lunch. How much juice is left? Show a number sentence.
  2. A board is 96 inches long. A carpenter cuts strips each \(\dfrac{3}{4}\) inch wide (with no waste). How many strips can the carpenter cut?
  3. A grocery store sells apples for $1.85 per pound. A customer buys 3.6 pounds of apples and pays with a $20 bill. How much change should they receive? Round to the nearest cent.
  4. A school orders 38 boxes of pencils. Each box contains 124 pencils. Each pencil costs $0.15. Find the total number of pencils and the total cost. Round to the nearest cent.
6. Volume of rectangular prisms 10 points
  1. Find the volume of a rectangular prism with length 8 cm, width 5 cm, and height 12 cm.
  2. A rectangular shipping carton has dimensions 30 in by 18 in by 12 in. Find its volume in cubic inches. Then convert to cubic feet (1 ft\(^3\) = 12 \(\times\) 12 \(\times\) 12 = 1728 in\(^3\)). Round to the nearest 0.01 ft\(^3\).
  3. A composite solid is made by stacking two rectangular prisms: the lower has dimensions 8 by 5 by 4, the upper has dimensions 4 by 3 by 5 sitting on top of the larger. Find the total volume.
  4. A fish tank is 24 in long, 12 in wide, and 16 in tall. The tank is filled to a depth of 10 inches. How many cubic inches of water are in the tank?
7. Numerical patterns and ordered pairs 10 points

Two number patterns are generated.

Pattern A starts at 0 and follows the rule "add 3."

Pattern B starts at 0 and follows the rule "add 6."

  1. List the first six terms of each pattern.
  2. Form ordered pairs \((A, B)\) using corresponding terms. List all six ordered pairs.
  3. Plot the ordered pairs on a coordinate plane. What do you notice about their arrangement?
  4. State the relationship between corresponding terms in words. (e.g., "Each term in B is ___ times the corresponding term in A.")
  5. If Pattern A's tenth term is 27, what is Pattern B's tenth term? Use the relationship from (d).
8. Measurement conversions and line plots 10 points
  1. Convert (show the conversion factor):

    (i) 5.2 km = ___ m    (ii) 4500 g = ___ kg    (iii) 3 hr 15 min = ___ min    (iv) 12 ft = ___ in    (v) 8 cups = ___ quarts

  2. Find the total and convert: A bottle holds 750 mL. Eight bottles hold ___ liters. (1 L = 1000 mL.)
  3. The lengths (in inches) of insects collected by a science class are: \( \dfrac{1}{4},\ \dfrac{3}{8},\ \dfrac{1}{2},\ \dfrac{1}{2},\ \dfrac{3}{8},\ \dfrac{5}{8},\ \dfrac{3}{4},\ \dfrac{1}{2},\ \dfrac{3}{8},\ \dfrac{1}{4},\ \dfrac{5}{8},\ \dfrac{1}{2}.\) Make a line plot for these lengths on a number line marked in eighths.
  4. From the line plot in (c), find the total combined length of all insects whose length is at least \(\dfrac{1}{2}\) inch. Show the addition of fractions.
9. Classifying two-dimensional figures 8 points
  1. Classify each polygon by the most specific name. Explain in one phrase what defining property pins down that name.

    (i) A four-sided figure with four right angles and four sides of equal length.
    (ii) A four-sided figure with two pairs of parallel sides but no right angles.
    (iii) A four-sided figure with exactly one pair of parallel sides.
    (iv) A three-sided figure with all sides equal.

  2. Explain in one or two sentences why every square is a rectangle but not every rectangle is a square. Use the language of "subcategory" or "set inside another set."
  3. Sort the following shapes into a category hierarchy: quadrilateral, parallelogram, rhombus, square, rectangle, trapezoid. Draw a tree diagram or Venn diagram showing which categories contain which.
10. Multistep problem — building a vegetable garden 17 points

A family is building a rectangular vegetable garden \(8\dfrac{1}{2}\) ft long and \(6\dfrac{1}{4}\) ft wide. They will install a low fence around the perimeter.

  1. Find the perimeter of the garden. Show the addition of mixed numbers.
  2. Fencing material is sold in 1-foot sections costing $2.50 each. The store does not sell fractional sections. How many sections are needed, and what does the fencing cost? Show the rounding-up step.
  3. Find the area of the garden in square feet (multiplication of mixed numbers).
  4. The family budgeted $80 for fencing. Are they within budget? By how much over or under?