Troy School District · Mathematics · Grade 4
Form A · 120 minutes · 100 points

Grade 4 Mathematics — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Grade 4 standards (4.OA, 4.NBT, 4.NF, 4.MD, 4.G) and is eligible to advance directly to Grade 5 Mathematics.

Student Name
Proctor
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Instructions

Part I · No Calculator · Problems 1–4 · 40 pts · ~60 min
1. Place value and multi-digit operations 10 points
  1. Write the number 47,506 in:

    (i) word form     (ii) expanded form     (iii) place-value chart with the value of each digit

  2. Compare using \(<\), \(>\), or \(=\):    (i) 32{,}458 ___ 32{,}548     (ii) 9{,}999 ___ 10{,}010     (iii) 605{,}004 ___ 605{,}040.
  3. Use the standard way (the algorithm) to find each answer. Show carries (regrouping) clearly.

    (i) \(4{,}632 + 8{,}547\)     (ii) \(7{,}004 - 2{,}856\)     (iii) \(63{,}500 - 28{,}769\)

  4. Round each number to the place named:

    (i) 4{,}567 to the nearest hundred    (ii) 28{,}349 to the nearest thousand    (iii) 154{,}998 to the nearest ten thousand

2. Factors, multiples, and patterns 8 points
  1. List all the factor pairs of 36. (Hint: pairs of numbers whose product is 36.) Then state whether 36 is prime or composite, and explain in one phrase.
  2. List the first six multiples of 7. Then decide whether each of the following is a multiple of 7: 49, 56, 60, 84.
  3. A pattern starts at 4 and follows the rule "add 5." List the first eight terms. Then state whether the pattern produces only even numbers, only odd numbers, or alternates. Explain in one sentence.
  4. A pattern starts at 1 and follows the rule "multiply by 2." List the first six terms. Find the sum of the first four terms.
3. Fractions — equivalence and operations 12 points
  1. Write four fractions equivalent to \( \dfrac{3}{4} \). For one of them, show the multiplication that produces it from \( \dfrac{3}{4} \).
  2. Are \( \dfrac{6}{8} \) and \( \dfrac{9}{12} \) equivalent? Explain by reducing each to simplest form (or by cross-multiplication).
  3. Compare each pair using \(<\), \(>\), or \(=\). Show your work — find a common denominator or use a benchmark like \(\dfrac{1}{2}\).

    (i) \( \dfrac{2}{3} \ ?\ \dfrac{3}{5} \)     (ii) \( \dfrac{5}{8} \ ?\ \dfrac{1}{2} \)     (iii) \( \dfrac{7}{12} \ ?\ \dfrac{3}{4} \)

  4. Add or subtract (like denominators only); write in simplest form:

    (i) \( \dfrac{2}{6} + \dfrac{3}{6} \)     (ii) \( \dfrac{7}{8} - \dfrac{3}{8} \)     (iii) \( 2\dfrac{1}{5} + 1\dfrac{2}{5} \)     (iv) \( 5 - 2\dfrac{3}{4} \)

  5. Multiply a fraction by a whole number: (i) \( 4 \times \dfrac{2}{5} \)    (ii) \( 6 \times \dfrac{3}{8} \). Show the multiplication and reduce if needed. Write improper fractions as mixed numbers.
4. Decimal notation for fractions 10 points
  1. Write each fraction as a decimal: (i) \( \dfrac{3}{10} \)    (ii) \( \dfrac{47}{100} \)    (iii) \( \dfrac{6}{10} + \dfrac{4}{100} \)    (iv) \( \dfrac{9}{100} \).
  2. Write each decimal as a fraction in simplest form:

    (i) 0.7     (ii) 0.25     (iii) 0.50     (iv) 0.08

  3. Compare using \(<\), \(>\), or \(=\):

    (i) 0.7 ___ 0.70     (ii) 0.4 ___ 0.39     (iii) 0.65 ___ \( \dfrac{2}{3} \)     (iv) 1.50 ___ 1.5

  4. Place each number on a number line drawn from 0 to 1, marked in tenths: 0.3, 0.85, \( \dfrac{1}{2} \), \( \dfrac{7}{10} \).
Part II · Calculator Permitted · Problems 5–10 · 60 pts · ~60 min
5. Multistep word problem 12 points

A school library has 4{,}287 books. The librarian buys 526 new books and donates 314 old books to a community center.

  1. How many books does the library have after these changes? Show your number sentence and computation.
  2. The library wants to organize the books into bookcases. Each bookcase holds 100 books. How many full bookcases can the library fill, and how many books are left over? (Use long division.)
  3. The librarian decides to give 4 books each to 87 students for summer reading. How many books does that take? Use multi-digit multiplication. Show the standard algorithm.
  4. After distributing the summer-reading books, how many books remain in the library? (Subtract the result of (c) from your answer in (a).)
  5. The library budget allots $5 per new book. How much did the 526 new books cost? Show the multiplication.
6. Multi-digit multiplication and division 10 points
  1. Find (standard algorithm; show partial products if helpful):

    (i) \(243 \times 6\)     (ii) \(1{,}472 \times 8\)     (iii) \(54 \times 36\)     (iv) \(238 \times 47\)

  2. Divide using long division. State the quotient and remainder:

    (i) \(486 \div 6\)     (ii) \(875 \div 7\)     (iii) \(2{,}943 \div 4\)     (iv) \(6{,}305 \div 9\)

  3. Estimate \(624 \times 25\) by rounding each factor to the nearest ten. Then find the exact product. State the difference between your estimate and the exact answer.
7. Perimeter, area, and conversions 10 points
  1. Find the perimeter and area of a rectangle that is 14 cm long and 9 cm wide.
  2. The perimeter of a rectangle is 50 m and its length is 17 m. Find its width. Show the equation you used.
  3. A composite figure is made of two rectangles: rectangle A is 8 ft by 5 ft; rectangle B is 4 ft by 3 ft, attached to one side of A. Sketch the figure (any valid attachment), then find the total area.
  4. Convert (use 1 ft = 12 in, 1 yd = 3 ft, 1 km = 1000 m, 1 kg = 1000 g):

    (i) 5 ft = ___ in     (ii) 4 yd = ___ ft     (iii) 3.5 km = ___ m     (iv) 7{,}200 g = ___ kg

  5. An athlete runs 4 laps around a rectangular track that is 100 m by 60 m. How many meters did they run? Convert to kilometers.
8. Angles, lines, and classification 10 points
  1. Tell whether each is a point, line, line segment, ray, or angle:

    (i) starts at one endpoint and extends in one direction forever    (ii) extends forever in two directions    (iii) has two endpoints    (iv) two rays sharing a common endpoint

  2. Classify each angle by its measure: 30°, 90°, 145°, 180°. Use the labels acute, right, obtuse, straight.
  3. Two angles together form a straight line. One measures 47°. Find the measure of the other. Show the equation.
  4. An angle is divided into two non-overlapping parts. The whole angle measures 88°; one part measures 35°. Find the other part.
  5. Use a protractor (if available) or a careful sketch: draw an angle of 65°. Label the vertex and the rays.
9. Classifying two-dimensional shapes & symmetry 8 points
  1. Classify each triangle by its angles (acute, right, obtuse) and by its sides (equilateral, isosceles, scalene). Sketch each:

    (i) angles 60°, 60°, 60°    (ii) angles 45°, 45°, 90°    (iii) angles 30°, 60°, 90°    (iv) angles 100°, 40°, 40°

  2. Sketch each two-dimensional shape and draw all lines of symmetry. State how many lines of symmetry each has:

    (i) square    (ii) equilateral triangle    (iii) rectangle (non-square)    (iv) regular pentagon

  3. Which capital letters of the alphabet have at least one line of symmetry? List as many as you can identify (sketches will help).
10. Line plot of fractions in measurement context 10 points

A fourth-grade class measured the lengths of pencils (in inches) at the start of the year. The pencils' lengths are:

\(7\dfrac{1}{4},\quad 6\dfrac{1}{2},\quad 7,\quad 5\dfrac{3}{4},\quad 7\dfrac{1}{4},\quad 6\dfrac{3}{4},\quad 7\dfrac{1}{2},\quad 7,\quad 6\dfrac{1}{2},\quad 7\dfrac{1}{4}\)

  1. Make a line plot for the data, with a number line marked in quarters from 5 to 8 inches. Place an "X" above each length for each pencil.
  2. How many pencils are at least \(7\) inches long?
  3. Find the difference (in inches) between the longest pencil and the shortest pencil. Show the subtraction of mixed numbers.
  4. If the 10 pencils were laid end-to-end (in a single line), what would be the total length? Add carefully — group whole numbers and fractions separately, then combine. Show your work.