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3B · Chapter 9 · Solutions

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Fractions of a Whole

Prerequisite

  1. 5 — There are four equal groups; place one counter into each tray repeatedly until all 20 are used. Each tray receives five counters, since 4 × 5 = 20.
  2. 6 — The denominator tells all equal pieces.
  3. No — A half must be one of two equal-sized parts.

Understanding checks

  1. 1/3 — The denominator counts all 3 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/3.
  2. 2/5 — The denominator counts all 5 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 2/5.
  3. 5/6 — The denominator counts all 12 equal parts; the numerator counts the 10 shaded parts. In simplest form this is 5/6.

Guided

  1. 3/5 — The denominator counts all 5 equal parts; the numerator counts the 3 shaded parts. In simplest form this is 3/5.
  2. 3/8 — The denominator counts all 8 equal parts; the numerator counts the 3 shaded parts. In simplest form this is 3/8.
  3. 2/5 — The denominator counts all 10 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 2/5.
  4. 1/6 — The denominator counts all 12 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 1/6.
  5. 9/10 — The denominator counts all 10 equal parts; the numerator counts the 9 shaded parts. In simplest form this is 9/10.

Independent

  1. 2/3 — The denominator counts all 3 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 2/3.
  2. 3/4 — The denominator counts all 4 equal parts; the numerator counts the 3 shaded parts. In simplest form this is 3/4.
  3. 11/12 — The denominator counts all 12 equal parts; the numerator counts the 11 shaded parts. In simplest form this is 11/12.
  4. 1/4 — The denominator counts all 4 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/4.
  5. 1/2 — The denominator counts all 8 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 1/2.
  6. 7/12 — The denominator counts all 12 equal parts; the numerator counts the 7 shaded parts. In simplest form this is 7/12.
  7. 7/8 — The denominator counts all 8 equal parts; the numerator counts the 7 shaded parts. In simplest form this is 7/8.
  8. 1/5 — The denominator counts all 5 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/5.

Challenge

  1. No — Equal fractions of differently sized wholes need not represent equal physical lengths.
  2. 5 selected out of 7 equal parts — The denominator gives the number of equal pieces in one whole; the numerator counts selected pieces.

Exit

  1. 2/3 — The denominator counts all 6 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 2/3.
  2. 3/4 — The denominator counts all 8 equal parts; the numerator counts the 6 shaded parts. In simplest form this is 3/4.
  3. 1/6 — The denominator counts all 6 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/6.

Fractions on a Number Line

Prerequisite

  1. 2/3 — The denominator counts all 6 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 2/3.
  2. 5/6 — The denominator counts all 12 equal parts; the numerator counts the 10 shaded parts. In simplest form this is 5/6.
  3. 2/3 — The denominator counts all 3 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 2/3.

Understanding checks

  1. 7/10 — Each interval represents 1/10. Moving 7 intervals from zero reaches 7/10, or 7/10 in simplest form.
  2. 4/5 — Each interval represents 1/5. Moving 4 intervals from zero reaches 4/5, or 4/5 in simplest form.
  3. 1/6 — Each interval represents 1/12. Moving 2 intervals from zero reaches 2/12, or 1/6 in simplest form.

Guided

  1. 2/5 — Each interval represents 1/5. Moving 2 intervals from zero reaches 2/5, or 2/5 in simplest form.
  2. 1/3 — Each interval represents 1/3. Moving 1 intervals from zero reaches 1/3, or 1/3 in simplest form.
  3. 1/4 — Each interval represents 1/4. Moving 1 intervals from zero reaches 1/4, or 1/4 in simplest form.
  4. 2/3 — Each interval represents 1/6. Moving 4 intervals from zero reaches 4/6, or 2/3 in simplest form.
  5. 7/8 — Each interval represents 1/8. Moving 7 intervals from zero reaches 7/8, or 7/8 in simplest form.

Independent

  1. 2/3 — Each interval represents 1/3. Moving 2 intervals from zero reaches 2/3, or 2/3 in simplest form.
  2. 1/4 — Each interval represents 1/8. Moving 2 intervals from zero reaches 2/8, or 1/4 in simplest form.
  3. 3/5 — Each interval represents 1/5. Moving 3 intervals from zero reaches 3/5, or 3/5 in simplest form.
  4. 1/2 — Each interval represents 1/6. Moving 3 intervals from zero reaches 3/6, or 1/2 in simplest form.
  5. 3/5 — Each interval represents 1/10. Moving 6 intervals from zero reaches 6/10, or 3/5 in simplest form.
  6. 4/5 — Each interval represents 1/10. Moving 8 intervals from zero reaches 8/10, or 4/5 in simplest form.
  7. 3/4 — Each interval represents 1/12. Moving 9 intervals from zero reaches 9/12, or 3/4 in simplest form.
  8. 5/12 — Each interval represents 1/12. Moving 5 intervals from zero reaches 5/12, or 5/12 in simplest form.

Challenge

  1. 1/2 — 7/14 is equivalent to 1/2 because half the equal intervals have been crossed.
  2. 7 equal intervals — To show 1/7, the full unit length must be divided into 7 equal intervals, not 6.

Exit

  1. 1/5 — Each interval represents 1/5. Moving 1 intervals from zero reaches 1/5, or 1/5 in simplest form.
  2. 3/4 — Each interval represents 1/8. Moving 6 intervals from zero reaches 6/8, or 3/4 in simplest form.
  3. 5/8 — Each interval represents 1/8. Moving 5 intervals from zero reaches 5/8, or 5/8 in simplest form.

Comparing Fractions with Like Denominators

Prerequisite

  1. 1/4 — Each interval represents 1/12. Moving 3 intervals from zero reaches 3/12, or 1/4 in simplest form.
  2. 3/5 — Each interval represents 1/5. Moving 3 intervals from zero reaches 3/5, or 3/5 in simplest form.
  3. 1/5 — Each interval represents 1/5. Moving 1 intervals from zero reaches 1/5, or 1/5 in simplest form.

Understanding checks

  1. < — Both fractions use pieces of size 1/12, so compare 3 pieces with 11 pieces: 3/12 < 11/12.
  2. < — Both fractions use pieces of size 1/10, so compare 3 pieces with 9 pieces: 3/10 < 9/10.
  3. < — Both fractions use pieces of size 1/3, so compare 1 pieces with 2 pieces: 1/3 < 2/3.

Guided

  1. > — Both fractions use pieces of size 1/6, so compare 5 pieces with 3 pieces: 5/6 > 3/6.
  2. < — Both fractions use pieces of size 1/5, so compare 1 pieces with 2 pieces: 1/5 < 2/5.
  3. > — Both fractions use pieces of size 1/4, so compare 2 pieces with 1 pieces: 2/4 > 1/4.
  4. < — Both fractions use pieces of size 1/6, so compare 1 pieces with 3 pieces: 1/6 < 3/6.
  5. > — Both fractions use pieces of size 1/10, so compare 6 pieces with 2 pieces: 6/10 > 2/10.

Independent

  1. < — Both fractions use pieces of size 1/12, so compare 2 pieces with 6 pieces: 2/12 < 6/12.
  2. > — Both fractions use pieces of size 1/12, so compare 8 pieces with 3 pieces: 8/12 > 3/12.
  3. > — Both fractions use pieces of size 1/8, so compare 6 pieces with 4 pieces: 6/8 > 4/8.
  4. < — Both fractions use pieces of size 1/6, so compare 2 pieces with 3 pieces: 2/6 < 3/6.
  5. < — Both fractions use pieces of size 1/6, so compare 1 pieces with 2 pieces: 1/6 < 2/6.
  6. > — Both fractions use pieces of size 1/4, so compare 3 pieces with 2 pieces: 3/4 > 2/4.
  7. > — Both fractions use pieces of size 1/10, so compare 6 pieces with 1 pieces: 6/10 > 1/10.
  8. < — Both fractions use pieces of size 1/5, so compare 2 pieces with 3 pieces: 2/5 < 3/5.

Challenge

  1. 5/10 — The pieces are equal-sized, so five of them exceed two.
  2. 3/9 > 3/10 — Each piece is larger when the whole is divided into fewer equal parts; taking three larger pieces gives more.

Exit

  1. < — Both fractions use pieces of size 1/8, so compare 2 pieces with 7 pieces: 2/8 < 7/8.
  2. > — Both fractions use pieces of size 1/5, so compare 3 pieces with 1 pieces: 3/5 > 1/5.
  3. < — Both fractions use pieces of size 1/8, so compare 2 pieces with 4 pieces: 2/8 < 4/8.

Comparing Fractions with Like Numerators

Prerequisite

  1. < — Both fractions use pieces of size 1/4, so compare 2 pieces with 3 pieces: 2/4 < 3/4.
  2. > — Both fractions use pieces of size 1/3, so compare 2 pieces with 1 pieces: 2/3 > 1/3.
  3. > — Both fractions use pieces of size 1/8, so compare 4 pieces with 2 pieces: 4/8 > 2/8.

Understanding checks

  1. > — There are 3 pieces in each fraction, but 1/4 is a larger piece than 1/6 when the whole is identical.
  2. < — There are 6 pieces in each fraction, but 1/8 is a larger piece than 1/10 when the whole is identical.
  3. < — There are 6 pieces in each fraction, but 1/10 is a larger piece than 1/12 when the whole is identical.

Guided

  1. < — There are 3 pieces in each fraction, but 1/6 is a larger piece than 1/8 when the whole is identical.
  2. < — There are 1 pieces in each fraction, but 1/4 is a larger piece than 1/10 when the whole is identical.
  3. > — There are 1 pieces in each fraction, but 1/8 is a larger piece than 1/10 when the whole is identical.
  4. > — There are 3 pieces in each fraction, but 1/6 is a larger piece than 1/10 when the whole is identical.
  5. > — There are 2 pieces in each fraction, but 1/3 is a larger piece than 1/5 when the whole is identical.

Independent

  1. < — There are 3 pieces in each fraction, but 1/5 is a larger piece than 1/8 when the whole is identical.
  2. > — There are 3 pieces in each fraction, but 1/5 is a larger piece than 1/8 when the whole is identical.
  3. > — There are 1 pieces in each fraction, but 1/4 is a larger piece than 1/8 when the whole is identical.
  4. > — There are 2 pieces in each fraction, but 1/3 is a larger piece than 1/6 when the whole is identical.
  5. > — There are 1 pieces in each fraction, but 1/3 is a larger piece than 1/10 when the whole is identical.
  6. < — There are 2 pieces in each fraction, but 1/3 is a larger piece than 1/4 when the whole is identical.
  7. > — There are 3 pieces in each fraction, but 1/6 is a larger piece than 1/12 when the whole is identical.
  8. < — There are 2 pieces in each fraction, but 1/8 is a larger piece than 1/12 when the whole is identical.

Challenge

  1. 5/6 — The pieces are equal-sized, so five of them exceed two.
  2. 3/5 > 3/6 — Each piece is larger when the whole is divided into fewer equal parts; taking three larger pieces gives more.

Exit

  1. > — There are 4 pieces in each fraction, but 1/5 is a larger piece than 1/8 when the whole is identical.
  2. < — There are 2 pieces in each fraction, but 1/4 is a larger piece than 1/6 when the whole is identical.
  3. > — There are 2 pieces in each fraction, but 1/5 is a larger piece than 1/10 when the whole is identical.

Practice

  1. 2/3 — The denominator counts all 3 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 2/3.
  2. 3/5 — Each interval represents 1/5. Moving 3 intervals from zero reaches 3/5, or 3/5 in simplest form.
  3. < — Both fractions use pieces of size 1/5, so compare 1 pieces with 2 pieces: 1/5 < 2/5.
  4. > — There are 1 pieces in each fraction, but 1/6 is a larger piece than 1/12 when the whole is identical.
  5. 1/8 — The denominator counts all 8 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/8.
  6. 5/8 — Each interval represents 1/8. Moving 5 intervals from zero reaches 5/8, or 5/8 in simplest form.
  7. > — Both fractions use pieces of size 1/10, so compare 5 pieces with 3 pieces: 5/10 > 3/10.
  8. > — There are 4 pieces in each fraction, but 1/10 is a larger piece than 1/12 when the whole is identical.
  9. 7/12 — The denominator counts all 12 equal parts; the numerator counts the 7 shaded parts. In simplest form this is 7/12.
  10. 2/3 — Each interval represents 1/3. Moving 2 intervals from zero reaches 2/3, or 2/3 in simplest form.
  11. < — Both fractions use pieces of size 1/10, so compare 5 pieces with 9 pieces: 5/10 < 9/10.
  12. > — There are 2 pieces in each fraction, but 1/5 is a larger piece than 1/10 when the whole is identical.
  13. 1/3 — The denominator counts all 3 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/3.
  14. 1/4 — Each interval represents 1/4. Moving 1 intervals from zero reaches 1/4, or 1/4 in simplest form.
  15. > — Both fractions use pieces of size 1/4, so compare 2 pieces with 1 pieces: 2/4 > 1/4.
  16. > — There are 1 pieces in each fraction, but 1/8 is a larger piece than 1/12 when the whole is identical.
  17. 7 equal intervals — To show 1/7, the full unit length must be divided into 7 equal intervals, not 6.
  18. No — Equal fractions of differently sized wholes need not represent equal physical lengths.

Chapter Review

  1. 2/3 — The denominator counts all 12 equal parts; the numerator counts the 8 shaded parts. In simplest form this is 2/3.
  2. 3/8 — Each interval represents 1/8. Moving 3 intervals from zero reaches 3/8, or 3/8 in simplest form.
  3. < — Both fractions use pieces of size 1/3, so compare 1 pieces with 2 pieces: 1/3 < 2/3.
  4. > — There are 4 pieces in each fraction, but 1/6 is a larger piece than 1/12 when the whole is identical.
  5. 3/5 — The denominator counts all 5 equal parts; the numerator counts the 3 shaded parts. In simplest form this is 3/5.
  6. 9/10 — Each interval represents 1/10. Moving 9 intervals from zero reaches 9/10, or 9/10 in simplest form.
  7. > — Both fractions use pieces of size 1/8, so compare 7 pieces with 2 pieces: 7/8 > 2/8.
  8. > — There are 6 pieces in each fraction, but 1/8 is a larger piece than 1/10 when the whole is identical.
  9. 1/2 — The denominator counts all 8 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 1/2.
  10. 1/8 — Each interval represents 1/8. Moving 1 intervals from zero reaches 1/8, or 1/8 in simplest form.
  11. < — Both fractions use pieces of size 1/12, so compare 4 pieces with 7 pieces: 4/12 < 7/12.
  12. > — There are 5 pieces in each fraction, but 1/8 is a larger piece than 1/10 when the whole is identical.
  13. 4/5 — The denominator counts all 5 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 4/5.
  14. 1/5 — Each interval represents 1/5. Moving 1 intervals from zero reaches 1/5, or 1/5 in simplest form.
  15. < — Both fractions use pieces of size 1/10, so compare 1 pieces with 2 pieces: 1/10 < 2/10.
  16. > — There are 2 pieces in each fraction, but 1/8 is a larger piece than 1/10 when the whole is identical.
  17. 1/10 — The denominator counts all 10 equal parts; the numerator counts the 1 shaded parts. In simplest form this is 1/10.
  18. 1/3 — Each interval represents 1/3. Moving 1 intervals from zero reaches 1/3, or 1/3 in simplest form.
  19. > — Both fractions use pieces of size 1/10, so compare 8 pieces with 7 pieces: 8/10 > 7/10.
  20. < — There are 1 pieces in each fraction, but 1/3 is a larger piece than 1/12 when the whole is identical.
  21. 7/8 — The denominator counts all 8 equal parts; the numerator counts the 7 shaded parts. In simplest form this is 7/8.
  22. 11/12 — Each interval represents 1/12. Moving 11 intervals from zero reaches 11/12, or 11/12 in simplest form.
  23. < — Both fractions use pieces of size 1/4, so compare 2 pieces with 3 pieces: 2/4 < 3/4.
  24. < — There are 1 pieces in each fraction, but 1/4 is a larger piece than 1/8 when the whole is identical.
  25. 5 equal intervals — To show 1/5, the full unit length must be divided into 5 equal intervals, not 4.
  26. No — Equal fractions of differently sized wholes need not represent equal physical lengths.

Practice Exam

  1. < — The suggested answer is not correct. Both fractions use pieces of size 1/10, so compare 4 pieces with 5 pieces: 4/10 < 5/10. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  2. 1/4 — The denominator counts all 8 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 1/4. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  3. > — Both fractions use pieces of size 1/8, so compare 7 pieces with 5 pieces: 7/8 > 5/8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  4. > — There are 1 pieces in each fraction, but 1/4 is a larger piece than 1/8 when the whole is identical. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  5. 1/2 — The suggested answer is not correct. Each interval represents 1/4. Moving 2 intervals from zero reaches 2/4, or 1/2 in simplest form. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  6. 1/3 — The denominator counts all 6 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 1/3. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  7. > — Both fractions use pieces of size 1/12, so compare 8 pieces with 7 pieces: 8/12 > 7/12. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  8. > — Both fractions use pieces of size 1/8, so compare 5 pieces with 1 pieces: 5/8 > 1/8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  9. 3/5 — The suggested answer is not correct. Each interval represents 1/10. Moving 6 intervals from zero reaches 6/10, or 3/5 in simplest form. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  10. 1/2 — Each interval represents 1/12. Moving 6 intervals from zero reaches 6/12, or 1/2 in simplest form. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  11. 5/8 — The denominator counts all 8 equal parts; the numerator counts the 5 shaded parts. In simplest form this is 5/8. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  12. < — There are 4 pieces in each fraction, but 1/8 is a larger piece than 1/12 when the whole is identical. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  13. < — The suggested answer is not correct. Both fractions use pieces of size 1/8, so compare 3 pieces with 7 pieces: 3/8 < 7/8. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  14. < — There are 2 pieces in each fraction, but 1/6 is a larger piece than 1/10 when the whole is identical. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  15. > — There are 1 pieces in each fraction, but 1/6 is a larger piece than 1/8 when the whole is identical. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  16. 11/12 — The denominator counts all 12 equal parts; the numerator counts the 11 shaded parts. In simplest form this is 11/12. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  17. < — The suggested answer is not correct. There are 5 pieces in each fraction, but 1/6 is a larger piece than 1/10 when the whole is identical. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  18. 3/4 — The denominator counts all 8 equal parts; the numerator counts the 6 shaded parts. In simplest form this is 3/4. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  19. 5/6 — Each interval represents 1/6. Moving 5 intervals from zero reaches 5/6, or 5/6 in simplest form. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  20. > — There are 3 pieces in each fraction, but 1/6 is a larger piece than 1/10 when the whole is identical. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  21. 3/4 — The suggested answer is not correct. Each interval represents 1/4. Moving 3 intervals from zero reaches 3/4, or 3/4 in simplest form. (3 points: 1 point for identifying the given answer as incorrect; 1 for explaining the error; 1 for giving the corrected solution and appropriate units.)
  22. 7/10 — Each interval represents 1/10. Moving 7 intervals from zero reaches 7/10, or 7/10 in simplest form. (2 points: 1 point for a mathematically valid calculation/model, 1 point for the correct final answer with units when appropriate.)
  23. 8 equal intervals — To show 1/8, the full unit length must be divided into 8 equal intervals, not 7. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)
  24. No — Equal fractions of differently sized wholes need not represent equal physical lengths. (3 points: 1 point for the correct claim; 1 for a mathematical justification; 1 for a clear diagram, example, or relevant calculation when applicable.)