Improper Fractions and Mixed Numbers
4A · Chapter 6 · Lesson 3 · MATH-4A-C06-L03
Learning goal. Recognize quantities greater than one in fraction strips and unit bars.
Before you begin
Recall: Grade 3B: fractions of equal wholes, equivalence, number lines and simplifying; Comparing and Ordering Fractions. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Order these fractions from least to greatest: 7/12, 5/8, 4/6.
Show answer & reasoning
Rename all three fractions using a common denominator, then compare numerators for equal-size parts.
Insert <, >, or =: 1/8 ___ 3/5.
Show answer & reasoning
Rename fractions using common denominator 40 and compare equal-size parts.
Insert <, >, or =: 1/2 ___ 4/10.
Show answer & reasoning
Rename fractions using common denominator 10 and compare equal-size parts.
Key vocabulary
- equivalent fractions
- different numerators and denominators representing the same quantity
- mixed number
- whole number plus proper fraction
- improper fraction
- fraction with numerator at least its denominator
- denominator
- number of equal parts in one whole
Understand the concept
An improper fraction has at least one whole when its numerator is at least its denominator. Seven quarters consist of four quarters in one whole and three more quarters, giving 1 3/4. A mixed number records whole units plus a remaining proper fraction. Draw equal whole bars so the representation is consistent.
Model it: partitioned unit bars extending beyond one whole. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
Write 36/8 as a mixed number in simplest form.
36 ÷ 8 gives 4 full wholes and 4 remaining parts of size 1/8. Simplify the fractional part to obtain 4 1/2.
Check your understanding 1
Write 17/3 as a mixed number in simplest form.
Show answer & reasoning
17 ÷ 3 gives 5 full wholes and 2 remaining parts of size 1/3. Simplify the fractional part to obtain 5 2/3.
Worked example 2
Write 8/5 as a mixed number in simplest form.
8 ÷ 5 gives 1 full wholes and 3 remaining parts of size 1/5. Simplify the fractional part to obtain 1 3/5.
Check your understanding 2
Write 15/6 as a mixed number in simplest form.
Show answer & reasoning
15 ÷ 6 gives 2 full wholes and 3 remaining parts of size 1/6. Simplify the fractional part to obtain 2 1/2.
Worked example 3
Write 16/6 as a mixed number in simplest form.
16 ÷ 6 gives 2 full wholes and 4 remaining parts of size 1/6. Simplify the fractional part to obtain 2 2/3.
Check your understanding 3
Write 43/8 as a mixed number in simplest form.
Show answer & reasoning
43 ÷ 8 gives 5 full wholes and 3 remaining parts of size 1/8. Simplify the fractional part to obtain 5 3/8.
Guided practice
Write 23/8 as a mixed number in simplest form.
Show answer & reasoning
23 ÷ 8 gives 2 full wholes and 7 remaining parts of size 1/8. Simplify the fractional part to obtain 2 7/8.
Write 22/4 as a mixed number in simplest form.
Show answer & reasoning
22 ÷ 4 gives 5 full wholes and 2 remaining parts of size 1/4. Simplify the fractional part to obtain 5 1/2.
Write 14/4 as a mixed number in simplest form.
Show answer & reasoning
14 ÷ 4 gives 3 full wholes and 2 remaining parts of size 1/4. Simplify the fractional part to obtain 3 1/2.
Write 56/10 as a mixed number in simplest form.
Show answer & reasoning
56 ÷ 10 gives 5 full wholes and 6 remaining parts of size 1/10. Simplify the fractional part to obtain 5 3/5.
Write 61/12 as a mixed number in simplest form.
Show answer & reasoning
61 ÷ 12 gives 5 full wholes and 1 remaining parts of size 1/12. Simplify the fractional part to obtain 5 1/12.
Independent practice
Write 7/3 as a mixed number in simplest form.
Show answer & reasoning
7 ÷ 3 gives 2 full wholes and 1 remaining parts of size 1/3. Simplify the fractional part to obtain 2 1/3.
Write 39/8 as a mixed number in simplest form.
Show answer & reasoning
39 ÷ 8 gives 4 full wholes and 7 remaining parts of size 1/8. Simplify the fractional part to obtain 4 7/8.
Write 26/6 as a mixed number in simplest form.
Show answer & reasoning
26 ÷ 6 gives 4 full wholes and 2 remaining parts of size 1/6. Simplify the fractional part to obtain 4 1/3.
Write 35/6 as a mixed number in simplest form.
Show answer & reasoning
35 ÷ 6 gives 5 full wholes and 5 remaining parts of size 1/6. Simplify the fractional part to obtain 5 5/6.
Write 59/12 as a mixed number in simplest form.
Show answer & reasoning
59 ÷ 12 gives 4 full wholes and 11 remaining parts of size 1/12. Simplify the fractional part to obtain 4 11/12.
Write 22/10 as a mixed number in simplest form.
Show answer & reasoning
22 ÷ 10 gives 2 full wholes and 2 remaining parts of size 1/10. Simplify the fractional part to obtain 2 1/5.
Write 19/5 as a mixed number in simplest form.
Show answer & reasoning
19 ÷ 5 gives 3 full wholes and 4 remaining parts of size 1/5. Simplify the fractional part to obtain 3 4/5.
Write 14/5 as a mixed number in simplest form.
Show answer & reasoning
14 ÷ 5 gives 2 full wholes and 4 remaining parts of size 1/5. Simplify the fractional part to obtain 2 4/5.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
Someone calls 7/4 seven whole numbers. Express 7/4 as a mixed number and explain.
Show answer & reasoning
Four fourths make one whole; three fourths remain.
Write 19/6 as a mixed number in simplest form.
Show answer & reasoning
19 ÷ 6 gives 3 full wholes and 1 remaining parts of size 1/6. Simplify the fractional part to obtain 3 1/6.
Exit check
Write 28/6 as a mixed number in simplest form.
Show answer & reasoning
28 ÷ 6 gives 4 full wholes and 4 remaining parts of size 1/6. Simplify the fractional part to obtain 4 2/3.
Write 27/5 as a mixed number in simplest form.
Show answer & reasoning
27 ÷ 5 gives 5 full wholes and 2 remaining parts of size 1/5. Simplify the fractional part to obtain 5 2/5.
Write 31/8 as a mixed number in simplest form.
Show answer & reasoning
31 ÷ 8 gives 3 full wholes and 7 remaining parts of size 1/8. Simplify the fractional part to obtain 3 7/8.