Exact fractions and mixed-number operations · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| improper fraction/ɪmˈprɒpə ˈfrækʃn/ | 假分数 | jiǎ fēn shù |
Can we keep the recipe exact?
- A recipe uses 1½ cups for one batch. How much remains after using ¾ of a cup, and why should the answer stay exact?
- This lesson studies improper fraction 假分数: A fraction whose numerator is at least as large as its positive denominator.
Choose the mathematical structure
- Convert mixed numbers to improper fractions. Add or subtract using a common denominator; multiply numerators and denominators; divide by a nonzero fraction by multiplying its reciprocal. Cancel common factors, not added terms. These rules also apply to negative fractions.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines improper fraction?
A fraction whose numerator is at least as large as its positive denominator.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
1½-¾=6/4-3/4=3/4. Also (-2/3)×(9/4)=-18/12=-3/2 and (3/4)÷(5/8)=(3/4)×(8/5)=6/5. A common denominator gives 5/6+3/4=10/12+9/12=19/12. Check division by multiplying 6/5 by 5/8 to recover 3/4. Exact multiples of π follow ordinary arithmetic: 3π+2π=5π and 6π/3=2π. Leave an answer such as 5π exact when requested; π≈3.14 would introduce approximation. For subtraction, 5/6-3/4=10/12-9/12=1/12; for a negative mixed number, -1½ means -(1+1/2)=-3/2.
Exact fractions and mixed-number operations
Convert mixed numbers to improper fractions
Compare the model with the worked case and explain one change.
Find 1½-¾.
6/4-3/4=3/4=0.75.
Test a tempting shortcut
- Adding denominators does not preserve the unit size: 1/2+1/3 is not 2/5. Cancel only factors of a whole numerator and denominator. A division by zero is undefined.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
To add two fractions, add their numerators and add their denominators. This claim is false. Explain which definition or assumption it violates.
Find (-2/3)×(9/4).
(-2×9)/(3×4)=-18/12=-1.5.
To add two fractions, add their numerators and add their denominators.
Adding denominators does not preserve the unit size: 1/2+1/3 is not 2/5. Cancel only factors of a whole numerator and denominator. A division by zero is undefined.
Interpret a new situation
- For a non-calculator question show the common denominator or reciprocal step. Convert 19/12 to 1 7/12 if a mixed number is requested; round only when the question explicitly needs a decimal.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find (3/4)÷(5/8).
Multiply by 8/5: 24/20=6/5=1.2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A fraction whose numerator is at least as large as its positive denominator. Choose the relationship, show the method, check its assumptions and interpret the result.