Solving An Arithmetic Problem With Fractions
Solving arithmetic problems with fractions follows a three-step procedure in Grade 6 Saxon Math Course 1: Shape (convert all values to improper fractions), Operate (multiply or divide), Simplify (reduce and convert back if needed). Before any multiplication, 2½ becomes 5/2 and a whole number like 4 becomes 4/1. Applying this systematic approach prevents the widespread error of multiplying whole and fractional parts separately, which produces incorrect answers.
Key Concepts
Property Step 1: Put the problem into the correct shape (if it is not already). Step 2: Perform the operation indicated. Step 3: Simplify the answer if possible.
Examples $1\frac{1}{2} \times \frac{2}{3} \rightarrow \frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1$ $2\frac{1}{2} \times 1\frac{1}{3} \rightarrow \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = 3\frac{1}{3}$ $4 \times 2\frac{1}{4} \rightarrow \frac{4}{1} \times \frac{9}{4} = \frac{36}{4} = 9$.
Explanation For multiplication, 'correct shape' means turning mixed numbers into improper fractions. It’s like a costume party for numbers where everyone must wear a fraction outfit, even whole numbers! This single step makes multiplying different types of numbers, like $3\frac{1}{2}$ and 5, super straightforward and prevents common errors. Get them dressed up first!
Common Questions
What is the Shape-Operate-Simplify procedure?
Shape: convert all mixed numbers and whole numbers to improper fractions. Operate: perform the multiplication or division. Simplify: reduce and convert back to a mixed number if appropriate.
How do you convert 2½ to an improper fraction?
(2 × 2 + 1)/2 = 5/2.
How do you convert the whole number 4?
Write it as 4/1. Its value is unchanged but it now has the fractional form required for multiplication.
Solve: 2½ × 1⅓
5/2 × 4/3 = 20/6 = 10/3 = 3⅓.
Why does converting first prevent errors?
Improper fractions capture the interaction between the whole and fractional parts correctly. Multiplying them separately ignores cross-products and gives wrong answers.