Fraction Insets — Equivalencies

The child sees that 1/2 and 2/4 and 5/10 fill the same space. The insets are the control — two pieces equivalent in area fit equally. Indirect aim: common denominators, operations on fractions, the concept of equivalence in all of mathematics.

• Framework: Montessori
• Unit: Mathematics
• Learning area: Fractions
• Level: Ages 7-9

Objectives

• Discover equivalent fractions by insetting pieces (1/2 = 2/4 = 3/6 = 4/8)
• Record equivalencies symbolically
• Build common denominators using equivalencies
• Prepare for addition and subtraction of unlike fractions

Materials

• Fraction Inset set (circles divided into 1, halves, thirds, fourths, fifths, sixths, sevenths, eighths, ninths, tenths)
• Fraction labels
• Recording paper
• Equivalencies chart

Instructions

1. Place 1/2 on a mat; fill it with fourths — two fourths fit exactly. 2. Record: 1/2 = 2/4. 3. Fill it with sixths — three sixths fit: 1/2 = 3/6. 4. Continue: 4/8, 5/10. 5. Notice the pattern: numerator and denominator each multiplied by the same factor. 6. Formalize: 1/2 = (1×n)/(2×n) for any n. 7. Apply to 1/3 = 2/6 = 3/9; 2/3 = 4/6 = 6/9. 8. File the equivalencies chart for future reference.

Educator tip

Equivalence is one of the deepest ideas in mathematics. Don't rush past the insight — let the child discover that a single quantity has infinitely many names.

Assessment

Observe accurate insetting, correct equivalence statements, and generalization of the multiply-both principle.

More from Mathematics

• Decimal Fraction Board — First Introduction
• Squaring Chain — Passage from Linear to Squared
• Cubing Chain of 5 — Linear to Cube
• Pythagorean Theorem — Sensorial Proof
• Algebraic Peg Board — Solving Linear Equations
• Fractions — Addition with Unlike Denominators
• Fractions — Multiplication
• Fractions — Division