Fraction Equivalence

Materials

• Fraction circles (or the paper-plate wedge families from the introduction lesson)
• A work mat or solid-color placemat
• Pencil and paper for recording the equivalences discovered
• Optional: a sheet of paper with a traced outline of the half piece, for laying trial pieces inside

No materials at home? Use the virtual Fraction Circles.

Before this lesson

• Introduction to Fractions

Aims

Presentation

1. 1 Lift one half out of its frame and lay it on the mat where the empty half-space in the frame stays visible.Here is one half. I wonder… can another family make one half exactly?
2. 2 Try the fourths. Lay fourth pieces on top of the half piece (or into its empty space in the frame), one at a time, until it is covered exactly.One fourth… two fourths. Two fourths cover one half exactly!
3. 3 Say the discovery clearly and write it in pencil where your child can see: 1/2 = 2/4.Two fourths and one half are worth the same. We say they are equivalent.
4. 4 Invite your child to try the sixths family the same way, counting each piece as it goes down, then the eighths and the tenths.How many sixths will it take? What do you think?
5. 5 Now try the fifths — this failure is the best part. Two fifths leave a sliver of the half uncovered; three fifths spill past its edge.Two fifths are too little, three fifths are too much. The fifths family cannot make one half.
6. 6 Read the finished record together: 1/2 = 2/4 = 3/6 = 4/8 = 5/10. Let your child hunt for a pattern in the numbers.Look at the top and bottom of each one. Do you notice anything?
7. 7 Another day, repeat the game with one third as the target and let your child run the investigation: sixths work (2/6), ninths work (3/9), and the rest do not.Which families do you predict can make one third?

Points of interest

• The suspense of the last piece — will it fit exactly or spill over?
• Fifths refusing to make a half, no matter how they are arranged
• The pattern hiding in 1/2, 2/4, 3/6, 4/8, 5/10: the top counts up by one while the bottom counts by two
• Different names — one half, two fourths, five tenths — for the very same amount of circle

Control of error

• The pieces themselves: an equivalent fill covers the target exactly, while a wrong family leaves a gap or overhangs the edge
• On the virtual material, the Check button confirms only what the pieces already show
• Recounting the pieces and rebuilding the fill settles any doubt

Vocabulary

• equivalent
• equivalence
• equal parts
• family (of fraction pieces)
• exactly

Variations

• Reverse the game: lay out three sixths and ask which single piece is worth the same
• A prediction game: before testing a family, your child writes down how many pieces they think it will take, then tests
• Use two halves, then try to rebuild one whole out of a single other family — every family can do it!

Extensions

• Make an equivalence chart: trace the half piece, then draw and label every fill that worked (2/4, 3/6, 4/8, 5/10)
• Investigate one fourth and one fifth as targets (2/8 works for a fourth; 2/10 works for a fifth)
• Ask the big question: why do fifths, sevenths, and ninths all fail to make a half? Let the pieces argue the answer

Follow-up work (pencil & paper)

• Print a fractions page with equivalence problems: your child fills in the missing number in pairs like 1/2 = ?/6, checking any doubtful one by rebuilding it with the pieces.Printable: Fractions
• Fold two same-size paper strips — one into halves, one into eighths — shade one half and four eighths, and hold the strips side by side to see the equivalence, then write it as symbols.
• Write each discovered equivalence chain from memory (1/2 = 2/4 = 3/6 = 4/8 = 5/10, and the chain for 1/3), then check the written chains against the circles.

What comes next

Once your child can find equivalent fills and record them in symbols, they are ready for Adding & Subtracting Fractions, where pieces from a single family are joined and taken away — and a sum grows past one whole circle.

Next in this strand: Adding & Subtracting Fractions