Adding & Subtracting Fractions
The first fraction arithmetic: joining and taking away pieces of one family. Because every piece is the same size, adding is simply counting pieces — and when a sum grows past a whole, the pieces visibly complete a full circle.
Materials
- Fraction circles — for sums past one whole you need extra pieces, so borrow from a second set or cut a spare paper-plate family (the virtual material has spares built in)
- A work mat or solid-color placemat
- Pencil and paper for writing each equation
- Slips of paper with same-denominator problems, e.g. 2/8 + 3/8, 5/6 − 2/6, 5/8 + 6/8
No materials at home? Use the virtual Fraction Circles.
Before this lesson
Aims
Direct
- Add fractions with the same denominator by joining and counting pieces
- Subtract fractions with the same denominator by taking pieces away
- Rename an improper sum as a mixed number: 11/8 is one whole and three eighths
Indirect
- Show that only the numerator counts pieces — the family (denominator) does not change when adding like pieces
- Prepare for adding unlike denominators, where equivalence work supplies the common family
- Prepare for abstract, pencil-only fraction arithmetic
Presentation
- 1 Write 2/8 + 3/8 = at the top of a paper. Build the first number on the mat with two eighths, leave a hand’s width of space, and have your child build three eighths beside it.Two eighths… plus three eighths. The plus sign tells us to put them together.
- 2 Slide the two groups together into one arc and count every piece aloud, touching each one.One, two, three, four, five — five eighths. Two eighths plus three eighths is five eighths.
- 3 Have your child write the answer: 2/8 + 3/8 = 5/8. Point at the numbers as you talk about them.Look — the bottom number never changed. We only counted pieces of the same family.
- 4 Now subtraction. Write 5/8 − 2/8 =, build five eighths, and have your child take two away and return them toward their frame.Five eighths, take away two eighths… count what is left. Three eighths.
- 5 The big moment: write 5/8 + 6/8 =. Build both numbers and slide them together — the pieces complete one full circle with pieces left over.Eleven eighths — and look, eight of them made a whole circle! Eleven eighths is one whole and three eighths.
- 6 Record the sum both ways: 5/8 + 6/8 = 11/8 = 1 whole + 3/8. Read both forms aloud together.Eleven eighths and one-and-three-eighths are two names for the same amount.
- 7 Give your child the slips of problems to work through alone: build, count, and write each equation in pencil. Stay nearby but let the pieces do the correcting.The pieces will always tell you if the answer is right.
Points of interest
- Watching a sum of flat pieces suddenly close into a complete circle
- The denominator stubbornly staying the same while the numerator does all the counting
- Saying big improper fractions out loud: "eleven eighths!"
- Subtraction physically undoing an addition, piece by piece
Control of error
- The answer is a count of real pieces — recounting them settles it without an adult
- A completed whole circle announces itself; the sum has passed one and must be renamed
- Taking away more pieces than are on the mat is physically impossible
Vocabulary
- plus, minus, equals
- sum
- difference
- numerator
- denominator
- mixed number
- improper fraction
Variations
- Roll a die twice to invent addition problems in a chosen family (e.g. 4/10 + 5/10), build them, and record them
- Play "make it whole": lay out 5/8 and ask how many more eighths are needed to finish the circle
- Chain problems: start with 3/6, add 2/6, subtract 4/6, adding and removing pieces as you go
Extensions
- Sums past one whole in other families: 7/10 + 6/10, 3/4 + 3/4 — build, rename as mixed numbers, and record
- After a sum like 4/8 + 2/8 = 6/8, use the equivalence work to rename the answer 3/4 with fourths pieces laid on top
- For an older child: try 1/2 + 1/4 with the pieces, and let the puzzle of unlike families point back to equivalence
Follow-up work (pencil & paper)
- Print a fractions page of same-denominator addition and subtraction: your child solves each problem in pencil, building any doubtful one with the circle pieces to check.Printable: Fractions
- Have your child write three "past a whole" problems of their own (like 5/8 + 6/8), solve them, and draw the completed circle plus leftover pieces beside each answer.
- Make a fact page for one family: all the ways to compose a whole from eighths (1/8 + 7/8, 2/8 + 6/8, …), written neatly in pencil and checked with the pieces.
What comes next
This closes the first fraction sequence. From here, practice mixed problems on paper until they are fluent; a child who also finishes the decimal-system work is ready for the Introduction to the Decimal Board, where tenths reappear as decimal fractions written with a point.