Fraction Insets (Fraction Circles)
Fraction insets are ten frames, each holding a circle 10 cm across: one whole, then one cut into halves, one into thirds, and so on to tenths. A child of about 4.5 who knows the geometric cabinet takes the pieces out, fits them back, and learns that two halves or four fourths fill the same circle.
Sourced from the texts, not yet presented at home
| Typical age | 4.5 to 7 years |
|---|---|
| Area | Math |
| Before this | |
| Materials | |
| Direct aim | To see and handle the equal parts of a whole, and to learn their names. |
| Control of error | The round opening. A circle is complete only when its pieces fill it with no gap and no overlap, and the child sees a gap at once. Counting the pieces back into each frame is a second check. |
| Leads to | Later lessons in this line are in preparation. |
| Time | About 10 minutes (our estimate). |
| Grouping | One child at a table |
| Parent prep | Low. Take out three frames before the child sits down. |
| Cost | Buy about $57 to $320. Make: possible to make with some effort. |
| Other names | Fraction circles, Metal fraction insets, Circular fraction insets |
What the work teaches
Direct aim. To see and handle the equal parts of a whole, and to learn their names.
Indirect aims, the ones you will not see at once
- The idea that equal parts of the same whole get smaller as there are more of them
- Equivalence felt in the hand: two fourths take the place of one half
- Preparation for writing fractions and for the operations with them in the elementary years
- Fine control of the pincer grip on small knobs
Set it out before you invite the child
Ten square frames, each with a round opening 10 cm across. The first holds a whole circle; the others hold circles cut into 2, 3, 4 and so on up to 10 equal parts, each piece with its own small knob.
Montessori's 1917 book describes them as ten metal plates. Metal sets are still the standard; Montessori Outlet and Alison's also sell wooden ones, Montessori Outlet's with red pieces in green frames.
A stand or two to hold the frames upright on the shelf. Alison's metal set includes stands; Nienhuis sells its stand separately.
A felt or a mat on the table, so the metal pieces don't skid or clatter.
For the first lessons, only the whole circle and the halves, with the fourths added the same day if all goes well.
Presentation, step by step
- 1Carry the frames to the table one at a time, the whole circle first and then the halves and fourths. Lay them in a row.
- 2Lift the whole circle out by its knob. Trace its edge with two fingers, then the edge of the empty frame, and put it back.
- 3Lift out the two halves and lay them below their frame. Trace each one, then fit them back together.
- 4Do the same with the four fourths.
- 5Name the parts in three periods, beginning with "This is one whole. This is one half."
- 6Take all the halves and fourths out, mix them on the mat, and invite the child to fill each circle again.
- 7On later days, add the other frames up to tenths, and try filling the halves' frame with fourths instead.
Details that hold the child's attention
Two fourths dropping into the space left by one half.
The tenths, ten thin slices of the same circle.
Counting the pieces in each frame and finding the count is the frame's number.
A circle filled with one half and two fourths, with no gap.
How the child sees a mistake
The round opening. A circle is complete only when its pieces fill it with no gap and no overlap, and the child sees a gap at once. Counting the pieces back into each frame is a second check.
The words come after the hands
| Words | Shown with |
|---|---|
| whole | The uncut circle. |
| half, halves | One of two equal parts; two make a whole. |
| third, fourth, fifth | Given once the halves are easy, and so on up to tenth. |
| equal | The same size. Every piece in one frame is equal to the others. |
| fraction | A part of the whole. Montessori gives the word with this material. |
The three-period lesson, scripted
You need the whole circle beside one half and one fourth.
| Period | What you do and say |
|---|---|
| 1. Naming | Hold up each in turn: "This is one whole." "This is one half." "This is one fourth." |
| 2. Recognition | "Give me one half." "Put one fourth in its frame." "Show me the whole." |
| 3. Recall | Point to a piece and ask "What is this?" |
Our suggestion. Montessori's 1917 book gives the names with the writing of fractions; with a child of 4 or 5 the spoken names come first.
Extensions and variations
- Filling one circle with mixed pieces, such as one half and two fourths
- Swapping small pieces for the largest one that fills the same space
- Labels for each piece: 1/2, 1/3, 1/4, written on slips, once the child writes numerals
- Tracing the pieces with a pencil, like the metal insets, and coloring the parts
- Commands from the 1917 book, for older children: take one fifth of 25 beads
Variations
- Halves and fourths only, for a child just past 4
- Cutting a paper circle or a pancake into halves and fourths at the kitchen table
Make it or buy it
| Material | Make it | Buy, approx. |
|---|---|---|
| Fraction Insets (Fraction Circles) | With effort | $57 to $320 |
Prices noted . Approximate US street prices for home-use versions; they vary by maker and change often, so recheck before buying.
Making it at home
Cut 10 cm circles from thick card and divide them with a protractor. Trace each whole circle onto a second card as its frame. Card works well for the first few frames.
What to check when buying a set
Check that every piece lifts out cleanly and that each circle fills with no gap. A stand saves table space.
Printables for this age
This work needs only the material, so it has no printable of its own. The shelf plan below places it among the other works for the same age.
- Shelf plan for age 4PDF, 1 page, 248 KBEvery work for this age on one page, in order, with a column to date each presentation.
Every printable for one age, by email
The shelf plan and the lesson printables for your child's age in one email, then three short emails on the first lesson, what to make or buy, and what comes next.
Signs the child is ready, and signs it is done
Ready for the first presentation
- Places the circles from the geometric cabinet with no trial fits.
- Counts to ten with quantities.
- Lifts a small knob with the thumb and two fingers.
- Notices when something is shared out unfairly at snack time.
Ready for the next lesson
- Names the parts from halves to tenths.
- Refills every frame from a mixed pile.
- Shows two ways to fill the halves' frame.
- Shows with the pieces that one third is bigger than one fourth.
Fitting it into a home, not a classroom
| If you have | Do this |
|---|---|
| A tight budget | Cut circles from thick card, 10 cm across, and divide them with a protractor: halves, thirds, fourths. Trace each circle onto a second card as its frame. |
| A toddler sibling | The tenths and their knobs are small. Work at a table and count the pieces back. |
| A child already past the names | Go to filling circles with mixed pieces. Swapping small pieces for big ones is where the thinking happens. |
| Snack time | Cut an apple or a flatbread into halves and fourths and name the parts out loud. |
When it does not go to plan
| When | Try this |
|---|---|
| Pieces from different frames get mixed up. | Take out only two frames at a time, as different as possible: halves and fifths. |
| The child says one fourth is bigger than one half because four is more. | Lay one half over two fourths and let the child see. Say nothing more; repeat another day. |
| Metal pieces stick in the frame. | Show lifting straight up by the knob. Wipe the frames dry; damp fingers make them grip. |
| Interest drops after the names. | Offer the mixed-piece game, or bring out the pencil and trace them. |
Hazards to check before each session
Metal frames are heavy. Carry one at a time, flat, with two hands.
The smallest pieces and the knobs are choking-size for a child under 3.
Where other sequences and standards place it
We have not bought the paid homeschool curricula, so they are not listed here.
| Framework | Where it sits |
|---|---|
| This site's Math sequence | Lesson 10 of 12 published, after Geometric Cabinet (Plane Geometric Insets). Checked in: the scope and sequence |
| Montessori, The Montessori Elementary Material (1917) | A second series of insets for fractions: ten metal plates with 10 cm circles, whole to tenths, used for equivalence, reducing to lowest terms and measuring angles. Checked in: Part IV, Equivalent, Identical and Similar Figures |
| Common Core, grade 1 and 2 geometry | 1.G.A.3: partition circles and rectangles into two and four equal shares and use the words halves, fourths and quarters. 2.G.A.3 adds thirds. Checked in: tasks.illustrativemathematics.org |
| Head Start Early Learning Outcomes Framework, preschool | P-MATH 9: the child identifies, describes, compares and composes shapes. Checked in: www.in.gov |
Fraction insets: questions parents ask
What are Montessori fraction circles?
Ten frames, each holding a circle 10 cm across. One circle is whole, and the others are cut into 2 to 10 equal pieces with small knobs. Children take the pieces out, fit them back, and learn the names of the parts.
What age are fraction insets for?
The naming work starts at about 4.5, after the geometric cabinet, and on this site it runs to 7. Montessori's 1917 book uses the same insets in the elementary years for operations with fractions.
Metal or wooden fraction circles?
Montessori described metal plates, and metal is still standard. Wooden sets cost less and do the same job for the naming lessons. Check that the pieces fill each circle with no gap.
Do I need all ten frames at once?
No. The first lessons use only the whole circle and the halves. Add the others a few at a time.
Sources cited by book and chapter
- Maria Montessori, The Montessori Elementary Material (1917)
Part IV, Equivalent, Identical and Similar Figures
The fraction insets are ten metal plates, each with a circular opening ten centimeters in diameter, holding one complete circle and circles divided into 2 to 10 equal parts.
The child can take the two half circles out of the circle and replace them with four quarter sectors, and so say that two halves are equal to four fourths.
Filling a circle with a half circle and two quarter circles shows in the inset itself that one half equals two fourths.
Replacing several small sectors with the largest piece that fills the same space initiates the child intuitively into reducing fractions to their lowest terms.
The divisions of the square give an intuitive idea of fractions, but the material for the study of fractions is another series of insets.
A series of commands guides the child's work, such as taking one fifth of 25 beads or one fourth of 36 counters.
Part IV, Plane Geometry
Work in fractions is made most interesting by the exercises with the circular insets, through which the real meaning of the word fraction becomes clear in the child's mind.
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What to offer next
Once the child chooses this work unprompted and finishes it without help, the next lesson to prepare is Hundred Board.
- Short bead chains4.5 to 6.5 years
- Tens boards4.5 to 6.5 years
- Fraction insets4.5 to 7 years
- Hundred Board5 to 7 years
- Addition strip board5 to 7.5 years