Cards and Counters: Odd and Even
Cards and counters is ten numeral cards and 55 red counters. A child of about 4 to 4.5 lays the cards 1 to 10 in a row, then places counters under each in pairs, side by side. Every number that leaves one counter alone at the bottom is odd; the rest are even. It checks the whole sequence from 1 to 10.
Sourced from the texts, not yet presented at home
| Typical age | 4 to 5.5 years |
|---|---|
| Area | Math |
| Before this | |
| Materials | |
| Direct aim | To show that the child knows the numerals 1 to 10 in order and the quantity each one stands for, without a box or a rod to help. |
| Control of error | The counters. There are exactly 55, so if any are left over or run short, a column is wrong. The pairs pattern also shows a slip at a glance: an odd number without a lone counter stands out. |
| Leads to | Later lessons in this line are in preparation. |
| Time | About 15 minutes (our estimate). |
| Grouping | One child |
| Parent prep | Low. The box of cards and counters and a clear table or rug. |
| Cost | Buy about $14 to $53. Make: easy to make at home. |
| Other names | Numerals and counters, Cut-out numerals and counters, Odd and even counters |
What the work teaches
Direct aim. To show that the child knows the numerals 1 to 10 in order and the quantity each one stands for, without a box or a rod to help.
Indirect aims, the ones you will not see at once
- Odd and even, seen in the pattern of pairs
- Each number is one more than the one before it
- Preparation for division by two
- Checking one's own work by counting
Set it out before you invite the child
Ten numeral cards, 1 to 10, cut out or printed on small wooden tiles.
Fifty-five red counters, exactly enough: 1 + 2 + 3 and so on up to 10 makes 55.
A rug or a table at least 80 cm wide, since ten columns spread out.
Montessori's 1912 book gives the same game with colored counters under the figures, in columns of two, and says the arrangement comes from Séguin. Her 1917 book adds red cubes for even numbers and blue for odd.
Presentation, step by step
- 1Take out the numeral cards, mixed. Find 1 and lay it at the left of the rug.
- 2Find 2 and lay it to the right of 1, leaving a gap. Carry on to 10, saying each number.
- 3Under 1, place one counter.
- 4Under 2, place two counters side by side.
- 5Under 3, place two side by side and the third below them, in the middle.
- 6Carry on to 10, always in pairs, any odd counter centered below the last pair. Count each column as you build it.
- 7Run a finger down between the pairs of each column. Where it reaches the bottom without meeting a counter, the number is even. Where a counter blocks it, the number is odd.
- 8Put the counters away by column, counting back, then invite the child to lay it out alone.
Details that hold the child's attention
The zigzag of lone counters across the bottom: 1, 3, 5, 7, 9.
A finger sliding cleanly down an even column.
Fifty-five counters, and none left over when every column is right.
Ten columns growing one taller each time.
How the child sees a mistake
The counters. There are exactly 55, so if any are left over or run short, a column is wrong. The pairs pattern also shows a slip at a glance: an odd number without a lone counter stands out.
The words come after the hands
| Words | Shown with |
|---|---|
| odd | A number with one left over when paired. |
| even | A number where every counter has a partner. |
| pair | Two counters side by side. |
| counter | One of the small red discs. |
The three-period lesson, scripted
You need the columns for 4 and 5.
| Period | What you do and say |
|---|---|
| 1. Naming | Run a finger down the 4: "Four is even." Down the 5, stopping at the lone counter: "Five is odd." |
| 2. Recognition | "Show me an even number." "Show me an odd one." |
| 3. Recall | Point to a column and ask "Odd or even?" |
Our suggestion. Montessori's 1917 version used red cubes for even numbers and blue for odd.
Extensions and variations
- Montessori's memory game: draw a folded numeral, keep it secret, fetch that many objects and lay them in pairs
- Odd and even with other things: shoes, spoons, chairs at the table
- Making two equal rows from an even column, then trying with an odd one
- Writing the odd numbers and the even numbers in two lists
- Laying the cards out of order on purpose and asking the child to fix them
Variations
- Cards and counters on a tray for a child who works at a small table
- Cards first one day, counters the next, for a child who tires before 55
Make it or buy it
| Material | Make it | Buy, approx. |
|---|---|---|
| Cards and Counters | Easy | $14 to $53 |
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
Write 1 to 10 on cards and count 55 buttons or beans into a jar. Matching counters make the pattern easier to see.
What to check when buying a set
Check that the set really has 55 counters, and count them on arrival. Wooden numerals stand up to years of use; card ones are fine too.
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
- Fills the spindle boxes correctly, including zero.
- Names the numerals 1 to 10 in any order.
- Counts out up to ten objects without skipping.
- Lines things up neatly by choice.
Ready for the next lesson
- Lays out all ten cards and 55 counters with none left over.
- Says which numbers are odd and which are even, and why.
- Spots a column with a wrong count before being asked.
- Moves easily between the cards and the beads. That's the cue for the bigger numbers.
Fitting it into a home, not a classroom
| If you have | Do this |
|---|---|
| No counters | Use 55 of anything the same: buttons, dried beans, red poker chips. Count them into a jar first so the total is right. |
| A toddler sibling | Counters are choking-size. Use bigger tokens, or do the work while the toddler naps. |
| Little floor space | A table runner or a strip of paper 80 cm long holds all ten columns. |
| Two children | One lays the cards, the other the counters, then they swap and check each other's columns. |
When it does not go to plan
| When | Try this |
|---|---|
| Counters are left over. | Count each column aloud together. A column with one too many is usually the culprit. |
| The cards go out of order. | Go back to the spindle boxes for a few days, where the order is printed for you. |
| Counters go in a line, not in pairs. | Present again, slower, setting each pair down together. The pairs are the point of the lesson. |
| The child finds it too easy. | Good: it's a check, not a struggle. Add the memory game for a challenge. |
Hazards to check before each session
Counters are small enough to swallow. Count all 55 back into the box with a toddler around.
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 6 of 12 published, after Spindle Boxes: Counting 0 to 9. Checked in: the scope and sequence |
| Montessori, The Montessori Method (1912) | Games placing colored counters under the figures in columns of twos, which make odd and even clear; the arrangement is credited to Séguin. Checked in: Chapter XIX; Chapter XX, Sequence of Exercises |
| Common Core, Grade 2 math | 2.OA.C.3: determine whether a group of objects (up to 20) has an odd or even number of members, by pairing objects or counting by 2s. Checked in: tasks.illustrativemathematics.org |
| Common Core, Kindergarten math | K.CC.B.4: understand the relationship between numbers and quantities; connect counting to cardinality. Checked in: tasks.illustrativemathematics.org |
Cards and counters: questions parents ask
What are Montessori cards and counters?
Numeral cards 1 to 10 and 55 red counters. The child lays the cards in order and puts the right number of counters under each, in pairs. It checks that the child knows the numerals and their quantities, and it shows odd and even.
Why are there 55 counters?
Because 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. With exactly that many, the counters check the work: a leftover counter or a short column means a mistake.
What age is cards and counters for?
About 4 to 4.5, after the spindle boxes, and on this site it runs to 5.5. Montessori wrote that odd and even shown in pairs was not difficult for children of five.
How does it teach odd and even?
The counters go in pairs. An even number's column is all pairs; an odd number's has one counter left alone at the bottom. A finger run down the middle of the column stops at that lone counter.
Sources cited by book and chapter
- Maria Montessori, The Montessori Method (1912)
Chapter XX, Sequence of Exercises
In the games the child places under the figures a corresponding number of coloured counters, arranged in columns of twos, which makes odd and even numbers clear; the arrangement is taken from Seguin.
Chapter XIX, Exercises for the Memory of Numbers
In the memory game the child draws a folded figure, keeps it secret, collects that many objects and arranges them in columns of two, putting an odd piece at the bottom between the last two.
Chapter XIX, Addition and Subtraction From One to Twenty
The numbers without an odd cube at the bottom are even, since they can be arranged in pairs and divided by two, and this is not difficult for children of five years.
- Maria Montessori, Dr. Montessori's Own Handbook (1914)
Arithmetic
Another exercise is to lay all the figures on the table and place below them the corresponding number of cubes, counters and so on.
- Maria Montessori, The Montessori Elementary Material (1917)
Part VII, Odd and Even Numbers
Even numbers are red and odd numbers blue; the child tries to make two equal rows of cubes and succeeds with the even numbers, while with the odd numbers a block is always left in the middle.
As the child puts each series of cubes into its number box he pronounces the number and adds odd or even.
Reviewed by
What to offer next
Once the child chooses this work unprompted and finishes it without help, the next lesson to prepare is Stamp Game: Static Addition.
- Spindle boxes4 to 5.5 years
- Teen boards4 to 6 years
- Cards and counters4 to 5.5 years
- Stamp game4.5 to 7.5 years
- Short bead chains4.5 to 6.5 years