The rekenrek, or arithmetic rack, was designed by Adrian Treffers, a
mathematics curriculum researcher at the Freudenthal Institute in
Holland, to support the natural development of number sense in children.
Smaller versions consist of two rows of 10 beads. Larger versions with ten
rows of ten beads are also available. Each row is made of five red beads
and five white beads. This allows students to make mental images of
numbers. Using 5 and 10 as anchors for counting, adding and subtracting
is obviously more efficient than one-by-one counting. This tool provides
learners with the visual models they need to discover number
relationships and develop a variety of addition and subtraction
strategies, including doubles plus or minus one, making tens, and
compensation, thereby leading to automaticity of basic facts.
- Meet the Rekenrek: Begin by asking children what they
notice about the rekenrek. Then introduce the ‘start position’(all
beads over to the far right) and have them practice sliding beads in
groups rather than one by one “Put your beads in start position. Now,
without touching the beads, count the first three beads in your
mind. On the count of three, slide all three beads at once across the
string. One… two…three!” Repeat with other numbers.
- Show me 0-10:
Say a number, or hold up a numeral card (0-10). Ask students to show
the given number by moving the beads with one push.
- Show me 11-20: As above but ask students to show the given number of beads using only two pushes.
This rekenrek, (advertised as an abacus) with 10 rows of 10 beads in color-coded groups of five, can lie flat on student desks.
- Quick Images: Push some beads across and display
them briefly before covering them with a piece of cloth or card. Ask,
“How many beads did you see? How do you know?” Asking children to draw
or write what they saw on a dry erase board
ensures that everyone is actively involved and serves as a quick
assessment. If using a 100 bead rack gradually add rows until you
are displaying quick images to 100. This can be extended by asking
students to show the number that is one more/one less/ten more/ten less
than/double the number flashed.
- Finding Different Ways to Make a Given Number: Initially
use only the top row of beads. Cover the bottom row with a folded sheet
of card or piece of fabric. Begin by sliding the red beads to the left
and the white beads to the right on the top row of the rekenrek. Choose a
number to build. “Let’s see how many ways we can build 6 by sliding
beads from each side to the middle. What if I slide 4 red beads from the
left and 2 white beads from the right. Does that make 6 beads? Can you
think of another way to make 6? Record the different ways 6 can be
This activity should be repeated many times using different numbers from
1-10. Once children are confident using the top row, combinations can
be found using both the top and bottom rows. Children can record the
different ways they find to build the given number.
- Building Missing Addends: Ask a student to be your
partner. Tell the class that you and your partner are going to build the
number 6 as a team. You will move beads on the top row
and your partner will move beads on the bottom row. “I am going to slide
4 beads to the left on the top row. Now in one move, you slide beads on
the bottom row to build the number 6.” Pair students up and have them
turn over the top card in a stack of numeral cards and work with their
to build that number in as many different ways as possible. Begin with
cards 1-10, later
increase to 1-20.
A ten row rekenrek can be used with students who are ready to
represent numbers larger than 20.
See the following links for prompt cards to use during mental math sessions. Print on cardstock and place on a ring for teacher reference or place in a math center for children to use when working with a partner.
a) Prompt Cards
b) Rekenrek Flash Cards (this set includes teen numbers (10-20), doubles and near doubles to 20
Possible Math Journal Activities:
- Show 5: How many different ways can you show 5 on your Rekenrek? (repeat with different numbers)
- Doubles: How many different doubles facts can you show on your Rekenrek? Record.
The above two work samples were completed by a Kindergarten
student. This student represented her work very clearly
on the blank pages in her Math Journal using pictures, numbers and
words. To begin with some Kindergarten students may find it difficult to
represent their work on the Rekenrek on paper. Having recording paper
available for children who choose to use it is one way to scaffold early
attempts. You may also like to leave a supply of this paper in
your Math Center to prompt children's recording during Math Center
Rekenrek Recording Paper
- Near Doubles: How many different ‘near doubles’ can you show? Record.
- Turn Around Facts: Show an addition fact. What would the turn around fact look like? Repeat.
- Number Stories:
Have children use individual arithmetic racks as a tool to solve
various types of addition and subtraction number stories. This may be
used as a journal or oral activity, with the focus on children
explaining their strategy for solving the problem.
Be sure to include open-ended problems that have more than one solution that children can model on the rekenrek, such as the following:
- There were 8 children on a bunk bed. Some were on the top bunk
and some were on the bottom bunk. How many children were on the top
bunk? How many children were on the bottom bunk? Show as many different
solutions as you can.
- There were 12 passengers on a double-decker bus. Some
passengers were on the top deck and some were on the bottom deck. How
many passengers were on the top deck? How many passengers were on the
bottom deck? Show as many different solutions as you can.
Interested in using Rekenreks in your class but don't have the budget
to buy a class set? See our instructions on how to make a class set for just a fraction of the price of store bought models.
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