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.
- 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.
- 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 partner 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.
- 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 built. 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.
See the following links for prompt cards and flashcards 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 images for 1-20)
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.
Possible Math Journal Tasks:
- 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.
- 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.
The above two work samples were completed by a Kindergarten student who represented her work very clearly on the unlined 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 rekenrek 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 sessions.