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Using the Rekenrek
as a Visual Model for
Strategic Reasoning
   in Mathematics

     by Barbara Blanke




                         QP857 0208
Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics


The Math Learning Center, PO Box 12929, Salem, Oregon 97309. Tel. 1 800 575–8130.
© 2008 by The Math Learning Center
All rights reserved.
Prepared for publication on Macintosh Desktop Publishing system.
Produced in the United States of America.

QP857
P0208

The Math Learning Center grants permission to classroom teachers to reproduce blackline masters in appropriate quanti-
ties for their classroom use.

The Math Learning Center is a nonprofit organization serving the education community. Our mission is to inspire and enable
individuals to discover and develop their mathematical confidence and ability. We offer innovative and standards-based
professional development, curriculum, materials, and resources to support learning and teaching. To find out more, visit us
at www.mathlearningcenter.org.
Using the Rekenrek as a Visual Model for Strategic
Reasoning in Mathematics
“…materials cannot transmit knowledge:the learner must construct the relationships” Gravemeijer, 1991


What is a Rekenrek?
Directly translated, rekenrek means calculating frame, or arithmetic rack. Adrian Treffers, a mathemat-
ics curriculum researcher at the Freudenthal Institute in Holland, designed it to support the natural
mathematical development of children and to help them generate a variety of addition and subtraction
strategies, including doubles plus or minus 1, making 10’s, and compensation. Students can use the rek-
enrek to develop computation skills or solve contextual problems. Once children understand the opera-
tions of addition and subtraction, and can model various situations, it is important that they automatize
the basic facts by finding and using patterns and relationships. Unlike drill and practice worksheets and
flashcards, the rekenrek supports even the youngest learners with the visual models they need to dis-
cover number relationships and develop automaticity.

The rekenrek looks like an abacus, but it is not based on place value columns or used like an abacus. In-
stead, it features two rows of 10 beads, each broken into two sets of five, much like the ten frames used
in Bridges in Mathematics.




                                                The rekenrek

Like ten frames, this tool helps students see the quantity of five as a whole and develop strategies to
solve equations like 5 + 2, 5 + 3, etc. The figure below shows how 7 + 8 would be set up on the reken-
rek. In working with this model, children might find the total by adding 7 + 7 + 1, or 5 + 5 + 2 + 3, or
10 + 5. They could also choose to count on, but the rekenrek is likely to stretch children to see groups of
five.




                                                   7+8




© The Math Learning Center                                                                          Rekenrek 1
Using the Rekenrek



Here are some strategies we’ve seen students use for solving 9+8 on the rekenrek.

                            2     4      6      8    9




                          10 11 12 13 14 15 16 17


                          Hannah’s Strategy: counts by 2’s to 8 then counts on by 1’s

                                                     9




                          10 11 12 13 14 15 16 17


                                 Ryan’s Strategy: sees 9 and counts on by 1’s

                                                     9




                                                         9


                         Ellie’s Strategy: doubles minus 1. Shows 9 + 9 and removes 1.

The rekenrek also allows students to develop a variety of subtraction strategies. Stanford professor
Robert McKim (1980) states that, “the thinker who fluently produces a number of possible solutions to
a problem is usually more successful than the thinker who settles for one solution”. The figure below
shows three different ways that children might solve 14 – 8 using the rekenrek.




                                          Various solutions for 14 – 8

2 Rekenrek                                                                               © The Math Learning Center
Using the Rekenrek



While students can use the rekrenrek to generate different strategies for solving basic facts, they can
also use it to solve story problems such as the ones below.

         There were 15 kids in the house. 9 were playing video games upstairs. How many were in the
         kitchen having snacks?




A student might show the 9 children upstairs with 9 beads in the top row, and then count on additional
beads in the bottom row until reaching 15. One can then see there were 6 children in the kitchen.

         There were 6 kids doing math Home Connections upstairs. 5 kids were downstairs reading books. 3
         more kids came home to study. How many were in the house?




                                   Various ways students solved this problem:
                                                 5+5+1+3=
                                                    10 + 4 =
                                                     9+5=
                                                   5+5+4=


How to Get Started Using the Rekenrek
Using 5 and 10 as anchors for counting, adding, and subtracting is far more efficient than one-by-one
counting. With the help of the rekenrek, kindergartners can start making the transition about the mid-
dle of the school year. The activities on the following pages show how. Each activity assumes familiarity
with rekenrek operations in a previous activity.




© The Math Learning Center                                                                         Rekenrek 3
Using the Rekenrek




4 Rekenrek           © The Math Learning Center
Using the Rekenrek




Activity 1

What the Heck is a Rekenrek? Getting Familiar with the Calculating Frame
Grade Levels                                             You Will Need
K–3                                                      H One rekenrek for teacher demonstration
                                                         H Something to “hide” the rekenrek’s bottom row of
Skills
                                                           beads (a folded 8˝ × 5˝ index card or piece of con-
H Familiarization with the calculating frame
                                                            struction paper, or a small piece of fabric)
H Subitize 5 and 10
H Develop vocabulary for rekenrek use


Instructions for What the Heck is a Rekenrek
As students watch, cover the bottom row of beads on your rekenrek with a folded index card or con-
struction paper or fabric. Then slide all the beads in the visible row over to the far right. (All beads over
to the right is the “start” position for this activity and others in this collection.)

Ask your students what they notice about today’s mathematical tool.

         Teacher What do you notice about the rekenrek?

         Jose I see red and white beads.

         Ellie I think there are ten.

         Teacher How can we prove or disprove Ellie’s thought?

         Hannah I counted five red and five white.

         Daniel That is right, because I saw that half were red and half were white.

         Marshall 5 plus 5 makes 10 total!

Explain that you’re going to slide some beads on your rekenrek from one side of the wire to the other.
Ask students to watch carefully and be ready to share how many you move.

• Slide 3 red beads to the left. Ask, “How many red beads do you see?”
• Return the beads to the right side of the rekenrek.
• Next slide 5 red beads to the left. Say, “Think provately. How many beads did I move? Quitely share
  with a friend why you think your number is right.”
• Return the beads to the right side of the rekenrek.
• Slide over 6 beads. Ask “What do you notice?”




© The Math Learning Center                                                                                 Rekenrek 5
Using the Rekenrek


Activity 1 What the Heck is a Rekenrek? (cont.)


At this point let several children share how they can prove there are 6 beads… 3 and 3, counting by
ones, five and 1 more, 1 and five more, 2 and 2 and 2 etc… Celebrate the multiple ways of “seeing” 6, es-
pecially the ones that did not involve one-by-one counting.
• Now slide 3 more beads over to the left for a total of 9. Ask, “How many do you see now?”

Keep soliciting different ways to prove there are 9 beads. Hopefully, someone will use the anchor of 10 say-
ing something like, “I know there are 10 beads, so one less than 10 is 9” or, “I know there are 5, and 5 more
would be 10, minus one is 9.” Your goal is to look for non-counting strategies that assist students in subitiz-
ing 5 and 10. We want these numbers to become anchors for the students that they quickly visualize.




6 Rekenrek                                                                                 © The Math Learning Center
Using the Rekenrek




Activity 2

See and Slide Getting Efficient Using the Rekenrek
Grade Level                                              You will need
K-2                                                      H One rekenrek for each child in the group
                                                         H 20 Colored Popsicle sticks (numbers 1–10 in one
Skills
                                                           color, numbers 11–20 in another color)
H Develop number sense strategies using 5 and 10 as
  anchor numbers                                         H 1 container to hold popsicle sticks

H Visualize the numbers 1–20
                                                         Note This activity can be extended on a ten row reken-
H Build numbers efficiently                               rek by using 100 colored popscicle sticks numbered from
                                                         1–100. Have students represent the numbers using the
                                                         fewest possible bead moves.




Instructions for See and Slide
Make sure each child has a rekenrek with all beads moved to the right side. (Do not cover up any beads
for this activity.)

Tell the students you are going to choose a number (from a container of popsicle sticks numbered 1–10).
They are to “slide” that number of beads on the rekenrek, using only one move.

         Teacher I chose number 8. Think how you will move eight beads on the rekenrek in only one move.
         Now slide the beads.

         Mike I slid 5 reds and 3 whites on the top row all in one move!

         Sandy I did the same thing but on the bottom row.




         Pat I moved 5 reds on top and 3 reds on the bottom.

         Merrie Is that one move? I thought we had to use only one move.

         Pat I used two fingers to make one move!

         Teacher All of you “saw” the number eight and made one slide to show that number on your rekenrek.



© The Math Learning Center                                                                              Rekenrek 7
Using the Rekenrek


Activity 2 See and Slide! (cont.)


Now include the numbered sticks 11-20 in the container. Tell students they are to use no more than two
slides to show numbers larger than 10.

         Teacher I chose the number 16. Think about that number. How will it look when we move the
         beads? Using one or two moves, how can I show 16 on the rekenrek?

         Henry If you move 8 beads on each row, that will make 16.

         Marcia Move 10 on the top, 5 on the bottom, and 1 more on the bottom.

         Mark        But that’s three moves. You would need to move 10 on the top and 6 on the bottom.




                                                       Doubles




                                                    Ten and More

Again, ask students to share their thinking. Continue the activity by asking students to choose and rep-
resent additional numbers. Discourage counting one by one by reminding them to use only one slide for
numbers 1-10, and at most two slides for numbers 11-20.




8 Rekenrek                                                                                  © The Math Learning Center
Using the Rekenrek




Activity 3

Sum it Up Multiple Ways to Find A Sum
Grade Level                                                   You Will Need
K–1                                                           H A rekenrek for each pair of children
                                                              H Journal/paper for recording number sentences
Skills
H Use the anchor numbers 5 and 10 to visualize numbers        H Pencil

H Use the “counting up” strategy to solve problems with       H Number cards 1–10 (page 25. Run on cardstock and
  missing addends                                               cut apart.)

H Build fluency with tens facts                                H Something to “hide” the rekenrek’s bottom row of
                                                                beads (a folded 8˝ × 5˝ index card or piece of con-
                                                                  struction paper, or small piece of fabric)


Instructions for Sum it Up
Explain that for this activity you are only going to use the top row of beads. (Cover the botton row with
a folded index card or a 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. Using the number cards, choose a number to build.




                                               How to start the activity

         Teacher I’ve chosen the number 7. Let’s see how many ways we can build it by sliding beads from
         each side into the middle. I wonder if there is more than one way? What if I slide 3 red beads from
         the left and 4 white beads from the right. Does that build 7?




         Carrie Yes, but you can also make 7 by sliding 2 reds and 5 whites.

         Teacher Let’s record all the ways we can build 7.



© The Math Learning Center                                                                                     Rekenrek 9
Using the Rekenrek


Activity 3 Sum it Up! (cont.)


Note record students’ combinations as noted below so they might discover the emerging pattern.




                                           2 reds + 5 whites = 7 beads
                                           3 reds + 4 whites = 7 beads
                                           4 reds + 3 whites = 7 beads
                                           5 reds + 2 whites = 7 beads

Continue the activity by choosing additonal numbers from the 1–10 number cards. Invite student pairs
to build the chosen number and record the different ways it can be built.




10 Rekenrek                                                                                © The Math Learning Center
Using the Rekenrek




Activity 4

It Takes Two to Build a Number Finding Missing Addends
Grade Level                                               You Will Need
K–2                                                       H One rekenrek for every two students
                                                          H A rekenrek for teacher demonstration
Skills
H Discover strategies to solve unknown or missing         H Student set of number cards 1–10 (page 25)
  addend problems                                         H Student set of number cards 1–20 (pages 25–26,
H Develop problem-solving strategies                        optional)

H Build fluency with facts to ten
                                                          Note Rekenrek activities llike this can be extended on the
                                                          ten row rekenrek by using the 1–100 number cards, pages
                                                          25-31. Have students represent the numbers using the
                                                          fewest possible bead moves.


Instructions for It Takes Two to Build a Number
Ask a student to be your partner. Tell the class that you and your partner are going to build the num-
ber 5 as a team. You will be in charge of moving beads on the top row of the rekenrek and your student
partner will move beads on the bottom row.

         Teacher to Partner I am going to slide 3 beads to the left on the top row. Now in one move, you
         slide beads on the bottom row to build the number 5.




         Shelby I slid 2 beads on the bottom row with one move!




         Teacher Now I would like all of you to pair up with a partner. I’ll choose a number from the num-
         ber cards. You will then build that number with your partner. (Begin with cards 1–10, or 1–20 if you
         are practicing facts to 20).




© The Math Learning Center                                                                                 Rekenrek 11
Using the Rekenrek


Activity 4 It Takes Two to Build a Number (cont.)


Choose a couple cards to further model this activity before inviting the class to build the numbers on
their rekenreks. Tell students the number of beads on the top row should be less than the number cho-
sen. For instance, if the number card chosen is 8, the first student might move 5 beads on the top. Her
partner would then move 3 beads on the bottom.

Optional: Building numbers 11-20
Use the 11–20 number cards to practice facts to 20. Help students see that certain row combinations re-
quire a minimal number of beads on one row to build numbers larger than 10.

          Teacher I’m going to move 4 beads on the top row. You slide beads on the second row to make the
          number 16.

          Michael That won’t work. I don’t have enough beads.

          Sharon You have to move more beads on the top.

          Kelly Move 6 beads on the top. Then the 10 on the bottom will make 16.

Extension If you have a ten row rekenrek, challenge those students who are ready to represent numbers larger than 20.
Have students work in pairs or small groups, taking turns drawing a number from the 1–100 number cards and building
the number together or individually.




12 Rekenrek                                                                                      © The Math Learning Center
Using the Rekenrek




Activity 5

Build a “Doubles” Number Looking at Symmetrical Numbers on the Rekenrek
Grade Level                                                                                  You Will Need
1–2                                                                                          H One rekenrek per student
                                                                                             H One rekenrek for teacher demonstration
Skills
H Add two digit numbers to find sums to 20                                                    H Even Ten Frame Cards 2, 4, 6, 8, 10 shown in different
                                                                                               combinations (page 32. Run on cardstock or a trans-
H Use 5’s and 10’s to anchor addition and subtraction
                                                                                                        parency and cut cards apart.)
  facts
                                                                                             H Optional Odd Ten Frame Cards 1, 3, 5, 7, 9 shown in
H See the usefulness of doubles
                                                                                               different combinations (page 33. Run on cardstock or
                                                                                               a transparency and cut cards apart.)

Instructions for Build a “Doubles” Number
Begin with all the beads to the right on the rekenrek. Display an even ten frame card and ask students
to build the number by moving beads to the left. Have students check and compare their results with
their neighbors. Then ask for volunteers to share different ways to build the number.

         Teacher Here is a Ten Frame Card. Think of different ways to show this number on your rekenrek.




                                                                    Eve n Ten Frame Ca rd




         Ben I moved 5 red beads and 3 white beads.

         Kathie I saw eight on the card so I made 4 red on top and 4 red on the bottom.

         Florence That’s a doubles problem. 4 + 4 = 8.

Repeat with several cards. Challenge students to build the numbers as doubles with equal numbers of
beads on both rows.

         Teacher Now I am going to display two Ten Frame cards. Can you make this a double using your
         rekenrek?




                                            Eve n Ten Frame Ca rd                           Eve n Ten Frame Ca rd




         Sharon I moved over eight beads on the top and six on the bottom. To make a double I slid one
         bead on top back to the right and slid one bead on the bottom over to the left. Now I have 7 plus 7
         which is the same as 8 plus 6!


© The Math Learning Center                                                                                                                  Rekenrek 13
Using the Rekenrek


Activity 5 Build a “Doubles” Number (cont.)


         Tom I slid six over on the top and six over on the bottom. I needed to slide two more over so I slid
         one on top and one on the bottom and saw the double 7 plus 7 equals 14.

         Glenda I saw that the cards had 8 and 6. To make them the same I took one away from the 8 and
         added it to the 6. That gave me 7 and 7. Then I moved 7 beads over on the top and 7 on the bottom.

         Gill I saw two 5’s, one 3, and one 1. That’s 14 altogether. I know that two 7’s make 14 so that’s how
         I made mine, 7 on top and 7 on the bottom. It’s symmetrical.

Optional Extension: Odd Ten Frame Cards
Use the Odd Ten Frame Cards to explore the concept of doubles plus one.

         Teacher Here’s a ten frame card that presents a new challenge. Can you build it as a double?




                                                  Eve n Ten Frame Ca rd




         Michael No. It has to be 6 or 8.

         Sharon The closest I can come is to put 3 on the top and 4 on the bottom.

         Kelly 7 is an odd number. Odd numbers can’t have doubles.

         Kip I think that odd numbers will always have doubles plus 1, like 3 plus 3 plus 1.

         Jessica Yes, one row will always be 1 more than the other.

         Teacher Let’s try another one....

When students seem comfortable with the Odd Ten Frames, mix them with the Even Ten Frame cards
and ask them to identify if a card is even or odd.




14 Rekenrek                                                                                © The Math Learning Center
Using the Rekenrek




Activity 6

Show the Sum How many ways?
Grade Level                                                       You Will Need
2–3                                                               H One rekenrek for each student
                                                                  H Optional rekenrek for teacher demonstration
Skills
H Develop and demonstrate computational fluency strat-             H Classroom whiteboard or chart to record various com-
  egies for addition facts to 20                                        putational strategies

H Add two 1-digit numbers                                         H 2 dice or wooden cubes labled with the following
                                                                    numbers—Die One: 5, 6, 7, 8, 9, 10; Die Two: 4, 5, 6,
H Use 5’s and 10’s as anchors to solve addition facts to
                                                                    7, 8, 9
  20
H Explore multiple strategies for adding two numbers


Instructions for Show the Sum
Invite a student volunteer to roll the two dice to determine two numbers to add. Students use their rek-
enrek to share their strategies for calculating the sum.

         Teacher Karen’s roll turned up 9 and 6. Show me 9 plus 6 on your rekenrek. What is the sum?

         Shelby I moved 5 red and 4 white on the top to show 9. Since I know 5 plus 1 is 6, on the bottom I
         moved all 5 red beads and one more white one. They all add up to 15.

Record the number sentences on a chart or whiteboard to represent each student’s strategy:

                                                           5+4=9
                                                            5+1=6
                                                           9 + 6 = 15

         Hannah I know 1 less than 10 is 9 so I moved all the beads except for 1 over. Then I moved all 10
         on the bottom row and slid four back to show 6. I think I made more slides than Shelby, but we got
         the same answer.

                                                           10 – 1 = 9
                                                           10 – 4 = 6
                                                           9 + 6 = 15

         Emmy I started with 10 on the top and 5 on the bottom. Because 9 is one less than 10 and 6 is one
         more than 5, I just moved one from the top to the bottom. 15 is the answer and what I started with!

                                                           10 – 1 = 9
                                                            5+1=6
                                                           9 + 6 = 15

Repeat the roll of the dice several times, making sure to have several students share various ways to cal-
culate a sum.

© The Math Learning Center                                                                                        Rekenrek 15
Using the Rekenrek




16 Rekenrek          © The Math Learning Center
Using the Rekenrek




Activity 7

Roll Three for 20 or Bust!
Grade Level                                                    You Will Need
2–3                                                            H One rekenrek for each student
                                                               H Optional rekenrek for teacher demonstration
Skills
H Develop and demonstrate computational fluency strat-          H Classroom whiteboard or chart to record various com-
  egies for addition facts to 20                                     putational strategies and number sentences

H Adding three 1-digit numbers                                 H Several sets of 3 dice or wooden cubes labled with
                                                                 the following numbers—Red Die: 0, 1, 2, 3, 4, 5; Blue
H Use 5’s and 10’s as anchors to solve addition facts to
                                                                 Die: 2, 3, 4, 5, 6, 7; Green Die: 4, 5, 6, 7, 8, 9
  20


Instructions for Roll Three for 20 or Bust!
The object of this game is to obtain a sum close to or equal to 20 without going over.

• Roll the three dice to determine the first three numbers to add.
• Tell students to use their rekenreks to share their strategy for calculating the sum of the three num-
  bers rolled.
• Students must then decide to stay put or roll 1, 2, or 3 dice to add more to the sum they have so far to
  come close or equal to 20 without going over.

Model the activity before inviting students to play this game in small groups. Roll the dice and ask the
class to help you determine the sum of the three numbers using their rekenrek. Ask student to share
their strategies. Use the whiteboard or a chart to record their thinking.

         Teacher I rolled a 6 and a 7 and a 4. How should we add these three numbers on the rekenrek?

         Ellie Move 6 and then 4 over. That makes 10. Then on the bottom move 5 red and 2 white which is
         7. 10 plus 7 equals 17. That is a fast ten fact!

                                                                             6           4




                                                                                 7


                                                       6 + 4 = 10
                                                        5 + 2 =7
                                                       10 + 7 = 17




© The Math Learning Center                                                                                        Rekenrek 17
Using the Rekenrek


Activity 7 Roll Three! For 20 or Bust! (cont.)


         Reba Slide 6 on top and double it by sliding 6 on the bottom. Then slide 4 over on top and 1 more
         on the bottom. That’s also 17.

                                                                          6       4




                                                                          6   1


                                                 6 + 6 + 4 + 1 = 17

         Ernesto I think it would be easy to move the beads over in fives. I know in my head that the an-
         swer is 17, so I would slide 5, and then 5 more to make 10, and then 5 more to make 15. You still
         need 2 more white beads to make 17.

                                                                      5           5




                                                                      5       2


                                                  5 + 5 + 5 = 15
                                                   15 + 2 = 17

         Teacher So we’ve determined that 6 + 7 + 4 equals 17. Do we want to roll more dice to get closer
         to 20?

         Carlin Yes, but only roll one and use the red one. It has the smallest numbers.

         Teacher Ok. We rolled a 4.

         Marci That’s too much! 17 + 4 is more than 20. We went bust!

         Have students break into pairs or small groups and play the game. Be sure there is a set of dice
         for each group. Watch for groups that get close to 20 and ask them to show you their sum on the
         rekenrek.




18 Rekenrek                                                                                © The Math Learning Center
Using the Rekenrek




Activity 8

Target Think Addition
Grade Level                                                You Will Need
2–3                                                        H Set of Number Cards 11-20 to use as Target Numbers
                                                             (pages 25 and 26, run on cardstock and cut apart.)
Skills
                                                           H Set of Number Cards 1-10 to use as Starting Numbers
H Use the “Think Addition” strategy to solve subtraction
                                                             (pages 25 and 26, run on cardstock and cut apart.)
  facts to 20
                                                           H Two containers—one for holding “Starting” number
H Combine numbers to find a target number
                                                             cards 1-10, one for holding “target” number cards 11-
H Decompose and compose numbers                              20
                                                           H One rekenrek per student
                                                           H Optional rekenrek for teacher demonstration


                                                           Note Subtraction as “Think Addition” is a significant strat-
                                                           egy for recalling subtraction facts.

Instructions for Target Think Addition
• Draw a Starting Number Card from a container of number cards 1-10. Ask students to build the num-
   ber on the rekenrek.
• Next draw a Target Number Card from a container of number cards 11-20. Tell students that number
   is the target they want to get to.
• Ask students to think of how many beads they must add to the Starting Number to equal the Target
   Number. Before they move any beads, ask students to share how many they will move. Then have
   them make the move to create the Target Number. Invite students to share their methods.

This mental activity leads to the use of a “think addition” strategy rather than a “count what is left” strat-
egy when solving subtraction problems. It reinforces the concept of anchor numbers and seeing combi-
nations rather than counting by ones.

         Teacher I’m going to draw our Starting Number—it’s 4. Build that number on your rekenrek.

         Teacher Now let’s choose our Target Number. We drew 19! Think for a moment before you move
         any beads. How many will you need to move over so that you have the target number 19? Remem-
         ber, we want to use as few moves as we can.

         Charles I would move the whole bottom 10 over first and that would make 14. Then I would move
         5 more over on the top and that would make the target number 19.

         Teacher That’s pretty good, only two moves. So how many beads did Charles move all together?

         Terri He moved 15. So we had to add 15 to 4 to hit the target 19. 4 + 15 is 19.



© The Math Learning Center                                                                                  Rekenrek 19
Using the Rekenrek


Activity 8 Target Think Addition! (cont.)




                                                           4
                                           N u m b e r C a rd
                                                                                               19
                                                                                     N u m b e r C a rd




                                          Starting Number                            Target Number
                                      4                                      +5




                                                                              + 10


Repeat this activity several times with new Starting and Target numbers.

Extend to Subtraction
To move students into subtraction, draw the cards in a differrent order. Start by drawing a Target Num-
ber (11-20). Ask the class to build it on their rekenrek. Then draw a Starting Number Card (0-10). Ask
students to think of how many beads they will need to remove from the Target Number (move back to
the right) to get back to the Starting Number.

         Teacher This time let’s start with a Target Number. I’ve drawn a target of 17. Build that on the rek-
         enrek. (Pause) Now let’s draw a Starting Number—9. Think for a moment. How many beads will
         you need to remove from the Target Number to get back to the Starting Number 9?

         Samantha To get to 9 I’m going to have to move 7 back on the bottom and 1 more on top, that’s 8 I
         have to move to get to 9. 17 – 8 is 9.

         Kelly I started with 17 as doubles plus 1. To get to 9, I took double 4s away. 17 - 8 is 9.


                                                                17
                                                    N u m b e r C a rd
                                                                                                               9
                                                                                              N u m b e r C a rd




                                               Target Number                          Starting Number

                                                                         1                                                                     8




                                                           7

                     17 – 7 – 1 = 9                                                                                17 - 8 = 9




20 Rekenrek                                                                                                                     © The Math Learning Center
Using the Rekenrek




Activity 9

Tens or Ones
Grade Level                                               You Will Need
2–3                                                       H Double set of number cards 11–19 (page 25 and 26,
                                                            run on cardstock and cut apart.)
Skills
                                                          H Run Away Ones Cards (page 35. Run on cardstock and
H Composing and decomposing numbers by tens or
                                                            cut apart.)
  ones
                                                          H Take Away Tens Cards (page 35. Run on cardstock and
H Fluency with addition and subtraction facts
                                                            cut apart.)
H Using strategies to demonstrate fluency
                                                          H Two containers from which to draw cards—one for
                                                              holding number cards 11-19, one for holding Take
                                                              Away Tens and Run Away Ones cards
                                                          H One rekenrek for every student

Instructions for Tens or Ones
Begin by drawing one of the Number Cards 11-19 from a container. Display the chosen card to the class
and ask them to build the number on their rekenreks.

Next draw a card from the second container holding the Take Away Tens and Run Away Ones Cards.
Have the class perform the action listed on the card by taking away either ten or the ones from the
number on their rekenreks. Be sure to look for different strategies and invite student sharing.

         Teacher Our first number card is 19, and it looks like each of you has it built on your rekenrek.
         Now let’s draw from the second container—a Take Away Tens Card. What should we do?

         Glenda Move ten beads back to the right! That leaves 9! I just moved the top row of ten and that
         left nine on the bottom! 19 – 10 is 9.

Repeat the process several times by drawing additional cards, asking students to build and subtract ei-
ther tens or ones from the number on their rekenrek. Play until all Number Cards are drawn.

         Teacher Carol has drawn our next number card—16. Everyone build 16.

         Charlotte I put 8 on top and 8 on the bottom. Doubles!

         Carl I slid 10 over on top and 6 on the bottom. I think that’s the quickest way.

         Teacher Carol, pull a card out of our second contatiner—it’s a Take Away Ones Card.

         Carol We have to take away all the ones from 16. That’s 6. So take 6 away.

         Charlotte I took 6 away on the bottom and that left 8 and 2. I know 8 and 2 is 10.

         Carl I just moved the 6 on the bottom and ten are left on top! We both got 10!

         Carol So our number sentence is 16 – 6 is 10.

© The Math Learning Center                                                                               Rekenrek 21
Using the Rekenrek




22 Rekenrek          © The Math Learning Center
Using the Rekenrek




Activity 10

Count and Compare
Grade Level                                             You Will Need
2–3                                                     H One rekenrek per student
                                                        H Optional rekenrek for teacher demonstration
Skills
H Fluency with addition and subtraction facts           H Count and Compare Gameboard (page 36. Run a half
                                                          class set on cardstock and on optional transparency.)
H Using the rekenrek model and mental strategies to
  solve addition and subtraction facts                  H Set of Count and Compare Cards (pages 37– 43. Run
                                                          a half class set on cardstock and on an optional trans-
                                                           parency, cut apart.)
                                                        H Half class set of clear overlay spinners or pencils and
                                                          paperclips to use as spinners

                                                        Note You may wish to model this game at the overhead
                                                        or play teacher against class. If so, run transparencies of
                                                        the items listed above. If you would like the students to
                                                        play in pairs, make copies of the gameboard and cards
                                                        for every two students. Below are the directions for the
                                                        student partner game.



Instructions for Count and Compare
Distribute the Count and Compare Gameboard and a set of Count and Compare Cards to each pair of
students. Ask them to shuffle their decks of cards and place the cards on top of the gameboard. Make
sure that each student has a rekenrek.

To begin the game, tell each student to draw one fact card from the top of the pile and use their reken-
rek to find the difference or sum for the card drawn.

Ask students to share their strategies for determining solutions for their cards. Once both partners have
agreed on their solutions, tell them to decide which card is more and which card is less. Have them
place each card in the appropriate More or Less box on the game board. If the solutions on both cards
are equal, tell them to place both cards in the equal box and draw two more cards until they have one
solution that is more and one that is less.

Using a clear spinner or pencil and paper clip, tell them to spin the More Or Less Spinner to decide who
gets to take both or all the cards. (If cards are in the equal box, the winner gets those cards as well.) If
the spinner lands on More, the person who drew the card with the greater sum or difference gets to take
the cards. If the spinner lands on Less, the person who drew the card with the smaller solution gets the
cards.




© The Math Learning Center                                                                                Rekenrek 23
Using the Rekenrek


Activity 10 Count and Compare (cont.)


Have students repeat the game until they are out of cards or until time runs out. Tell each student to
count the number of cards they each collected. Then tell the players with the most cards to place their
stacks of cards in the More box and those with the fewest cards to put their cards in the Less box. Have
them spin the spinner one more time to decide who wins the game!




24 Rekenrek                                                                           © The Math Learning Center
© The Math Learning Center
                             Number Card
                                           1             2
                                               Number Card
                                                                       3
                                                             Number Card
                                                                                     4
                                                                           Number Card




                                       5                 6             7             8
                                                                                         Number Cards page 1 of 7




                             Number Card       Number Card   Number Card   Number Card
                                                                                                                    Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




                                    9 10 11 12
                             Number Card       Number Card   Number Card   Number Card




                                   13 14 15 16
                             Number Card       Number Card   Number Card   Number Card




Rekenrek 25
26 Rekenrek
                                    17 18 19 20
                             Number Card   Number Card   Number Card   Number Card




                                    21 22 23 24
                                                                                     Number Cards page 2 of 7




                             Number Card   Number Card   Number Card   Number Card
                                                                                                                Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




                                  25 26 27 28
                             Number Card   Number Card   Number Card   Number Card




                                  29 30 31 32
                             Number Card   Number Card   Number Card   Number Card




© The Math Learning Center
© The Math Learning Center
                                  33 34 35 36
                             Number Card   Number Card   Number Card   Number Card




                                  37 38 39 40
                                                                                     Number Cards page 3 of 7




                             Number Card   Number Card   Number Card   Number Card
                                                                                                                Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




                                  41 42 43 44
                             Number Card   Number Card   Number Card   Number Card




                                  45 46 47 48
                             Number Card   Number Card   Number Card   Number Card




Rekenrek 27
28 Rekenrek
                                  49 50 51 52
                             Number Card   Number Card   Number Card   Number Card




                                  53 54 55 56
                                                                                     Number Cards page 4 of 7




                             Number Card   Number Card   Number Card   Number Card
                                                                                                                Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




                                   57 58 59 60
                             Number Card   Number Card   Number Card   Number Card




                                   61 62 63 64
                             Number Card   Number Card   Number Card   Number Card




© The Math Learning Center
© The Math Learning Center
                                  65 66 67 68
                             Number Card   Number Card   Number Card   Number Card




                                  69 70 71 72
                                                                                     Number Cards page 5 of 7




                             Number Card   Number Card   Number Card   Number Card
                                                                                                                Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




                                  73 74 75 76
                             Number Card   Number Card   Number Card   Number Card




                                  77 78 79 80
                             Number Card   Number Card   Number Card   Number Card




Rekenrek 29
30 Rekenrek
                                    81 82 83 84
                             Number Card   Number Card   Number Card   Number Card




                                  85 86 87 88
                                                                                     Number Cards page 6 of 7




                             Number Card   Number Card   Number Card   Number Card
                                                                                                                Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




                                  89 90 91 92
                             Number Card   Number Card   Number Card   Number Card




                                  93 94 95 96
                             Number Card   Number Card   Number Card   Number Card




© The Math Learning Center
Using the Rekenrek Blackline Run on cardstock and cut apart as needed.




Number Cards page 7 of 7


 97 98 99 100                Number Card
                             Number Card
                             Number Card
                             Number Card




© The Math Learning Center                                               Rekenrek 31
32 Rekenrek
                             Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card
                                                                                                                     Even Ten Frame Cards




                             Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card
                                                                                                                                            Using the Rekenrek Blackline Run on cardstock or transparency and cut cards apart.




                             Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card




                             Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card   Even Ten Frame Card




© The Math Learning Center
© The Math Learning Center
                             Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card
                                                                                                                 Odd Ten Frame Cards




                             Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card
                                                                                                                                       Using the Rekenrek Blackline Run 2 transparencies and cut cards apart.




                             Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card




                             Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card   Odd Ten Frame Card




Rekenrek 33
34 Rekenrek
                               Run Run Run Run
                              Away Away Away Away
                                                                                                                 Run Away Ones Cards




                              Ones Ones Ones Ones
                             Run Away Ones Card   Run Away Ones Card   Run Away Ones Card   Run Away Ones Card
                                                                                                                                       Using the Rekenrek Blackline Make one transparency or run on cardstock, cut cards apart.




                               Run Run Run Run
                              Away Away Away Away
                              Ones Ones Ones Ones
                             Run Away Ones Card   Run Away Ones Card   Run Away Ones Card   Run Away Ones Card




© The Math Learning Center
Take Take Take Take




© The Math Learning Center
                               Away Away Away Away
                                                                                                                     Take Away Tens Cards




                               Tens Tens Tens Tens
                             Take Away Tens Card   Take Away Tens Card   Take Away Tens Card   Take Away Tens Card
                                                                                                                                            Using the Rekenrek Blackline Make one transparency or run on cardstock, and cut cards apart.




                               Take Take Take Take
                               Away Away Away Away
                               Tens Tens Tens Tens
                             Take Away Tens Card   Take Away Tens Card   Take Away Tens Card   Take Away Tens Card




Rekenrek 35
Using the Rekenrek Blackline Run a half class set and one transparency. Transparency is optional.




                               More
                               Equal




                                                                                                                     more
                                                                                                    less
   Count & Compare Gameboard

                               Less




36 Rekenrek                                                                                                © The Math Learning Center
© The Math Learning Center
                                11 11 11 11
                               –8 –7 –6 –5
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card
                                                                                                                         Count & Compare Cards, page 1 of 7




                                12 14 12 12
                                                                                                                                                              Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




                               –7 –5 –5 –4
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card




Rekenrek 37
38 Rekenrek
                                13 13 13 12
                               –6 –5 –4 –8
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card
                                                                                                                         Count & Compare Cards, page 2 of 7




                                12 14 13 13
                                                                                                                                                              Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




                               –7 –5 –8 –7
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card




© The Math Learning Center
© The Math Learning Center
                                15 15 14 14
                               –7 –6 –9 –8
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card
                                                                                                                         Count & Compare Cards, page 3 of 7




                                16 16 15 15
                                                                                                                                                              Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




                               –9 –7 –9 –8
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card




Rekenrek 39
40 Rekenrek
                                17 8 8 8
                               –8 +8 +7 +6
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card
                                                                                                                         Count & Compare Cards, page 4 of 7




                                17 6 6 10
                                                                                                                                                              Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




                               –9 +7 +5 +5
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card




© The Math Learning Center
© The Math Learning Center
                                 8 11 11 11
                               +5 +6 +5 +4
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card
                                                                                                                         Count & Compare Cards, page 5 of 7




                                 6 10 10 11
                                                                                                                                                              Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




                               +4 +7 +5 +8
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card




Rekenrek 41
42 Rekenrek
                                 6 9 9 8
                               +8 +7 +6 +9
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card
                                                                                                                         Count & Compare Cards, page 6 of 7




                                11 9 9 9
                                                                                                                                                              Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




                               +7 +9 +7 +9
                               ___ ___ ___ ___
                             Count & Compare Card   Count & Compare Card   Count & Compare Card   Count & Compare Card




© The Math Learning Center
Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional.




Count & Compare Cards, page 7 of 7


     8
   +8
   ___

                                                                                             9
                                                                                           +8
                                                                                           ___
                                                                    Count & Compare Card




                                                                                                                     Count & Compare Card




© The Math Learning Center                                                                                      Rekenrek 43
Using the Rekenrek




References
Blanke, Barbara. 2003. Number Corner: Grade 3. The Math Learning Center: Salem, OR

Burk, Donna and Snider, Allyn. 1999. Bridges in Mathematics, Grades K–2. The Math Learning Center: Sa-
lem, OR.

Fosnot, C., Dolk, M. 2001. Young Mathematicians at Work: Constructing Number Sense, Addition and Sub-
traction. Heinemann: New York.

Gravemeijer, K.P.E. 1991. “An instruction-theoretical reflection on the use of manipulatives.” In Realistic
Mathematics Education in Primary School, ed. L. Streefland. Utrecht: Utrecht University (CD-(beta) series
on research in education: 9).

Hanson-Powell, Pia. 2003. Bridges in Mathematics: Grade 3. The Math Learning Center: Salem, OR.

McKim, R. H. 1980. Experiences in Visual Thinking. 2nd ed. Brooks/Cole Publishing Company: Monterey, CA.

Van de Walle, J. and Lovin, L. H. 2006. Teaching Student-Centered Mathematics, Grades K–3, Pearson Edu-
cation, Inc: Boston, MA.


Where to get a rekenrek
The Math Learning Center
PO Box 12929
Salem, Oregon 97309–0929
1 800 575–8130
www.mathlearningcenter.org

Or

Jegro BV
Koopman Heeresweg 14
8701 PR Bolsward
Holland




44 Rekenrek                                                                            © The Math Learning Center

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Rekenrek

  • 1. Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics by Barbara Blanke QP857 0208
  • 2. Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics The Math Learning Center, PO Box 12929, Salem, Oregon 97309. Tel. 1 800 575–8130. © 2008 by The Math Learning Center All rights reserved. Prepared for publication on Macintosh Desktop Publishing system. Produced in the United States of America. QP857 P0208 The Math Learning Center grants permission to classroom teachers to reproduce blackline masters in appropriate quanti- ties for their classroom use. The Math Learning Center is a nonprofit organization serving the education community. Our mission is to inspire and enable individuals to discover and develop their mathematical confidence and ability. We offer innovative and standards-based professional development, curriculum, materials, and resources to support learning and teaching. To find out more, visit us at www.mathlearningcenter.org.
  • 3. Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics “…materials cannot transmit knowledge:the learner must construct the relationships” Gravemeijer, 1991 What is a Rekenrek? Directly translated, rekenrek means calculating frame, or arithmetic rack. Adrian Treffers, a mathemat- ics curriculum researcher at the Freudenthal Institute in Holland, designed it to support the natural mathematical development of children and to help them generate a variety of addition and subtraction strategies, including doubles plus or minus 1, making 10’s, and compensation. Students can use the rek- enrek to develop computation skills or solve contextual problems. Once children understand the opera- tions of addition and subtraction, and can model various situations, it is important that they automatize the basic facts by finding and using patterns and relationships. Unlike drill and practice worksheets and flashcards, the rekenrek supports even the youngest learners with the visual models they need to dis- cover number relationships and develop automaticity. The rekenrek looks like an abacus, but it is not based on place value columns or used like an abacus. In- stead, it features two rows of 10 beads, each broken into two sets of five, much like the ten frames used in Bridges in Mathematics. The rekenrek Like ten frames, this tool helps students see the quantity of five as a whole and develop strategies to solve equations like 5 + 2, 5 + 3, etc. The figure below shows how 7 + 8 would be set up on the reken- rek. In working with this model, children might find the total by adding 7 + 7 + 1, or 5 + 5 + 2 + 3, or 10 + 5. They could also choose to count on, but the rekenrek is likely to stretch children to see groups of five. 7+8 © The Math Learning Center Rekenrek 1
  • 4. Using the Rekenrek Here are some strategies we’ve seen students use for solving 9+8 on the rekenrek. 2 4 6 8 9 10 11 12 13 14 15 16 17 Hannah’s Strategy: counts by 2’s to 8 then counts on by 1’s 9 10 11 12 13 14 15 16 17 Ryan’s Strategy: sees 9 and counts on by 1’s 9 9 Ellie’s Strategy: doubles minus 1. Shows 9 + 9 and removes 1. The rekenrek also allows students to develop a variety of subtraction strategies. Stanford professor Robert McKim (1980) states that, “the thinker who fluently produces a number of possible solutions to a problem is usually more successful than the thinker who settles for one solution”. The figure below shows three different ways that children might solve 14 – 8 using the rekenrek. Various solutions for 14 – 8 2 Rekenrek © The Math Learning Center
  • 5. Using the Rekenrek While students can use the rekrenrek to generate different strategies for solving basic facts, they can also use it to solve story problems such as the ones below. There were 15 kids in the house. 9 were playing video games upstairs. How many were in the kitchen having snacks? A student might show the 9 children upstairs with 9 beads in the top row, and then count on additional beads in the bottom row until reaching 15. One can then see there were 6 children in the kitchen. There were 6 kids doing math Home Connections upstairs. 5 kids were downstairs reading books. 3 more kids came home to study. How many were in the house? Various ways students solved this problem: 5+5+1+3= 10 + 4 = 9+5= 5+5+4= How to Get Started Using the Rekenrek Using 5 and 10 as anchors for counting, adding, and subtracting is far more efficient than one-by-one counting. With the help of the rekenrek, kindergartners can start making the transition about the mid- dle of the school year. The activities on the following pages show how. Each activity assumes familiarity with rekenrek operations in a previous activity. © The Math Learning Center Rekenrek 3
  • 6. Using the Rekenrek 4 Rekenrek © The Math Learning Center
  • 7. Using the Rekenrek Activity 1 What the Heck is a Rekenrek? Getting Familiar with the Calculating Frame Grade Levels You Will Need K–3 H One rekenrek for teacher demonstration H Something to “hide” the rekenrek’s bottom row of Skills beads (a folded 8˝ × 5˝ index card or piece of con- H Familiarization with the calculating frame struction paper, or a small piece of fabric) H Subitize 5 and 10 H Develop vocabulary for rekenrek use Instructions for What the Heck is a Rekenrek As students watch, cover the bottom row of beads on your rekenrek with a folded index card or con- struction paper or fabric. Then slide all the beads in the visible row over to the far right. (All beads over to the right is the “start” position for this activity and others in this collection.) Ask your students what they notice about today’s mathematical tool. Teacher What do you notice about the rekenrek? Jose I see red and white beads. Ellie I think there are ten. Teacher How can we prove or disprove Ellie’s thought? Hannah I counted five red and five white. Daniel That is right, because I saw that half were red and half were white. Marshall 5 plus 5 makes 10 total! Explain that you’re going to slide some beads on your rekenrek from one side of the wire to the other. Ask students to watch carefully and be ready to share how many you move. • Slide 3 red beads to the left. Ask, “How many red beads do you see?” • Return the beads to the right side of the rekenrek. • Next slide 5 red beads to the left. Say, “Think provately. How many beads did I move? Quitely share with a friend why you think your number is right.” • Return the beads to the right side of the rekenrek. • Slide over 6 beads. Ask “What do you notice?” © The Math Learning Center Rekenrek 5
  • 8. Using the Rekenrek Activity 1 What the Heck is a Rekenrek? (cont.) At this point let several children share how they can prove there are 6 beads… 3 and 3, counting by ones, five and 1 more, 1 and five more, 2 and 2 and 2 etc… Celebrate the multiple ways of “seeing” 6, es- pecially the ones that did not involve one-by-one counting. • Now slide 3 more beads over to the left for a total of 9. Ask, “How many do you see now?” Keep soliciting different ways to prove there are 9 beads. Hopefully, someone will use the anchor of 10 say- ing something like, “I know there are 10 beads, so one less than 10 is 9” or, “I know there are 5, and 5 more would be 10, minus one is 9.” Your goal is to look for non-counting strategies that assist students in subitiz- ing 5 and 10. We want these numbers to become anchors for the students that they quickly visualize. 6 Rekenrek © The Math Learning Center
  • 9. Using the Rekenrek Activity 2 See and Slide Getting Efficient Using the Rekenrek Grade Level You will need K-2 H One rekenrek for each child in the group H 20 Colored Popsicle sticks (numbers 1–10 in one Skills color, numbers 11–20 in another color) H Develop number sense strategies using 5 and 10 as anchor numbers H 1 container to hold popsicle sticks H Visualize the numbers 1–20 Note This activity can be extended on a ten row reken- H Build numbers efficiently rek by using 100 colored popscicle sticks numbered from 1–100. Have students represent the numbers using the fewest possible bead moves. Instructions for See and Slide Make sure each child has a rekenrek with all beads moved to the right side. (Do not cover up any beads for this activity.) Tell the students you are going to choose a number (from a container of popsicle sticks numbered 1–10). They are to “slide” that number of beads on the rekenrek, using only one move. Teacher I chose number 8. Think how you will move eight beads on the rekenrek in only one move. Now slide the beads. Mike I slid 5 reds and 3 whites on the top row all in one move! Sandy I did the same thing but on the bottom row. Pat I moved 5 reds on top and 3 reds on the bottom. Merrie Is that one move? I thought we had to use only one move. Pat I used two fingers to make one move! Teacher All of you “saw” the number eight and made one slide to show that number on your rekenrek. © The Math Learning Center Rekenrek 7
  • 10. Using the Rekenrek Activity 2 See and Slide! (cont.) Now include the numbered sticks 11-20 in the container. Tell students they are to use no more than two slides to show numbers larger than 10. Teacher I chose the number 16. Think about that number. How will it look when we move the beads? Using one or two moves, how can I show 16 on the rekenrek? Henry If you move 8 beads on each row, that will make 16. Marcia Move 10 on the top, 5 on the bottom, and 1 more on the bottom. Mark But that’s three moves. You would need to move 10 on the top and 6 on the bottom. Doubles Ten and More Again, ask students to share their thinking. Continue the activity by asking students to choose and rep- resent additional numbers. Discourage counting one by one by reminding them to use only one slide for numbers 1-10, and at most two slides for numbers 11-20. 8 Rekenrek © The Math Learning Center
  • 11. Using the Rekenrek Activity 3 Sum it Up Multiple Ways to Find A Sum Grade Level You Will Need K–1 H A rekenrek for each pair of children H Journal/paper for recording number sentences Skills H Use the anchor numbers 5 and 10 to visualize numbers H Pencil H Use the “counting up” strategy to solve problems with H Number cards 1–10 (page 25. Run on cardstock and missing addends cut apart.) H Build fluency with tens facts H Something to “hide” the rekenrek’s bottom row of beads (a folded 8˝ × 5˝ index card or piece of con- struction paper, or small piece of fabric) Instructions for Sum it Up Explain that for this activity you are only going to use the top row of beads. (Cover the botton row with a folded index card or a 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. Using the number cards, choose a number to build. How to start the activity Teacher I’ve chosen the number 7. Let’s see how many ways we can build it by sliding beads from each side into the middle. I wonder if there is more than one way? What if I slide 3 red beads from the left and 4 white beads from the right. Does that build 7? Carrie Yes, but you can also make 7 by sliding 2 reds and 5 whites. Teacher Let’s record all the ways we can build 7. © The Math Learning Center Rekenrek 9
  • 12. Using the Rekenrek Activity 3 Sum it Up! (cont.) Note record students’ combinations as noted below so they might discover the emerging pattern. 2 reds + 5 whites = 7 beads 3 reds + 4 whites = 7 beads 4 reds + 3 whites = 7 beads 5 reds + 2 whites = 7 beads Continue the activity by choosing additonal numbers from the 1–10 number cards. Invite student pairs to build the chosen number and record the different ways it can be built. 10 Rekenrek © The Math Learning Center
  • 13. Using the Rekenrek Activity 4 It Takes Two to Build a Number Finding Missing Addends Grade Level You Will Need K–2 H One rekenrek for every two students H A rekenrek for teacher demonstration Skills H Discover strategies to solve unknown or missing H Student set of number cards 1–10 (page 25) addend problems H Student set of number cards 1–20 (pages 25–26, H Develop problem-solving strategies optional) H Build fluency with facts to ten Note Rekenrek activities llike this can be extended on the ten row rekenrek by using the 1–100 number cards, pages 25-31. Have students represent the numbers using the fewest possible bead moves. Instructions for It Takes Two to Build a Number Ask a student to be your partner. Tell the class that you and your partner are going to build the num- ber 5 as a team. You will be in charge of moving beads on the top row of the rekenrek and your student partner will move beads on the bottom row. Teacher to Partner I am going to slide 3 beads to the left on the top row. Now in one move, you slide beads on the bottom row to build the number 5. Shelby I slid 2 beads on the bottom row with one move! Teacher Now I would like all of you to pair up with a partner. I’ll choose a number from the num- ber cards. You will then build that number with your partner. (Begin with cards 1–10, or 1–20 if you are practicing facts to 20). © The Math Learning Center Rekenrek 11
  • 14. Using the Rekenrek Activity 4 It Takes Two to Build a Number (cont.) Choose a couple cards to further model this activity before inviting the class to build the numbers on their rekenreks. Tell students the number of beads on the top row should be less than the number cho- sen. For instance, if the number card chosen is 8, the first student might move 5 beads on the top. Her partner would then move 3 beads on the bottom. Optional: Building numbers 11-20 Use the 11–20 number cards to practice facts to 20. Help students see that certain row combinations re- quire a minimal number of beads on one row to build numbers larger than 10. Teacher I’m going to move 4 beads on the top row. You slide beads on the second row to make the number 16. Michael That won’t work. I don’t have enough beads. Sharon You have to move more beads on the top. Kelly Move 6 beads on the top. Then the 10 on the bottom will make 16. Extension If you have a ten row rekenrek, challenge those students who are ready to represent numbers larger than 20. Have students work in pairs or small groups, taking turns drawing a number from the 1–100 number cards and building the number together or individually. 12 Rekenrek © The Math Learning Center
  • 15. Using the Rekenrek Activity 5 Build a “Doubles” Number Looking at Symmetrical Numbers on the Rekenrek Grade Level You Will Need 1–2 H One rekenrek per student H One rekenrek for teacher demonstration Skills H Add two digit numbers to find sums to 20 H Even Ten Frame Cards 2, 4, 6, 8, 10 shown in different combinations (page 32. Run on cardstock or a trans- H Use 5’s and 10’s to anchor addition and subtraction parency and cut cards apart.) facts H Optional Odd Ten Frame Cards 1, 3, 5, 7, 9 shown in H See the usefulness of doubles different combinations (page 33. Run on cardstock or a transparency and cut cards apart.) Instructions for Build a “Doubles” Number Begin with all the beads to the right on the rekenrek. Display an even ten frame card and ask students to build the number by moving beads to the left. Have students check and compare their results with their neighbors. Then ask for volunteers to share different ways to build the number. Teacher Here is a Ten Frame Card. Think of different ways to show this number on your rekenrek. Eve n Ten Frame Ca rd Ben I moved 5 red beads and 3 white beads. Kathie I saw eight on the card so I made 4 red on top and 4 red on the bottom. Florence That’s a doubles problem. 4 + 4 = 8. Repeat with several cards. Challenge students to build the numbers as doubles with equal numbers of beads on both rows. Teacher Now I am going to display two Ten Frame cards. Can you make this a double using your rekenrek? Eve n Ten Frame Ca rd Eve n Ten Frame Ca rd Sharon I moved over eight beads on the top and six on the bottom. To make a double I slid one bead on top back to the right and slid one bead on the bottom over to the left. Now I have 7 plus 7 which is the same as 8 plus 6! © The Math Learning Center Rekenrek 13
  • 16. Using the Rekenrek Activity 5 Build a “Doubles” Number (cont.) Tom I slid six over on the top and six over on the bottom. I needed to slide two more over so I slid one on top and one on the bottom and saw the double 7 plus 7 equals 14. Glenda I saw that the cards had 8 and 6. To make them the same I took one away from the 8 and added it to the 6. That gave me 7 and 7. Then I moved 7 beads over on the top and 7 on the bottom. Gill I saw two 5’s, one 3, and one 1. That’s 14 altogether. I know that two 7’s make 14 so that’s how I made mine, 7 on top and 7 on the bottom. It’s symmetrical. Optional Extension: Odd Ten Frame Cards Use the Odd Ten Frame Cards to explore the concept of doubles plus one. Teacher Here’s a ten frame card that presents a new challenge. Can you build it as a double? Eve n Ten Frame Ca rd Michael No. It has to be 6 or 8. Sharon The closest I can come is to put 3 on the top and 4 on the bottom. Kelly 7 is an odd number. Odd numbers can’t have doubles. Kip I think that odd numbers will always have doubles plus 1, like 3 plus 3 plus 1. Jessica Yes, one row will always be 1 more than the other. Teacher Let’s try another one.... When students seem comfortable with the Odd Ten Frames, mix them with the Even Ten Frame cards and ask them to identify if a card is even or odd. 14 Rekenrek © The Math Learning Center
  • 17. Using the Rekenrek Activity 6 Show the Sum How many ways? Grade Level You Will Need 2–3 H One rekenrek for each student H Optional rekenrek for teacher demonstration Skills H Develop and demonstrate computational fluency strat- H Classroom whiteboard or chart to record various com- egies for addition facts to 20 putational strategies H Add two 1-digit numbers H 2 dice or wooden cubes labled with the following numbers—Die One: 5, 6, 7, 8, 9, 10; Die Two: 4, 5, 6, H Use 5’s and 10’s as anchors to solve addition facts to 7, 8, 9 20 H Explore multiple strategies for adding two numbers Instructions for Show the Sum Invite a student volunteer to roll the two dice to determine two numbers to add. Students use their rek- enrek to share their strategies for calculating the sum. Teacher Karen’s roll turned up 9 and 6. Show me 9 plus 6 on your rekenrek. What is the sum? Shelby I moved 5 red and 4 white on the top to show 9. Since I know 5 plus 1 is 6, on the bottom I moved all 5 red beads and one more white one. They all add up to 15. Record the number sentences on a chart or whiteboard to represent each student’s strategy: 5+4=9 5+1=6 9 + 6 = 15 Hannah I know 1 less than 10 is 9 so I moved all the beads except for 1 over. Then I moved all 10 on the bottom row and slid four back to show 6. I think I made more slides than Shelby, but we got the same answer. 10 – 1 = 9 10 – 4 = 6 9 + 6 = 15 Emmy I started with 10 on the top and 5 on the bottom. Because 9 is one less than 10 and 6 is one more than 5, I just moved one from the top to the bottom. 15 is the answer and what I started with! 10 – 1 = 9 5+1=6 9 + 6 = 15 Repeat the roll of the dice several times, making sure to have several students share various ways to cal- culate a sum. © The Math Learning Center Rekenrek 15
  • 18. Using the Rekenrek 16 Rekenrek © The Math Learning Center
  • 19. Using the Rekenrek Activity 7 Roll Three for 20 or Bust! Grade Level You Will Need 2–3 H One rekenrek for each student H Optional rekenrek for teacher demonstration Skills H Develop and demonstrate computational fluency strat- H Classroom whiteboard or chart to record various com- egies for addition facts to 20 putational strategies and number sentences H Adding three 1-digit numbers H Several sets of 3 dice or wooden cubes labled with the following numbers—Red Die: 0, 1, 2, 3, 4, 5; Blue H Use 5’s and 10’s as anchors to solve addition facts to Die: 2, 3, 4, 5, 6, 7; Green Die: 4, 5, 6, 7, 8, 9 20 Instructions for Roll Three for 20 or Bust! The object of this game is to obtain a sum close to or equal to 20 without going over. • Roll the three dice to determine the first three numbers to add. • Tell students to use their rekenreks to share their strategy for calculating the sum of the three num- bers rolled. • Students must then decide to stay put or roll 1, 2, or 3 dice to add more to the sum they have so far to come close or equal to 20 without going over. Model the activity before inviting students to play this game in small groups. Roll the dice and ask the class to help you determine the sum of the three numbers using their rekenrek. Ask student to share their strategies. Use the whiteboard or a chart to record their thinking. Teacher I rolled a 6 and a 7 and a 4. How should we add these three numbers on the rekenrek? Ellie Move 6 and then 4 over. That makes 10. Then on the bottom move 5 red and 2 white which is 7. 10 plus 7 equals 17. That is a fast ten fact! 6 4 7 6 + 4 = 10 5 + 2 =7 10 + 7 = 17 © The Math Learning Center Rekenrek 17
  • 20. Using the Rekenrek Activity 7 Roll Three! For 20 or Bust! (cont.) Reba Slide 6 on top and double it by sliding 6 on the bottom. Then slide 4 over on top and 1 more on the bottom. That’s also 17. 6 4 6 1 6 + 6 + 4 + 1 = 17 Ernesto I think it would be easy to move the beads over in fives. I know in my head that the an- swer is 17, so I would slide 5, and then 5 more to make 10, and then 5 more to make 15. You still need 2 more white beads to make 17. 5 5 5 2 5 + 5 + 5 = 15 15 + 2 = 17 Teacher So we’ve determined that 6 + 7 + 4 equals 17. Do we want to roll more dice to get closer to 20? Carlin Yes, but only roll one and use the red one. It has the smallest numbers. Teacher Ok. We rolled a 4. Marci That’s too much! 17 + 4 is more than 20. We went bust! Have students break into pairs or small groups and play the game. Be sure there is a set of dice for each group. Watch for groups that get close to 20 and ask them to show you their sum on the rekenrek. 18 Rekenrek © The Math Learning Center
  • 21. Using the Rekenrek Activity 8 Target Think Addition Grade Level You Will Need 2–3 H Set of Number Cards 11-20 to use as Target Numbers (pages 25 and 26, run on cardstock and cut apart.) Skills H Set of Number Cards 1-10 to use as Starting Numbers H Use the “Think Addition” strategy to solve subtraction (pages 25 and 26, run on cardstock and cut apart.) facts to 20 H Two containers—one for holding “Starting” number H Combine numbers to find a target number cards 1-10, one for holding “target” number cards 11- H Decompose and compose numbers 20 H One rekenrek per student H Optional rekenrek for teacher demonstration Note Subtraction as “Think Addition” is a significant strat- egy for recalling subtraction facts. Instructions for Target Think Addition • Draw a Starting Number Card from a container of number cards 1-10. Ask students to build the num- ber on the rekenrek. • Next draw a Target Number Card from a container of number cards 11-20. Tell students that number is the target they want to get to. • Ask students to think of how many beads they must add to the Starting Number to equal the Target Number. Before they move any beads, ask students to share how many they will move. Then have them make the move to create the Target Number. Invite students to share their methods. This mental activity leads to the use of a “think addition” strategy rather than a “count what is left” strat- egy when solving subtraction problems. It reinforces the concept of anchor numbers and seeing combi- nations rather than counting by ones. Teacher I’m going to draw our Starting Number—it’s 4. Build that number on your rekenrek. Teacher Now let’s choose our Target Number. We drew 19! Think for a moment before you move any beads. How many will you need to move over so that you have the target number 19? Remem- ber, we want to use as few moves as we can. Charles I would move the whole bottom 10 over first and that would make 14. Then I would move 5 more over on the top and that would make the target number 19. Teacher That’s pretty good, only two moves. So how many beads did Charles move all together? Terri He moved 15. So we had to add 15 to 4 to hit the target 19. 4 + 15 is 19. © The Math Learning Center Rekenrek 19
  • 22. Using the Rekenrek Activity 8 Target Think Addition! (cont.) 4 N u m b e r C a rd 19 N u m b e r C a rd Starting Number Target Number 4 +5 + 10 Repeat this activity several times with new Starting and Target numbers. Extend to Subtraction To move students into subtraction, draw the cards in a differrent order. Start by drawing a Target Num- ber (11-20). Ask the class to build it on their rekenrek. Then draw a Starting Number Card (0-10). Ask students to think of how many beads they will need to remove from the Target Number (move back to the right) to get back to the Starting Number. Teacher This time let’s start with a Target Number. I’ve drawn a target of 17. Build that on the rek- enrek. (Pause) Now let’s draw a Starting Number—9. Think for a moment. How many beads will you need to remove from the Target Number to get back to the Starting Number 9? Samantha To get to 9 I’m going to have to move 7 back on the bottom and 1 more on top, that’s 8 I have to move to get to 9. 17 – 8 is 9. Kelly I started with 17 as doubles plus 1. To get to 9, I took double 4s away. 17 - 8 is 9. 17 N u m b e r C a rd 9 N u m b e r C a rd Target Number Starting Number 1 8 7 17 – 7 – 1 = 9 17 - 8 = 9 20 Rekenrek © The Math Learning Center
  • 23. Using the Rekenrek Activity 9 Tens or Ones Grade Level You Will Need 2–3 H Double set of number cards 11–19 (page 25 and 26, run on cardstock and cut apart.) Skills H Run Away Ones Cards (page 35. Run on cardstock and H Composing and decomposing numbers by tens or cut apart.) ones H Take Away Tens Cards (page 35. Run on cardstock and H Fluency with addition and subtraction facts cut apart.) H Using strategies to demonstrate fluency H Two containers from which to draw cards—one for holding number cards 11-19, one for holding Take Away Tens and Run Away Ones cards H One rekenrek for every student Instructions for Tens or Ones Begin by drawing one of the Number Cards 11-19 from a container. Display the chosen card to the class and ask them to build the number on their rekenreks. Next draw a card from the second container holding the Take Away Tens and Run Away Ones Cards. Have the class perform the action listed on the card by taking away either ten or the ones from the number on their rekenreks. Be sure to look for different strategies and invite student sharing. Teacher Our first number card is 19, and it looks like each of you has it built on your rekenrek. Now let’s draw from the second container—a Take Away Tens Card. What should we do? Glenda Move ten beads back to the right! That leaves 9! I just moved the top row of ten and that left nine on the bottom! 19 – 10 is 9. Repeat the process several times by drawing additional cards, asking students to build and subtract ei- ther tens or ones from the number on their rekenrek. Play until all Number Cards are drawn. Teacher Carol has drawn our next number card—16. Everyone build 16. Charlotte I put 8 on top and 8 on the bottom. Doubles! Carl I slid 10 over on top and 6 on the bottom. I think that’s the quickest way. Teacher Carol, pull a card out of our second contatiner—it’s a Take Away Ones Card. Carol We have to take away all the ones from 16. That’s 6. So take 6 away. Charlotte I took 6 away on the bottom and that left 8 and 2. I know 8 and 2 is 10. Carl I just moved the 6 on the bottom and ten are left on top! We both got 10! Carol So our number sentence is 16 – 6 is 10. © The Math Learning Center Rekenrek 21
  • 24. Using the Rekenrek 22 Rekenrek © The Math Learning Center
  • 25. Using the Rekenrek Activity 10 Count and Compare Grade Level You Will Need 2–3 H One rekenrek per student H Optional rekenrek for teacher demonstration Skills H Fluency with addition and subtraction facts H Count and Compare Gameboard (page 36. Run a half class set on cardstock and on optional transparency.) H Using the rekenrek model and mental strategies to solve addition and subtraction facts H Set of Count and Compare Cards (pages 37– 43. Run a half class set on cardstock and on an optional trans- parency, cut apart.) H Half class set of clear overlay spinners or pencils and paperclips to use as spinners Note You may wish to model this game at the overhead or play teacher against class. If so, run transparencies of the items listed above. If you would like the students to play in pairs, make copies of the gameboard and cards for every two students. Below are the directions for the student partner game. Instructions for Count and Compare Distribute the Count and Compare Gameboard and a set of Count and Compare Cards to each pair of students. Ask them to shuffle their decks of cards and place the cards on top of the gameboard. Make sure that each student has a rekenrek. To begin the game, tell each student to draw one fact card from the top of the pile and use their reken- rek to find the difference or sum for the card drawn. Ask students to share their strategies for determining solutions for their cards. Once both partners have agreed on their solutions, tell them to decide which card is more and which card is less. Have them place each card in the appropriate More or Less box on the game board. If the solutions on both cards are equal, tell them to place both cards in the equal box and draw two more cards until they have one solution that is more and one that is less. Using a clear spinner or pencil and paper clip, tell them to spin the More Or Less Spinner to decide who gets to take both or all the cards. (If cards are in the equal box, the winner gets those cards as well.) If the spinner lands on More, the person who drew the card with the greater sum or difference gets to take the cards. If the spinner lands on Less, the person who drew the card with the smaller solution gets the cards. © The Math Learning Center Rekenrek 23
  • 26. Using the Rekenrek Activity 10 Count and Compare (cont.) Have students repeat the game until they are out of cards or until time runs out. Tell each student to count the number of cards they each collected. Then tell the players with the most cards to place their stacks of cards in the More box and those with the fewest cards to put their cards in the Less box. Have them spin the spinner one more time to decide who wins the game! 24 Rekenrek © The Math Learning Center
  • 27. © The Math Learning Center Number Card 1 2 Number Card 3 Number Card 4 Number Card 5 6 7 8 Number Cards page 1 of 7 Number Card Number Card Number Card Number Card Using the Rekenrek Blackline Run on cardstock and cut apart as needed. 9 10 11 12 Number Card Number Card Number Card Number Card 13 14 15 16 Number Card Number Card Number Card Number Card Rekenrek 25
  • 28. 26 Rekenrek 17 18 19 20 Number Card Number Card Number Card Number Card 21 22 23 24 Number Cards page 2 of 7 Number Card Number Card Number Card Number Card Using the Rekenrek Blackline Run on cardstock and cut apart as needed. 25 26 27 28 Number Card Number Card Number Card Number Card 29 30 31 32 Number Card Number Card Number Card Number Card © The Math Learning Center
  • 29. © The Math Learning Center 33 34 35 36 Number Card Number Card Number Card Number Card 37 38 39 40 Number Cards page 3 of 7 Number Card Number Card Number Card Number Card Using the Rekenrek Blackline Run on cardstock and cut apart as needed. 41 42 43 44 Number Card Number Card Number Card Number Card 45 46 47 48 Number Card Number Card Number Card Number Card Rekenrek 27
  • 30. 28 Rekenrek 49 50 51 52 Number Card Number Card Number Card Number Card 53 54 55 56 Number Cards page 4 of 7 Number Card Number Card Number Card Number Card Using the Rekenrek Blackline Run on cardstock and cut apart as needed. 57 58 59 60 Number Card Number Card Number Card Number Card 61 62 63 64 Number Card Number Card Number Card Number Card © The Math Learning Center
  • 31. © The Math Learning Center 65 66 67 68 Number Card Number Card Number Card Number Card 69 70 71 72 Number Cards page 5 of 7 Number Card Number Card Number Card Number Card Using the Rekenrek Blackline Run on cardstock and cut apart as needed. 73 74 75 76 Number Card Number Card Number Card Number Card 77 78 79 80 Number Card Number Card Number Card Number Card Rekenrek 29
  • 32. 30 Rekenrek 81 82 83 84 Number Card Number Card Number Card Number Card 85 86 87 88 Number Cards page 6 of 7 Number Card Number Card Number Card Number Card Using the Rekenrek Blackline Run on cardstock and cut apart as needed. 89 90 91 92 Number Card Number Card Number Card Number Card 93 94 95 96 Number Card Number Card Number Card Number Card © The Math Learning Center
  • 33. Using the Rekenrek Blackline Run on cardstock and cut apart as needed. Number Cards page 7 of 7 97 98 99 100 Number Card Number Card Number Card Number Card © The Math Learning Center Rekenrek 31
  • 34. 32 Rekenrek Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Cards Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Using the Rekenrek Blackline Run on cardstock or transparency and cut cards apart. Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card Even Ten Frame Card © The Math Learning Center
  • 35. © The Math Learning Center Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Cards Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Using the Rekenrek Blackline Run 2 transparencies and cut cards apart. Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Odd Ten Frame Card Rekenrek 33
  • 36. 34 Rekenrek Run Run Run Run Away Away Away Away Run Away Ones Cards Ones Ones Ones Ones Run Away Ones Card Run Away Ones Card Run Away Ones Card Run Away Ones Card Using the Rekenrek Blackline Make one transparency or run on cardstock, cut cards apart. Run Run Run Run Away Away Away Away Ones Ones Ones Ones Run Away Ones Card Run Away Ones Card Run Away Ones Card Run Away Ones Card © The Math Learning Center
  • 37. Take Take Take Take © The Math Learning Center Away Away Away Away Take Away Tens Cards Tens Tens Tens Tens Take Away Tens Card Take Away Tens Card Take Away Tens Card Take Away Tens Card Using the Rekenrek Blackline Make one transparency or run on cardstock, and cut cards apart. Take Take Take Take Away Away Away Away Tens Tens Tens Tens Take Away Tens Card Take Away Tens Card Take Away Tens Card Take Away Tens Card Rekenrek 35
  • 38. Using the Rekenrek Blackline Run a half class set and one transparency. Transparency is optional. More Equal more less Count & Compare Gameboard Less 36 Rekenrek © The Math Learning Center
  • 39. © The Math Learning Center 11 11 11 11 –8 –7 –6 –5 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Cards, page 1 of 7 12 14 12 12 Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. –7 –5 –5 –4 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Rekenrek 37
  • 40. 38 Rekenrek 13 13 13 12 –6 –5 –4 –8 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Cards, page 2 of 7 12 14 13 13 Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. –7 –5 –8 –7 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card © The Math Learning Center
  • 41. © The Math Learning Center 15 15 14 14 –7 –6 –9 –8 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Cards, page 3 of 7 16 16 15 15 Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. –9 –7 –9 –8 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Rekenrek 39
  • 42. 40 Rekenrek 17 8 8 8 –8 +8 +7 +6 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Cards, page 4 of 7 17 6 6 10 Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. –9 +7 +5 +5 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card © The Math Learning Center
  • 43. © The Math Learning Center 8 11 11 11 +5 +6 +5 +4 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Cards, page 5 of 7 6 10 10 11 Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. +4 +7 +5 +8 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Rekenrek 41
  • 44. 42 Rekenrek 6 9 9 8 +8 +7 +6 +9 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Cards, page 6 of 7 11 9 9 9 Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. +7 +9 +7 +9 ___ ___ ___ ___ Count & Compare Card Count & Compare Card Count & Compare Card Count & Compare Card © The Math Learning Center
  • 45. Using the Rekenrek Blackline Run a half class set on cardstock and cut cards apart. Transparency is optional. Count & Compare Cards, page 7 of 7 8 +8 ___ 9 +8 ___ Count & Compare Card Count & Compare Card © The Math Learning Center Rekenrek 43
  • 46. Using the Rekenrek References Blanke, Barbara. 2003. Number Corner: Grade 3. The Math Learning Center: Salem, OR Burk, Donna and Snider, Allyn. 1999. Bridges in Mathematics, Grades K–2. The Math Learning Center: Sa- lem, OR. Fosnot, C., Dolk, M. 2001. Young Mathematicians at Work: Constructing Number Sense, Addition and Sub- traction. Heinemann: New York. Gravemeijer, K.P.E. 1991. “An instruction-theoretical reflection on the use of manipulatives.” In Realistic Mathematics Education in Primary School, ed. L. Streefland. Utrecht: Utrecht University (CD-(beta) series on research in education: 9). Hanson-Powell, Pia. 2003. Bridges in Mathematics: Grade 3. The Math Learning Center: Salem, OR. McKim, R. H. 1980. Experiences in Visual Thinking. 2nd ed. Brooks/Cole Publishing Company: Monterey, CA. Van de Walle, J. and Lovin, L. H. 2006. Teaching Student-Centered Mathematics, Grades K–3, Pearson Edu- cation, Inc: Boston, MA. Where to get a rekenrek The Math Learning Center PO Box 12929 Salem, Oregon 97309–0929 1 800 575–8130 www.mathlearningcenter.org Or Jegro BV Koopman Heeresweg 14 8701 PR Bolsward Holland 44 Rekenrek © The Math Learning Center