Upload
ezahnita-ilias
View
235
Download
0
Embed Size (px)
Citation preview
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 1/32
REKENREK
A Resource for Teachers
A Tool for Children
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 2/32
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 3/32
Sekilas Pandang
• Dalam sesi ini kita akan menguji Rekenrek,benda yg ringkas, tapi berkuasa untuk membina
kemahiran berfikir dan merangsang pemahaman
Matematik.
• Khususnya , kita akan melihat… – Rasional penggunaan Rekenrek
– Rekenrek dalam Matematik
– Aktiviti Rekenrek yang boleh meningkatkan kemahiran
berfikir murid;kefahaman dan kepakaran menambahdan menolak nombor berdasarkan sistem asas
sepuluh.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 4/32
Pengenalan
• Solve th.• As you come up with the answer, be aware of the
strategy that you used to determine the answer.
• Most likely, your brain will make adjustments on
these numbers very quickly, and you will use aninformal strategy to find the result. Of course,some of you will know these by rotememorization as well.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 5/32
8 + 7 = ?
• What mental adjustments did you make as you
solved this problem?
– Double 8, subtract 1? (8 + 8 = 16; 16 - 1 = 15)
– Double 7, add 1? (7 + 7 = 14; 14 + 1 = 15)
– Make 10, add 5? (8 + 2 = 10; 10 + 5 = 15)
– Make 10 another way? (7 + 3 = 10)
– Other strategies?
• Next problem…
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 6/32
5 + 8 = ?
• What mental adjustments did you make as yousolved this problem? – Make 10, add 3? (5 + 5 = 10; 10 + 3 = 13)
– Make 10 another way? (8 + 2 = 10; 10 + 3 = 13)
– Use another fact? (If 8 + 4 = 12, then 8 + 5 = 13)
– Other strategies?
• Next problem…
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 7/32
9 + 7 = ?
• What mental adjustments did you make as yousolved this problem? – Make 10, add 6? (9 + 1 = 10; 10 + 6 = 16)
– Other strategies?
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 8/32
Questions… • If we use these strategies as adults, do we teach
them explicitly to young children?• Should we?
• If so… how?
With the Rekenrek • The Rekenrek is a powerful tool that helps children
subitize (see “inside” numbers), develop cardinality(one-to-one correspondence), work flexibly with
numbers by using decomposition (part-part-whole)strategies, anchors of 5 and 10, and informalstrategies s.a. doubling, halving, one/two more, andone/two less.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 9/32
What is the Rekenrek?
• Developed by Adri Treffers• The rekenrek combines key features of othermanipulative models like counters, the numberline, and base-10 models.
• It is comprised of two strings of 10 beads each,strategically broken into groups of five.
• The rekenrek therefore entices students to thinkin groups of 5 and 10.
• The structure of the rekenrek offers visual picturesfor young learners, encouraging them to “see”numbers within other numbers… to see groups of5 and 10.
• For example…
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 10/32
Visual Model• With the rekenrek, young learners learn quickly
to “see” the number 7 in two distinct parts: One
group of 5, and 2 more.
A group of 10
3 more
• Similarly, 13 is seen as one group of 10
(5 red and 5 black), and three more.
5 2
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 11/32
Constructing a Rekenrek
• What do we need?• A small cardboard rectangle
• String
• 20 beads
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 12/32
• Tie a knot at the end of each string.
• Slip the ends of the string through the slits on the
cardboard so that the beads are on the front of
the cardboard, and the knot of the string is on theback side.
Constructing a Rekenrek
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 13/32
Flash Attack• Objectives
– To help students begin to “subitize” -- to see acollections of objects as one quantity rather thanindividual beads
– To help students develop visual anchors around 5
and 10 – To help students make associations between
various quantities. For example, consider the way apupil might make the connection between 7 and 10.
“I know there are ten beads in each row. Therewere three beads left in the start position. So…there must be 7 in the row because 10 - 3 = 7.”
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 14/32
Examples• Time two secs (so they do not have time to count each bead
individually)
• Push over beads and ask “How many? and ask for
explanations.
Listen for …”I knew there were reds, and /less onemore” (encourage this identification of 5 reds/ ten beads as
central to the solution)
• Expand to two rows. Ask “How many beads are there in the
top row? The bottom row? Altogether? (Reasoning)
• Look for patterns as they determine the quantities being
flashed (how they know without counting each bead)
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 15/32
Basic Combinations 0-10• Modeling the activity:
– Use both rows to keep the addends clearly visible – Suggested sequence (begin with 5 on the top row)
• “Let’s make 8. I start with 5. How many more?”
• “Let’s make 9. I start with 6. How many more?”
• “Let’s make 6. I start with 5. How many more?”
• “Let’s make 6 again. I start with 3. How many more?”
– E.g., Build on previous relationship. Use doubles.
– Listen to the thinking/explanations of students.
– Expand to Combinations of 10-20
Activity:
Extend to Subtraction
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 16/32
Basic Combinations 0-10
• Lesson Objectives
– Relational View of Equality
• One of the strengths of the rekenrek is its connections to
other forms of mathematical reasoning. For example,
equality .• Our children are only used to seeing problems like…
» 4 + 5 = ? 6 + 3 = ?
– Part-Part-Whole relationships
– Missing Addend problems
– Continue to build informal strategies and means for
combining numbers
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 17/32
Doubles• Objectives
– Help students visualize doubles (e.g., 4+4; 6+6)
– Help students use doubles in computation
• The visualization is key:
1+1=2 2+2=4 3+3=6 4+4=8
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 18/32
Developing Understanding of the Doubles
– Ask students what they notice about these visualizations.
They might see them as vertical groups of two… as two
horizontal lines of the same number of beads… evennumbers… etc.
– As students are ready, teachers should include the doubles
between 6 and 10, following a similar teaching strategy. In
this case, students should know to use their knowledge of adouble of 5 (two groups of 5 red beads = 10) to compute
related doubles.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 19/32
7 + 7 = 14
Seen as, perhaps…
2 groups of 5, plus 4
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 20/32
Almost a Double • Students should use their understanding of
doubles to successfully work with “near doubles”,
e.g., 7 + 8
• Students can begin to recognize the difference
between even numbers (even numbers can be
represented as a pair of equal numbers) and oddnumbers (paired numbers plus one).
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 21/32
Near Doubles…
1 2+1 = 3 4+1 = 5 6+1 = 7 8+1 = 9
2 + 1 3 + 2 4 + 3 5+ 4
Use a pencil
to separate
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 22/32
Near Doubles: Developing Ideas
• Develop this idea by doing several additionalexamples with the Rekenrek. Ask pupils to use theRekenrek to “prove” whether or not the following aretrue.
• Have students visually identify each component ofthese statements:
• Does 6 + 7 = 12 + 1?
• Does 3 + 2 = 4 + 1? • Does 4 + 5 = 8 + 1? • Does 8 + 9 = 16 + 1?
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 23/32
Part-Part-Whole
• To develop an understanding of part-part-wholerelationships in number problems involvingaddition and subtraction.
• To develop a relational understanding of theequal sign.
• To develop confidence and comfort with“missing addend” problems.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 24/32
Developing Ideas• Push some beads to the left. Cover the remaining beads.
• Ask students: “How many beads do you see on the top row?”
• Ask: “How many beads are covered (top row)?”
• Listen for answers like the following: – “5 and 1 more is 6. I counted up to 10 … 7, 8, 9, 10.
4 are covered.”
– “I know that 6 + 4 = 10. I see 6, so 4 more.”
– “I know that there are 5 red and 5 white on each row. I only see onewhite, so there must be 4 more.”
• Next… move to both rows of beads.
Cover
Remaining
Beads
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 25/32
Show the Sum: How many ways?
• Get two numbers (maybe from the rolling of two dice)..
• Teacher: “The dice turned up 9 and 6. What is the sum?”
• Get students to share their strategies for calculating the sum.
• Eg.: I moved 5 red and 4 white on the top to show 9. Since Iknow 5 plus 1 is 6, on the bottom I moved all 5 red beads andone more white one. They all add up to 15.
• (Record the number sentences to represent each student’sstrategy)
A t ti it
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 26/32
Automaticity:
Math Facts Objectives in Learning the Math Facts
Quick recall, yes. But… with understanding, and with astrategy!
• To develop fluency with the addition number facts through 20.
• To reinforce anchoring on 10 and using doubles as helpful strategiesto complete the math facts through 20.
• Students can model the number facts on one row of the rekenrek
(like 5 + 4), or model facts using both rows (which they have to do
when the number get larger, e.g., 8 + 7)
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 27/32
• To promote automaticity, it is important to use
the Rekenrek to visually represent the numberfacts for children.
• It is important to reinforce the various strategies
that the pupils used.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 28/32
• With other number facts, other compensation strategies
should be encouraged. E.g: 8 + 4 = 12
• While some students might see doubles, it is more likelyin this instance that pupils will anchor on 10 by mentally
sliding two additional beads to the leaft on the top row,
and then compensating on the bottom row by removing
two beads.
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 29/32
Tens or Ones
• Draw one of the number cards.
• Ask them to build the number on their rekenrek
• Next draw from the second container (the Take
Away Tens and Run Away Ones cards)
• Perform the action and look for different
strategies and invite student sharing.
• Let’s try
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 30/32
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 31/32
The Rekenrek and a Relational
View of Equality
• Take, for example, the = sign
What belongs in the box?
8 + 4 = + 5
How do children often answer this problem?
Discuss in pairs… (Activity)
7/22/2019 Rekenrek (Eng Version
http://slidepdf.com/reader/full/rekenrek-eng-version 32/32
8 + 4 = + 5
• 3 research studies used this exact problem• No more than 10% of US students in grades 1-6 in
these 3 studies put the correct number (7) in the box.In one of the studies, not one 6th grader out of 145 put
a 7 in the box.• The most common responses?
• 12 and 17
• Why?• Students are led to believe through basic factexercises that the “problem” is on the left side, and the“answer” comes after the = sign.
• Rekenrek use in K 3 mitigates this misconception