Mengembangkan Penalaran Proporsional Siswa oleh: DR. FAHINU, M.Pd

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1 Mengembangkan Penalaran Proporsional Siswa oleh: DR. FAHINU, M.Pd

2 Mohon dikerjakan Kura-kura menempuh jarak 1 km dalam 2 jam, Berapakah jarak yang ditempuh bila 8 jam? Manakah hubungan proporsional untuk mengerjakan situasi ini? Mengapa? Jelaskan! Use the Handout in Trainer Notes with the proportion shown. Turn the card 90 degrees and read the new proportion. Turn the card again 90 degrees---read the proportion Turn the card one last time 90 degrees. Why does this work? (You are maintaining the multiplicative relationship that exists within and between the numbers in the relationship.) Hold up the copy of the proportion in the handouts folder. Once the proportion is set up correctly, the paper can be turned to show the between and within proportions. Non-example: x/8 = 2/1 The unknown number of lawns can be related to the number of lawns that could be mowed in 8 hours, but not in the same way that 2 hours relates to 1 lawn. The multiplicative relationship between and within is not maintained.

3 Berpikir Aljabar & Geometrik
Kuantitas Berpikir Aljabar & Geometrik Bahasa Proportional Reasoning Numerasi Bentuk Bilangan Kesamaan Dasar 10 Komponen Number Sense

4 Proportional Reasoning
Penalaran proporsional adalah meliputi hubungan matematis antara dua kuantitas. Penalaran proporsional merupakan aktivitas mental dalam mengkordinasikan dua kuantitas yang berkaitan dengan relasi perubahannya. . Have participants read this slide.

5 Idea Kunci pada Penalaran Proporsional
Rasio merupakan perbandingan dari dua kuantitas/ukuran(pecahan, persentase, peluang, kecepatan, , dll). Proporsi merupakan pernyataan kesetaraan antara dua rasio. We will explore these key ideas in the following slides.

6 Masalah rasio dan proporsi
Gunakan pemahaman bahwa hubungan perkalian adalah sama untuk setiap rasio dalam proporsi dalam mencari nilai yang tidak diketahui berikut. Have participants note the within the ratio relationship. (multiply by 5) Equality is maintained when this multiplicative relationship is maintained.

7 Masalah rasio dan proporsi
Hubungan perkalian antara dua rasio dalam proporsi. 2:10 = 4 : x Jika kita mengalikan kedua bilangan dalam rasio 2:10 oleh 2, kita menemukan nilai dalam rasio kedua. Notice that we are just multiply by 2/2 which is one. Here is another chance to reinforce the Identity Property for Multiplication.

8 Mengapa kita mempelajari penalaran proporsional?
Use the Handout in Trainer Notes with the proportion shown. Turn the card 90 degrees and read the new proportion. Turn the card again 90 degrees---read the proportion Turn the card one last time 90 degrees. Why does this work? (You are maintaining the multiplicative relationship that exists within and between the numbers in the relationship.) Hold up the copy of the proportion in the handouts folder. Once the proportion is set up correctly, the paper can be turned to show the between and within proportions. Non-example: x/8 = 2/1 The unknown number of lawns can be related to the number of lawns that could be mowed in 8 hours, but not in the same way that 2 hours relates to 1 lawn. The multiplicative relationship between and within is not maintained.

9 Penalaran proporsional: Dasar matematika tingkat tinggi
Tidak selalu dikerjakan dengan representasi simbolik Menyelesaikan masalah rasio dan proporsi dalam kehidupan sehari-hari. Mempunyai empat level strategi peny: 0 (gunakan strategi penjumlahan, solution diperoleh karena keberuntungan) 1 (gunakan gambar, model, manipulasi) 2 (gunakan strategi level 1 and strategi perkalian/pembagian ) 3 (gunakan perkalian silang atau kesamaan rasio) Levels Level 0: Student is able to make a lucky guess or find that by adding, the proportion works for the first time they try to establish the relationship. (Example: 6 Students/1 Principal) If a student can see that adding five to the number of principals will produce a rule for the first situation in a T-Chart, then they might infer that this rule will work for all cases. Level 1: Students may use pattern blocks to show that 3 hexagons: 2 squares, then they can demonstrate that 6 hexagons is proportional to 4 squares. They may be able to represent this relationship with pictures as well. Level 2: Students begin to see multiplicative relationship between and within their proportions. Remind participants this works for division as well since dividing by 5 is the equivalent to multiplying by 1/5. Level 3: The use of equal ratios– ½ = 2/4 and the cross multiplication procedure is where teachers traditionally begin exploration of this concept. Cross-multiplication is not proportional reasoning, but becomes the most efficient way for students to solve most proportions. Establishing a conceptual base for proportional reasoning is critical to Algebraic and Geometric Thinking! Say no to “Just Do It!”.

10 Karakteristik Pemikir Proporsional (Lamon,1999)
Memiliki pemahaman ttg kovariasi: memahami hubungan dua kuantitas yang mempunyai variasi bersama dan dapat melihat kesesuaian antara dua variasi berbeda. Mengenali hubungan proporsional dan non-proporsional dalam dunia nyata. Mengembangkan banyak strategi untuk menyelesaikan masalah proporsi. Memahami rasio sebagai entitas tersendiri yang menyatakan hubungan antar kuantitas. Read through these student issues.

11 Bagaimana penalaran proporsional siswa dan Level Strateginya?

12 Malasah (Perbandingan senilai):
Siswa kelas IV membutuhkan 5 helai daun setiap hari untuk memberi makan 2 ekor ulat. Berapa helai daun untuk memberi makan 12 ulat? Participants may draw pictures, diagrams, etc to solve this problem. They should do so intuitively as fourth grade students have not had any experience with formal proportional reasoning. This is what Deizman and English refer to as diagram literacy. Possible correct answer: For every 5 leaves I can feed two caterpillars, so if I have 12 caterpillars that is six groups of 2 caterpillars. So six groups of five leaves equals 30. Participants could diagram this proportional relationship by showing 2 caterpillars with 5 leaves coming off the caterpillars. Six groups of 2 caterpillars would give students 12 caterpillars---so they could then count the leaves or add groups of five or multiply 6 x 5 until all six groups have been accounted for.

13 Solusi (Level 0)

14 Solusi (Level 1) x 1 2 3 4 5 6 7 8 10 12 9 15 18 16 20 24 25 30 30? 36 Jadi jumlah helai daun untuk memberi makan 12 ekor ulat adalah 30 lembar daun

15 Solusi (Level 2): 5/x = 2/12 5/x = 1/6 =5/30 X = 30 Jadi jumlah helai daun untuk memberi makan 12 ekor ulat adalah 30 helai daun. Biasanya kesulitan karena tidak mendapat kan pecahan senilai sehingga menggunakan diagram

16 Solusi (Level 3): 5/x = 2/ = 2.x 60 = 2x X = 60/2 = 30 Jadi jumlah helai daun untuk memberi makan 12 ekor ulat adalah 30 helai daun.

17 Masalah (Kesebangunan):
Ukuran lebar dan tinggi sebuah slide berturut-turut adalah 36 mm dan 24 mm. Jika lebar pada layar 2,16 m, tentukan tinggi pada layar.

18 Solusi (level 0), salah/tidak dapat mengerjakannya karena mereka menekankan pada penjumlahan
Solusi (Level 1), kesulitan membuat gambar/tabel karena harus membuat 36 kali kolom … 96 72 48 24/36 106 142

19 Solusi (Level 2): 36/24=2160/x x =24 . 60 =1440 mm = 1,44 m 60 x0
24/36 72 108 144 … 2160 60 x0 36/24=2160/x x = =1440 mm = 1,44 m

20 Solusi (Level 3): 36/2160=24/x 1/60 =24/x x =1440 mm = 1,44 m

21 Solusi (level 0), salah/tidak dapat mengerjakannya karena mereka menekankan pada penjumlahan
1 10

22 Masalah (Reduksi): Sebuah lantai ruangan berukuran 8 m x 10 m ingin digambar pada kertas gambar dengan skala 1 m : 10 cm. Tentukanlah ukuran lantai ruangan pada kertas gambar tersebut.

23 Solusi level 0, tidak dapat mengerjakan
Solusi level 1: 80 cm x 100 cm 10

24 Solusi (Level 2/3) (8 m /10 cm) x (10 m / 10 cm)
(800 cm/10 cm) x (1000 cm/10 cm) 80 cm x 100 cm. Jadi ukuran lantai ruangan pada kertas gambar adalah 800 cm x 100 cm.

25 Masalah (Penguk. tak langsung):
Sebuah pohon mempunyai tinggi 180 cm dengan bayangannya 240 cm. Berapakah tinggi bangunan yang mempunyai bayangan 1200 cm? Level 0 & 1, Salah/sulit Solusi (Level 2/3): 180/x = 240/ /x =1/5 180(5)/x = 1(5)/5 x = 900 cm

26 Alasan tentang situasi proporsional Mencari nilai satuan unit
Peluang yang dibutuhkan siswa untuk memahami masalah (Level 0, 1, dan 2): Alasan tentang situasi proporsional Mencari nilai satuan unit Mengkonstruksi rasio equivalen Menghubungkan rasio, persen, and pecahan Relates to the caterpillar and leaves NAEP problem, as well the Paper Clip Chains. How much per item? ½ = 2/4 What is the connection between these forms of the number? Students are intuitive about the relationships, so let them explore.

27 Menghubungkan Level 3 Memerlukan pemahaman yang kuat dari komponen penalaran proporsional: Perubahan antara rasio equivalen (bukan penjumlahan) Perubahan perkalian adalah konstan Hubungan antara rasio adalah faktor skala (Jika mengalikan suatu rasio dengan faktor skala, hasilnya adalah rasio yang baru) Have participants read.

28 Masalah Sebuah bola besi berjari-jari 3 cm, dimasukkan ke dalam tabung berisi air sehingga permukaan air dalam tabung naik. Jari-jari alas tabung 10 cm, Berapa cm kenaikan air dalam tabung tersebut? Have the participants discuss: How will students solve this problem involving proportional reasoning? (Perhaps by cross multiplying depending on their experience.) Is it just an exercise in cross-multiplication? (No, students should seize the opportunity to think about the fact that they will be traveling less than 2 hours extra to travel the extra 40 miles. This type of problem can reinforce time measurement. What if students see that for every 60 miles it will take 120 minutes, which means that if I double the miles, I double the minutes. So 40 additional miles will take 80 minutes more for a total of 200 minutes which is 3 hours and 20 minutes.) This is making sense of the concept rather than 60:2 = 40:x and cross multiplying to get 1 1/3 hours and then adding in the 2 hours and then oh my goodness what is a third of an hour??? You will still arrive at 3 hours and 20 minutes, but without the understanding involved in the intuitive strategy that students need to feel comfortable with at this stage of the concept.

29 Solusi (hanya Level 3): Volume air yang naik = volume bola
.r22.t =4/3..r13 3,14(10)2.t =4/3(3,14)(3)3 t = 36/100 = 0,36 Jadi kenaikan air dalam tabung adalah 0,36 cm.

30 Mengembangkan Penalaran proporsional
Kegiatan informal untuk mengembangkan penalaran proporsional (Van De Walle:2008): Mengidentifikasi hubungan perkalian Pemilihan rasio equivalen Perbandingan rasio Pembuatan skala berdasarkan tabel rasio Konstruksi dan pengukuran. Selanjutnya banyak latihan Problem solving

31 PROBLEM SOLVING

32 Cognitive Processes in Problem Solving Richard E. Mayer
Translating Integrating Planning Executing Remember these are the four steps involved in Mayer’s problem solving process. We will trace these processes through the area and perimeter lesson.

33 Masalah Seorang petani mempunyai 200 m kawat untuk memagari suatu kebun berbentuk persegi panjang. Petani tersebut menginginkan luas kebunnya mempunyai luas yang maksimum. Berapakah panjang dan lebar kebun petani tersebut?

34 Translating Translating involves the language we use in talking about the math, both semantic and linguistic. We must find ways to help students translate and internalize the language of mathematics.

35 Discussion of the Concrete
Prototype Mengkonstruksi Pengetahuan 2 1 Luas (Squared Measure) vs. Keliling (Linear Measure) L= p x l=10 P= 2p + 2l=14 In this slide we can see how we can view the quantity, math structure and the symbols involved in a understanding area and perimeter. Kuantitas: Konsep konkrit Math Structure: Discussion of the Concrete Symbols Record Keeping!

36 Frayer Model for Linguistics
Definition (in own words) Characteristics Pengukuran satu dimensi Melibatkan penjumlahan Melibatkan problem solving Jarak keliling suatu objek (linear measure) Keliling Let’s see how the model would work with “perimeter”. Click through the transitions. Non-Examples Examples 10 cm 2cm The picture shows an Luas = 20 cm². Keliling = 24 cm

37 Frayer Model for Linguistics
Definition (in own words) Characteristics Melibatkan pengukuran dua dimensi Imelibatkan perkalian Melibatkan problem solving Ukuran interior of a figuresuatu gambar (squared measure) Luas And for area— Click through the transitions. Non-Examples Examples 10 cm 2cm Keliling = 24 cm. Luas = 20 cm²

38 Problem Solving-Translating
Mengkonversi masalah ke dalam gambaran mental. Saya mempunyai 200 m kawat, dibuat persegi panjang. Saya tahu bahwa persegi panjang mempunyai 2 pasang sisi yang sejajar. Saya tahu bahwa untuk menghitung keliling menggunakan penjumlahan dan menghitung luas menggukan perkalian. Saya tahu bahwa banyak ukuran pasang sisi yang berbeda yang dapat dibuat. These are some of the concepts that must be talked about with students through discussion and modeling with regard to perimeter. We must teach students to expand their thinking about a concept through these rich discussions and modeling.

39 Problem Solving-Translating
Saya tahu bahwa luas adalah ukuran persegi pada interior gambar persegi panjang. Saya tahu bahwa luas yang diinginkan adalah maksimal. These are some key ideas that the teacher can illicit from students or model in the translating process.

40 Integrating

41 Problem Solving-Integrating
Membangun model mental problem. Schematic knowledge is needed to tie the information together. Jika Keliling adalah 200 m, persegi panjang mempunyai luas maksimum?

42 Planning

43 Problem Solving-Planning
Konkrit (Kuantitas) Representasi (Verbal Discussion) Abstrak (Symbolic) Strategic Knowledge is needed here to create the plan. At the beginning of the concept—the concrete/quantity must be addressed.

44 Konkrit Semua persegi panjang yang terjadi disusun dari persegi satuan sisi10 m

45 Representasi Bagaimana kita mengorganisasi data?
Persegi panjang manakah yang mempunyai luas terbesar bila kelilingnya tertentu? What questions other than these can we ask to connect students to what they already know about area and perimeter? We can begin to address our Components of Number Sense questioning as we flow through the lesson. See next slide.

46 Executing

47 Organisasi Data p l K= 2p + 2l L =p x l 1 9 20 units 9 sq. units 2 8
3 7 21 sq. units 4 6 24 sq. units 5 25 sq. units 20units Note that this chart becomes a representation of the concrete data that was collected.

48 Symbolic 1 x 9 = 9 square units 2 x 8 = 16 square units
5 x 5 = 25 square units (largest area!!!)

49 Jadi panjang kebun adalah 5 x 10 m = 50 m, lebar kebun adalah 5 x 10 m = 50 m, serta luas kebun maksimal adalah 2500 m2

50 Penutup Penalaran proporsional merupakan dasar matematika tingkat tinggi sehingga penalaran proporsional siswa perlu dikembangkan secara kontinu. Mengembangkan penalaran proporsional siswa perlu memperhatikan level penalaran proporsional. Melatih siswa mengembangkan strategi melalui problem solving