Introducing Multiplication …With Understanding

The Conceptual Understanding Pentagon (as I like to call it!) suggests that students should build and represent multiplication using models, connect those models to words and translate them to pictures, write a multiplication number sentence and describe a real-world situation to match. There is great power in being able to translate between these representations; mathematicians do this kind of thinking seamlessly. Although students will not be asked to represent every multiplication sentence in 5 different ways, it is important that they use 2-3 of the 5 most of the time, particularly as they are making sense of multiplication in the early years.

There are several games for practice that will support students in continuing to make meaning in multiplication. The first is called “Circles and Stars“. It’s a classic game from Marilyn Burns that focuses on the idea of multiplication as “groups of” something. Students roll a dice twice – or a double die just once. The first number tells how many circles to draw. The second number rolled tells how many starts to drawn inside those circles. Students should write a number sentence to match their picture and then solve the equation.

When playing Multiplication BINGO, students all use the same card. Like traditional Bingo, the idea is to complete a line (diagonal, vertical or horizontal). Each child in turn rolls the double dice (or 2 dice) and find the product of the numbers rolled. Students then cover the matching product on their own game card and pass the dice to the next player, who rolls just for themselves. Students should check each others work…! For example, if I roll a 2 and a 4, then I can cover the 8 in either the I column (corresponding to the 2 facts) or in the G column (which corresponds to the 4′s facts) on my own card. My partner rolls 2 numbers and finds the product on their own card and play continues in this way. To keep it within the bounds of the curriculum – and to ensure the game cards work! – cover the 6 with a sticker or tell students to roll again if a 6 comes up.

Backwards Multiplication BINGO focuses on the commutative property – that is, that 3 x 2 is the same as 2 x 3… The game comes with 4 different cards (cards A through D). Students each take a card and some counters. Like traditional Bingo, the idea is to complete a line (diagonal, vertical or horizontal). In this game, a double die is rolled (or a single die is rolled twice). To keep it within the bounds of the curriculum – and to ensure the game cards work! – cover the 6 with a sticker or tell students to roll again if a 6 comes up… The product is read aloud. Students then find the multiplication sentence that matches the product. For example, if “12″ is called out as a product, students could cover the 3×4 or the 4×3 spaces on their cards… Students will learn to be strategic as they play this game. Likewise, they will begin to see relationships between products and factors – an important idea in early division!

The BEAM game called Mice invites students to roll 2 dice and then to choose a number to cover, using either addition, subtraction or multiplication to create the numbers on the grid. The winner creates a line of three in their colour. Clearer instructions are included on the form itself, taken from the BEAM – Maths Of the Month site… I love that e-resource!

The game called “Around the World”, or “I have… who has…?” game is a great one to practice the facts and to make connections to visual representations of number. I have included two versions of the game – a game that matches arrays with their corresponding multiplication sentences, and a more abstract version which uses only numbers. Distribute all the cards and have one child read theirs aloud. In response to the question “Who has…?”, students should look at their own cards to find the match, then read their statement and ask their question. Play is over when you’ve gone “Around the World” (or rather around the room!) and ended with the same person who started the game. For the simpler version with the arrays, students should name the picture and the multiplication sentence using the language of arrays: “I have 3 rows of 4. Who has 2 rows of 3?”

It is important that students have contextualized experiences with multiplication. It is likewise important that they practice their knowledge and deepen their understanding through games and meaningful tasks – but not through timed drill. Memorization of the multiplication facts to 5×5 is not intended in the grade 3 curriculum. Spend time instead working to support your learners with mastering the ideasbehind multiplication and developing fluency in meaningful and engaging ways…

Memperkenalkan Algebra di Sekolah Rendah

Pelajar di Sekolah Rendah di Malaysia tidak diajar matapelajaran algebra secara lansung tetapi mereka telah didedahkan kepada elemen-elemen yang ada di dalam algebra dalam kemahiran-kemahiran matematik yang tertentu. Pada peringkat ini mereka boleh dikatakan mempelajari matapelajaran pra-algebra dimana melalui kemahiran-kemahiran matematik yang mereka pelajari semasa didalam kelas ada diselitkan unsur-unsur algebra di dalamnya.

Pada peringkat sekolah rendah penekanan terhadap pemikiran algebra lebih diutamakan. Perkara ini sangat penting kerana pada peringkat ini cara pelajar tersebut berfikir secara algebra adalah lebih penting berbanding kebolehan pelajar tersebut menjawab soalan matematik yang melibatkan algebra. Dalam proses memperkenalkan algebra kepada pelajar sekolah rendah iannya dimulakan dengan menghubungkaitkan proses arimetik yang mereka pelajari didalam kelas di dalam bentuk algebra. Sebagai contoh dalam operasi tambah, tolak, darab dan bahagi, pelajar boleh dilatih untuk menjawab soalan yang berbentuk proses sonsangan.

Sebagi satu contoh operasi yang sebelum ini di ajar dalam bentuk 8 + 5 = _____ boleh ditukar kepada bentuk ___+ 5 = 13 atau 8 + ____ = 18. Melalui latihan bentuk begini, kita sudah mula membina pemikiran algebra kepada pelajar tersebut dimana pelajar mula berfikir dengan cara yang lain daripada kebiasaan mereka untuk mencari jawapan terhadap soalan yang dikemukakan. Dalam proses membina pemikiran algebra terhadap pelajar di peringkat sekolah rendah ianya hendaklah dimulakan dari pewakilan nombor yang bersifat konkrit dahulu sebelum pergi kepada membina pemikiran algebra pelajar ke arah yang lebih abstrak.

Apabila pelajar telah menguasai ataupun telah membina pemikiran algebra dalam diri mereka, barulah mereka boleh dilatih kearah pemikiran algebra yang lebih abstrak itu dengan memberi pelajar soalan yang melibatkan situasi tertentu contohnya seperti berikut :-

i) Apabila satu nombor ditambah kepada 5, ia memberikan jawapan 11. Apakah nombor tersebut?

ii) Berapakah yang perlu ditolak dari 25 supaya jawapannya menjadi 8?

iii) Ali ada 5 batang pensil. Ahmad pula ada dua kali ganda pensil dari Ali. Berapakah bilangan pencil mereka?

Blanton dan Kaput(2003) telah mencadangkan, untuk menggalakkan pelajar berfikir secara algebra guru hendaklah sentiasa mengemukakan pertanyaan kepada mereka contohnya seperti berikut :-

i) Boleh beri tahu cikgu apakah yang sedang kamu fikirkan?

ii) Adakah kamu boleh menyelesaikan masalah ini dengan cara yang lain?

iii) Bagaimanakah kamu tahu cara yang kamu gunakan tersebut betul?

iv) Adakah cara yang kamu gunakan tersebut selalunya betul?

Melalui beberpa pertanyaan seperti berikut ia akan dapat meransang fikiran pelajar untuk berkir secara algebra ke arah mencari penyelesaian ke atas sesuatu masalah matematik yang dihadapi.