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Taking the Fear out of Math next #3 Multiplication Using Tiles © Math As A Second Language All Rights Reserved next Tiles and Multiplication The Egyptians had a numeral system that was similar to Roman numerals. However, unlike the Romans who would represent the numbers one through nine by writing… © Math As A Second Language All Rights Reserved next the Egyptians recognized that by arranging the tally marks in geometric patterns, it was easier to recognize the number. Thus, they might represent the whole numbers from one to nine by writing them as… © Math As A Second Language All Rights Reserved next Using patterns to represent numbers led to many interesting connections between number theory, arithmetic, and geometry. To make the results more visual for young learners it is helpful to replace the tally marks by equally-sized square tiles. © Math As A Second Language All Rights Reserved next In this way the tally mark representations of the numbers one through nine would become… © Math As A Second Language All Rights Reserved next The above representations make it rather easy to see how the tiles can be used to help students better internalize multiplication, division, and factoring. Key Point © Math As A Second Language Given any number of tiles, they can always be arranged to form a rectangle, often in many different ways. All Rights Reserved next For example, given 1 × 12 = 12 12 tiles we can arrange 1 row of 12 tiles each them in a rectangular way in any one of six ways, and each of the 12 × 1 = 12 6 ways is a segue to 12 rows of multiplication, division 1 1 tile each and factorization… note Notice that even though 1 × 12 = 12 × 1, one row of twelve tiles does not look like one column of 12 tiles. They are equal in the sense that the cost is the same if you buy 1 pen for $12 or 12 pens for $1 each, but the two transactions are quite different. 1 © Math As A Second Language All Rights Reserved next Other rectangular combinations for 12 tiles would be… 2 × 6 = 12 2 rows of 6 tiles each 6 × 2 = 12 6 rows of 2 tiles each 3 × 4 = 12 3 rows of 4 tiles each 4 × 3 = 12 4 rows of 3 tiles each © Math As A Second Language All Rights Reserved next Visualizing What a Factor Is When we write 3 × 4 = 12, the traditional vocabulary is to refer to 12 as the product of 3 and 4, and to refer to 3 and 4 as factors of 12. However, by agreeing to represent whole numbers in terms of square tiles, there is a more visual way to view what a factor of a number is. © Math As A Second Language All Rights Reserved next Visualizing What a Factor Is For example… 4 is a factor of 12 because 12 tiles can be arranged into a rectangular array that consists of 4 rows (or columns) each with 3 tiles. 7 is a factor of 28 because 28 tiles can be arranged into a rectangular array that consists of 7 rows (or columns) each with 4 tiles. © Math As A Second Language All Rights Reserved next Two Special Cases 1 is a factor of any number because any number can be arranged in a rectangular array that consists of only 1 row. For example, 13 tiles can be arranged in 1 row that consists of all 13 tiles. 1 13 © Math As A Second Language All Rights Reserved next Two Special Cases Every number is a factor of itself. For example, 12 tiles can be arranged into a rectangular array consisting of 12 rows, each with 1 tile. 12 1 © Math As A Second Language All Rights Reserved next Note The fact that n = n × 1 = 1 × n for any number n gives us an interesting insight as to how numbers can be written as a product of different factors. 1 tile already exists in the form of a rectangular array. Namely, it is already an array that has 1 row that consists of a single tile. © Math As A Second Language All Rights Reserved next Note Other than for 1, every other set of tiles can be arranged as a rectangular array in at least 2 ways, namely either as 1 row or 1 column (in the language of factors and products this simply states that n = n × 1 = 1 × n). However, for some sets of tiles these are the only 2 ways in which they can be arranged in a rectangular array, and for other sets of tiles there are additional ways. © Math As A Second Language All Rights Reserved next Note For example… As a rectangular array, 2 tiles can be arranged only as 2 rows or 2 columns each with 1 tile. On the other hand, while 6 tiles can be arranged as 6 rows or 6 columns each with 1 tile; they can also be arranged as 3 rows or columns, each with 2 tiles. © Math As A Second Language All Rights Reserved next next Definition A number greater than 1 is called a prime number if its only factors are 1 and itself. In terms of tiles, a number is prime if the tiles can be represented as a rectangular array in exactly 2 ways.2 note 2 The tile definition of prime number eliminates 1 from being a prime number because 1 can be represented in only 1 way as a rectangular array. © Math As A Second Language All Rights Reserved next next Definition A number greater than 1 that is not a prime number is called a composite number. In terms of tiles, a number is composite if the tiles can be represented as a rectangular array in more than 2 ways. The number 1 is neither prime nor composite. It is referred to as a unit. © Math As A Second Language All Rights Reserved next We will talk about prime and composite numbers in greater detail in future presentations. For now the important point is that by introducing properties of numbers in terms of tiles, even the youngest students can begin to internalize important mathematical concepts before they are encountered more abstractly later. © Math As A Second Language All Rights Reserved 2×2