Models+and+Tools

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 * __Models and Tools __**

Models and Tools are used to represent students' thinking. The examples below are tools and models that students use to help explain their thinking of various strategies for problem solving.

=**// Grades K-2 //**=

** The Math Rack or Rekenrek **


The rekenrek, or arithmetic rack, was designed by Adrian Treffers, a mathematics curriculum researcher at the Freudenthal Institute in Holland, to support the natural development of number sense in children.There are different versions of the math rack. Smaller versions, used in our primary grades consist of two rows of 10 beads. Larger versions with ten rows of ten beads are also available. Each row is made of five red beads and five white beads. This allows students to make mental images of numbers. Using 5 and 10 as anchors for counting, adding and subtracting is obviously more efficient than one-by-one counting. This tool provides learners with the visual models they need to discover number relationships and develop a variety of addition and subtraction strategies, including doubles plus or minus one, making tens, and compensation, thereby leading to automaticity of basic facts. ( [|K-5 Math Teaching Resources] )

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**Ten Frames **
Ten frames and dot cards can be used to develop students’ subitizing skills, the ability to “instantly see how many”. This skill plays a fundamental role in the development of students’ understanding of number. Two types of subitizing exist. Perceptual subitizing is closest to the original definition of subitizing: recognizing a number without using other mathematical processes. For example, a child as young as two might “see 3” without using any learned mathematical knowledge. Conceptual subitizing is being used when a person sees an eight dot domino and “just knows” the total number. The number pattern is recognized as a composite of parts and as a whole. The domino is seen as being composed of two groups of four and as “one eight”. ( [|K-5 Math Teaching Resources] )

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** Open Number Line with Addition and Subtraction **
The empty number line, or open number line as it is sometimes referred to, was originally proposed as a model for addition and subtraction by researchers from the Netherlands in the 1980s. A number line with no numbers or markers, essentially the empty number line is a visual representation for recording and sharing students’ thinking strategies during the process of mental computation. Before using an empty number line students need to show a secure understanding of numbers to 100. Prior experiences counting on and back using number lines, recall of addition and subtraction facts for all numbers to ten and the ability to add/subtract a multiple of ten to or from any two-digit number are all important prerequisite skills. ( [|K-5 Math Teaching Resources] )

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=//** Grades 3-5 **//=

** Open Array with Multiplication and Division **
An array is a rectangular arrangement of a set of objects or numbers often in rows or columns used to express multiplication and division situations. Real world examples of arrays are used to help introduce these concepts. Examples of arrays are found in everyday life such as a box of chocolates, a crate of oranges, rows of chairs in an auditorium. An open array, like the open number line, is a model for multiplication and division. Unlike the closed array, the open array does not explicitly show each object in the row or column but is a pictorial representation of larger numbers. It lends itself to modeling the partial product strategy and the distributive property of multiplication.

Open Array for Multiplication
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Open Array for Division
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** Partial Product and Partial Quotient **
Using partial products to multiply is a strategy that demonstrates an understanding of the Distributive Property. The array is a way to visually represent that strategy. The following is an example of how students would use partial quotients to solve a “long division” problem. The strategy relies on using multiples of ten to “partition” the number. It is the same as the "traditional" (using compatible numbers) with the quotient recorded on the right side rather than on top of the problem.

** Partial Product **
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**Partial Quotient **
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