Course Guide - USAID Teacher Education Project

• There is an additional model for division: sharing or distribution. For youngchildren, introducing this model could be as simple as a story about sharing 10sweets amongst 5 children. Each child would receive 2. Thesharing/distribution model also can raise the issue of remainders, for examplesharing 11 sweets amongst 5 children.e) Division as factoring vs. division as sharing: The sharing/distribution model isless of a visual model (such as in the array and area models) and more of a dynamicmodel. Children can act out story problems (such as 11 sweets for 5 children) andequations (11 ÷ 5 = ?) to help them understand how a quantity can be divided amongsta group. Ensure that pre-service teachers notice the difference between "dividing" aquantity into parts vs. "whole number factoring."f) Nature of the Remainder: The sharing/distribution model, with its ability togenerate remainders, allows children to make real life connections for division and toconsider how to express the remainder.For example, suppose you had 11 sweets to share amongst 5 children. When actingthis out children will probably interpret the remainder of 1 as meaning each child willreceive 2 sweets and there will be 1 left over. Whereas there may be a discussionabout what to do with the leftover sweet (e.g., give it to the teacher), no young childwill consider it sensible to unwrap a small sweet, try to cut it into 5 equal sized pieces,and give 1/5 or 0.2 of the remaining sweet to each of the 5 children! In this context,the remainder is simply "1."However, in other contexts (and in abstract mathematical terms), the answer to 11 ÷ 5= ? as either a fraction (2 1/5) or decimal (2.2) would be both sensible andappropriate.g) Symbolic Representation for Multiplication and Division: While addition has its+, and subtraction has its –, multiplication and division have multiple symbolicrepresentations. The expression 6 times the quantity 3 could be written as 6 x 3, 6*3(especially on a calculator), and in algebra, 6 ⋄ 3 or 6(3).Similarly, 18 divided by 3 could be written as 18 ÷ 3, 18/3 (often found on acalculator), 18 (with a horizontal fraction bar), or by using the long division "box":32. How do children think about these concepts?a) After having used the array and area models for multiplication, youngsters shouldnotice a connection between "the whole" (the product) and its factors. Building onwhat they already know, connecting these models to division is the next step.b) However, confusion may occur when symbolic notation related to multiplicationand division is introduced: How does 6 x 3 = 18 relate to 18 ÷ 6 = 3 ? This is why itis important to build on two concepts children learned earlier: