Students are introduced to the concept of addition through the construction of addition stories. Addition stories are used to help children associate addition with the part-whole concept. At the next stage, they are introduced to the symbols “+” and “=.” As children become more familiar with the number sequence and the “count on” strategy, they will rely less on visual aids. The numbers that make up 10 are reviewed again and again. Games and repeated practices of the number facts are used to help reinforce these addition facts. |

Students are learning to add numbers up to three digits where renaming occurs. Children will also learn mentally add ones and tens to numbers within 1000. The learning task problems are rewritten vertically. Students are learning all the steps of adding with renaming in step by step sequence. At this stage, students apply mental math techniques to the addition or subtraction of three-digit numbers with renaming. The goal is for students to develop flexibility in working with numbers through experimenting with different ways of regrouping. Students develop their own criteria for which problems are easier to solve using the formal algorithm and which can be solved mentally or with alternative strategy. |

The goal at this stage is for students to understand multiplication, not on memorizing the multiplication facts. Later bar models aid multiplication. It is associated with the part-whole concept. Given the number of equal parts and the number in each part, we can multiply to find the whole (the total). |

Students are introduced to mental strategies, which are built on previous methods of addition. Learning and using mental calculation strategies encourages flexibility in thinking about numbers and helps student develop a strong number sense. Flexibility is a key. Students are encouraged to develop, utilize, and share their own strategies. The goal is for students to be able to add mentally without writing out the number bonds. |

In previous word problems students were given both steps to solve word problems. At this point, students will have to determine the intermediate steps themselves. Modeling with part-whole, comparison and a combination of both are used to solve the problems. Students are also encouraged to write the equations and determine the answer without actually drawing the model. |

Students use units in a fraction bar to represent fractions of a whole, instead of a whole number, and the total bar is 1 instead of a multiple of the value of unit. Later, students will use fraction bars to illustrate fractions of numbers larger than 1. |

Students review addition and subtraction of numbers up to 4-digits and the use of bar models. Then students learn some new mental math strategies. Models are used to diagram word problems in order for it to be easier to see what operation needs to be used. |

Students learn to extend the formal algorithm of division to decimals. At this stage, students already know the basic division facts and understand place value. Students will be solving many problems by using mental calculations. |

Students learn to solve word problems that involve finding fraction of a fraction. Several methods are shown. Students should understand each method. They only need to use one method when solving word problems, however students should be aware that other strategies are possible, each of which gives the same answer. |

Students learn to compare two quantities by finding their ratio. Students also compare two measurements. The measurements must always be in the same units. Students learn to use ratio to compare equal groups of differing quantities. The equal groups are also represented as equal bar units. Students solve corresponding word problems. |