Learn more about Graham's Foundation of Fractions Workshop and get registered.

Check out Graham's Building Fact Fluency Toolkit for addition and subtraction.

 Strategy Stage Descriptions

Stage 0Emergent Students at the Emergent stage are unable to consistently count a given number of objects because they lack knowledge of counting sequences and/or the ability to match things in one-to-one correspondence.
Stage 1One to One Counting The One to One Counting stage is characterized by students who can count and form a set of objects up to ten but cannot solve simple problems that involve joining and separating sets, like 4 + 3.
Stage 2Counting From One on Materials Students at the Counting From One on Materials stage rely on counting physical materials, like their fingers. They count all the objects in both sets to find an answer to a joining or separating of sets problem.
Stage 3Counting From One by Imaging The Counting from One by Imaging stage is characterized by students counting all of the objects. Students at this stage are able to image visual patterns of the objects in their mind and count them.
Stage 4Advanced Counting Students at the Advanced Counting stage understand that the end number in a counting sequence measures the whole set and can relate the addition or subtraction of objects to the forward and backward number sequences by ones, tens, etc.
Stage 5Early Additive At the Early Additive stage, students have begun to recognize that numbers are abstract units that can be treated simultaneously as wholes or can be partitioned and recombined. This is called part-whole thinking.
Stage 6Advanced Additive Students at the Advanced Additive stage are learning to choose appropriately from a repertoire of part-whole strategies. They see numbers as whole units in themselves but also understand that “nested” within these units is a range of possibilities for subdivision and recombining.
Stage 7Advanced Multiplicative Students at the Advanced Multiplicative stage are learning to choose appropriately from a range of part-whole strategies to solve and estimate the answers to problems involving multiplication and division. These strategies require one or more of the numbers involved in a multiplication or division to be partitioned, manipulated, then recombined.
Stage 8Advanced Proportional Students at the Advanced Proportional stage are learning to select from a repertoire of part-whole strategies to solve and estimate the answers to problems involving fractions, proportions, and ratios. These strategies are based on finding common factors and include strategies for the multiplication of decimals and the calculation of percentages.

Explanation of the Strategy Stages -click here to see students solution samples aligned to each strategy stage.