Models for operations with fractions, decimals and percents

Number strings depend on particular models that are powerful tools for student thinking.  Students should always be introduced to models through a context.  Models begin as representations of a context, then representations of strategies, and then finally become tools for thinking.

Here are some critical models for fractions, decimals and percents.

Double Number Line

This number line is developed in the early elementary grades through investigations with linear measurement[1] that gradually transition to a generalized open number line (only marked where you use it). For rational numbers, the number line can have equivalencies, as in a race, where the halfway point is 50% and 1/2. It is developed through the context of a bike race[2] or a fuel gauge,[3] or a cross country trip.[4]

Array

This powerful tool for multiplication and division is all around us: fruit boxes, windows of skyscrapers, tiles on the floor. For division with the array, try the soda machine problem in The Teachers’ Lounge.[5] For multiplication with fractions, try Exploring Parks and Playgrounds. [6]

Ratio table

The ratio table is a flexible tool that kids can use in understanding rational numbers, algebra, etc. Once it is a tool for thinking, kids can use it to make moves that make sense for them, often beginning with doubling or halving and then gradually making bigger moves. In The Big Dinner,[7] kids in grades 3 are introduced to the ratio table, including decimals, when they plan a big dinner. The model is further developed in Field Trips and Fund-Raisers[8]and Best Buys, Ratios and Rates.[9] It is a very helpful tool for multiplication, and division.

Money

Using money to help kids see equivalencies is a powerful strategy for operations with rational numbers.  It is one of the focus models in adding and subtracting fractions.[10]

Clock

Time can be another powerful tool in building kid’s understanding of equivalencies: How much of an hour is 30 minutes?  What is 1/3rd of an hour?  It is also a model for adding and subtracting fractions.[11]


[1] Fosnot, C. T. (2008). Measuring for the Art Show: Addition on the Open Number Line. Heinemann Press.

[2] Fosnot (2007). Field Trips and Fund-Raisers; Introducing Fractions. Heinemann. (Two week unit)

[3] Jacob & Fosnot (2007). Best Buys, Ratios and Rates. Heinemann. (Two week unit)

[4] Jacob & Fosnot (2007). Best Buys, Ratios and Rates. Heinemann. (Two week unit)

[5] Fostnot & Natale (2007). The Teachers’ Lounge; Place Value and Division. Heinemann. (Two week unit)

[6] Tarlow-Hellman & Fosnot (2007). Exploring Parks and Playgrounds; Multiplication and Division with Fractions . Heinemann. (Two week unit)

[7] Fosnot (2007). The Big Dinner; Multiplication with the Ratio Table. Heinemann. (Two week unit)

[8] Fosnot (2007). Field Trips and Fund-Raisers; Introducing Fractions. Heinemann. (Two week unit)

[9] Jacob & Fosnot (2007). Best Buys, Ratios and Rates. Heinemann. (Two week unit)

[10] Imm, Fosnot, & Uittenbogaard (2007). Minilessons for Operations with Fractions, Decimals and Percents. Heinemann.

[11] Imm, Fosnot, & Uittenbogaard (2007). Minilessons for Operations with Fractions, Decimals and Percents. Heinemann.

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