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 1/3^{rd} 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.