# Equivalence

What do students need to understand?

*Main Idea: Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value. *

*Some examples include:
*

Numbers and Numeration:

- Numbers can be decomposed in an infinite number of ways. For example, 15 can be broken down into 3×5, or 13 + 2, etc.
- Numerical expressions can be named in an infinite amount of ways. For example, 42 x 3= 3 (40 + 2), or 5 +3 = 4+4, etc.
- Numbers can be named in an infinite number of ways using different place values. For example 2 hundreds and 3 tens is equal to 23 tens or 230 ones.

Number Theory and Fractions

- Every composite number can be expressed as the product of prime numbers in exactly one way, disregarding the order of factors. (Fundamental theorem of Arithmetic)
- Every fraction/ratio can be represented by an infinite set of different but equivalent fractions/ratios.

Algebraic Expressions and Equations

- algebraic expressions can be named in an infinite number of different but equivalent ways. For example 2 (4x-3)= 8x – 6

- An equation can be represented in an infinite number of different ways that have the same solution. For example 3x – 5 = 16 and 3x = 21 because both solve out to x= 7

Measurement

- Measurements can be represented in equivalent ways using different units. For example, 2 feet, 3 inches = 27 inches.
- A given time of day can be represented in in more than one way.
- For most money amounts, there are different, but finite combinations of currency that show the same amount. For example, 5 nickels = 1 quarter. The number of coins in two sets does not necessarily indicate which of the two sets has the greater value. For example, 2 quarters is greater than 5 nickels.

(Randall, 2005)

What resources are available for teaching Equivalence?

Pan Balance – an interactive scale from NCTM illuminations. It uses shapes to explore equivalence through a balance. Students can add different shapes and see how it tips the balance. For older students, there is a similar application that uses a pan balance to compare equivalent expressions. Expression pan balance

Other similar interactive activities for different concepts using a pan balance can be found at the illuminations website. This is a great resource for many interactive links relating to mathematical concepts.

Students can also practice equivalence using money. The money site creates problems for students to solve. Students can use different coins to make a certain amount. They can see how different coin combinations work to make the desired, equivalent amount.

Equivalent fractions can be explored using an application that allows you to create an equivalent fraction to the fraction given.

Another all around neat math website is cool math 4 kids. It has all kinds of games on all kinds of subjects.

Other useful tools for teaching about equivalence include pattern blocks, Cuisenaire rods, fraction circles, fraction bars, and fraction towers, or a fraction chart. These can be bought and used in the classroom, or you can find virtual versions online. One good resource is the National Library of Virtual Maniuplatives. For more information about how to effectively use manipulatives in the classroom, you can visit the OTEC website.

References:

Randall, C.I. (2005).Big ideas and understandings as the foundation for elementary and middle school mathematics. *Journal of mathematics education leadership*. *7*,