Understanding Equivalence in Mathematics
The term equivalent introduces students to the idea that different representations can have the same value, even if they look different. This concept of “different but equal in value” is particularly important when working with fractions, measurements, and algebraic expressions.
This concept begins in the primary grades with number relationships and continues through more advanced ideas like fractions, equivalent expressions, and converting measurement units. Understanding equivalence supports deeper thinking about comparison, balance, efficiency, and mathematical reasoning.
Key Ideas for Teaching Equivalence
Equivalence is a big idea that shows up across many areas of elementary math: fractions, expressions, measurement, and even early number relationships. Helping students build a strong understanding of equivalence means giving them hands-on, visual, and meaningful experiences in each of these contexts.
While the representations may differ, the core idea remains the same: different forms can represent the same value. Below are some ways to explore and teach equivalence in different areas of math, with practical examples to support student understanding.
Equivalent Fractions
The term equivalent is most commonly and formally introduced when students begin working with fractions. Equivalent fractions are fractions that have different numerators and denominators but represent the same value. It’s important for students to explore this concept using concrete tools like area models, fraction strips, and number lines to see that different-looking fractions can cover the same amount of space or name the same point. For example:
Partitioning Rectangles
Give students two same-sized rectangles and ask them to divide one into 2 equal parts and shade 1 part to represent ½. Then have them divide the other into 8 equal parts and shade 4 parts to represent 4/8. Encourage students to compare their shaded areas and talk about what they notice. Guide them to see that even though the partitions look different, the shaded areas are the same—showing the fractions are equivalent. Repeat with more examples of equivalent fractions.
Exploring Equivalent Fractions with Receipt Paper
Give students a strip of receipt paper to represent one whole. Have them mark the beginning with 0 and the end with 1. Ask them to fold the strip in half and mark the fold to show ½. Then, have them fold it in half again to create fourths, marking each fold. Fold once more to show eighths, and mark those as well. As students label each crease, encourage them to look closely and notice which folds line up—revealing that some locations on the strip have more than one fraction name. This helps them see how equivalent fractions can name the same location but in a different way.
Equivalent Measurements
Equivalence also shows up in measurement, especially when converting between units. Help students understand that 12 inches = 1 foot, 100 centimeters = 1 meter, and 60 minutes = 1 hour. These conversions are examples of equivalent values. For example:
Measurement Match-Ups
Create matching card sets where students pair measurements that are equivalent (e.g., 16 ounces = 1 pound, 4 cups = 1 quart). Use visuals or actual measurement tools when possible.
Liquid Volume Equivalence with Containers
Set up a station with clear containers labeled with units (cups, pints, quarts, gallons). Let students pour water from one container into another to discover how many cups fill a pint, how many pints fill a quart, etc.
Measurement Number Line
Use a large floor or wall number line marked in inches. Have students place “1 foot,” “2 feet,” and “3 feet” cards at the 12-inch, 24-inch, and 36-inch marks. This visual connection helps reinforce that multiple units can represent the same length, depending on how the whole is partitioned.
Equivalent Expressions
Equivalence in expressions begins in the primary grades, when students learn that different combinations of numbers can have the same value, like 3 + 2 and 1 + 4. This helps them understand the equal sign as a symbol of balance, not just a signal that “the answer comes next.” Later, in third grade, students apply this understanding as they break apart multiplication problems into simpler but equivalent expressions to build fluency and flexibility.
Use activities that allow students to build and compare expressions physically, to see that different combinations can result in the same total. For example:
Balance Scale with Cubes
Use a physical or drawn balance scale. On one side, place 3 red cubes and 4 blue cubes. On the other side, place 5 green cubes and 2 yellow cubes. Ask students whether the scale is balanced and why. This shows that 3 + 4 = 5 + 2 and that equivalent expressions keep the scale balanced.
Build and Match Expressions with Counters
Give students counters and small whiteboards. Ask them to build one expression (e.g., 4 + 3) and then challenge them to build another expression that shows the same total using a different combination (e.g., 2 + 5). This shows that 4 + 3 and 2 + 5 are equivalent because they result in the same amount.
Equation Cards and True/False Sort
Give students a stack of equation cards like 3 + 2 = 4 + 1, 5 + 0 = 2 + 3, and 4 + 1 = 5 + 2. Ask them to sort the cards into “true” and “not true” piles based on whether both sides are equal. Follow up with a discussion on what made the expressions equivalent and how they knew.
Expression Number Bonds
Use number bond mats and have students show two different ways to make the same number. For example, show 7 as 6 + 1 and also as 3 + 4. Then write those as equations: 6 + 1 = 3 + 4. This connects visual part-whole thinking with formal equivalence.
Building Math Vocabulary Around What It Means To Be ‘Equivalent’
The language we use and encourage students to use around the word equivalent can significantly shape their understanding of the concept. It’s important to avoid framing equivalence as “is the same as,” which can lead to misconceptions. Instead, emphasize that equivalent means “has the same value as,” even if the representations are different. For example, a model showing ³⁄₆ and ²⁄₄ look different, but they represent the same quantity.
Encourage students to use precise language, such as saying, “has the same value as”, or, “is at the same location as” or even, “has the same area as” instead of simply stating, “is the same as,” when discussing equivalent fractions. This helps build a deeper understanding of equivalence as a relationship based on value, not appearance.
