Exponents: Just the Beginning

The exponent of a number shows how many times to multiply it; it is repeated multiplication. In Grade 5, students are introduced to exponents while writing big numbers in expanded form. At this level, they are required to use whole number exponents to denote powers of 10, use the new terminology (base, power, squared, and cubed), and practice moving between exponential notation, expanded notation, and standard notation.

This learning experience is part of the unit plan titled "Numbers" under the central idea "Numbers can be written and represented in different forms."

The Inquiry Prompt. Prompts are mathematical statements that promote students' curiosity, develop mathematical knowledge, challenge them to discover new concepts, and provide opportunities for different forms of thinking, including inductive and deductive approaches (as defined by Andrew Blair, @inquirymaths, inquirymaths.com).

Learners were divided into groups and each group took an inquiry prompt similar to the following. All the prompts were about the power of 10, with a different power for each group. Learners were asked to write what they noticed at three different stages.

Learners demonstrated a deep inquiry process, connecting their prior knowledge to develop their understanding of the mathematical structure of the prompt. They interpreted the three cards of the prompt and wondered if they represent the same value. The learners were able to conclude that the yellow and the blue cards are equal. However, a common mistake came to some learners' minds when finding the product of the yellow card: they wrote 100 × 100 equals 200. Teachers were ready with their scaffolding questions, so they asked learners to reflect and go back to the definition of multiplication as repeated addition and to how we multiply by the multiples of 10. Therefore, learners connected the yellow card to the blue one and started wondering about the meaning and the value of the green one.

At the final stage, learners drew the following conclusions: the "small number" (power) represents the number of zeros in the number written in word form; all the cards are equal, but they are written in different ways; and repeated multiplication is written in different forms.

They also raised their own questions: What do we call repeated multiplication? What is the value of 100? If 1,000,000,000 is equal to a power of 10, is 9,000,000,000 too? Why do we use exponents? Can we do calculations (add, subtract, multiply, or divide) with exponents? Do we use power only with base 10, or could it be applied to other numbers? Is an exponent written only in standard form? Why do we learn about exponents and where can we apply them in real life?

Teachers updated their planning based on the learners' wonderings and proceeded with the cycle of inquiry until the learners reached the understanding of the required generalizations: exponents are used to represent repeated multiplication, and exponential notation simplifies the writing of numbers in expanded form.

The Assessment. In order to assess the learners' understanding, teachers provided them with a WODB (Which One Doesn't Belong?) activity where learners apply their mathematical and reasoning skills to decide which of the four items does not belong, and then justify why their choice is valid. The task surfaced not only what learners knew about exponents, but how flexibly they could reason across representations.

Fifth grade teachers Noor Taweel, Farah Darazi, Nadine Assaf, Nancy Karout, Zeina Zaatary, Sasha Ghosn, and Rasha Hamzawi collaborated very well to implement this engagement in their classrooms.