First, students will learn the etymology and definition of polynomial. Then they'll apply the definition to determine whether some expressions are polynomials. They'll also learn how to determine the degree of univariate polynomials, and refer to them as monomial, binomial, trinomial, etc. Then they'll learn how to identify coefficients, terms, leading terms, leading coefficients, and constant terms. Further, they'll learn how to identify the degree of terms and polynomials. The terminology introduced here is important, as students will be seeing it extensively in the future.
Conclude by giving your students these challenges:
Problem: Imagine the colored rectangular prisms, pictured below, as wooden blocks, which can be physically moved. What equation can be seen from this picture?
One way you can lead students is to cover all but the bottom row. From this, students should see 1, 2, 3, 4. From here, you can lead them to \((1 + 2 + 3 + 4)^2\) as one way to represent the picture. To help them with the right side of the equation, you can tell them they must move the blocks in some way to find a second way of describing the picture. You should only provide hints when students are totally lost. Remember that struggle is essential to learning.
Solution: Hopefully, at least some of your students will arrive at$$(1 + 2 + 3 + 4)^2 = 1^3 + 2^3 + 3^3 + 4^3$$
Once your students have found the solution, or have struggled significantly, show them this animation.