Students have learned how to distribute with problems like 2(3 +2x). Now they combine this with powers to distribute expressions like 2x(3x+2x²). Although they are now acquiring new knowledge and skills, it is not unusual that they will forget some of the basics such as x∙x = x². Students are being presented with new information and it is not always obvious to some that they are building on skills they have already learned.
I will therefore see how they did on the previous night’s homework, and take time to go over problems they found difficult. The most common mistakes are failing to raise numbers to the power outside the parentheses; e.g., for (4a²b)³, the 4 must be raised to the third power.
This particular lesson is helpful in further cementing in the concepts and procedures of the rules of exponents for multiplication.
Warm-Ups.
Problems 1-3 are for review. The introduction of new information that builds on fundamentals can sometimes cause students to lose track of the original foundation. In particular, I’ve seen students who forget that x∙x = x². Problem 1 is a reminder of what the fundamentals are. Problems 2 and 3 emphasize the difference between adding and multiplying two terms that are the same, which students now start to do. I will see 3x² + 3x² expressed as 6x⁴, for example. Problem 5 is a segue into today’s lesson. Prompts for this problem: “You’re multiplying x by 2 and x.” “What is x times x? Look at Problem 1.”
Distribution with Rules of Exponents. Having gone over the Warm-Ups (as well as the previous night’s homework), I bring the students’ attention to Problem 5. “In this problem what we did was multiply x by 2 and x. We combined the distributive property with one of the rules of exponents for multiplication. Let’s look at another.”
I write on the board:
I write down the intermediate computations:
They now work in their notebooks as I go around offering advice, hints and so on:
They usually pick up on this right away. I then show how this same distribution can be represented by finding the area of a rectangle that is split into two. The area is equal to the sum of the areas of the other two rectangles:
Summary of Steps. The procedure is broken down simply:
Step 1: Use the distributive property to multiply each term of the polynomial by the monomial.
Step 2: Add the products.
This last I would likely work at the board with student input as I go through each step.
Common Errors. I typically still see students who will misinterpret an expression like 3(– y) as 3 – y. Also, for expressions like (3xy)² some students will continue to not raise the number 3 to the second power.
Homework. Homework consists of problems like these, interspersed with problems from the last two lessons.