This opening (simplifying_radicals_launch.pdf) will get students to think more conceptually about what a square root means. Have students turn and talk about what sqrt(25) actually means. Most students will say that the answer is 5. Push them to explain why. The idea is that 5*5 is 25.
Then draw a picture of a square on the board and tell students that the area of the square is 25. Have them determine the length of each side of the square. Explain to students that this is where the term "square root" comes from. Have them determine the side length for a square with an area of 16. Lastly, ask them to turn and talk about how to find the side length of a square with an area of 10. Let students share their ideas and critique each other's reasoning (MP3). Guide students towards the understanding that the length of the sides will need to be a little larger than 3 but will not be equal to 4. The next portion of the lesson will help students to make accurate approximations of square roots.
This portion will be mostly direct instruction based on (simplifying_radicals_direct.pdf).
The idea is to first help students understand that radical expression are numbers too! By thinking about "nearby" radicals that simplify to whole numbers, students can get a decent approximation as to the quantity that a radical expression represents. This will be key in students interpreting the roots of quadratic functions that are in simplest radical form.
Now that students are beginning to understand that you can take the square root of "non-perfect squares", we want to talk more about estimating. I start with sqrt(8) and ask students to make an estimate for the value. Many students will say that the answer should be a little less than 3 (around 2.8). Encourage students to explain their reasoning.
The next example is sqrt(12). Now I ask students to offer some ideas for this value once they realize it is between 3 and 4. I typically will get answers like 3.4, 3.5, 3.6. I record all of these on the board and then ask students to think about which makes the most sense. I have them do a brief turn and talk to decide. As students discuss, I am listening for students who can verbalize that the answer should be 3.4 because 12 is closer to 9 then it is to 16. We can have a similar conversation for sqrt(27). In each case, encourage students to find the "actual" answer using the calculator to see how close they are able to get with their estimate.
Slide 2 and 3
Both of these slides will require a mostly "direct teaching" approach. Both techniques for simplifying radicals are skill based and require students to follow the appropriate steps. Here is an instructional video that explains the techniques of "two and out" and "using perfect squares" for writing radicals in simplest radical form. The key to either technique is that students practice MP6 by ensuring that their answer in simplest radical form makes sense. They can do this by typing the original expression and their simplified expression into the calculator and ensuring that the decimals match.
This slide will allow students to practice the procedure they have just learned. I intentionally crafted these questions to be slightly different and an extension from those the students have just seen. The idea is that students will attempt to write the expressions in simplest radical form using a common sense approach (e.g. multiply the number outside the radical by a number that is being taken out). By using their calculator, they will know whether their attempt is correct or incorrect and they can revise their work accordingly (MP1). Try not to be too helpful during this portion of the lesson. Encourage students to make a hypothesis about how to simplify the expression, test their hypothesis, and revise if needed.
Students will work with their partners on this practice portion of the lesson (simplifying_radicals_practice.pdf). I threw some unfamiliar forms of radical expressions (#2,#3) in on purpose because I want student to think critically and logically about how they would simplify them (MP1). Try not to be too helpful and encourage students to take risks and try to figure out how to simplify these expressions. The important point to drive home for students is that they have a calculator which will allow them to compare the decimal equivalents of the original expression and their simplification to check for accuracy! (MP6). If the decimal equivalents don't match, they can go back and try again.
This ticket out (simplifying_radicals_exit.doc) will serve as a quick diagnostic check of student's understanding. Encourage students to show work so that you can see their thinking. Because this is multiple choice, you will be able to grade these quickly and get good feedback on which students have grasped the day's lesson.
NOTE: At this point in the year, my district was utilizing one-to-one computing and so I gave this ticket out to students using the Socrative learning platform. For more information go to www.socrative.com. This allowed me to get instantaneous feedback on how students did with each question.