Radical Radicals

I finished up my unit on Systems of Equations before the Thanksgiving break and was left with a two-week window before the district said I had to give a Mid-Year Exam. The pacing guide had us starting polynomials and quadratics, but that was not a unit I wanted to break up over the Christmas break, so I decided to spend these two weeks on Radicals. In the past two years, I have crammed in Radicals right before we start reviewing for the EOC, so I liked having the unit at this time instead because I didn't feel so rushed. Coming off Systems of Equations, this was also a chance for students to catch their breath with an easier concept -- before we head into quadratics and really blow their minds ;)

We started with a quick review of the Laws of Exponents and this free foldable. I added some notes for students to fill in summarizing the rule for each step before I sent it to the copier.  

I showed them how to expand each problem and then asked them to come up with the rule, which helps them to visualize what's going on, and gives them a strategy to go back to when they forget the rule.

Negative and Zero exponents always throw students for a loop, so they needed their own sheet of notes. I loved this post from Scaffolded Math and Science about Negative Exponents. We did these three problems and then I asked students to find the rule. We continued the pattern to see that what happens when you have negative exponents. 

Again, the students summarized the rule and had a pretty good understanding. So I put them to the test with this puzzle. 

I love listening to students talk about these problems and watching their understanding grow. 

We finished up with a word problem about exponential growth. I love how this problem has students interpret what the negative exponents mean in the context of this word problem - I was proud that students in each class came to the conclusion that negative exponents would indicate time before the experiment started. It also helps them solidify the idea that a zero exponent doesn't equal zero, because it represents the starting point.

Day 2, it was time for radicals! Like last year, I was determined not to teach a trick. I showed them both how to simplify with prime numbers and perfect squares. I write out a lot of steps, and often students find ways to simplify and shorten once they understand what they are doing. I also made a point of explaining every step. "The square root of 2 squared is 2, so I can simplify it as a whole number outside the radical. But the square root of 5 is an irrational number, so I leave that inside the radical." By repeating this step (what seemed like a million times), I didn't have students trying to bring the number with its exponent outside the radical. I did, however, still have some students tricked by this check all that apply.

That last problem always leads to a debate, but I had far less students thinking it was correct that in years past. We practiced simplifying radicals with this coloring activity. I love that the answer bank allows students to self-check as they go. There is nothing worse than practicing a skill incorrectly, so I like when they have to stop and find their mistake when their answer doesn't match one of the choices. Plus the coloring provides a nice brain break!

For Day 3, I put a problem up on the board with variables inside the radical and asked students to decide what to do. I've been focusing a lot on hooking everything to their prior knowledge. Someone suggested expanding x^3, and then students could see variables were really no different than dealing with factors. I love these Check All That Apply question types for this topic.

After a few examples, they practiced with a puzzle. Students had to be careful to "attend to precision" because several of the problems and answer choices were similar. This forced students to really focus on their exponents and see the difference between having x^3 inside a radical and x^4. It also meant that students didn't have to keep re-creating a factor tree for each problem. I heard great discussions about what happens when the radical has a coefficient in front of it too.

Next we moved on to Operations on Radicals - adding, subtracting and multiplying them, which I introduced with these fun {FREE} Interactive Notes. Students practice applying the operations on radicals to find the area and perimeter of shapes - and the shapes are super fun to color! 

We also did some regular practice problems on the neighboring page. 

Students practiced with a Versatiles activity and then completed an exit ticket/ mini quiz. Here is a link to the exit tickets if you want to use them.

That left us with one day left for review/wrap up. I love having an extra catch-up day like this at the end of the unit. The exit ticket provided me with some great data for who needed remediation (students who had been absent for a few days were super confused) and who was ready for some enrichment. I reminded students that so far we had been dealing with square roots, which have an index of 2, and then I presented a problem with an index of 3 and asked students how they thought it would be different. Then I did one with an index of 4 - they knew right away what to do. 

Next students completed a variety of assignments -  those who needed help completed this maze. I love how they receive instant feedback by finding their answer in one of the arrows. This maze was perfect for those students still struggling with the overall concept or students who had missed a couple classes over the two -week period. 

Some students wrapped up the Simplify Radicals Coloring Activity from earlier in the week. Those who were up for a challenge took on the Operations on Radicals Coloring Activity. Again with the answer bank, this time in the flower petals. 

Students with extra time answered some pennant problems - these are my go-to when I have a few minutes left at the end of class that needs to be filled. Plus students love decorating them and seeing their work on display on the walls and halls.

I also gave students the option to retake the Exit Ticket/ Mini Assessment if they were unhappy with their grade or thought they could do better. I love giving students the opportunity to challenge themselves to get a better score, and to see improvement. Sometimes just one class period makes all the difference in their understanding - like this student who went from a 50% to a 100% :) Totally RADICAL!