The radical sign is like a prison. Twelve can be expressed as a product of prime factors so √12 = √(2×2×3). The 2’s pair up and try to break out. Sadly, only one of them survives the escape. √12 becomes 2√3.
That’s how I was taught to simplify radicals. No joke.
I imagined the numbers yelling “All in the name of liberty! Got to be free! JAlLBREAK!” as they scaled the prison walls. To this day, I can’t get this song out of my head when I teach this topic.
Many students are shown this method, albeit without the prison imagery. Write the prime factorization of the number. Circle the pairs. Write/multiply circled numbers outside the radical sign. There is real math behind this procedure. By definition, √2 × √2 = 2. However, I found that students who were taught this method couldn’t tell me why √(2×2×3) = 2√3. Where did the other 2 go?
Instead, I asked students to evaluate √12, then 2√3, using their calculators. Why are they equivalent? Students factored √12 as √4 × √3 (with some scaffolding for some). They understood where the 2 came from. Some began by factoring √12 as √6 × √2. Correct, but not helpful. The importance of finding factors that are perfect squares was discussed.
Marc Garneau shared with me his visual approach to simplifying radicals.
Consider a square with an area of 24. The side has length √24.
This square can be divided into 4 smaller squares, each with an area of 6. The sides of these smaller squares have length √6. Two of these lengths make up the side length of the large square, so √24 = 2√6.
24 can also be divided into 3 rectangles, each with an area of 8. Again, correct, but not helpful. How to simplify √45 as 3√5 and √72 as 6√2 are also shown above. Again, factors that are perfect squares are key.
I think it would be interesting to try this out. Some students may prefer this method, but most students will likely move towards simplifying radicals without drawing pictures. But by drawing pictures as they are learning this skill, students will be connecting mathematical ideas and building conceptual understanding. New learning (simplifying radicals in Math 10) will be connected to prior learning (concept of a square root introduced in Math 8). Students will have a more solid understanding of why perfect squares are used.
Chris Hunter 
I used the approach “How much would you pay to get out of a bad movie?” Square roots weren’t so awful, it only costs two bucks to get out. so the square root of 24 would be 2^2 * 2*3.
That first guy has two bucks and gives it up on the way out.
Cube roots were even worse movies. This was always good for a laugh and a way to always have the kids thinking of prime factorization.
Joseph,
Sure it’s fun, but does it enhance understanding or is it just distracting? When students are asked to explain why root 12 is equal to 2 root 3, I want to hear something about 12 having a perfect square factor of 4. Or, thinking of prime factors, I want to hear something about the square root of 2 squared being 2 itself. Do students think that ‘one is shot/pays money on the way out’ passes for mathematical reasoning and sense making? Do cute stories or mnemonics contribute to students viewing math as a set of rules to be memorized?
Chris
I would argue that if the focus is on the mathematical thought processes, it i possible to be fun and memorable without being distracting. It’s about focus and retention. We know that memory requires hooks. I find I can effectively link real-life thought processes with mathematical thought processes to bring in fun, but I find it’s only effective if the analogy is fairly tight. Instead of movie theaters, I effectively use a simple analogy of two-factors becoming one when they are square rooted and two becoming one in a marriage, but don’t make a huge deal out of it. And when moving onto cube roots, the students see that the same thought process comes into play when they remove factors in trios. I don’t think all math procedures need to be isolated from other thought processes or must be sterile and boring just because we math teachers tend to be left brained. We must be able to connect with and instill confidence in people who find math intimidating. When we can relate mathematical though processes to thinking processes we do naturally, we can do both if our focus is well balanced.
Not anti-fun, just pro-sense-making. I’m a fun guy. Honest.
I’m not sure how the two become one marriage analogy relates to two factors becoming one. The square root function causes both factors to lose their own identities? What are these same thought processes that come into play when simplying cube roots? A guess: if square-rooting takes two numbers and makes them one, then cube-rooting makes one number out of three. The procedure can be extended, without any understanding of cube-rootyness.
If students rely on hooks (especially ones that are not especially tight) to memorize rather than make sense of the mathematics at hand, I’d argue it’s a false confidence. Or that it breaks down.
Sense-making doesn’t have to be sterile and/or boring. Check out MathyCathy’s introduction to the distributive property: http://www.mathycathy.com/blog/2013/08/simplest-inquiry-strategy-ever-discovered-completely-by-accident/
Here, the analogy of purchasing a wardrobe is naturally connected to the distributive property. Beats being told 2(3 + 4) = 2*3 + 2*4 by those left-brainers.
The deal I was most interested in was prime factorization: once that was in place all kinds of things were possible. Simplifying roots and GCF and LCM were easy beans after this. I dealt with high school kids.
I love this visual approach, and I will use it the next time I teach simplification of radicals. Thanks for sharing it!
Here’s an analogy that my kids came up with today that ties in the math concepts/vocab:
If you’re a “perfect” citizen, they let you out of jail, but anyone “imperfect” gets left behind. So as long as you’re “perfect” (a perfect square), you can get out of jail (the radical). Once you’re out of prison, you’ve gone through a lot and are a new person (take the square root). So when you factor the number inside the radical, look for perfect squares (perfect citizens) so it can get out (take the square root).
I love, love, love this and I definitely agree that getting students to *understand why* things work is the goal. I’ve used/stole it to create a lesson I’ll be using next week: bit.ly/simplifyingradicalsvisually and tweeted a thank you: https://twitter.com/WarrenJSchnack/status/1187738194810408960 . I’ll be perusing your blog and twitter feed moving forward. Everything in mathematics except notation/communication is figure-out-able. Thank you for spreading the word pushing understanding over memorization.