A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card)

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.

“Under the M… the square root of 12”

On this blog, sometimes I share my thoughts about transforming math education. This is not one of those times.

Here, I’m using my blog as a digital filing cabinet.

One activity that my students enjoyed was MATHO (and its variations FACTO and TRIGO).

Have students select and place answers from the bottom of each column to fill up their MATHO cards. In some versions, I pulled prepared questions from a hat. In other versions, I translated answers to questions on the fly. For example, if I grabbed 2√3, I called out “Under the M… the square root of 12”. After a student shouts “MATHO!” ask potential winners to read aloud their numbers. (Remember to keep track of answers you have called.)

Nothing revolutionary here – just a fun way to review content.

Squares & Square Roots
Exponent Laws
Simplifying Radicals
Rational Exponents
Factoring Trinomials x^2+bx+c
Factoring Special Products
Trig Functions

By the way, if you are looking to read about changing things, please check out Sam Shah’s recent post, The Messiness of Trying Something New.

A pictorial representation that will have you running naked through the streets

The sum of the first consecutive odd numbers is a square number.

Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.

Some of us (okay, me) may have presented something like this:

1     +     3     +     5 + … + (2n – 1)
(2n – 1) + … + 5     +     3     +     1

The sum of each column is 2n. We have n columns. The total is then × 2= 2n². We added the sum twice so 2n² ÷ 2 = n².

Can you see what perfect squares have to do with odd numbers? Me neither.

Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.

Inspired by this pictorial representation, I created this poster below.

 ¹ Lockhart might say it’s not the fact that perfect squares are made up of odd numbers which can be represented as L-shapes. What matters is the idea of chopping the square into these nested shapes.