Are the rest of you feeling the end of the semester like I am? It’s so close! All this grading has given me a chance to think about my next topic… and I could write about a million different things… But I think I want to continue with **simplifying radicals**, the natural segue from my last post on Factor Cakes & Ladders. There’s a chance that some of you out there haven’t quite reached simplifying radicals in your Algebra 1 curriculum this year, so I’ll give it a shot.

Before we begin, let’s lay the groundwork with a few key **division vocab words**, especially if you’re like me and you do mental gymnastics any time you come across words you don’t use regularly:

**Dividend**: The number “inside the house” when you’re dividing**Divisor**: The number that’s dividing, or breaking up, the other number**Quotient**: The answer that results from dividing two numbers.

And pictorially (*since, really, I’m a much more visual person…*):

#### Now for Simplifying Radicals:

Simplifying radicals is SO easy… for kids who have mastered their multiplication facts and perfect squares, and who have a great sense of integer division. In the traditional way of doing things:

- Yes, you can have students
**memorize their perfect squares and cubes***(In fact, maybe you should! I have been known to quiz my classes on the “list”, at times with a calculator and at times without. It really depends on the circumstances – the class, the individual ability levels, etc. We then have a conversation when those perfect squares and cubes come up in our answer and the radical completely disappears.)* - Students then
**divide their integers**until they figure out which perfect square or cube divides the original number nicely. - They write the root of the perfect square/cube in front (
*the divisor*), and the root of the quotient last, simplifying the first root if they haven’t already.

This takes a whole lot of **trial and error** for those who struggle with basic math facts. It’s possible, but I’ve found a much better method that work every single time. If you’ve read my entry on Solving Equations *the river method*, *h**ave you noticed that I tend to prefer the methods that ALWAYS WORK? * *Especially for my struggling students?*

Let’s dive right in to the most complicated type of example! The same basic rules apply when you’re working with easier radicals.

**(1)** Create a Factor Ladder/Cake to **list all prime factors of 540**: I generally default to ladders because they allow me to work down the page.

**(2)** **Circle or draw arrows next to the pairs of factors**. In this case, there’s a pair of 2s and a pair of 3s, leaving behind a 3 and a 5. My key catch phrase here is: *Couples go out on dates, singles stay home*.*

***Yes, I’m probably a terrible teacher for ingraining this “fact” in my students’ minds! But… I deal with high schoolers. It helps them to remember, and they mostly find it funny. Or corny. *Anyway*, I spend time talking about that one couple they know that’s ALWAYS together. YOU know, you could practically morph their names into one, like Ginny + Ronaldo = Ginaldo. Therefore, you **write only ONE 2 and ONE 3 out front of the radical, because they’re each one couple.**

**Just wait. IT GETS WORSE! I then tell my high school students that **couples go out and *** multiply, *and

**so do singles who stay home**. (

*gasp!*) The singles call one another up and hang out together at the same house. And if there’s a “couple that’s already out” (here, the -7), it multiplies with the other couples. The students that pick up on the innuendo find it even more hilarious than the whole dating scenario. I recommend knowing your audience before you dive right in to these jokes, but I’m telling you, they work as far as memory tricks are concerned. 😀

***Another more PG-friendly version of this is to say “*your parents won’t let you out of the house unless you go with a friend – stick together!*” etc, etc. This also works really well when you start working with cube roots.

**(3) Write out the variable factors: **

Here, we have 2 xs and 5 ys. To be consistent with my factor ladders, I write the factors vertically, although technically kids can write out the xs and ys however they’d like.

**(4)** **Pair up the variables**: I ask “how many pairs of x do we have?” One. So one x goes out front, with none remaining “at home.” “How many pairs of ys do we have?” Two. So two ys, or y squared, are in front of the radical, with one y left “in the house.”

**(5)** **Simplify!** Multiply the inside of the square root, and multiply the coefficient of the square root. *And voila*! You’re done.

**(6)** If the question happens to have all perfect squares (or all perfect cubes), like √(144a²), we talk about how all factors are paired up, so everyone can “leave the house.” Therefore, there’s no longer a need for our square root sign. The answer is just 12a.

You might ask…

### How does this differ from simplifying cube roots??

It doesn’t. Well, it does, but only slightly. In this case, students should be looking for groups of 3 instead of pairs. *The students’ parents are only letting them out of the house to go to the mall in groups of 3 this time.*

Since there’s one group of 3 and one group of ys, we write one 3 and one y in front of the cube root. Everything else is stuck “at home.” Emphasize the point that students MUST write the 3 on the “house” so we know their parents’ rules. Then they can simplify.

### Resources:

You can find a power point I created a while ago, along with the guided notes for FREE(!), on Teachers Pay Teachers. It’ll give you a sense of how I do my guided notes – I leave blanks so students can’t rush ahead and make silly mistakes.

- This lesson usually takes 1.5 blocks
- 1st block: #1 – 18
- 2nd half: #19 – 27
- Some *we do* questions, some *you do* questions

- This is the first time most of my students have seen the ladder method of prime factoring.
**If yours have worked with it before, You shouldn’t need to focus on the first half of the lesson as much.** - I photocopy pages 1 & 2 front to back
- I make HALF the copies of pages 3 & 4 front to back, and then cut those pages in half. This is intended to save paper.

You don’t have to use the notes “as is,” but they give you a good starting point.

Happy Simplifying!

– *Ms. Elsie*

LOL! Love your, “Couples go out on dates … “. Great way to remember multiplying by an already existing coefficient (which they always forget). I always call the radical a prison and you need a buddy to escape or else you’re stuck. Kind of depressing compared to yours! Ha!

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