The first time I taught
radical estimation in my 8

^{th}grade classes I used organized steps and number patterns to introduce the lesson. I modeled, answered questions, and allowed for individual practice time. However, my students looked at me like I had two heads. I went home and thought carefully about how to make the lesson more concrete. Thus the Making Sense of the Irrational discovery worksheet was born (out of necessity for my kiddos). Describing distance between numbers using pictorial representations of fractions made a huge difference for my visual learners. Now I introduce my lesson using a discovery approach.
Provide Context

First, demonstrate
why we need to estimate these radicals.
Have students plot rational numbers on a number line. Note that it’s possible to be exact or very
accurate. Then ask students to plot a
few irrational numbers on a number line.
This can open up a good discussion about how these numbers are estimated
when placed on the number line because they are non-repeating, non-terminating
decimals. This demonstration creates a
good segway; although we do not have exact decimals for these numbers, we can
estimate the value to the nearest tenth to give us an idea of the value of the
number.

Introduce with Visuals

Now that the
background knowledge has been established, use concrete models to help students
visualize the distance between whole numbers.

Here’s how it works:

1)
Identify the two perfect squares the radicand is
between.

2)
Take the square root of each of the perfect
squares.

(The
estimation is between these.)

3)
Find the distance between the two perfect
squares. Draw this many open circles.

4)
Find the distance between the smaller perfect
square and the radicand. Color in this
many circles.

5)
The pictorial representation can facilitate a
good guestimate for the decimal.

Draw Conclusions & Practice

At this point in the
lesson, some students will continue with the concrete models and some will
express “aha” moments and find shortcuts on their own. Making the lesson concrete first can help set
a strong foundation for shorter, more efficient methods that students will more
deeply understand. Consider allowing
students to Think-Pair-Share about the methods they can use. Now they are ready to practice, practice,
practice to make the skill permanent.

4 More Fun & Effective Practice Activities

Scavenger Hunt –
Students move around the room as they practice estimating radicals. Each answer
leads them to another station. A teacher and student favorite!!

Hands-On Number Line Activity – This hands-on activity allows students to work in groups to order
real numbers. This activity can facilitate great discussions – especially because
many of the numbers are close together.

Traditional Notes & Practice – This resource could make a great activity for day two to reinforce
the procedures previously discovered.
The best part is that it is differentiated based on readiness!

FREE Practice – I created
this free practice just for YOU!
Students estimate radicals then plot the estimations on number lines.

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