This week your student will be working with the relationship between the side length and area of squares. We know two main ways to find the area of a square:

- Multiply the square’s side length by itself.
- Decompose and rearrange the square so that we can see how many square units are inside. For example, if we decompose and rearrange the tilted square in the diagram, we can see that its area is 10 square units.

But what is the side length of this tilted square? It cannot be 3 units since **square root**. We write “the square root of 10” as

9⎯⎯√=3 because32=9 16⎯⎯⎯⎯√=4 because42=16 10⎯⎯⎯⎯√ is the side length of a square whose area is 10 square units, and(10⎯⎯⎯⎯√)2=10

Here is a task to try with your student:

If each grid square represents 1 square unit, what is the side length of this titled square? Explain your reasoning.

Solution:

The side length is

This week your student will work with the Pythagorean Theorem, which describes the relationship between the sides of any right triangle. A right triangle is any triangle with a right angle. The side opposite the right angle is called the hypotenuse, and the two other sides are called the legs. Here we have a triangle with hypotenuse

We can use the Pythagorean Theorem to tell if a triangle is a right triangle or not, to find the value of one side length of a right triangle if we know the other two, and to answer questions about situations that can be modeled with right triangles. For example, let’s say we wanted to find the length of this line segment:

We can first draw a right triangle and determine the lengths of the two legs:

Next, since this is a right triangle, we know that

Here is a task to try with your student:

- Find the length of the hypotenuse as an exact answer using a square root.
- What is the length of line segment
p ? Explain or show your reasoning. (Each grid square represents 1 square unit.)

Solution:

- The length of the hypotenuse is
50⎯⎯⎯⎯√ units. With legsa andb both equal to 5 and an unknown value for the hypotenuse,c , we know the relationship52+52=c2 is true. That means50=c2 , soc must be50⎯⎯⎯⎯√ units. - The length of
p is25⎯⎯⎯⎯√ or 5 units. If we draw in the right triangle, we have legs of length 3 and 4 and hypotenusep , so the relationship32+42=p2 is true. Since32+42=25=p2 ,p must equal25⎯⎯⎯⎯√ or 5 units.

This week your student will learn about cube roots. We previously learned that a square root is the side length of a square with a certain area. For example, if a square has an area of 16 square units then its edge length is 4 units because

Even without the useful grid, we can calculate that the edge length is 4 from the volume since

Cube roots that are not integers are still numbers that we can plot on a number line. If we have the three numbers

Here is a task to try with your student:

Plot the given numbers on the number line:

Solution:

Since

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.