In this activity, students find the areas of “tilted” squares and use tracing paper, a ruler, or insights from the Warm-up to estimate their side lengths (MP5). They will see other techniques for estimating side lengths in later lessons. 

Monitor for students who use these strategies for the second question:

• Reason that the side lengths of Squares E and F are between 5 and 6 units because their areas are between 25 and 36 square units, respectively (MP7)

• Measure using tracing paper or a ruler

The goal of this discussion is to establish that if the area of a square is in between the areas of two other squares, then its side length must also be in between the side lengths of the two other squares. This reasoning strategy can be verified with a measuring strategy using tracing paper or a ruler. 

Display 2–3 approaches from previously selected students for all to see. Invite students to briefly describe their approach. Introduce the class display listing perfect squares and refer to as needed. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?” (Both approaches show that Squares E and F have side lengths between 5 and 6. The approaches are different because one uses measurements.)

• “Did anyone solve the problem the same way but would explain it differently?” (Answers vary.)

• “Are there any benefits or drawbacks to one approach compared to another?” (Answers vary.)

It is not necessary at this time to be more precise than knowing that the side lengths of Squares E and F are somewhere between 5 and 6. More precise strategies for estimating the side lengths will be explored in later lessons.

The purpose of this activity is for students to estimate the side length of a square using a geometric construction that relates the side length of the square to a point on the number line. Students then verify their estimate using techniques from an earlier lesson.

Once students connect the side length to a point on the number line, they learn that this number has a name and a special notation to denote it: square root and the square root symbol. Students will have many opportunities to deepen their understanding of square roots and practice using square root notation in later activities and lessons.

The purpose of this discussion is to introduce students to square roots and the square root symbol in the context of area and side length of squares.

Display the image from the second problem and ask students, “What do you think the actual side length of the square is? That is, what is the number that when squared is equal to 29?” As students share, take each guess and square it, noticing out loud how some come very close to 29. For example, point out that is 28.09, and is 28.6225, a number closer to 29.

Tell students that some squares like this one have areas that are whole numbers, but their side lengths are not whole numbers. Explain that there is a point on the number line (the -axis is a number line) that corresponds to these side lengths and can be imagined by rotating the square about the origin so that its sides line up with the - and -axes. 

Tell students that the exact side length of a square with area 29 square units is called “the square root of 29,” which can be written as “,” and means that .