Understanding Perfect Squares and Square Roots

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Objective

SWBAT understand what it means to square a number, understand a number is not a perfect square, and how to estimate square roots.

Big Idea

Explore with manipulatives why some numbers are perfect squares and others are not then how this translates into square roots!

Launching the Lesson

10 minutes

My students arrive today knowing that we are about to begin a unit on exponents, but I remind them of this as they arrive. I will have written 2^2 (2-squared) on the board before students arrive. Once all of the students are seated I will ask a student to read this expression out loud. Asking a student to read the expression helps to remind my students that expression describes a process using mathematical notation. Next, I will ask several students to explain the meaning of the expression. I'll say something like, "When you hear the phrase 2-squared, what does it mean?"

I expect that some (perhaps many) of my students already understand the mathematical meaning of squaring a number like 2. By asking them to explain what they know I am both activating students as owners of their own learning and indicating that explaining the meaning of exponent notation is an expectation of our work in this unit. I want my students to both be able to share their understanding and to be resources for one another as we learn to explain and to apply exponentiation. 

After one or more students share an explanation of 2-squared, I will write the expression √16 on the board and we will repeat the same sequence of prompts. If students were quiet (or too quick to answer) during the explanation of the first expression, I may ask students to write their answers on a slip of paper. Then, students can read what they wrote, or, I may collect them and choose slips to read based on the answer that is given or at random. For example, if I sense that my students understand things pretty well, I may choose partial explanations and ask students to respond to them.

Beginning the Perfect Square Activity

1 minutes

As we begin the main activity I group students into cooperative groups of two to three students and give each group a bag or handful of square tiles that I have prepared before the lesson. I give every student a copy of the Perfect Squares Tile Activity. I ask students to write today's learning target at the top of the page is:

I will understand what it means to square a number, be a perfect square, and take a square root of a number.

For my students this is an explicit statement of the learning goals implied during the opening of the lesson. I always like to clarify the learning goals before beginning the main activity of a lesson. I think that this is particularly important at the start of a new unit. 

Once students have recorded the Learning Target on their handout, I let them begin working in their groups on Questions 1 - 8 on the Tiles Activity handout. As they work I will visit groups and provide feedback to keep the learning moving forward. I will begin with the groups that I know have less prior experience with exponents or are less likely to persevere when they encounter difficult. During my visits to the groups, I am making note of different approaches (strategies, explanations) among the groups. I try to identify specific strategies or explanations that I want students to share during the Mini Wrap-up at the end of today's class. I often tell groups ahead of time which questions they will present, indicating what I observed and liked in their work.

After about 10-15 minutes of group work on the Activity, we will reconvene for the Mini-Wrap Up. We will discuss Questions 1-8, even if all of the students have not completed all 8 questions. Students can learn from hearing other students answers, as well as completing the tasks on their own. For today I want to keep the class moving at a pace that challenges most students.  

Teacher's Notes:

  • Before the lesson I prepared square tiles that are either one inch square or one centimeter square. I used craft foam from the Dollar Tree and a die cutter to create many tiles in a variety of colors.
  • A time-saving idea for this lesson is to carry around an iphone or ipad and take pictures of the perfect squares you want groups to present.  Then, as students are speaking I project the picture of their work for the class to see. This proves to be much faster than working with tiles under the document camera.

After the students present I will ask students to look at Question 9. In this task students will complete a table. My students are sometimes confused about what to add to the table, so I like to complete the first row as a class:

The smallest perfect square was 1 square unit so only one tile was used.  The dimensions of this square for column two were 1 X 1 and the square root of 1 is 1 for the third column. 

After students understand how to complete the table, I ask my students to complete Questions 9 and 10.  If we run short on time, these two questions will be assigned for homework. The Activity concludes in tomorrow's lesson