As students arrive in class, I give each group the square tiles used on Day 1. Once all of the groups are seated I will ask students to work together to represent each of the following numerical expressions using the tiles:
42 62 √16
As students work on this task, I will walk about the room taking pictures of different student work samples. As I take pictures, I will ask each group to explain their representation. I also ask students to evaluate the expression (if they do not). Because some groups will have realized that they can represent all three expressions with two models, I like to ask, “Why do you only have two squares represented when I have three expressions on the board?” Of course I want my students to understand that a 4 X 4 square is representing "four squared", and, it can also be used to show the "square root of 16". With respect to our learning goals, I want to make sure that students are able to provide a better explanation than, "Its the same thing."
Once the groups have considered each expression, we will have a whole class discussion about the use of exponentiation and the use of radicals, using squaring and square roots as our example. The use of the tiles helps to ground the discussion. Its always clear that we are talking about the area of a square, and the side lengths of the same square. But, we are also considering the properties of the numbers. The discussion of the opening three expressions sets the stage for looking at three more numerical expressions.
When we reach a lull in our initial conversation, I will erase the three expressions on the board and write these three in their place:
32 √9 √12
I will casually say, "Let's practice with three more," although I know that there is a new obstacle that students will encounter. Again, I will move about the room, this time a little more quickly trying to get a sense of what students do when they try to represent the square root of 12. As I observe students struggling to represent this quantity, I have one question that I want them to consider: Why is it so hard to represent this number with the tiles?" When students make observations that relate to some of the other numbers we have worked with like 9 and 16, I will ask, “Does this mean that there numbers that are not perfect squares?” When we debrief their work, we will focus on the different ideas that students came up with for representing the square root of 12.
In some cases it may make sense to use the discussion time as an opportunity to create a list of perfect square numbers on the board. I like to make such a list vertically on the board, then discuss the gaps between the numbers. I find that the list helps some of my students to follow the conversation and participate by pointing out observations that they make while looking at the list on the board. I also like to ask my students to try to create a perfect square with five tiles. With only five they can work as we talk, gain a sense that there is a problem. Then, I like to say, "If you have to make a square, what is the largest that you can make using 5 square tiles?" When they suggest a 2x2 square, I like to start the conversation about estimating the square root of 5. I'll say, "So, do you think that the square root of 5 is pretty close to the square root of 4?" This question helps students to begin thinking about non-perfect squares. In Question 11 of the Perfect Squares Tile Activity, students will be asked to estimate the numerical value of radical expressions with non-perfect square radicands.
Although we are continuing an investigation from yesterday, I will begin today's lesson by clarifying the learning goals for today. The final discussion from the Lesson Opener is a good example of our objective:
Extend your understanding of squares and square roots to include strategies for estimating the square root of non-perfect squares.
After identifying the objective I will lead a discussion of the answer to Question 10. In this conversation we will focus on the relationship between squares and square roots. I will scribe observations on the board to provide students with a visual reference as the discussion builds. During the discussion I make sure students have access to their square tiles so that they can try out ideas.
Following our discussion, of Question 10 students will work in their cooperative groups to complete questions 11 – 12. Again, I move about the room assessing students as they work. I am making sure that groups are on target, and I am looking for examples that students will be able to share during the Mini Wrap-up. Today, I try to assign each group something to present. Today's work is a good opportunity for multiple groups to present on the same problems, if their strategies are different. All students will benefit from seeing a variety of methods and thinking. I tell the groups what I would like them to present, why their work is noteworthy. This may include asking students to share ideas or hints that we discussed, so that other groups can benefit from the learning that occurred. I find that Mini Wrap-ups provide an opportunity for students to develop confidence explaining mathematical ideas, so I regularly give students who I helped to become experts the opportunity to share.
The primary content standard addressed by this lesson is 8.EE.A.2:
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Several mathematical practices were developed and/or demonstrated over the two days of the lesson. During collaboratively work time, students become resources for one another and worked together to resolve problems:
MP1 Make sense of problems and persevere in solving them.
As I questioned student groups and during the Mini Wrap-ups, students worked to use language with precision and to construct (or critigue arguments:
MP3 Construct viable arguments and critique the reasoning of others (click below to watch a short video on MP3)
During this lesson my students were also modeling numbers and numerical expressions and thinking about the structure of the representation of a number (MP4, MP7). Although from year to year the implementation of this lesson may result in different mathematical practices or some discussion of different content, the above are the primary ones that I observe and try to motivate during the lesson.