Squares have roots? Concepts: Square roots and the structure of the integers Description of activity: The activity segment reinforces the geometric interpretation of the square root of a number n as the length of the side of a square with area n and investigates the distribution of the square numbers in the integers. Anticipatory set: What is a square number? What is a square root? What kinds of patterns can we find in the integers? Materials: Some sort of blocks or tokens that the students can arrange into squares. State standards: NO 1.7.6 Recognize subsets of the real number system, NO 3.7.5 Represent and solve problem situations using square roots, A4.7.3 Interpret and write a rule for a two operation function table. Prerequisite skills: Students should be proficient with integer arithmetic, but that‟s really about all. Key Questions: What is the definition of a square? What is the only property that distinguishes one square from another? Can numbers that are not square have square roots? Management and organization suggestions: This lesson gets a little abstract towards the end, so it is absolutely imperative that you keep the children engaged. Ask them for their opinions frequently. Include them in the discovery process. Procedure: Begin by asking students for the definition of a square. They often immediately say “a shape with four sides” or “a shape with four corners”, but usually have to stop and think before giving the full definition as a figure with four equal sides and four right angles. If you can, draw an example on the board of a shape that isn‟t a square but that does satisfy their incomplete definitions. For example, if they define the square just to be a shape with four right angles, draw a rectangle on the board (a shape with four right angles that is not a square). This bit helps illustrate the importance of good definitions, and can be used to introduce the idea that the search for good definitions underpins a large portion of research mathematics. Now that they have a firm definition of the square, ask them what information one needs to know about a given square in order to be able to produce it. All plane figures are made up of only two components, lines and angles, each of which has only two properties, number present and measure. Our definition prescribes the number of sides and angles and also the measure of the angles. The only thing about a square we can choose is the length of the sides. Thus, if one knows the length of a square‟s side, one knows everything there is to know about the square. So if we have only a line segment of a given length we can “grow” it into a square, so it actually makes sense to call the side the root of the square. Make a note to leave somewhere on the board that “square roots tell us lengths of sides”. You might do an example or two on the board, such as showing that a square of area four has a side of length two. Now you can discuss square roots a bit more formally, addressing such issues as terminology (radical, radicand, etc.) and the correct way to write square roots Now tell them that you are going to give each of them a bag of tokens and you ask them how they might use these tokens to find some square roots. Of course, they will have to arrange the tokens into squares of various areas and find the length of the sides for each. The can combine their tokens with their neighbor‟s to make larger squares and find larger square roots. We made a race out of it, with students coming to the board and writing any new square roots they found (e.g. 81 = 9). Let them work at this for fifteen minutes or so, and you will probably have all the square roots up to ten on the board. At this point, you can stop them and start looking for patterns. One obvious pattern is that they only came up with the square roots for square numbers. If the students are unfamiliar with square numbers, now is the time to develop the idea of a square number as a quantity of things that can be arranged in a square (e. g. four is a square number because four objects can be arranged into two rows of two, a square). Clearly the fact that they only discovered the square roots of square numbers is a consequence of the method they used to determine the roots, but ask them what they think. Could it be that only square numbers have square roots? Was there something about their method of discovery that would skew the results this way? Another pattern that is fairly easy to see is that, if one orders the square numbers from lowest to highest, the position of a number in line is just its square root. This is a consequence of the fact that the nth square number is just n2. Make another note on the board, directly below “square roots tell us lengths of sides”, that “square roots tell us the number‟s place in line”. Finally, we can ask how the square numbers are distributed among the integers. You can pose this question as follows: “if I start at some square number, can I tell how far I‟m going to have to count before I get to the next one?” Mathematically, this is asking for the difference of any two consecutive square numbers. Here you can give them some examples of simple patterns in the integers. For instance, multiples exhibit about the simplest pattern: the same distance separates any two consecutive numbers of the sequence (e.g. 3, 6, 9, 12, …). For a slightly more complicated pattern, consider a sequence such as (1, 3, 4, 6, 7, 9, 10, 12, …), which moves one unit then two then one then two, etc. The sequence of Fibonacci numbers (2, 3, 5, 8, 13, …) which moves one then two then three, etc. shows another level of complexity. But the square numbers don‟t exhibit any pattern so obvious, so the students will probably conjecture that there is no pattern to them at all. Having the students fill out a function table will help them see the skeleton of the pattern, that the distance between two consecutive squares increases as we move down the line (in fact it is a function of position in line). The key here is geometric, however. Observe that, to make nine from four I must add an „L‟ of dots to a square of area four (see fig 1). In fact, this is true of any consecutive pair of squares, and illustrates that there is indeed a pattern to be found. Now they can formulate a numerical rule to describe this pattern, if they can figure out how many dots each „L‟ contains. Observe in each case, we must add a number of dots equal to the length of the vertical side, a number of dots equal to the horizontal side (the same number as that for the vertical side, since we have a square), and then one more. That is we must add two times the length of a side and then add one more. Remember, you have on the board that the square root tells us both the length of the side and the number‟s place in the sequence if squares. So it follows that if we start with any square and want to know how far ahead we will find the next square, we find where we are in line (i.e. take the square root), multiply our answer by two, and then add one to it. Assesment plan: You can give them square numbers and ask how far until the next one, the one after that, the one after that, etc. A good indicator is to look around the room when you‟ve finished up the lesson and count the number of glazed over eyes. If it‟s less than half, you‟ve succeeded. Extension and Enrichment: You can ask if any fractions can be square numbers, or if there are numbers of other shapes (yes in both cases). The fact that numbers that aren‟t square have square roots leads pretty naturally to irrational numbers, since the square root of a non-square number is always irrational. The formula uncovered can be written as (n+1)2 – n2 = 2n + 1, from which follows the algebraic identity (n+1)2 = n2 + 2n +1. The more general identity (a + b)2 = a2 + 2ab + b2 can then be introduced. Also the distance between any two consecutive squares is a linear function of the position, so you can talk about linear relationships here as well. Also, there are some very, very old but still interesting algorithms for calculating square roots that students might find interesting. The Babylonians had a very simple one, and the ancient Chinese had an excellent (though more complicated) one. A demonstration of either or both of these, as well as a discussion on how they work could be interesting. Reflection: This went surprisingly well. I lost a few students near the end, I think, but overall, I think the students saw and understood some real math. My teaching partner agreed.
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