Mathematics Florida Standards (MAFS): 2.G.1.2, 2.OA.3.4

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1 1 U n t er r ich t splan Partitioning a Rectangle into Squares Altersgruppe: Grade 2, Grade 3 Texas - TEKS: G2.9.GM.F, G3.4.N O.K, G3.6.GM.C Riverside USD Scope and Sequence: 2.G.2 [2.6], 2.OA.4 [2.2] Oklahoma Academic Standards Mathematics: 2.N.2.6, 2.N.3.2 Virginia - Mathematics Standards of Learning (2009): 3.9d Common Core: 2.G.A.2, 2.OA.C.4 Mathematics Florida Standards (MAFS): 2.G.1.2, 2.OA.3.4 Alaska: 3.M D.9a, 3.M D.9b, 3.M D.9c Minnesota: , , Fairfax County Public Schools Program of Studies: 3.9.d.1, 3.9.d.2 Nebraska Mathematics Standards: M A f, M A b South Carolina: 2.AT O.4, 2.G.2 Indiana: 3.M.5, 3.M.6 Georgia Standards of Excellence: M GSE 2.G.2, M GSE 2.OA.4 Online-Ressourcen: F ai r and Sq uare Opening Teacher present s Students play Class discussion ZIELE: E xpe rii e nc e partitioning rectangles into equal-sized squares Learn to create the idea of an array without tiling the entire

2 2 rectangle De vel o p a connection between creating an array and finding the number of squares (area) using multiplication Ope ni ng 15 Start with a brief conversation about the attributes of rectangles. A key attribute for this lesson will be the congruence of opposite sides. Then, draw a rectangle on the board. Say: Suppose you want to split this rectangle into equal-sized pieces. How can we do that? Typically, students will assume this to mean split into t wo pieces. This is a fine place to start. Either have a student come to the board and draw in a partitioning line segment, or have a student describe a way and draw it in yourself. Ask: Is there another way to split this rectangle into equal-sized pieces? Again, it is very likely that another way will be given for partitioning the rectangle into two pieces. Ask: What if we wanted to split the rectangle into more than two equal-sized pieces? This question can yield a variety of responses, so ensure that the students understand why some partitions do create equalsized pieces, while others do not.

3 3 Creates equal-sized pieces Does not create 4 equal-sized pieces Then, introduce the idea of splitting or covering a rectangle by fixed-sized squares. Perhaps start by reding the students that squares are a special type of shape, because all of the side lengths are the same. Using cut-outs with magnets (for a magnetic whiteboard) or tape/sticky tack (for a non-magnetic blackboard), have several rectangles and many equal-sized squares. The rectangles should be measured such that they can be covered by the squares. Choosing one of the rectangles to start, ask: How many squares will it take to cover the rectangle? Have one or several students come to the board and place squares on the rectangle until it is covered completely. Ask: How many squares did we use? As a first method, simply count the squares. It may be worthwhile to ask if anyone had a different strategy for finding the total number of squares. Repeat this process with another rectangle, this time point out that a completely covered rectangle has rows and columns of squares. Further note that each row has the same number of squares as the other rows, and each column has the same number of squares as the other columns. Again, the number of squares can be counted, but try to hint at the usage of the array to find a more efficient method of detering

4 4 the number of squares. The students will hopefully arrive at the conclusion (perhaps with some nudging), that the total number of squares can be found using a sum of equal addends (either squares in each row or each column). Sumg equal addends serves as the basis for multiplication, while partitioning a rectangle into rows and columns of squares leads to area. T e ac he r prese nt s F ai r and Sq uare: Area o f a R e c t angl e 12 Present Matific s episode F ai r and Sq uare - Area o f a R e c t angl e to the class, using the projector. The goal of this episode is to partition (or cover) rectangles using equal-sized squares, ultimately laying the groundwork for finding area. Some questions ask how many squares are needed to cover the rectangle. Note that covering the rectangle is equivalent to partitioning the rectangle. The students can drag squares from the bank on the left in order to cover the rectangle. The students can count the number of squares needed either as they are being placed, or after the squares are in place. Once the correct answer is

5 5 entered, the episode will animate the rows and columns. This would be a good opportunity to tie in the idea of creating an array. Then, connect the total number of squares in the array to sumg equal addends (either rows or columns). For the example below, this means 3 equal rows of 2 squares OR 2 equal columns of 3 squares. Either way, of course, the result is 6 the same as simply counting all of the squares. Other questions provide visuals with squares already placed along one row and one column. The first will be slightly more explicit, using an outer row and column. The second will use an inner row and column. Ask the students if they have an ideas about finding the total number of squares without needing to cover the entire rectangle. The success of this depends, of course, on the strength of the comprehension of the array concept in the first two questions. Especially for the first screen of this type, take the time to walk

6 6 through the idea of covering the rectangle row by row or column by column, showing the sum of equal addends on the board as you proceed. Again, emphasize the fact that sumg rows and sumg columns yields the same result. Even if some students say the correct answer early on, be sure they understand the reasoning and can explain the process. Other questions provide visuals with some squares already placed. Although the number and placement of these squares appears random, they will be placed in ways that allow you to count how many squares are needed for a row and for a column. Note, however, that this is no t obvious, nor should it be the first method used to answer the question. Instead, this method can be primarily cultivated while students are working on their own devices. Some students may struggle to visualize this.

7 7 St ude nt s pl ay F ai r and Sq uare: Area o f a R e c t angl e 10 Have the students play F ai r and Sq uare - Area o f a R e c t angl e on their personal devices. Circulate, answering questions. Continue to support the utilization of arrays. Be sure to connect the total number of squares to strategies more efficient than simply counting, such as sumg the rows or the columns. Sumg equal addends is the foundation of multiplication, hence its importance. Partitioning a rectangle into equal-sized squares is a lead-in for area, hence its importance. Cl ass di sc ussi o n 5 Now that the students have practiced other methods for finding the total number of squares, revisit the rectangle and square props from the opening. This time, set up a scenario similar to the last two questions of the episode, where some squares are already placed. Starting with squares along an outer row and column, ask: How many squares are needed to cover this rectangle? After all of the discussions about arrays, the students should now discuss the strategies of sumg equal addends. Then, move one square to another location and ask the same question. Some students may say that the same number of squares are needed. Be sure to get some clarification as to why they believe

8 8 that. Of course, the rectangle has not changed, so the number of squares has not changed, but there still needs to be a connection to the idea of the array. The number of squares has not changed because the number of rows and columns has not changed! Then, move a few more squares and ask the same question. Again, be sure to tie the answer back to rows and columns (the array).