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Math Essentials, High School Level
By Frances McBroom Thompson
John Wiley & SonsCopyright © 2005 John Wiley & Sons, Inc.
All right reserved.
Chapter OneSection 1
ALGEBRAIC THINKING AND APPLICATIONS
Objective 1: Simplify Algebraic Expressions Involving One or Two Variables
Students have great difficulty recognizing the differences among linear, quadratic, and constant terms in algebraic form. Exponents seem insignificant to them. Viewing each type of term as an area helps students visualize the role each term plays in an expression. The following activities provide experience with such visualization in the combining of like terms. It is assumed that students have already mastered the four operations with integers.
Packets of variable and unit tiles (described in step 1 below) Worksheet 1-1a Legal-sized plain paper or light tagboard (for building mats) Regular paper and pencils
1. Give each pair of students a packet of tiles, two copies of Worksheet 1-1a, and a sheet of plain paper or tagboard (approximately 8.5 inches by 14 inches) for a building mat. If preferred, laminate the mats to make them more durable. Mats define a specific space on which to represent a problem being solved. If teacher-made tiles are used, each packet should contain the following in different colors of laminated tagboard: 8 square (quadratic) variable tiles, each 3 inches by 3 inches (color #1); 8 square variable tiles, each 3.25 inches by 3.25 inches (color #2); 12 rectangular (linear) variable tiles, 0.75 inches by 3 inches (color #1); 12 rectangular variable tiles, 0.75 inch by 3.25 inches (color #2); and 20 unit tiles, 0.75 inch by 0.75 inch (color #3). Each tile should have a large X drawn on one side to show the inverse of that tile. Use tagboard that is thick enough so that the X will not show through to the other side. Commercial tiles are also available for two different variables, but a large X must be drawn on one of the largest faces of each tile in order to represent the inverse of that tile when the X faces up.
2. The meaning of a large square tile needs to be connected to a long rectangular tile of the same color. Have students place a rectangular variable tile of color #1 (call it variable A) horizontally on the mat. Then have them place two more variable tiles below and parallel to the first tile on the mat. Ask: "If a single variable tile A is considered to cover an area of 1 by A, or A, how can we describe the arrangement indicated by these tiles on the mat?" ("3 rows of A.") "What product or area is this?" ("3A.") Ask: "How can we show A rows of A on the mat if we do not know what the value of A is?" Show students how to build several rows of one variable tile each, using one variable tile A as the multiplier, or "ruler," that indicates when to stop putting tiles in the product on the mat (see the illustration below). When the product is finished, the multiplier tile should be removed from the mat. Depending on the dimensions used to make the tiles, whether commercial or teacher-made, the width across several rectangular tiles placed with their longer sides touching may or may not match the length of the longer side of the same type of tile. Such a match is not important and should be deemphasized since the variable tile A is not considered to have a specific length or value in unit tiles. Therefore, although the width of 4 of the variable tile A may appear to match to one variable tile length as shown on the mat below, do not allow students to say that 4 rows of A equal 4A.
3. Ask: "Is there another single block that will cover the same surface area on the mat that the product A of A, or A(A), covers?" ("Yes. The large square tile in color #1; its side length equals the length of the variable tile A.") Again, discuss the idea that the large square tile in color #1 may or may not fit perfectly on top of the "A rows of A" tile arrangement; it will be close enough. Since both the square and rectangular tiles in color #1 are representing variables without known values, we want to maintain their variable nature as much as possible. Physical models like the tiles naturally have specific dimensions that affect or limit areas being built with the tiles, but for our purpose, we will assume that only the unit tiles may be used to represent exact amounts of area. We will now assign the large square tile in color #1 the name of A-squared, or [A.sup.2]. Hence, A rows of A equal [A.sup.2]. From now on, whenever A rows of A are needed, the large square tile will be used to show that amount of area on the mat.
4. Similarly the areas of the square and long rectangular variable tiles in color #2 might be described as B-squared, or [B.sup.2], and B, respectively. If an X appears on the top side of a variable tile, the inverse or opposite of the tile's area will be indicated. For product example, a B-squared tile with X on top will be called "the opposite or inverse of B-squared" and written as (-[B.sup.2]). Each small square tile in color #3 represents an area of 1 by 1, or 1 square unit of area. If a given set of unit tiles all have an X showing-for example, 5 tiles with X-then the tile value will be the "negative of 5 square units of area" and written as (-5). Note that area itself is an absolute measure, neither positive nor negative. Area, however, can be assigned a direction of movement in real applications; hence, we can consider the opposite or negative of a given area.
5. After the area of each type of tile is identified, have students do the exercises on Worksheet 1-1a. For each exercise, they should place a set of tiles on the building mat to show the first expression. Then they will either add more tiles to this initial set or remove some tiles from the set according to the second expression of the exercise.
6. After combining tiles that have the same amount of area, students should record an expression for the total or remaining area on the worksheet.
7. Discuss an addition exercise and a subtraction exercise with the class before allowing students to work the other exercises independently.
Consider Exercise 1 on Worksheet 1-1a for addition: (3[A.sup.2] + 2A-5) + (-[A.sup.2] + 3A + 2) Use color #1 variable tiles with the color #3 unit tiles.
Have students place 3 large square (quadratic) variable tiles, 2 long rectangular (linear) variable tiles, and 5 negative unit tiles on the building mat to represent the first expression. Any such group of tiles is called a polynomial, that is, a combination of variable tiles and/or unit tiles. Leaving this set of tiles on the mat, have students place additional tiles on the mat below the initial tiles to represent the second expression. The second set should contain a quadratic variable tile with X showing, 3 linear variable tiles, and 2 unit tiles:
Ask: "Can any 0-pairs be made through joining, then removed from the mat?" (One 0-pair of the large quadratic tiles and two 0-pairs of the small unit tiles should be formed and removed from the building mat.) "Can you now describe the total in tiles still on the mat?" Since tiles for two of A-squared, 5 of A, and -3 remain on the mat, students should complete the recording of Exercise 1 on Worksheet 1-1a: (3[A.sup.2] + 2A -5) + (-[A.sup.2] + 3A + 2) = 2[A.sup.2] + 5A -3
Now consider Exercise 2 on Worksheet 1-1a for subtraction: (4[A.sup.2] -3A + 4)- ([A.sup.2] + 2A -2). Again, use color #1 variable tiles with the unit tiles.
Have students place tiles on their mats to show the first group. There should be 4 of the quadratic variable tile, 3 of the linear variable tile with the X-side showing for the inverse variable, and 4 positive unit tiles on the building mat. Discuss the idea that the subtraction symbol between the two polynomial groups means to remove each term in the second group from the first group. Ask: "Can we remove one quadratic variable tile from the original four? ("Yes; 3 quadratic variable tiles, or 3[A.sup.2], will remain.") "Can 2A be removed from -3A?" Since only inverse variable tiles are present initially, 0-pairs of A and -A tiles will need to be added to the mat until two of the variable A are seen. Then 2A can be removed, leaving 5 of -A on the mat. Similarly, -2 will be removed from +4 by first adding two 0-pairs of +1 and -1 to the mat. Then -2 can be removed from the mat, leaving +6. The mat arrangement of the initial tiles and the extra 0-pairs of tiles is shown here before any tile removal occurs. Have students complete Exercise 2 on Worksheet 1-1a by writing an expression for the tiles left on the mat: (4[A.sup.2] -3A + 4)-([A.sup.2] + 2A -2) = 3[A.sup.2] -5A + 6.
Remind students that when they use 0-pairs of a tile and remove one form of the tile (for example, positive), then the other form (for example, negative) remains to be added to the other tiles on the mat. Show students that when they needed to remove 2A from the mat earlier, two 0-pairs of A and -A were placed on the mat. After 2A was removed to show subtraction, the two inverse variable tiles, -2A, still remained on the mat to be combined with the other tiles for the final answer. Hence, a removal of a tile from the mat is equivalent to adding the inverse or opposite of that tile to the mat.
To confirm this, have students place the original group of tiles (4[A.sup.2] -3A + 4) on the mat again. The opposites needed (-[A.sup.2], -2A, and +2) should then be placed on the mat and combined with the original tiles. See the illustration below. Remove any 0-pairs formed, leaving tiles for 3[A.sup.2], -5A, and +6 on the mat as the answer. Finally, have students write another equation below Exercise 2 on Worksheet 1-1a, this time showing the alternate method that uses addition: (4[A.sup.2] - 3A + 4) + (-[A.sup.2]-2A = 3[A.sup.2]. - 5A + 6. Encourage students to use whichever of these two methods seems comfortable to them.
In the answer key for Worksheet 1-1a, when the coefficient of a final variable is 1, the number 1 will be written with the variable. This approach seems to be helpful to many students. Nevertheless, discuss the idea with the class that the 1 in such cases is often not recorded but simply understood as being there.
Answer Key for Worksheet 1-1a
1. 2[A.sup.2] + 5A - 3
2. 3[A.sup.2]-5A +6; alternate: (4[A.sup.2]-3A + 4 + (-[A.sup.2] - 2A + 2) = 3A.sup.2] -5A + 6
3. [3B.sup.2] + 2; alternate: (5[B.sup.2] + 3) + (-2[B.sup.2] - 1) = 3[B.sup.2] + 2
4. 2[A.sup.2] + 1A-4
5. 1[B.sup.2] + 2A - 8
6. 3A + 1[A.sup.2] + 5; alternate: 3A + ([A.sup.2] + 5) = 3A + 1[A.sup.2] + 5
7. I[A.sup.2]; alternate: (5[B.sup.2] - 4) + (-5[B.sup.2] + 4 + 1[A.sup.2])
8. 3A + 1[A.sup.2]-13+ 1B
Building Sums and Differences Name ___________________ with Tiles Date ___________________
Build each polynomial exercise with tiles. Different variables require different tiles. Record the result beside the exercise. For each subtraction exercise, also write the alternate addition equation below the subtraction equation.
1. (3[A.sup.2] + 2A - 5) + (-[A.sup.2] + 3A + 2)
2. ([4A.sup.2] - 3A + 4) - ([A.sup.2]) + 2A - 2) =
3. (5[B].sup.2] + 3) - (2[B.sup.2] + 1) =
4. 2[A.sup.2] - 3A + 1) + (4A-5) =
5. (4A-2+[B.sup.2]) + (-6-2A)
6. 3A-(-[A.sup.2]-5)= 7. (5B.sup.2]-4)-(5[B.sup.2]-4[-[A.sup.2]) =
8. 3A-2[A.sup.2]-5+B-8+3[A.sup.2] =
Pictorial Stage Materials
Worksheet 1-1b Regular paper and pencil
1. Give each student a copy of Worksheet 1-1b. Have students work in pairs, but they should draw the diagrams separately on their own worksheets. Large squares will be drawn for the quadratic variable, a long rectangle whose length equals an edge length of the large square will be drawn for the linear variable, and a small square will represent the integral unit. A large X should be drawn in the interior of a shape to show the inverse of that shape. If an exercise involves two different variables, letters need to be written on the drawn shapes to identify the different variables. The product of two different variables, for example, A and B, should be shown as a large rectangle similar in size to the quadratic squares and labeled as AB. The notation AB simply means A rows of B, or the area AB.
2. For addition exercises, students should draw the required shapes and connect any two shapes that represent a 0-pair. The remaining shapes will be recorded in symbols to show the sum.
3. For subtraction exercises, students will be asked to use either the removal method or the alternate method, which involves addition of inverses. To remove a shape, students should mark out the shape. When needed, two shapes should be drawn together as a 0-pair. For the alternate method, inverses of the subtrahend expression should be drawn and combined with the first expression to produce a sum. The result will be recorded symbolically.
4. When checking students' work after all are finished, allow time for students to explain their steps; do not just check for answers. Students need to practice expressing their ideas mathematically. Such verbal sharing is also very beneficial to auditory learners.
5. Discuss Exercises 1 and 2 on Worksheet 1-1b with the class before allowing partners to work together on their own.
Consider Exercise 1: (-3[B.sup.2] + B + 2) + ([B.sup.2]-4B + 1)Students should draw the necessary shapes on their papers to represent each polynomial group. The shapes for the first polynomial group may be drawn in a row from left to right following the order of the given terms. The shapes for the second polynomial group should be drawn as a second row below the first row, but students may rearrange the shapes and draw them below other like shapes in the first row. Since only one variable is involved, no labeling is needed for the shapes. Any 0-pairs should be connected. Remaining shapes will then be counted and recorded as the answer. A sample drawing is shown here:
The final equation will be as follows and should be recorded on Worksheet 1-1b: (-3[B.sup.2] + B + 2) + ([B.sup.2] - 4B + 1) = -2[B.sup.2] -3B + 3. At this point, begin to encourage students to record the terms of a polynomial with their exponents in decreasing order.
Now consider Exercise 2: ([A.sup.2] + 5A - 3) - (2[A.sup.2] + 3A + 2). Since the removal process is required for this exercise, students should draw shapes for the first polynomial group and then draw any 0-pairs below that group, which will be needed in order to mark out the shapes shown in the second group. The shapes remaining or not marked out in the finished diagram will be the difference. Here is the completed diagram:
The final equation should be recorded on Worksheet 1-1b as follows: ([A.sup.2] + 5A - 3)- (2[A.sup.2] + 3A + 2) = -[A.sup.2] + 2A - 5. It may be helpful for some students to write -1[A.sup.2] instead of -[A.sup.2] . This is acceptable notation.
Answer Key for Worksheet 1-1b
Only symbolic answers may be given.
1. -2[B.sup.2] + 3 - 3B + 3(see sample diagram in text)
2. -[A.sup.2] + 2A - 5 (see sample diagram in text)
3. 5[A.sup.2]-2A + 6
4. -3A-2B +1
5. 2[A.sup.2] + 3[B.sup.2]-2
Excerpted from Math Essentials, High School Level by Frances McBroom Thompson Copyright © 2005 by John Wiley & Sons, Inc.. Excerpted by permission.
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