6th Grade - Relationships In Geometry

 
     
 
     
 
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6th
Geometry
Relationships in Geometry
Express in symbolic form simple relationships arising from geometry.
Use variables (symbols/letters), along with formulas to write expressions or equations to show relationships in geometric shapes (polygons and circles). Understand that a set formula will always be used to solve problems involving: perimeter, area or circumference. Memorize and use the formulas for perimeter; area of a circle, square, rectangle, and triangle; and circumference of a circle. Use these formulas and the relationships between the parts of the one geometric figure or congruent and similar figures to write expressions or equations to show relationships. Precisely describe, classify, and understand relationships among types of geometric figures using their defining properties. Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar and congruent objects. Create and analyze expressions and equations concerning geometric ideas and relationships, such as congruence, similarity. Understand and use academic vocabulary for relating geometric figures: congruent – two or more figures that are the same size, shape, and angle measures similar – two or more figures that have the same shape and angle measures, but different sizes corresponding sides – two proportional sides of congruent or similar figures with relative positions on the figures. corresponding angles – two proportional angles of congruent or similar figures with relative positions on the figures.
 

Sample Problems

(1)

A rectangle can be cut into two congruent triangles by drawing a straight line diagonally from a top corner to a bottom corner. Knowing this, if a triangle has the area of n, write an algebraic expression to show how to find the area of one of the triangles.

(n/2)

(2)

Triangle 1 is similar to triangle 2. If triangle 1 has a perimeter measuring x, and triangle 2 is 3 times larger, what expression could be used to find the perimeter of triangle 2?

(a) P = 2 x 3 (b) P = 2x (c) P = 3x

(c)

(3)

A rectangle has the width measuring, w. The length of the rectangle is 2 times the width. If the width measures w, write an expression to find the length.

(2w)

(4)

Two congruent squares have sides labeled x and y. The area of square x can be found by using the equation x2. Write an equation to show the area for square y in relationship to square y.

(y2)

(5)

A trapezoid has base 1 with a measurement of n. Base 2 is twice as large as base 1. The height of the figure is 4. Use this information to write an expression to solve for the area of the trapezoid.

((n+2n) x 4/2 or 1/2(n+2n) x 4)

Learning Tips

(1)

Formulas are algebraic expressions. These expressions use variables to take the place of unknown numbers. The variables chosen usually match the beginning letter of a word. For example, w, stands for width and l stands for length. So, when looking at a shape, you can find the value for w and input it into the formula. Formulas are used so that different sizes of the same shape can be easily found. So, if there are two similar rectangles; the first with a width of w and a length of l and the second that similar, but five times larger; the area of both rectangles a can used following the same steps in the same formula. Since the A = l x w, will use this to show the area for the first rectangle. However, since the second rectangle is five times larger, its area formula would be A = 5lw (5 x length x w).

(2)

In order to be successful in writing algebraic expressions from word form, your child will need to use semantics. In other words, he/she needs to think of the words in a real-world situation to make sense of how they are related. This is especially true for subtraction and division. Though the variable and numbers can switch places to get he same result for addition and multiplication problems, this is untrue for subtraction problems. For example, 5 -3 = 2, but 3 – 5 = -2. For this reason, the placement of the variable in a subtraction or division problem is important. While working with the sentence “10 less than a number x”, many children will write 10 – x. This is incorrect. In this sentence, 10 is being taken away from x and the correct expression is x – 10. However, if the sentence say, “9 decreased by a number t” the problem would be written 9 – t because t is decreasing or taking away from the 9. For division expressions such as, “23 divided by a number z”, the expression would be written 23 r. Relatively, “the quotient m and 17”, would be p 17.

(3)

Use key words to determine the operation when converting a sentence or word problem into an algebraic expression. Here is a list of some of the words that may be used to determine which operation to use.

Addition: added to, more than, increased

Subtraction: decreased by, less than, subtracted from, taken away

Multiplication: times, multiplied by, product of, times as many, twice the size

Division: quotient of, divided by, half

For example, if relating the sides of a triangle where a = side one and side two is 4 less than a and side three is 2 greater than a, you equation to show the values of side two (x) and side three (y) would be as follows;
x = a – 4

y = a + 2

(4)

Perimeter is the measure around the outside of a shape. The word rim is inside the word perimeter, which may help some children remember the definition. The quickest and easiest way to find the perimeter of any polygon (2-D shape with 3 or more sides) is to add up all the sides. So, if a triangle has its three sides measuring 4, 5, and 6, to find the perimeter you would add 4 + 5 + 6 to get a perimeter of 15. The formula P = a + b +c can be used to solve for the perimeter of a triangle, with a, b, and c representing each side. However, there are short cuts that can be taken. For example, a square will always have four equivalent sides, hence the perimeter can be found by adding up all four sides or by multiplying the length of one side by 4. This formula can be used to show relationships of figures or sides of figures. For example, if you have two similar triangles with triangle one having sides labeled a, b, and c and a perimeter of 60 and you know that triangle two is ½ the size of triangle one. You can use the information to write an equation for the perimeter of the second triangle. Since the perimeter of triangle one is a + b + c = 60 and triangle 2 is half the size you would use the following equation to represent triangle two: (a + b + c) 2 = P or (a + b + c)/2 = 30.

(5)

There are different formulas to find the area of a polygon. The area of a polygon is the measure of the inside of the shape. The capital letter A stands for area. Below is a list of formulas organized by shape.

Square: A = s x s (side x side) or A = S2

Rectangle: A = l x w (length x width)

Parallelogram: A = b x h (base x height)

Triangle: A = ½ b x h or A = b x h 2

Trapezoid: A = ½ (b1 + b2) x h. A = ½ (base 1 + base 2) x height or

A = (b1 + b2) x h

Circle: A = r 2

It is important that sixth graders memorize these formulas. You may want to make flash cards or work together to create pneumonic devises to aid memorization. In writing the final answer for the area of a figure, the units must be squared.

The area formulas may be used to define relationships among similar or congruent shapes. For example, if trapezoid x is twice the size of a similar trapezoid y, the formula could be used to show this relationship, as follows;

A of x = [( b1 + b2) x h/ 2] x 2

(6)

Since a circle is a special shape, you cannot find its area the way you would with a polygon. Also, a circle does not have a perimeter, it has what is called the circumference. The circumference is the measure around the outside of a circle. Since a circle is round, it is difficult to measure around the outside of it using a ruler. Since the circle is so different, its formulas are different too. The formulas for area and circumference both use a symbol to represent pi. The symbol or pi is . Its rounded value is 3.14 as a decimal and 22/7 as a fraction equivalent. When solving a problem with other fractions, the fraction would be used. The decimal form of pi can be used when working with whole numbers and decimals. The formulas for circles are as follows:

Circumference: C = d

Area: Circle: A = r 2

In these formulas, d stands for diameter. The diameter of a circle is the measure from one edge of the circle to the opposite edge. The r stands for radius. The radius is the measure from the center of the circle to the outside edge. The radius of a circle is ½ the size of the diameter.

An example of using relationships to write an expression can be seen for example in solving to find the circumference when the radius = r. As you can see, the circumference formula asks for diameter. We only know r is the problem above, so we will need to think of the relationship between diameter and radius. The radius of a circle is always ½ the side of its diameter. This means that it takes to radii to make one diameter. We could use this information to write an expression; C = x r x 2.

(7)

You may need to use corresponding angles or sides to solve some problems with similar or congruent shapes. For example, triangle x has a side, m, that is three times the size of the corresponding side, s, of triangle y. If we are asked to find the size of the corresponding side of triangle x, we will use this information to write the either the equation: m = s x 3 or m 3 = s.

Extra Help Problems

(1)

There are two similar rectangles. The first with a width of W and a length of L. The second triangle is 3 times larger than the first. Write an expression to show how to find the perimeter of the first triangle.

(2W + 2L)

(2)

There are two similar rectangles. The first with a width of W and a length of L. The second triangle is 4 times larger than the first. Write an expression to show how to find the perimeter of the second triangle.

3 (2W + 2L)

(3)

There are two similar rectangles. The first with a width of W and a length of L. The second triangle is ½ the size of the first. Write an expression to show how to find the area of the first triangle.

(A=L x W)

(4)

There are two similar rectangles. The first with a width of W and a length of L. The second triangle is ½ the size of the first. Write an expression to show how to find the area of the second triangle.

(A= ½ L x W)

(5)

A circle has the diameter, d. Write a formula to show how to find the area of this circle.

(C d)

(6)

A circle has the radius, r. Express algebraically how to find the circumference of this circle.

(C x 2 x r)

(7)

There is a trapezoid with the following measures:

base 1 - s; base 2 – is 3 times larger than base 1, height – h. Which formula could be used to find area?

(a) ½ (s + 3s) x h (b) ½ (s + 3)h (c) s x 3 x h

(a)

(8)

Write an expression to find the perimeter for a trapezoid where b1 = a, b2 = b, h= ½ the size of a.

((a + b)x 1/2a/2 or ½(a + b) x 1/2a)

(9)

There are two congruent squares. Square One has a side = x and Square two has a size = y. Write an equation to compare the areas of both squares.

(x2 = y2 )

(10)

A rectangle has a width (w) ¼ the size of its length (L). Which formula could be use to show perimeter?

(a) P = ¼ w + L (b) P = (1/4 L 2) + L2 (c) (2L) (2w 4)

(c)

(11)

The base of a triangle is 5 times its height. Write an expression to show how to find the area.

(1/2 (5h x h)

(12)

A rectangle has a length that is twice as great as its width. Write an expression to solve for area.

(W x 2W)

(13)

The sides of a triangle are w = side one; side two is 10 less than w; and side three is 11 greater than w, write an equation to show the value of side two, if side two is x.

(x = w – 10)

(14)

A rectangle has a perimeter of 100. Write a formula to show perimeter, if length is 9 units more than the width.

(2w + (9w x 2) = 100)

(15)

The sides of a triangle are w = side one; side two is 10 less than w; and side three is 11 greater than w, write an equation to show the value of side three, if side three is y.

(y = w + 11)

(16)

A scalene triangle has a side, b. A similar triangle has a corresponding side, e. Side e is 1/3 the side of side b. Write an expression find the measure of side b.

(3e = b)

(17)

A scalene triangle has a side, b. A similar triangle has a corresponding side, e. Side e is 1/3 the side of side b. Write an expression find the measure of side e.

(1/3b = e)

(18)

A trapezoid has an angle, m. A similar trapezoid has a corresponding angle, p. Angle m is 4 times the size of angle p. Write an expression to show the measure of angle m.

(1/4p)

(19)

A trapezoid has an angle, m. A similar trapezoid has a corresponding angle, p. Angle m is 4 times the size of angle p. Write an expression to show the measure of angle p.

(4m)

(20)

A rectangle has the perimeter of n. Dave is going to cut the rectangle into two equal triangles. Write an expression to show how to find the area of the triangles.

((a x b)/2 for one triangle, a x b for both.)

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