6th Grade - Use Variables In Formulas

 
     
 
     
 
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6th
Algebra
Use Variables in Formulas
Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = 1 - 2bh, C = Pi*d - the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
Use variables (symbols/letters), along with the correct formula to write expressions for: perimeter, area and circumference in algebraic terms. Understand that a set formula will always be used to solve problems involving: perimeter, area or circumference. Memorize and use the formula for perimeter; area of a circle, square, rectangle, and triangle; and circumference of a circle.
 

Sample Problems

(1)

t

s

What is the area of the triangle above? Express the answer algebraically.

(t x s 2)

(2)

A square has four sides with the value of s. How would you solve to find the perimeter?

(s + s + s + s, 4s)

(3)

If a circle has the diameter of x, what is its circumference?

(C x)

(4)

Which is the correct algebraic representation for the area of a circle?

(a) A = d (b) A = rr (c) A = r2

(b)

(5)

A rectangle has width (w). Its length is 4 times more than its width. Find the perimeter of the rectangle. (Your answer will be expressed in terms of w. )

(w + w + 4w + 4w or 2w + 2(4w))

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 rectangles; the first with a width of 4 and a length of 5 and the second with a width of 20 and a length of 25; the area of both rectangles a can used following the same steps in the same formula. Since the A = l x w, you just need to find the value of each length and width and then multiply them together. So the area of the first rectangle is 20, where the area of the second triangle is 500.

(2)

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.

(3)

To teach the concepts behind the formula for perimeter to the visual/kinesthetic learner, you can use sidewalk chalk to draw different polygons on the ground outside. Begin by talking about how the letter P can stand for perimeter. Inform your child that you will be using his/her foot as the unit of measure. Next, have your child walk around the shape very carefully counting steps. It is important that there are no gaps or overlapping between steps. So, if he/she steps with the right foot first, the left heel will touch the right big toe when the second step is taken. First, have your child walk around the outside of the shape and count his/her steps. Next, have him/her walk one side at a time. After each side of the shape is walked, have your child write how many steps he/she took for that section only. For example, if the first shape is a rectangle and your child starts on the left side and counts 10 steps, have him/her write a 10 next to the left line. If your child ends on part of a step, it is okay to round. For example, if he/she takes 10 ½ steps, you can round it up to 10 steps. These steps will be repeated for each side of the shape. Finally, have your child add up the sides. He/she should find that the same amount of steps were taken both times. Have your child write the letter p to represent perimeter and record the numerical value for that shape. If the child took 30 steps, he/she would write p = 30 steps above the shape. Finally, have your child create a formula for what he/she did to find perimeter. You can do this by having him/her label each side with a variable (i.e. a, b, c, d). Then he would think about the operation needed in finding perimeter. An example of a formula that could be written is: P = a + b + c + d. However, your child may notice that in working with rectangles, there are two pairs of congruent (same) sides. So, if he or she labeled both long sides with an L and the short sides with a W a formula could be written as P = 2L + 2W. Often times there is more that one way to write a formula. This is why it’s so important for children to understand the concept behind the formula. You can repeat this activity for squares, triangles and other polygons. Challenge your child to write as many formulas as possible for each shape. For instance, the square can be written as P = a + b + c + d, P = 4s, P = s + s + s + s, because all four sides of a square are equal.

(4)

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.

(5)

To teach the concepts behind the formulas for area to the kinesthetic/visual learner, you can use sidewalk chalk using an activity similar to the one explained above for perimeter. In fact, it would best best to use the same shapes as used for the last activity. This will allow children to see the true difference between finding the perimeter of a shape and finding its area. Begin by talking about how the letter A can stand for area. Remind your child that you will be using his/her foot as the unit of measure. However, this time instead of having your child walk around the shape, you will have him/her walk up and across the inside of the shape. While doing this, have your child imagine that he/she has paint on his/her shoes and tell him/her that the goal would be to color in the inside of the shape completely. Be sure to reinforce the idea that area is the measure inside a shape. Make sure the total steps taken to fill the inside of your polygon in written down as A = _____. So, if your child walked 23 steps, he/she would write A = 23. Now, to use this idea to write a formula for a rectangle, you will have to have your child count the steps going up or down one side of the polygon and across (left to right) the other side. So, if he/she walks up 4 steps and across 6 steps, you will need talk about what operation you could using 6 and 4 to almost get the answer 23. Ask questions, like would 6 + 4 be 23? Since 6 x 4 = 24, it will be made clear that you need to multiply to find an area. Now have your student use the variables along the sides of the shape to create a formula for the area of that shape. So if the rectangle had one side labeled x and the other y, the formula would be A = x x y. You can then compare this to the actual area formula.

(6)

When working on the area for a triangle, be sure to discuss how two triangles make one rectangle. This will make it easier to comprehend why the area formula has ½ or must be divided by two at the end.

***It would be helpful to insert a drawing of this here.***

(7)

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. However, a simple activity that can be done to find the circumference of a circle using a string and a ruler. First, draw or trace a circle. Next place a piece of string around the outside of the circle and cut it when both sides of the string touch. Now measure the string and you have the circumference. 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 (This can be remembered by cd.)

Area: Circle: A = r 2 (This can be remembered by ar.)

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.

(8)

Visual learners will also benefit from drawing shapes on grid paper. They can then use these shapes and the grid squares to quickly count and find perimeter and area.

Extra Help Problems

(1)

A rectangle has the width of x and length of y, which best algebraic representation to find area?

(a) A = x + y (b) A = xy (c) A = 2x + 2y

(b)

(2)

How many ways could you write the formula to find the perimeter of a square with sides that have a measure of b?

(two; b+b+b+b or 4b)

(3)

An equilateral triangle has sides that measure y, which best algebraic representation to find perimeter?

(a) 3y (b) 2y (c) y + y + y + y

(a)

(4)

A right triangle has a base m and height n. How could this be written as a formula to show the area?

(m x n 2 or ½ mn)

(5)

A circle has the diameter, d, which formula could be used to find circumference?

(a) r 2 (b) d2 (c) r (d) d

(d)

(6)

A triangle has a base with a measure of b and a height of h. Which formula could be used to show area?

(a) b x h 2 (b) b x h (c) b x h x 2 (d) b x h + ½

(a)

(7)

There is a trapezoid with the following measures:

base 1 - s; base 2 – t, height – h. Which formula could be used to find area?

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

(b)

(8)

Write an expression to find the perimeter for a trapezoid where b1 = a, b2 = b, h=4

((a + b) 4 2or ½ (a + b) 4)

(9)

A square has a side = x. Write a formula to find the area for this square.

(xx)

(10)

A rectangle has a width (w) ½ the size of its length (L). Which formula could be use to show perimeter?

(a) P = ½ w + L (b) P = (½ L 2) + L2 (c) 2L 2w1/2

(b)

(11)

A circle has a radius worth, y. Write a formula to solve for area.

(A y 2 )

(12)

A circle has the radius worth r, which formula would show how to use this to find circumference?

(a) (r 2) (b) r 2) (c) r 2

(a)

(13)

A circle has a diameter, measuring d. Write a formula to find the area.

(A (d2) 2 )

(14)

A rectangle has a perimeter of 60. Write a formula to show perimeter, if length is 5 times greater than the width.

(60 = 2w + 2 (wx5))

(15)

A triangle has a base of 10 cm and height of 4 cm. Use the correct formula to find its area.

(A 20 cm2 )

(16)

An equilateral triangle has sides (s) worth 6 inches. Write a formula and solve to find its perimeter.

(P = 3s, P = 18 in.)

(17)

A square has sides worth 5 yd. What is its area and perimeter? Explain how the formulas differ for each.

(A = 25 yd2 , P = 20 yd)

(18)

The measures of a rectangle are w = 12 mm, l = 20 mm. Use this information to solve for perimeter and area. Explain your process in solving each.

(P = 64 mm, A = 240 mm2 )

(19)

The measures of a triangle have a base of 11 ft and height of 12 ft. Use this information to solve for area. Could you use the information in this problem to find the perimeter? Explain.

(A= 66 sq. ft. No, you can’t find perimeter, you need to know the measures of the sides of the triangle to find the perimeter.)

(20)

A circle has a diameter measuring 10 inches. Use formulas to find the area and circumference of the circle. Explain the difference between the steps to solve this problem.

(C = 31.4 in, A = 78.5 in2 )

(21)

Jake is making a frame for his grandparents. The photo he needs to frame has a width of 6 inches and a length of 8 inches. What is the minimum amount of wood Jake needs to build his frame?

(28 in.)

(22)

Angelica is building a fence around her circular garden. The radius of the garden is 3 inches. How much fencing should Angelica buy?

(18.84 in.)

(23)

Timmy has a rectangular patio that he’d like to fill with tiles. It has a length of 6 cm and width of 4 cm. What formula should he use to find out how many centimeters of tiles are needed to fill it?

(24 cm2 )

(24)

Lisa has a square frame. She’d like to fill the inside of the frame with a decorative fabric. The sides of the frame are 3 yards. How much fabric will Lisa need to fill the frame?

(9 yards)

(25)

Dave is making a triangular wall hanging. The base of the triangular frame is 6 inches, and the height 7 inches. If Dave wants to fill the inside with aluminum, how much will he need?

(21 inches2)

 

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