Angles and Measurement - 6th Grade

• Study notes below for class 6 Mathematics

Angles in a Circle

• Dividing a circle into equal parts using set squares is a fundamental concept
• Dividing a circle into equal parts using corners of set squares works

Angle Measurement

• Class 5 showed how a set square divides a circle into three equal parts
• Corners of a set square can create three, four, or six parts in a circle
• Creating five equal parts requires an alternative method
• Measuring line lengths starts with millimeters and centimeters
• Measuring different angles uses a small angle as a standard
• A circle divided into 360 equal parts is used to measure angles
• The measure of this angle is 1 degree, written as 1°
• An angle twice as large measures 2°, and so on
• A protractor is used to measure angles
• A protractor has two sets of numbers at the end of each line

Using a Protractor

• The numbers indicate the degree measure of angles made by lines from the bottom line marked 0
• The bottom line and the line just above form a 10° angle
• The bottom line and the line marked 40 form a 40° angle
• An angle of 40° consists of four 10° angles
• Another set of numbers is available on top for measuring angles from the left
• A line is drawn, and another line is then drawn straight up using the square corner of a set square
• An angle at a square corner measures 90° and is a right angle

Right angles

• A right angle is marked with a special symbol in figures
• Angles can be estimated as less than 90°, more than 90°, or exactly 90° without direct measurement

Drawing Angles

• Start by drawing a line (AB) in GeoGebra
• Select the "Angle with Given Size" tool
• Click on points B and A
• In the dialogue box, enter the desired angle
• Click "OK" to create a new point B’
• Connect points A and B’ to form the angle

Circle Division and Degrees

• A circle has 360 equal parts
• An angle of 1° is the base
• Drawing two 1° degree angles at the centre makes 2° angles with 180 equal parts
• Taking three angles yields the value of each angle and the number of equal parts of the circle
• Dividing a circle into 4 equal parts results in angles of 90° each
• Other corners of set squares create equal parts in a circle
• An angle at this corner of the set square is 45°
• 360÷5=72 divides the circle into five equal parts and the size of the angle at the centre

Polygons

• Angles of 72° at the centre can be used to draw a 5 sided polygon inside circle
• The number of sides of the polygon in side the circle dictates the angles

Averages

• A decision being made about how much each child should donate to purchase books means everyone gives the same amount
• Divide the total amount needed by the number of children
• $1000 with 40 children means each child donates on average $25 (1000 / 40 = 25)
• Raising $1200 for a medical fund with 30 children will give you an average amount given

Problem Solving with Averages

• Average problems involve situations where the amounts may vary
• If Manikuttan supplies 56 L milk over 7 days = average is 8 L per day
• An average of 8L doesn't mean exactly 8 L given on each day

Applying Averages

• On 5 days spent $300, $250, $270, $280 and $290 = must find the average daily spending
• Get average expenditure by dividing total amount spent by the number of days
• Using the average length, calculate total cloth needed for 23 boys
• Total length needed for 5 boys: 600 cm and needs 120 cm each

Examples and Applications

• Using the average length, calculate total cloth needed for 23 boys
• Total length needed for 5 boys: 600 cm and needs 120 cm each
• Electricity used by Madeed in January, February, March, April and May means need to find average usage per month
• Some months will be above average
• Average kid in group of 10 that weigh 35 kilograms means total kilograms is 350 (10 x 355 =350)

Average Word problems

• The incomes for five days of a trade has an average
• Add each given day value
• Use the average values to estimate future performance

Fractional Multiplication

• A bottle holds 250 ml. 3 bottles require 3 x 250 = 750 ml
• "3 times as much as 250 ml" is 750 ml
• A packet holds 500 g. We need 4 packs so= 4 x 500 g = 2000 g
• "4 times as much as 500 grams" is 2000 g
• 2000 g is 2 kg
• 500 g is 1/2 a kg
• Four times as much as 1/2 a kg is 2 kg
• Four times as much as 1/2 is 2

Fractional Parts and Multiplication

• Writing "4 times 1/2" as 4 x 1/2
• Our water problem is in liters instead of milliliters
• 250 milliliters means a quarter litre
• Thrice a quater makes 3 /4
• 3 times litre is 3 ( 3* 1/4 = 3/4) 4
• We say five strings 1 meter long 4

Geometry and multiplication

• We show with an equilateral triangle within the larger triangle
• We find the number of coloured triangles by stating 3 * 6/25
• If given four peices we use 5x 1 to solve ( 5* quater metre = 1,1 ) 4 4

Dividing parts

• If a 6 metre long piece is cut in two equal pieces = 3 m each
• Half written as 1 2
• 1 of six metres is 3
• Part is also written as Multiplication 2*2 , 4 is is split , one peice is 2 of 3 ( 2/3)
• Another rectangle with a length of 2 and equal widths
• Can divide into three equal parts

Fractions and Sharing

• Equal sharing problem of liters of milk between people require division
• Multiplying ( and writing as multiplication) 3x 1 4
• 21 in Geogebra means to divide 21 / 4 etc

Division of items with geometry

• A rectange of seven scm is split into 3 equal parts, what is the are of one, it is 7/3
• What if split into 4? It is 7/4
• What if split into , how should be done by listing of items in the next section 12/ equal groups