7th Maths Study notes
Chapter 1: Parallel Lines
- Definition of Parallel Lines: Lines that do not meet, maintaining the same distance between them. They can be drawn with a scale and a set square. Two lines drawn at the same slant to a given line are parallel.
- Parallelogram: A quadrilateral where the two pairs of opposite sides are parallel.
- Angles Formed by Intersecting Lines: When one line crosses another, four angles are formed.
- If the lines are perpendicular, all angles are 90°.
- If one line is tilted, two small angles and two large angles are formed.
- The two small angles are of the same measure.
- The two large angles are of the same measure.
- The sum of a small angle and a large angle is 180°.
- Angles Formed by a Line Intersecting Two Parallel Lines:
- A line intersects two parallel lines at angles of the same measure.
- Corresponding Angles: Angles in the same position (e.g., top right, bottom right, top left, bottom left) when two parallel lines are cut by another line are called corresponding angles, and they measure the same.
- Alternate Angles: Angles in opposite positions when two parallel lines are cut by another line are called alternate angles, and they measure the same.
- Co-interior Angles: Interior angle pairs (e.g., inner right, inner left) formed when two parallel lines are cut by another line. The sum of the angles in each such pair is 180°.
- Co-exterior Angles: Exterior angle pairs (e.g., outer left, outer right) formed when two parallel lines are cut by another line. The sum of the angles in each such pair is 180°.
- If the intersecting line is perpendicular to one parallel line, it is perpendicular to the other, and all angles are right angles (90°).
- Triangle Sum Property: The sum of all angles of a triangle is 180°.
- If you subtract the measure of one angle of a triangle from 180°, you get the sum of the other two angles.
Chapter 2: Fractions
- Multiplication of a Number by a Fraction (Whole Number by Fraction):
- "N times a fraction (e.g., N × 1/D)" can be thought of as adding the fraction N times (e.g., 1/D + 1/D + ... N times).
- In general, N × (A/B) = (N × A) / B.
- Division as a Fraction: When 'A' is divided into 'B' parts, it can be written as A ÷ B or A/B.
- Fraction of a Number:
- "Half of 6 metres" is 1/2 of 6 metres, which is 3 metres. This can be written as a product: 1/2 × 6 = 3.
- In general, (A/B) of N can be calculated as (A × N) / B.
- Part of a Part (Multiplication of Fractions):
- When one part of a rectangle (e.g., 1/2) is further divided into parts (e.g., 1/3), each resulting smaller part represents a multiplication of the fractions (e.g., 1/3 × 1/2 = 1/6).
- The product of two fractions (A/B × C/D) is (A × C) / (B × D).
- Multiplication with Mixed Fractions:
- Mixed fractions can be converted to improper fractions before multiplication (e.g., 4 × 1 1/2 = 4 × 3/2 = 6).
- Alternatively, the whole number part and fractional part can be multiplied separately and then added (e.g., 3 × 2 1/4 = (3 × 2) + (3 × 1/4) = 6 + 3/4 = 6 3/4).
- Fractional Area: The area of a rectangle with fractional side lengths is still the product of the lengths of its sides (Length × Breadth)....
- Example: Area of a rectangle with sides 1/2 cm and 1/3 cm is 1/6 sq cm.
Chapter 3: Triangles
- Equilateral Triangle: A triangle where all sides are equal. To draw one, use compasses to mark the third corner at the intersection of two circles with radius equal to the side length, centered at the ends of the base.
- Drawing Triangles with Unequal Sides: Similar to equilateral triangles, draw the base, then use compasses with radii equal to the other two side lengths, centered at the ends of the base, to find the third corner.
- Relation between Angles and Opposite Sides: In any triangle, the angles and their opposite sides have sizes in the same order (the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle).
- Triangle Inequality Theorem: For three lengths to form the sides of a triangle:
- The greatest length must be less than the sum of the other two lengths.
- Alternatively, the sum of the lengths of the two smaller sides of a triangle is more than the length of the greatest side.
- More generally, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
- Drawing Triangles with Specified Angles:
- The sum of angles in a triangle is 180°. Therefore, only two angles can be freely fixed.
- To draw a specific triangle, not only two angles but also the length of the side on which they stand must be specified.
- Triangles with the same side length and the same two angles on that side are essentially the same triangle, possibly flipped or turned.
- Drawing Triangles with Two Sides and the Included Angle: Once the lengths of two sides of a triangle and the angle between them are specified, the triangle is determined.
- Drawing Triangles with Two Sides and a Non-Included Angle:
- Drawing a triangle with two specified side lengths and an angle not between them can sometimes result in two possible triangles, one single triangle, or no triangle at all, depending on the lengths and angle....
Chapter 4: Reciprocals
- Comparing Quantities as "Times" and "Part": Quantities can be compared by stating how many times one quantity is larger than another, or what part one quantity is of another....
- Reciprocals: To reverse the "times" and "part" relation, fractions are turned "upside down" (interchanging the numerator and denominator).
- Example: 2/3 and 3/2 are reciprocals of each other.
- Division Using Reciprocals: Division by a number is the same as multiplication by its reciprocal.
- Example: A ÷ B = A × (1/B).
- To find "what number multiplied by X gives Y?", you calculate Y multiplied by the reciprocal of X....
Chapter 5: Decimal Methods
- Decimal Form of Fractions with Powers of 10 Denominators: Fractions with denominators that are powers of 10 (10, 100, 1000, etc.) can be directly written as decimals by shifting the decimal point.
- Example: 43/10 = 4.3; 439/100 = 4.39; 4391/1000 = 4.391.
- The positional values diminish to a tenth, and places shift one place to the right when multiplying by 1/10.
- Fractional Form of Decimals: A decimal can be converted to a fraction by placing the digits after the decimal point over a power of 10 corresponding to the number of decimal places.
- Example: 327.45 = 32745/100; 327.045 = 327045/1000.
- Multiplying Decimals by Whole Numbers:
- Convert the decimal to a fraction, multiply the numerator by the whole number, then convert back to decimal (e.g., 3 × 4.25 = 3 × 425/100 = 1275/100 = 12.75).
- This is equivalent to multiplying the numbers ignoring the decimal point, then placing the decimal point in the product based on the number of decimal places in the original decimal.
- Multiplying Decimals by Decimals:
- Convert both decimals to fractions, multiply the fractions, then convert the resulting fraction to a decimal (e.g., 8.5 × 6.5 = 85/10 × 65/10 = 5525/100 = 55.25).
- The number of decimal places in the product is the sum of the number of decimal places in the numbers being multiplied.
- Dividing Decimals by Whole Numbers:
- Convert the decimal to a fraction, divide the fraction by the whole number, then convert back to decimal (e.g., 10.4 ÷ 2 = 1/2 × 104/10 = 104/20 = 52/10 = 5.2)....
- Converting Fractions to Decimals:
- Find an equivalent fraction with a denominator that is a power of 10 (10, 100, 1000, etc.).
- Example: 1/2 = 5/10 = 0.5; 1/4 = 25/100 = 0.25; 1/8 = 125/1000 = 0.125....
- Dividing Decimals by Decimals:
- Convert both decimals to fractions, then use the reciprocal method for division (e.g., 5.25 ÷ 0.75 = 525/100 ÷ 75/100 = 525/100 × 100/75 = 525/75 = 7)....
Chapter 6: Ratio
- Ratio Definition: Expresses the relationship between two quantities. "A to B" is written as A : B.
- Simplifying Ratios: Ratios are usually expressed using the smallest possible natural numbers.
- Maintaining Ratio: When the size of a picture or quantities in a mixture are changed, the ratio between its dimensions or components should remain the same to preserve the shape or consistency....
- Aspect Ratio: The height to width ratio of a rectangular picture is called its aspect ratio.
- Application of Ratios: Ratios can express relationships for any two measures (lengths, capacities, counts, etc.)....
- Dividing in a Ratio: To divide a total quantity according to a given ratio, find the sum of the ratio parts, divide the total quantity by this sum to find the value of one part, then multiply by each ratio number....
Chapter 7: Shorthand Math (Algebra)
- Algebraic Language: A method of denoting relations between measures or numbers using letters.
- Conventions in Algebraic Expressions:
- Products are often written without the multiplication sign (e.g., 4s instead of 4 × s).
- In products involving a number and a letter, the number is written first (e.g., 2n instead of n2).
- Division is usually written as a fraction (e.g., x/1 instead of x ÷ 1).
- Basic Algebraic Identities (Examples):
- n + n = 2n.
- 2x + x = 3x.
- 3x + 2x = 5x.
- x × 1 = x.
- x ÷ 1 = x (or x/1 = x).
- (x + y) - x = y.
- x + 0 = x.
- x - 0 = x.
- x - x = 0.
- x × 0 = 0.
- x ÷ x = 1 (for x ≠ 0).
- Associative Property for Addition: Adding numbers one after another, or adding their sum, gives the same result: (x + y) + z = x + (y + z). Brackets are important to show the order of operations when order matters.
- Property for Sequential Subtraction: Subtracting two numbers one after the other from a number, or subtracting the sum of these two numbers from the first number, both give the same result: (x - y) - z = x - (y + z).
- Combined Addition and Subtraction (Rule 1): Starting with a number, if you add a larger number and then subtract a smaller number, or add the difference of the smaller number from the larger, either way you get the same result: (x + y) - z = x + (y - z), for any three numbers x, y, z with y > z. This can also be used in reverse: x + (y - z) = (x + y) - z.
- Combined Subtraction and Addition (Rule 2): Starting with a number, if you subtract a larger number and then add a smaller number, or subtract the difference of the smaller from the larger, either way you get the same result: (x - y) + z = x - (y - z), for any x, y, z with y > z. This can also be used in reverse: x - (y - z) = (x - y) + z.
- Distributive Property (Multiplication over Addition): Multiplying a sum by a number, or multiplying each number in the sum separately and adding, either way gives the same result: (x + y)z = xz + yz.
- Distributive Property (Multiplication over Subtraction): Multiplying a difference by a number, or multiplying each number in the difference and subtracting, either way gives the same result: (x - y)z = xz - yz.
- These distributive properties can also be used in reverse: xz + yz = (x + y)z and xz - yz = (x - y)z.
