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Year 8 Maths Knowledge Map – Spring Term Key Word Integer Sum Product Ratio Definition An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component. The result of adding two or more numbers. The answer when two or more numbers are multiplied together. A ratio shows the relative sizes of two or more values. Ratios can be shown in different ways. Using the ":" to separate example values, or as a single number by dividing one value by the total. Equivalent Having the same value. Numerator Proportion Fraction The top number in a fraction. Shows how many parts we have. The bottom number in a fraction. Shows how many equal parts the item is divided into. Proportion says that two ratios (or fractions) are equal. A fraction is a part of a whole. Equivalent fraction Equivalent Fractions have the same value, even though they may look different. Denominator Example 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, and √2 are not. 9 is the sum of 2, 4 and 3 (because 2 + 4 + 3 = 9). 8 is the product of 4 and 2. If there is 1 boy and 3 girls you could write the ratio as: 1:3 (for every one boy there are 3 girls). 1/4 are boys and 3/4 are girls. 0.25 are boys (by dividing 1 by 4). 25% are boys (0.25 as a percentage). 1 Dollar is equivalent to 100 cents. 120 seconds is equivalent to 2 minutes. In the fraction 3/7, the numerator is 3. In the fraction 3/7, the denominator is 7. 1/3 = 2/6 2/3 of a pizza. The top number says how many slices we have. The bottom number says how many slices the pizza is cut into. These fractions are really the same: 1 2 Linear equation A linear equation is an equation for a straight line Equation of a line The equation of a straight line is usually written this way: y = mx + c. m represents the gradient of the line, c = 2 4 = 4 8 All linear equations: y = 2x+1 5x = 6+3y y/2 = 3 - x If y = 3x -7, the gradient is 3 and the y-intercept is -7 Variable represents the y-intercept. A symbol for a number we don't know yet. It is usually a letter like x or y. In x + 2 = 6, x is the variable. Estimate A close guess to the actual value, usually with some thought or calculation involved. Alex estimated that there were 10000 sunflowers in the field by counting rows. Simultaneous equation A simultaneous equation is when we have two or more equations working together. Each equation contains two variables. x+y=6 and −3x + y = 2 are simultaneous equations both containing variables x and y. Substitution In Algebra "Substitution" means putting numbers where the letters are. Nth term The 'nth' term is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next. 'n' stands for the term number so to find the 50th term we would just substitute 50 in the formula in place of 'n'. What is x + x/2 when x=5? Put "5" where "x" is: 5 + 5/2 = 5 + 2.5 = 7.5 For the sequence 6, 11, 16, 21, … Where there is a common difference between terms. nth term = dn + (a - d) Where d is the difference between the terms, a is the first term and n is the term number. e.g. 6, 11, 16, 21, ...For this sequence d = 5, a = 6 So the formula is nth term = 5n + (6 - 5) which becomes nth term = 5n + 1 Common difference The difference between each number in an arithmetic sequence. Congruence The same shape and size. Similarity Two shapes are congruent when you can Turn, Flip and/or Slide one so it fits exactly on the other. Two shapes are Similar when the only difference is size (and possibly the need to move, turn or flip one around). The sequence {3, 5, 7, 9, 11 ...} is made by adding 2 each time, and has a "common difference" of 2 (there is a difference of 2 between each number) When two shapes are similar, then: corresponding angles are equal, and the lines are in proportion. Quadrilateral A flat shape with four straight sides. Plane figure A 2-dimensional shape. Has width and breadth, but no thickness. Inverse proportion When one value decreases at the same rate that the other increases. Square, rectangle, trapezium, rhombus, kite, parallelogram Examples are square, rectangle, triangles and octagon. 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it? (Assume everyone works at the same rate) It is an Inverse Proportion: As the number of people goes up, the painting time goes down. As the number of people goes down, the painting time goes up. Direct proportion Directly proportional: as one amount increases, another amount increases at the same rate. ∝ The symbol for "directly proportional" is ∝ You are paid £20 an hour. How much you earn is directly proportional to how many hours you work. Work more hours, get more pay; in direct proportion. This could be written: Earnings ∝ Hours worked If you work 2 hours you get paid £40. If you work 3 hours you get paid £60. etc ... Sum of angles in a triangle Sum of angles in a polygon The three angles always add to 180° Sum of Interior Angles = (n-2) × 180° Each Angle (of a Regular Polygon) = (n-2) × 180° / n What about a Regular Decagon (10 sides) ? Sum of Interior Angles = (n-2) × 180° Where n = the number of sides = (10-2)×180° = 8×180° = 1440° And it is a Regular Decagon so: Sequence A list of numbers or objects in a special order. Each interior angle = 1440°/10 = 144° 3, 5, 7, 9, ... is a sequence starting at 3 and increasing by 2 each time.