Lernzettel: Fundamentals of Data and Geometry

📋 Course Outline

1. Stem and leaf plots
2. Histograms and data displays
3. Measures of center
4. Measures of spread and box plots
5. Probability representations
6. Probability calculations
7. Indices and square numbers
8. Pythagoras theorem
9. Angles and polygons
10. Similarity and trigonometry

📖 1. Stem and leaf plots

🔑 Key Concepts & Definitions

• Stem and leaf plot : A stem and leaf plot is a chart that splits each value into a stem and a leaf to show ordered data.
• Stem : The stem is the part of each value used to group numbers that share the same leading digits.
• Leaf : The leaf is the remaining digit(s) written on the same row to represent each data value.
• Ordered leaves : Ordered leaves are written in increasing order within each stem row so the plot can be read quickly.

📝 Essential Points

• To find the data value, combine the stem and the leaf in the same row using place value.
• Leaves are usually kept in ascending order so the plot reads left to right by size.
• If a value like 37 is shown with stem 3, the leaf records the 7 on that row.
• A stem can repeat across rows if the data need more than one row per stem value.

💡 Memory Hook

Stem holds the tens, leaf holds the units.

📖 2. Histograms and data displays

🔑 Key Concepts & Definitions

• Histogram : A histogram is a bar chart for continuous data where bars show frequencies in class intervals.
• Class interval : A class interval is a range of values grouped together for one bar in a histogram.
• Frequency : Frequency is the count of data values in a class interval.
• Bar height : Bar height is the frequency density scale used to represent how much data lies in that interval.

📝 Essential Points

• In a histogram, the x-axis shows class intervals and the y-axis represents frequency (or frequency density).
• Adjacent bars touch because each class interval covers a continuous range with no gaps.
• To read a histogram, locate the interval for the value range and use its bar height to get the frequency.
• If class widths differ, frequency density must be used so area represents frequency.

💡 Memory Hook

Histogram: area matters (bar area gives frequency).

📖 3. Measures of center

🔑 Key Concepts & Definitions

• Mean : The mean is the average found by adding all values and dividing by how many values there are.
• Median : The median is the middle value when all data are ordered from smallest to largest.
• Mode : The mode is the value that occurs most often in the data set.

📝 Essential Points

• If there is an even number of values, the median is the average of the two middle values.
• If there are multiple values with the same highest frequency, the data set can be multimodal.
• The mean can be pulled toward extreme values more than the median.
• For ordered data, the median can be read directly from the central position.

💡 Memory Hook

Median splits the list in half; mean is the balance point of all values.

📖 4. Measures of spread and box plots

🔑 Key Concepts & Definitions

• Range : The range is the difference between the largest and smallest values in a data set.
• Interquartile range : The interquartile range (IQR) is the spread between the first quartile and the third quartile.
• Box plot : A box plot is a diagram that summarizes a data set using five-number summary.
• Five-number summary : The five-number summary lists the minimum, first quartile, median, third quartile, and maximum.

📝 Essential Points

• Range = maximum − minimum.
• IQR = Q3 − Q1.
• In a box plot, the box edges are Q1 and Q3 and the line inside is the median.
• The whiskers extend to the minimum and maximum values shown (or to limits if outliers are treated separately).

💡 Memory Hook

Box plot: box gives the middle 50% (Q1 to Q3).

📖 5. Probability representations

🔑 Key Concepts & Definitions

• Event : An event is an outcome or set of outcomes that you are interested in measuring probability for.
• Venn diagram : A Venn diagram uses overlapping circles to represent events and their overlaps.
• Tree diagram : A tree diagram shows outcomes of multi-step experiments as branches.
• Probability line (washing line) : A probability line represents event likelihood by marking positions along a line from 0 to 1.

📝 Essential Points

• A probability value P(E) must satisfy 0 ≤ P(E) ≤ 1.
• The probability of the impossible event is 0.
• The probability of the certain event is 1.
• In a tree diagram, probabilities multiply along a path for multi-step outcomes.

💡 Memory Hook

Tree paths multiply; Venn regions represent overlap.

📖 6. Probability calculations

🔑 Key Concepts & Definitions

• Complement : The complement of an event E is the event that E does not happen.
• Mutually exclusive events : Mutually exclusive events are events that cannot both occur in the same trial.
• Independent events : Independent events are events where the outcome of one event does not change the probability of the other.
• Two-way table : A two-way table organizes counts by two categories for probability calculations.

📝 Essential Points

• P(not E) = 1 − P(E).
• If A and B are mutually exclusive, P(A or B) = P(A) + P(B).
• If A and B are independent, P(A and B) = P(A)·P(B).
• From a two-way table, probability can be found by dividing the relevant count by the total count.

💡 Memory Hook

Complement flips to 1 − x.

📖 7. Indices and square numbers

🔑 Key Concepts & Definitions

• Index : An index is the small number that shows how many times a base is multiplied by itself.
• Power (exponent form) : A power is written in exponent form as a^b to show repeated multiplication of a.
• Square number : A square number is a number that can be written as n^2 for some whole number n.
• Prime factor : A prime factor is a prime number used in writing an integer as a product of primes.

📝 Essential Points

• n^2 means n multiplied by itself.
• Square numbers grow as you square successive integers: 1^2, 2^2, 3^2, and so on.
• Factor trees break a number into prime factors using repeated splitting.
• In prime factor form, a^b corresponds to repeated factors that can be regrouped by index rules.

💡 Memory Hook

Square number = square the root.

📖 8. Pythagoras theorem

🔑 Key Concepts & Definitions

• Right-angled triangle : A right-angled triangle is a triangle with one angle equal to 90°.
• Hypotenuse : The hypotenuse is the side opposite the 90° angle and it is the longest side.
• Legs : The legs are the two sides that meet at the 90° angle in a right-angled triangle.
• Pythagoras’ theorem : Pythagoras’ theorem links the hypotenuse and legs of a right-angled triangle using a^2 + b^2 = c^2.

📝 Essential Points

• For a right-angled triangle, hypotenuse^2 = leg1^2 + leg2^2.
• If you are finding the hypotenuse, square the legs and add, then take the square root.
• If you are finding a shorter leg, rearrange to leg^2 = hypotenuse^2 − other leg^2.
• Use c as the hypotenuse and a, b as the two legs consistently.

💡 Memory Hook

Pythagoras: c^2 = a^2 + b^2.

📖 9. Angles and polygons

🔑 Key Concepts & Definitions

• Interior angle : An interior angle is the angle inside a polygon formed by two adjacent sides.
• Exterior angle : An exterior angle is formed by extending one side of a polygon and measuring the outside angle.
• Polygon : A polygon is a closed 2D shape with straight sides.
• Angle sum of a polygon : Angle sum is the total of all interior angles when moving around a polygon.

📝 Essential Points

• A triangle has 180° interior angle total.
• A quadrilateral has 360° interior angle total.
• For a regular polygon, each interior angle is the angle sum divided by the number of sides.
• Angles in polygons can be found by using angle sums and splitting shapes into triangles.

💡 Memory Hook

Triangles add to 180°, quadrilaterals add to 360°.

📖 10. Similarity and trigonometry

🔑 Key Concepts & Definitions

• Similar shapes : Similar shapes have the same angles and proportional corresponding side lengths.
• Scale factor : The scale factor is the multiplier that changes one shape into a similar shape.
• Trigonometric ratios : Trigonometric ratios (sin, cos, tan) relate angles in a right triangle to side lengths.
• Sine cosine tangent : Sine, cosine, and tangent are ratios formed from specific right-triangle sides relative to an angle.

📝 Essential Points

• For similar triangles, corresponding sides are in the same ratio given by the scale factor.
• In a right triangle, sin(angle) = opposite/hypotenuse, cos(angle) = adjacent/hypotenuse, and tan(angle) = opposite/adjacent.
• When finding unknown sides with trig ratios, identify opposite and adjacent relative to the chosen angle.
• When finding an unknown angle with trig ratios, use the inverse ratio method (e.g., sin^-1) after computing the ratio.

💡 Memory Hook

SOH: sin = opposite/hypotenuse; CAH: cos = adjacent/hypotenuse; TOA: tan = opposite/adjacent.

⚠️ Common Pitfalls & Confusions

1. Confusing the stem and leaf can swap place value and produce wrong data values.
2. Reading histogram frequencies as heights when class intervals have different widths can give incorrect totals.
3. Using the mean instead of the median when the data are skewed can shift the “central” value wrongly.
4. For box plots, mixing up Q1 and Q3 swaps the direction of spread and gives the wrong IQR.
5. In Pythagoras, using a non-hypotenuse side as c leads to an incorrect equation.
6. In trigonometry, using the wrong opposite/adjacent sides relative to the given angle gives the wrong ratio.
7. In probability, mixing up event and complement (or using 1 − P(E) incorrectly) leads to probabilities outside 0 to 1.

✅ Exam Checklist

1. Construct a correct stem and leaf plot from numerical data values.
2. Read values from a stem and leaf plot by combining the stem and leaf in the right order.
3. Draw and interpret histograms by using class intervals and bar heights (or frequency density).
4. Explain what a histogram’s bar area/frequency represents for the given class intervals.
5. Compute the mean by summing all values and dividing by the number of values.
6. Find the median from ordered data, including the even-data case.
7. Identify the mode from a data set.
8. Calculate range and interquartile range and state what they measure.
9. Interpret a box plot using the five-number summary (min, Q1, median, Q3, max).
10. Identify and use probability bounds (0 and 1) and event complements.
11. Calculate probabilities using complements, addition for mutually exclusive events, and multiplication for independent events.
12. Use a two-way table to find probabilities from counts.
13. Use indices notation to interpret powers and square numbers to compute $n^2$.
14. Apply Pythagoras’ theorem to find a missing side in a right-angled triangle.