Integers
What is the difference between a positive integer and a negative integer? Give one real-life example of each.
A positive integer is greater than zero; a negative integer is less than zero. Examples: a temperature of +5°C and a temperature of −5°C, or money in credit and debt.
What does the word percent mean? Explain what 25% means in everyday language.
Percent means “per hundred.” 25% means 25 out of every 100, or one quarter of the whole.
Question: What is statistics used for? Give an example of a question that could be answered using statistical data.
Statistics is used to collect, organise, represent, and interpret data. Example: How do students in our class travel to school?
In statistics, what is a characteristic (Merkmal)?
Give one example of a characteristic that could be recorded for every student in a class.
A characteristic is a feature or property being investigated.
Examples include eye colour, height, age, or how a student travels to school.
What was the original reason people celebrated Halloween?
It was to celebrate the end of summer and the beginning of winter. People offered their gods food to get protection them from the coming winter months.
What do we call two integers that are the same distance from zero but lie on opposite sides of zero? Give an example.
Aditive inverse, or Gegenzahlen. For example, +7 and −7.
In percentage calculations, what are the base (Grundwert), percentage amount/part (Prozentwert) , and percentage rate (Prozentsatz)? Explain what each one represents.
Why is a bar chart useful when comparing the favourite sports of students in a class? What information should the axes communicate?
A bar chart makes it easy to compare categories. One axis should identify the categories, such as sports, and the other should show the number or frequency of students.
If “means of transport to school” is the characteristic, name four possible values (Ausprägungen) it could have.
Walking, cycling, travelling by bus, and travelling by car. Also, travelling by train, tram, or public transport.
Where did the name Halloween come from?
The first of November is called All Hallows' Day (Hallow=holy or blessed) or All Saints Day (Nov 1). The night before was called All Hallows' Eve and over time was shortened to Halloween.
What does the absolute value of an integer tell us? Why can an absolute value never be negative?
It tells us the distance from zero on the number line. A distance cannot be negative. For example, the absolute value of −6 is 6.
Two classes have the same percentage of students (25%) who cycle to school. Class A has 24 students, and Class B has 30 students. But Class B has more students who cycle to school. Explain how this is possible, and identify which percentage-calculation values are the same and which are different.
Both classes have the same percentage rate, but Class B has a larger base amount. Therefore, its percentage amount (the number of students who cycle) is greater.
Explain the difference between the mean, the median, and the mode of a data set.
Mean: add all values and divide by the number of values.
Median: the middle value when the data is ordered.
Mode: the value that occurs most frequently.
Explain the difference between an ordinal characteristic and a nominal characteristic. Give one example of each.
Why do people wear costumes at Halloween?
Long ago, people thought that ghosts walked around on Halloween night. They began wearing masks to trick them into leaving them alone. Later, people started wearing costumes with their masks.
Imagine that you are standing at −3 on a number line. Explain what happens when you subtract −4. Do you move left or right, and why?
You move four places to the right because subtracting a negative number is equivalent to adding its positive opposite. You arrive at +1.
A schoolbag is on sale for 30% of its original price. The sale price is €21. What was the original price of the schoolbag?
Explain how you can work it out mentally, without using a calculator.
The original price was €70. 30% is three times 10%. If 30% equals €21, then 10% equals €7. Therefore, 100% equals €70.
Six students recorded the number of minutes they spent reading each day:
20, 35, 25, 40, 30, ?.
The mean reading time for the six students was 30 minutes. How many minutes did the sixth student read?
The sixth student read for 30 minutes.
You want to find out whether students in your class usually sleep for fewer or more than eight hours per night. What characteristic would you record, and what values might it have?
The characteristic is the number of hours slept per night.
Values could be 6 hours, 7.5 hours, 8 hours, or 9 hours.
It is an ordinal characteristic.
What vegetable were Jack o' Lanterns made out of before they were made out of pumpkins?
Clue: in Austria, sugar is made from a similar vegetable.
Turnips (Rübe). They are hard roots. Pumpkins are easier to cut into and the orange colour makes the lanterns look brighter.
Without writing anything down, calculate (−3) × (−4). Explain the sign of your answer using the multiplication rules for integers.
+12. Multiplying two negative integers gives a positive result.
Question: Two students each calculate 20% of an amount. One gets €10 and the other gets €50. Can both answers be correct? Explain what must be different in their examples.
Yes. The base amount (B/G) must be different. For example, 20% of €50 is €10, while 20% of €250 is €50.
A survey asks only the students in the school football team which sport is most popular. Why might the results be misleading if we use them to describe the favourite sport of the whole school?
The sample is biased because it includes only football-team members, who may not represent the whole school. A more representative sample should include students with different interests.
Question: A teacher collects the following information from students in a class:
Classify each characteristic as nominal, ordinal or metric. Explain your reasoning for each choice.
1. Favourite school subject: Nominal. The categories have no natural order. The mode can be used.
2. Satisfaction with the school cafeteria: Ordinal. The categories can be ranked in a meaningful order. You can use the median.
3. Travel time in minutes: Metric. The values are numerical, and differences between them are meaningful. You can calculate the average from the values.
Why do we go trick-or-treating?
a) Long ago people put food outside their homes to distract the bad spirits that might otherwise try to get in. Over time, people began to eat the food themselves.
b) People gave each other small cakes in return for prayers for the dead family members on All Soul's Day (Nov 2). Later children began visiting their neighbours for treats.