Week 11
Name 3 important ideas in assessing quality of graphs.
1. Make sure the intervals are even (increase by only 2's or only 5's. etc) on both axes. 2. If you're using percentages, make sure they're out of 100%. 3. Make sure it starts at 0.
Explain how metric measures are derived via the properties of natural objects. What are 5 reasons that the metric system is an improvement over the English or Customary system? What is the prefix for thousand? thousandth?
1. Meter (the measure of distance) is 1/10,000,000 distance from the equator to the North Pole. 2.Liter (the measure of capacity) is measured by building a cube which is 1/10 of a meter or a decimeter. The amount of water that fills the container is a liter. Gram (measure o weight) is the weight of water that fills 1cm cube of water. The metric system is based on powers of 10 so it is much easier to convert units, often just by moving the decimal point. The metric system is an international standard used by nearly all the countries of the world. There are only 3 countries (i.e. United States, Liberia and Burma) that are still using the English system. The prefix for a thousand is kilo and the prefix for thousandth is centi
Find the range of dates for Thanksgiving (4th Thursday in November)..
22, 23, 24, 25, 26, 27, 28.
What is the need for leap year? Give all the leap years between 1776 and 1812. How can you tell if any year is a leap year?
A leap year is important because it adds an additional day to the entire year, so instead of living a 5 hour 48 minute and 46 second day at the end of each year, at the end of 4 years, we simply have an extra day in our year (because 5 hours 48 minutes and 46 seconds times four is equal to 24 hours, or one day). Leap years between 1776 and 1812: 1776, 1780, 1784, 1788, 1792, 1796, 1804, 1808, and 1812. A year is a leap year if it is divisible by 4.
Define one year. The earth takes 365 days 5 hours 48 minutes and 46 seconds to make one complete revolution around the sun. What is the difference between the exact time and 365.25 days? What difference would this make in 10 years? 100 years? 1000 years?
A year is the orbital period of the Earthmoving in its orbit around the Sun. For an observer on the Earth, this corresponds to the period it takes the Sun to complete one course throughout the zodiac along the ecliptic. The average length of the year in the Gregorian (modern) calendar is 365.2425 days (taking account of the century rule for leap years). In astronomy, the Julian year is a unit of time, defined as 365.25 days of 86400 SI seconds each (no leap seconds), or 31557600 seconds total. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by changes in weather, the hours of daylight, and consequently vegetation and fertility. The Julian calendar assumes a full year is 365.25 days, whereas it is actually 11 minutes less. In 10 years, we would have lost 110 minutes. In 100 years, we would have lost 1100 minutes. In 1000 years, we would have lost 11,000 minutes!
Why is your birthday on a different day of the week each year?
Any day of the year (including your birthday) moves one day later each year (making it a different day of the week each year) or sometimes two days later if the year is a leap year, because the remainder of 365 divided by 7 (the amount of days in a week) is 1 (the leap day).
What happens to the perimeter of a square if the length of its side is doubled? To the area of a square if the length of its side is doubled? What happens to the volume of a cube if the length of the side is doubled? To the surface area?
Doubled. Quadrupled. Increase by 8. Quadrupled.
What causes seasons? Define the beginning of each season in terms of daylight. How does this show that the earth, like an equation, is in balance?
Earth's axis is an imaginary pole going right through the center of Earth from "top" to "bottom." Earth has seasons because its axis doesn't stand up straight. So instead of rotating with its axis straight up and down, it leans over a bit. As Earth orbits the sun, its tilted axis always points in the same direction. So, throughout the year, different parts of Earth get the sun's direct rays. Sometimes it is the North Pole tilting toward the sun (around June) and sometimes it is the South Pole tilting toward the sun (around December). The summer solstice day has the longest period of daylight, a hemisphere's winter solstice occurs on the shortest day and longest night of the year, when the sun's daily maximum elevation in the sky is at its lowest. September and March equinox: night and day are nearly exactly the same length - 12 hours - all over the world.
Give an example of non-decimal measurement units
Examples include feet, inches, meters, yards, etc.
Test the hypothesis that the circle creates the largest area/perimeter (circumference). What shape represents the greatest volume for a given surface area? Why are most items packaged in rectangular solids rather than spheres?
Given: Perimeter /circumference = 24. Make three rectangles that have a perimeter of 24. What is the area of each? Consider a circle with the perimeter/circumference of 24.
What is the difference between a histogram and a bar graph?
In a bar graph, the data in the bars is not connected, while in a histogram the data in the bars is connected. When the bar graphs of data are connected, like in a histogram, this means the data is on-going, it's related to one another. In a bar graph, each data bar represents something else and isn't necessarily related to the previous bar. Bar graphs are categories such as color, pets; histograms are continuous data in which the designer sets the frequency.
Two rectangles are similar. One is 14 x 10. Give the dimensions of the other. About how many similar rectangles are there? How can math problems about similar figures be considered productive multiplication practice. Proportions and cross-multiplication
Make a proportion. There are an infinite number of possibilities. Proportions and cross-multiplication
Give examples to show that measurement drives progress.
Measurement is linked to many mathematical and science developments. You could use an example like the sundial. As the sun moves across the sky, shadows change in direction and length, so a simple sundial can measure the length of a day. We now have much more accurate timepieces.
109. What are the measures of central tendency? What are the measures of variability?
Measures of central tendency: mean, median, and mode. Measures of variability: range, interquartile range (IQR), variance, and standard deviation.
For what momentous calendar reform is Pope Gregory famous? What questions might people have about such reform? Discuss the difference between the Julian and the Gregorian calendar.
The Julian calendar assumes a full year is 365.25 days whereas it is actually 11 minutes less. The Julian calendar has a regular year of 365 days divided into 12 months. A leap day is added to February every four years. The Julian year is, therefore, on average 365.25 days long. However, the Gregorian calendar was able to make up for the 11 minutes by making 3 out of 4 centurial years common instead of leap years. More clarification: Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the year 2000 is. Pope Gregory also dropped 10 days from the calendar.
The calendar is an important part of humankind's need for quantitative thinking. You may be asked to expand upon this statement.
The calendar follows numerical patterns: how many days in a month, leap years, etc. (we did a class activity in which pairs of partners explained the social, economic, political, religious and personal uses for calendars)
When was the metric system invented? Name one of the mathematicians involved in designing the metric measurement or SI system.
The metric system was officially adopted in 1791. One mathematician involved in designing the metric measurement was Napoleon Bonaparte. Lagrange
How many time zones are in the world? Why do we need time zones? Why is the International Date Line located in the Pacific Ocean? What happens when you cross it?
There are 24 standard time zones in the world. We need time zones because dawn and dusk occur at different times at different places because of the Earth's rotation. For example, it helps express that midnight on one side of the world is midday on the other side. The International Date Line is located in the Pacific Ocean so that it runs through the least inhabited part of the globe, thus avoiding potential problems with one half of a country (or even a city) being in different days of the week. If you are on the East side of the line, you would move ahead into the next day of the calendar, and if you crossed West, you would go back one day.
Explain the following division problems by using measurement, visual representation, decimal, complex fraction and multiplication of reciprocal:
a) 1 ½ divided by ½ .5/1.5=3 1½/½=3 1½x½=3 b) ¼ divided by ½ .5/.25=.5 ¼/½=½ ¼x½=½
Solve problems using the Pythagorean Theorem.
a^2 + b^2 = c^2, so if you're given two legs of a triangle (a and b), you have to find the hypotenuse c. Ex: legs of triangle are 5 and 10, you would do 5^2 + 10^2 = 25 + 100 = 125. Then take the square root of 125 to find the hypotenuse, so c= 11.18.
