# Elementary Mathematics (K-6) Explained

## Video Lectures

Displaying all 19 video lectures.

Lecture 1Play Video |
Counting using the grid planeWe start this course in elementary mathematics suited for a general audience by introducing geometry in the simplest possible way: by using a sheet of graph paper as the basic model, and then doing some simple arithmetic in a geometrical context. We introduce the horizontal and vertical lines of the grid plane, grid points, segments and rectangles, along with the idea of measuring their sizes. This is a good place to start practicing counting: and we count a number of interesting things associated to a rectangle, including numbers of sub rectangles of a certain size, paths from one corner to another, and tilings by 1x2 or 2x1 dominoes. |

Lecture 2Play Video |
Arithmetic with rectanglesWe start our study of arithmetic in a very primitive, naive fashion suitable for Kindergarten level students. A number for us is a row rectangle: a row of unit squares or cells. We show how to use such `numbers' for counting. Then the definitions of addition and multiplication are relatively simple. We are able to use our Kindergarten numbers to talk about the sizes of collections, and to compare two sizes. |

Lecture 3Play Video |
Number systems throughout historyAlmost every ancient culture had a system for counting. In this video we discuss the number systems of the ancient Babylonians, the Egyptians, the Indus valley culture, the Chinese, the Greeks and the Romans. In each case, a more primitive system (perhaps of strokes, twigs or pebbles) was refined by introducing larger units, or denominations, often from a base 5 or base 10 foundation. This is an excellent starting point for understanding our rather sophisticated system: the Hindu-Arabic number system. |

Lecture 4Play Video |
The Hindu-Arabic number systemThe Hindu-Arabic number system is the most important single advance in the history of science and mathematics. In this video we start from a naive Kindergarten approach to numbers, and then move to a simplified Roman numeral system, as a stepping stone to the Hindu-Arabic system. Along the way we introduce the usual names and symbols for the numbers 1-9, mention the big role that the number zero (0) plays, and discuss the world's earliest and most popular calculator---the abacus. |

Lecture 5Play Video |
Laws of arithmetic using geometryThe commutative laws for addition and multiplication have a nice geometric aspect. This video introduces simple motions in the grid plane: translations and reflections, and then uses them to understand commutativity. |

Lecture 6Play Video |
Fun with polyominoesPolyominoes are shapes formed formed unit squares (cells) in the grid plane, connected in such a way that we can go from any one square to another via common edges. Polyominoes with three squares are called trominoes, with four squares they are tetrominoes (popularized by the game Tetris) and with five square they are pentominoes (popularized by the game Blokus). In this video we count polyominoes of different forms, and investigate some nice tiling problems. Some of these puzzles are very challenging, but fun even for elementary primary school students. |

Lecture 7Play Video |
Addition and the names of numbersWe begin studying addition, the simplest and most important operation. First we concentrate on adding numbers in the range 1-5, then we move on to the range 1-10. For this latter step, we need to introduce the names for the numbers 11-20. Unfortunately the English names for these are somewhat adhoc and not entirely logical, putting students at a disadvantage. An alternate numbering system is described. Finally we organize things into addition tables, and note some pleasant properties of these. |

Lecture 8Play Video |
Addition in practiceTo generate examples for addition, we recall the associative and commutative laws. This also introduces some pleasant counting problems relating to the use of brackets. Even and odd numbers are introduced arithmetically and geometrically. The even-odd addition rule is also explained. |

Lecture 9Play Video |
Multiples, and more names of numbersAfter the even numbers, the multiples of 10 are particularly important. Their names are discussed, at least up to 100. Then we consider also multiples of 5, of 4 and of 3. While generated by addition (to get multiples of 3, just add 3 repeatedly), sequences of multiples naturally form a bridge to multiplication. The names for all the numbers up to 200 are given, and we also discuss a simplified naming system that is worth thinking about. We also emphasize the important of representing multiples geometrically, using rectangles of a fixed height. |

Lecture 10Play Video |
Word problems using additionWe introduce word problems involving elementary addition. The emphasis is on correct calculation and clear exposition. The lecture treats a good range of example problems, giving complete solutions, as well as some practice exercises at the end. |

Lecture 11Play Video |
Elementary projective (line) geometryElementary projective geometry is just the geometry of a line, or straightedge. It was introduced by Pappus around 300 A.D., and is ideally suited for K-6 education. This video introduces some basic notions: the join of two points, and the meet of two lines, and definitions of side, vertex, triangle and trilateral. Then we examine a remarkable correspondence between quadrangles and quadrilaterals. The three diagonal points and diagonal sides feature prominently in this construction. You will want to follow along with these constructions, on your own piece of paper with a pen or pencil and ruler. |

Lecture 12Play Video |
Pappus and PascalContinuing with our introduction to elementary projective geometry, meant for primary school students, we discuss two of the most famous theorems in mathematics: one due to Pappus of Alexandria around 300 A D and one due to Blaise Pascal in the 1600's. The first result only requires a piece of paper, a pen and a straightedge, or ruler, to appreciate. Pascal's theorem requires also a circle. Both theorems really ought to be more widely known in primary school mathematics education. |

Lecture 13Play Video |
Logical reasoning with tic-tac-toeTic-tac-toe is a great game for kids trying to learn mathematics---because it introduces them gently to logical reasoning, planning ahead, and the symmetries of a square. In this video we analyse the game, giving a three step approach to the right strategy. Two other games, Dots and Sprouts (the latter invented by John Conway) are also introduced, and you are challenged to figure out how to play simple versions. |

Lecture 14Play Video |
The multiplication tableEveryone learning mathematics ought to learn the multiplication table. Here we start with a simple form, involving just the numbers 1 through 5. We give some properties of the multiplication table, relating to multiples along rows and columns, mention the important square numbers running down the diagonal, and give some hints on how to commit this to long term memory. Do not rely on calculators---multiplication must be memorized! |

Lecture 15Play Video |
More multiplication: The 10x10 tableA rite of passage for all young people: learning the multiplication table! After having mastered the 5x5 table in the previous video, we are here set to extend to the rather bigger 10x10 table. This video offers some orientation towards this major goal. We emphasize that rote memorization must be combined with understanding of what is being memorized; that proceeding one row at a time (multiples of 10, then multiples of 2, then multiples of 5 etc) can help, as can the commutative law, which makes the table symmetric about the diagonal. Also paying special attention to the squares appearing down the diagonal is useful. |

Lecture 16Play Video |
Some tricks to help with multiplicationUnderstanding and memorizing the multiplication table is a necessary and important step for all young people when they are learning mathematics. In this lecture we give a few small tricks or lessons that can help. These deal with multiplying by ten, the important distinction between even and odd, and multiplying by five, three and nine. Then we give some typical sample problems involving multiplication, including some word problems. Of course the student ought to do many more such exercises! |

Lecture 17Play Video |
Area problems using multiplicationIn this video we use the correspondence between multiplication and areas of rectangles to solve some area problems arising from figures formed in the grid plane. We mention the historical connection between multiplication and rectangles going back to the ancient Greeks. These kinds of problems allow us to practice using both our addition and multiplication skills! |

Lecture 18Play Video |
The time scale of a human lifeNumbers are important for both counting and measurement, and today we start discussing measurement in the familiar context of time and the phases of a human life. This is not a bad place for young people to start learning about a scale and different units to measure something. We begin by explaining how the fundamental units of a day, month and year come about from the relative motions of the earth, moon and sun. Then we lay out a linear scale representing 100 years, the rough potential life span of a human (although some people do live beyond that). On this time scale we can point out various important times and stages of life. This is a good way for students to engage with numbers and their relation to their own lives and others around them. It is also a place to mention some interesting facts related to life spans of very long-lived creatures (like the bowhead whate), and also to talk about average life expectancies, such as 18 years for a domestic cat. |

Lecture 19Play Video |
An introduction to measuringWe introduce measurement in a general sense, discussing in particular length/distance, weight, area, volume and temperature. We discuss pro's and con's of the Imperial and Metrical systems, and then focus on length: creating rulers and using them to measure things around us. |