Math 6A, Lesson 6, Fall 2020, 10/25/2020

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1. Opposite Numbers (additive inverse): is a number that when added to a given number yields 0. The opposite number for any number x is -x. Note that x may be positive or negative.

(-5) + 5 = 0

x + (-x) = 0

On the number line, the numbers 5 and -5 are located at the same distance from zero. We say that the numbers 5 and -5 are opposites. We can also say that -5 is opposite to 5, and 5 is opposite to -5 on the number line.

2. Addition of integers:

  • Addition of a positive number is taken as a movement to the RIGHT of the number line
  • Addition of a negative number is taken as a movement to the LEFT of the number line

(-3) + 4

2 + 3

(-1) + (-3)

2 + (-5)

0 + (-4)

3. Rule for addition:

  • If the signs of the integers being added are the same, the sum has the same sign as the integers and we add the absolute values of the integers

For any a > 0, and b > 0,

a + b = a + b;        3 + 5 = 8

(-a) + (-b) = -(a+b);         (-3) + (-5) = – (3 + 5) = -8

  • If the signs of the integers being added are different,, the sum takes the sign of the integer with the greater absolute value and we find the difference of the absolute values of the integers.

For any a > 0, and b > 0,

a + (-b) = +(a – b) if a >=b;        9 + (-6) = +(9 – 6) = +3 = 3

a + (-b) = – (b -a) if b > a;         15 + (-20) = -(20 -15) = -5

-a + b = -(a – b) if a >= b;         -18 + 12 = -(18 -12) = -6

-a + b = +(b -a) if b > a;          -23 + 27 = +(27 – 23) = +4 = 4

4. Home Work:

  • Handout: one page
  • Workbook: page 7: 4

Math 6A, Lesson 5, Fall 2020, 10/18/2020

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1. Negative Numbers: we often across quantities that have opposite directions or meanings. For example,  traveling due east and west, the rise and fall in price, profit and loss, and being above or below sea level. When we assign a certain meaning of quantity to be positive, a value of the quantity that has the  opposite meaning may be considered as negative and is represented with a “-” sign.

  • Positive numbers: 1, 2, 3, 0.8, ⅜, …
  • Negative numbers: -1, -2, -3, -⅔, -0.0025, …
  • Meanings of positive and negative numbers: teaching on whiteboard in the class.

2. The (horizontal) number line:

How to draw a number line:

  • Draw a line, and mark the zero point on it;
  • Choose a unit length, e.g., 1 cm, to mark the points 1, 2, 3, … at equal unit intervals on the right of 0, and the points -1, -2, -3, … on the left of 0;
  • Draw an arrow at each end.

Inequality signs:

  • > greater than
  • < less than
  • >= greater or equal
  • <= less than or equal

On a horizontal number line:

  • All the positive numbers are to the right of 0
  • All the negative numbers are to the left of 0
  • Numbers are arranged in ascending (increasing) order from left to right
  • Every number is smaller than any number on its right and greater than any number on its left

3. Absolute Value: the absolute value of a number is the distance that number is from 0 on the number line. Both 3 and -3 are the same distance from 0. The absolute value of a number is never negative.

|3| = 3, or |x| = x if x >= 0

|-3| = 3, or |x| = -x if x < 0

4. Home Work:

  • Handout: two pages
  • Workbook: page 7: 1, 2, 3, 5

Math 6A, Lesson 4, Fall 2020, 10/04/2020

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1. Quiz

2. Square Roots

3*3 = 9, 9 is called the square of 3, we also say that 3 is the positive square root of 9.

3. Perfect Squares: the number 1, 4, 9, 16, 25, … whose square roots are whole numbers are called perfect squares. We can find the square root of a perfect square by using prime factorization. Teaching on whiteboard in the class.

4. Cube Roots

2*2*2 = 8, 8 is called the cube of 2, we also say that 2 is the cube root of 8.

5. Perfect Cubes: the number 1, 8, 27, 64, … whose cube roots are whole numbers are called perfect cubes. We can find the cube root of a perfect cube by using prime factorization. Teaching on whiteboard in the class.

6. Home Work:

  • Handout: three pages
  • Workbook:
    • page 2: 8, 9
    • Page 4: 18, 19, 20

Math 6A, Lesson 3, Fall 2020, 9/27/2020

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1. Least Common Multiple (LCM):

The least common multiple of a group of numbers is the smallest positive integer which is divisible by all the numbers in the group.

The multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

The multiple of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

24 and 48 are the first two common multiples of 6 and 8. Since 24 is the least of all common multiples, we say the least common multiple (LCM) of 6 and 8 is 24.

2. Methods to find LCM

  • Using prime factorization. LCM is obtained by multiplying the highest power of each prime factor of the given numbers. Teaching on whiteboard in the class.
  • Continuous division. Teaching on whiteboard in the class.
  • Venn diagram. Teaching on whiteboard in the class.

3. Why LCM?

  • adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. Ex. 1/15 + 1/18 + 1/21
  • Word problems: Teaching on whiteboard in the class

4. Home Work:

  • Handout: two pages
  • Workbook:
    • page 2: 7
    • Page 3: 13, 15(a), 15(b)(ii), 17(a), 17(b)(ii)
    • Page 4: 23
    • Page 5: 24

Math 6A, Lesson 2, Fall 2020, 9/20/2020

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1. Greatest Common Factor (GCF): the largest common factor of a group of numbers is the largest positive integer that can divide all the numbers in the group.

The factors of 18: 1, 2, 3, 6, 9, 18

The factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

1, 2, 3, and 6 are called the common factors of 18 and 24. The largest of these common factors is 6. Thus, 6 is the greatest common factor of 18 and 24.

HCF: the highest common factor, same as GCF

GCD: the greatest common divisor, same as GCF

Relatively Prime or Mutually Prime: when two numbers, such as 15 and 16, have no common factors greater than 1, their GCF = 1 and the numbers are said to be relatively prime, or mutually prime.

2. Methods to find GCF

  • Using prime factorization. GCF is obtained by multiplying the lowest power of each common factor of the given numbers. Teaching on whiteboard in the class.
  • Continuous division. Teaching on whiteboard in the class.
  • Venn diagram. Teaching on whiteboard in the class.

3. Why GCF?

  • Reduce a fraction to its lowest terms: 18/48 = (18 / 6) / (48 /6) = ⅜
  • Algebra, GCF of the coefficients: 18x*x + 48x = 6x(3x + 8)
  • Word problems: Teaching on whiteboard in the class

4. Home Work:

  • Handout: one page
  • Workbook:
    • page 2: 6, 10(a), 10(b), 11
    • Page 3: 12, 14, 15(a), 15(b)(i), 17(a), 17(b)(i)
    • Page 4: 21, 22
    • Page 5: 26, 27, 28, 29(b)

Math 6A, Lesson 1, Fall 2020, 9/13/2020

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1. Factors: the number 12 can be expressed as the product of two smaller whole numbers as follow:

12 = 1 x 12

12 = 2 x 6

12 = 3 x 4

1, 2, 3, 4, 6, and 12 are called factors of 12. We say that 12 is divisible by each of its factors.

2. Multiplies: when a number is multiplied by a non-zero whole number, we get multiple of the number:

The multiple of 3 are: 3×1, 3×2, 3×3, 3×4, 3×5, …

That is: 3, 6, 9, 12, 15, …

The multiple of 4 are: 4×1, 4×2, 4×3, 4×4, 4×5, …

That is: 4, 8, 12, 16, 20, …

3. Link between factors and multiplies:

Recall 3 and 4 are factors of 12. Interestingly, 12 is a multiple of both 3 and 4. The number 12 is a multiple of each of its factors 1, 2, 3, 4, 6, and 12.

4. Prime number, composite number, prime factor

  • Prime number: a prime number is a whole number greater than 1 that has only two factors. 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, …
  • Composite number: a composite number is a whole number greater than 1 that has more than two factors.
  • Prime factor: a prime number that is a factor of a composite number is called a prime factor of the composite number.

5. Prime factorization methods:

6. Exponential notation:

Squared, cubed, fourth power, … teaching on whiteboard in the class

7. Home Work:

  • Handout: three pages
  • Workbook page 1: 1, 2, 3, 4, 5

Letter to Math 6A Class Parents 9/7/2020

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Dear NCLS Math 6A parents,

Welcome to a new NCLS school year! My name is Li Zhen, the teacher for Math 6A class at Newton Chinese School. I am excited to meet your children this coming Sunday afternoon!

First thing first: each student needs to set up a gmail account and please email me his/her gmail address so I can share my Google Classroom with him/her. Please do so no later than the end of day on Friday , 9/11/2020.

What does each student need to bring to the first class? Pencils and/or pens; three note books — one for taking notes in class and class work and two for homework; and most importantly, a Can-Do Attitude.

I’d like to have regular communication with students, parents and my wonderful and very capable Teaching Assistants Jessie Wang. You can find a lot of information on my blog and you can find out what we have learned in the class on any given Sunday, and what the homework is for that week:

http://blog.newtonchineseschool.org/zhenli/

Please feel free to email me or call me (in evenings) to talk about your concerns, things like the materials we have covered in class; home work load; or just chat like parents. My older daughter has graduated from college, and my younger daughter would be a sophomore at Harvard University this fall. Here is my contact info:

Li Zhen Phone: 617-785-6137 email: math_lizhen@yahoo.com

In the past, we had some 4th graders, 5th graders and mostly 6th graders in class. We’ll get to know each other in the first class. A great group of beautiful children for sure, and they are at the perfect age to learn basic math skills, and most importantly, to shape up their problem-solving abilities. It is my deep belief that every child is smart, every child can learn and every child will exceed our expectations! My ultimate goal is, to encourage and to help our students to develop the love of (the beauty of ) math and the confidence of solving many problems in real world.

With that in mind, I will give a lot of homework this year, not only do I believe the students can do it, but also because this is the only way that they can master a certain skill by practicing a lot. In addition to the class handouts and the exercise in the Work Book, I also encourage the students to try some Math Contest problems each week, for fun and challenge.  If the load is too much for a student, the latter can be optional.

My TA will take attendance and correct all the homework. Quiz and exams are given on a regular basis. The students’ progress and grades will be recorded in each class. Final report will be distributed to each family.

It’s a privilege to work with your child, and I thank you for that!

(PS. Please come to the classroom on the first day, or at the very least, reply to this email that you have read and been fully aware of our course load.)

Thank you very much!

Li Zhen

Math 6A, Lesson 16, Spring 2020, 6/21/2020

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Go over the Final Exam, and class celebration party – share your story and your thoughts.  All of our students have worked very hard and have made so much progress. I’m so proud of you! At this unprecedented time, we have had 14 ZOOM classes together with an almost perfect attendance rate! Your patience, resilience, self-motivated-learn and your humorous comments and laughter in the classroom, are the inspiration for me. I love you all!

Keep up your hard work! Have a great summer!

Math 6A, Lesson 15, Final Exam Spring 2020, 6/14/2020

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Final Exam Spring 2019, 6/9/2019. The exam will cover all the materials we have covered in Spring 2019 term, namely chapter 5 to chapter 8:

Math 6A, Lesson 14, Spring 2020, 6/6/2020

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1. Perpendicular Bisectors and Angle Bisectors

  • Use of compass: center, radius, pin leg, and drawing leg
  • To draw a circle
  • To mark off or copy a line segment
  • How to draw a perpendicular bisector of a line segment?
  • Any point on the perpendicular bisector of a line segment is equidistant from the two  end points of the segment.

2. Angle Bisectors

  • A ray AZ divides <BAC into two equal angles, <BAZ and <CAZ. The ray is called the angle bisector of <ABC
  • How to draw an angle bisector?
  • Any point on the angle bisector of an angle is equidistant from the two sides of the angle.

3. Class work

  • Construct/draw circles, triangles, angles, equal line segments
  • construct /draw perpendicular bisectors of line segments
  • construct/draw angle bisectors of angles

4. Classification of Triangles

  • The number of equal sides in the triangle: scalene triangle – no equal sides; isosceles triangle – two equal sides; equilateral triangles – three equal sides
  • The type of angles of the triangle: acute-angled triangle – all angles are acute; right-angled triangle – one of the angles is a right angle; obtuse-angled triangle – one of the angles is an obtuse angle
  • Is an equilateral triangle also an isosceles triangle?
  • Is it Possible to draw a triangle with more than one obtuse angle?
  • Can a scalene triangle be an acute-angled, right-angled or obtuse-angled triangle?
  • All the three angles in a scalene triangle are different size
  • The angles opposite the equal sides of an isosceles triangle are equal
  • All the three angles in an equilateral triangle are equal in size

5. Quadrilaterals

  • A closed plane figure with four straight sides joined by four vertices is called a quadrilateral
  • Vertices, diagonals
  • Properties of special quadrilaterals
  • Parallelogram: 2 pairs of parallel and equal opposite sides
  • Rectangle: all angles are right angles
  • Rhombus: all sides are equal, diagonals are perpendicular to each other
  • Square: all sides are equal, all angles are right angles
  • Trapezoid: 1 pair of parallel sides

6. Home Work:

  • Handout:
    • Two pages
  • Workbook:
    • Page 48-50: 12, 13, 14,15,16