Download Classwork 6. 10/30/2016

Math 4. Class work 6.
Algebra.
1. Properties of addition and multiplication. Compute by the most
convenient way:
a. 4 ∙ 23 =
b. 17 ∙ 15 =
c. 13 ∙ 18 =
d. 12 ∙ 17 + 35 ∙ 13 + 17 ∙ 23 =
e. 41 ∙ 80 − 25 ∙ 41 + 55 ∙ 29 =
f. What is the last digit of the sum of all three digits’ numbers?
2. Addition and subtraction of fractions with unlike denominatoers.
2
2
. What should we do? Why do we need to bring both fractions
9
3
to the same denominator? We can add together only similar objects: apples to apples
2
2
and oranges to oranges. Are two fractions and similar objects?
9
3
Let’s try to add
and
2 1 1
= + ,
3 3 3
2 1 1
= +
9 9 9
How we can add together
2
9
2
1
1
1
1
3
9
9
3
3
+ = + + +
To be able to add two fractions we have to be sure that they
1
have the same denominator. Each is exactly the same as
3
3
2
6
and =
9
3
9
2
2 3 2×3
×1 = × =
3
3 3 3×3
If we multiply both numerator and denominator by the same number the fraction will not
change. To bring 2 fractions to the same denominators we have to multiply the
numerators and the denominators of both fractions by two different numbers to get a
common multiple as the denominator for both fractions. There are many common
multiples of 2 numbers. Of course, one of them is their product, but is not always the
simplest one. Usually, it is convenient to find LCM of these 2 (or, sometimes more than
2) numbers.
Compute:
2 1
− =
3 4
7 1
− =
15 5
5
4
+
=
12 15
In a zoo there are birds ( they have 2 legs) and mammals with 4 legs. How many
birds and mammals were there in the zoo, if they had 6000 legs and 2500 heads
altogether.
Exercises.
1. Proof that among any 3 natural numbers there are always 2 numbers sum of
which is even number.
2. Proof that if the sum of 2 natural numbers is less than 17, then the product
of them will not be greater than 64.
3. How will the product change if
a. one factor increases two times
b. one factor is three times smaller
c. one factor increases 2 times and second factor decreases 8 times
d. one factor increases 2 times and another factor increases 3 times
e. one factor decreases 2 times and another factor decreases 3 times
Geometry.
Triangle.
Triangle is a closed figure
consisting of three line
segments linked end-toend.
Acute triangle has all
acute angles, not only 60.
Obtuse triangle has an
obtuse angle. Can a
triangle have more than
one obtuse angle?
Isosceles triangle has two
equal sides. Equilateral triangle has all three sides equal. Right triangle has a right
angle.
The line segment from a vertex of the triangle to
the line containing the other two vertices and
perpendicular to that line is called the altitude
(the height). The length of this segment is also
called the height of a triangle relative to its base.
Three angles of any triangle sun to a straight angle.
Line l is parallel to line AC. Angles (3) are
equal as vertical angles, angles (2) are equal
and angles (1) are equal because line l is parallel
to line AC.
Area of the triangle.
1
𝑆∆ = ℎ×𝑎
2
The area of a triangle is equal to half of the
product of its height and the base, corresponding
to this height.
For the acute triangle it is easy to see.
𝑆 = ℎ×𝑎 = 𝑥×ℎ + 𝑦×ℎ
1
𝑆∆𝐴𝐵𝑋 = ℎ×𝑥,
2
1
𝑆∆𝑋𝐵𝐶 = ℎ×𝑦,
2
𝑆∆𝐴𝐵𝐶 = 𝑆∆𝐴𝐵𝑋 + 𝑆∆𝑋𝐵𝐶
1
1
1
1
𝑆∆𝐴𝐵𝐶 = ℎ×𝑥 + ℎ×𝑦 = ℎ(𝑥 + 𝑦) = ℎ×𝑎
2
2
2
2
For an obtuse triangle, for one out of the three
heights, it is not so obvious.
1
𝑆∆𝑋𝐵𝐶 = ℎ×𝑥,
2
1
𝑆∆𝑋𝐵𝐴 = ℎ×𝑦
2
1
1
𝑆∆𝐴𝐵𝐶 = 𝑆∆𝑋𝐵𝐶 − 𝑆∆𝑋𝐵𝐴 = ℎ×𝑥 − ℎ×𝑦
2
2
1
1
= ℎ×(𝑥 − 𝑦) = ℎ×𝑎
2
2