1. Ratios and Scale
UMI: July 20, 2016

2. Ratios
A ratio compares values: A ratio says how much of one thing there is compared to another thing
3:5
There are 3 diamonds to 5 hearts

3. Ratios can be shown in different ways:
Using the ":" to separate the values: 3 : 5
Instead of the ":" we can use the word "to": 3 to 5
Or write it like a fraction: 𝟑 𝟓

4. 3:5 A ratio can be scaled up:
Here the ratio is also 3 diamonds to 5 hearts even though there are more diamonds and hearts

5. Using Ratios
The trick with ratios is to always multiply or divide the numbers by the same value.
Examples:
3 : 5 is same as 𝟑×𝟐:𝟓×𝟐=𝟔:𝟏𝟎
𝟔:𝟏𝟎 is same as 𝟔÷𝟐:𝟏𝟎÷𝟐=𝟑:𝟓

6. "Part-to-Part" and "Part-to-Whole" Ratios:
Example: There are 20 students, 5 are girls, and 15 are boys
Part-to-Part:
The ratio of boys to girls is 15:5=3:1 or 3/1
The ratio of girls to boys is 5:15=1:3 or 1/3

7. Part-to-Whole:
The ratio of boys to all students is 15:20=3:4 or 3/4
The ratio of girls to all students is 5:20=1:4 or 1/4

8. Practice 1:
The ratio of boys to girls at the basketball game is 8:5. There are 30 girls.
How many more boys are there than girls?

9. Practice:
Ben and Matt received votes in the ratio 2:3. The total number of votes cast was 60.
How many votes did Ben get?

10. Ratio and Scale
A scale drawing is a drawing of a real object in which all the dimensions are proportional to the real object.
A scale drawing can be larger or smaller than the object it represents.
The scale is the ratio of the drawing size to the actual size of the object

11. Scale Factor
The ratio of the length of the scale drawing to the corresponding length of the actual object is called as scale factor.
The ratio of any two corresponding lengths in two similar geometric figures is called as scale factor

12. Example:
On a map, 5 inches represents 100 miles. How many inches would there be between two cities that are 2,500 miles apart?
Solution :
5𝑖𝑛𝑐ℎ𝑒𝑠 100 𝑚𝑖𝑙𝑒𝑠 = 𝑥 2500𝑚𝑖𝑙𝑒𝑠
𝑥= 5𝑖𝑛𝑐ℎ𝑒𝑠 100 𝑚𝑖𝑙𝑒𝑠 ×2500𝑚𝑖𝑙𝑒𝑠=125𝑖𝑛𝑐ℎ𝑒𝑠

13. Practice 1:
Joe I'S traveling from one city to another. He is looking at a map and realizes that 4cm represents 25 miles. He then uses a ruler to measure the distance on the map between the two cities that he is traveling. The distance between them is 160cm. How far, in miles, are the two cities apart?

14. Practice 2:
On a map, 4 inch represent 12 miles. The distance between two cities on that map is 3 1/4 inches. What is the actual distance, in miles, between the two cities?
Solution :

15. Pythagorean Theorem
UMI: July 20, 2016

16. Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
𝑐 2 = 𝑎 2 + 𝑏 2
Definition: The longest side of the triangle is called the "hypotenuse".

17. History and Proofs
The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.?-500 B.C.?), who was perhaps the first to offer a proof of the theorem. We do not know for sure how Pythagoras himself proved the theorem that bears his name because he refused to allow his teachings to be recorded in writing.

18. History and Proofs
More than 4000 years ago, the Babylonians and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle. There are over hundreds ways to proof Pythagorean Theorem. We are going to introduce some here.

19. Euclid's Proof
Let ACB be a right triangle with right angle CAB.
On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order.
From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
Join CF and AD, to form the triangles BCF and BDA

20. Euclid's Proof
∠𝐹𝐵𝐴=∠𝐷𝐵𝐶=∟, 𝑠𝑜 ∠𝐹𝐵𝐶=∠𝐷𝐵𝐴
𝐹𝐵=𝐴𝐵, 𝐷𝐵=𝐶𝐵,so ∆𝐵𝐶𝐹≅∆𝐵𝐷𝐴
𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐵𝐶𝐹= 1 2 𝐹𝐵×𝐹𝐺= = 𝐴𝐵 2
𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐵𝐷𝐴= 1 2 𝐵𝐷×𝐷𝐿
Since Congruent triangles have the same area, therefore
1 2 𝐵𝐷×𝐷𝐿= 𝐴𝐵 2 ,so 𝐵𝐷×𝐷𝐿= 𝐴𝐵 2
Similarly, we can proof 𝐶𝐸×𝐿𝐸= 𝐴𝐶 2

21. Euclid's Proof
𝐴𝐶 𝐴𝐵 2 = 𝐵𝐶 2

22. Zhou Yuan Zhi’s Proof 邹元治证明
Let a, b, c denote the legs and the hypotenuse of the given right triangle.
Consider the two squares in the accompanying figure, each having a+b as its side.

23. Zhou Yuan Zhi’s Proof 邹元治证明
Left figure gives 𝑎+𝑏 2 = 𝑎 2 + 𝑏 2 +4× 1 2 𝑎𝑏
Right figure gives 𝑎+𝑏 2 = 𝑐 2 +4× 1 2 𝑎𝑏
Therefore 𝑎 2 + 𝑏 2 +4× 1 2 𝑎𝑏= 𝑐 2 +4× 1 2 𝑎𝑏
𝑎 2 + 𝑏 2 = 𝑐 2

24. Zhao Shuang‘s Proof 赵爽证明
Let a, b, c denote the legs and the hypotenuse of the given right triangle.
4 congruent right triangles arranged as the picture.
𝑐 2 =4× 1 2 ×𝑎𝑏+ 𝑏−𝑎 2
𝑐 2 =2𝑎𝑏+ 𝑏 2 + 𝑎 2 −2𝑎𝑏= 𝑏 2 + 𝑎 2

25. US president James Garfield’s Proof
Let a, b, c denote the legs and the hypotenuse of the given right triangle.
Construct the trapezoid by three right triangles as the picture
Use the formula for the area of trapezoid, we have
𝐴= 1 2 𝑎+𝑏 × 𝑎+𝑏
On other hand, 𝐴=2× 1 2 𝑎𝑏+ 1 2 𝑐 2
1 2 𝑎 2 + 𝑏 2 +2𝑎𝑏 =𝑎𝑏 𝑐 2
𝑎 2 + 𝑏 2 = 𝑐 2

26. Practice 1:
Only one of these triangles is really a right triangle. Which one?

27. Practice 2:
A 3m ladder stands on horizontal ground and reaches 2.8 m up a vertical wall. How far is the foot of the ladder from the base of the wall?
Solution :

28. Thank You References: A problem solving approach to mathematics
for elementary school teachers by Billstein,
Libeskind, Lott
Math Playground:
Math is Fun :