Unidad didáctica para la memoria de profesor DNL Matemáticas 2012
UNIDAD 8. Proporcionalidad numérica.
CURSO: 2º ESO
DISTRIBUCIÓN TEMPORAL: 10 sesiones.
Conocimientos previos: Fracciones, números decimales.
1. OBJETIVOS.
Se pretende que los alumnos sean capaces de:
• Determinar si dos razones forman proporción.
• Distinguir si dos magnitudes son directamente proporcionales.
• Resolver problemas reales que impliquen el uso de una regla de tres simple directa o de la reducción a la unidad.
• Determinar si dos magnitudes son inversamente proporcionales.
• Resolver problemas reales que impliquen el uso de una regla de tres simple inversa o de la reducción a la unidad.
• Hallar el tanto por ciento de una cantidad.
• Calcular aumentos y disminuciones porcentuales.
2. NÚCLEOS DE CONTENIDOS.
CONCEPTOS.
• Razón y proporción.
• Magnitudes directamente proporcionales.
• Regla de tres simple directa y método de reducción a la unidad.
• Magnitudes inversamente proporcionales.
• Regla de tres simple inversa y método de reducción a la unidad.
• Tanto por ciento de una cantidad.
• Aumentos y disminuciones porcentuales.
PROCEDIMIENTOS, DESTREZAS Y HABILIDADES
• Distinción entre magnitudes directa o inversamente proporcionales.
• Interpretación de mensajes que contengan informaciones sobre cantidades y medidas.
• Construcción de tablas de proporcionalidad directa e inversa.
• Resolución de problemas mediante reglas de tres simples (directas e inversas) y por reducción a la unidad.
• Resolución de problemas de cálculos de porcentajes.
• Utilización de la calculadora para facilitar los cálculos de tipo numérico.
ACTITUDES.
• Incorporación al lenguaje cotidiano de términos relacionados con la proporcionalidad numérica, directa e inversa.

• Orden en la resolución y la presentación de los cálculos y soluciones en problemas de proporcionalidad.
• Curiosidad, interés y valoración crítica ante las informaciones y los mensajes de naturaleza numérica.
• Confianza en las propias capacidades para afrontar problemas, comprender las relaciones matemáticas y tomar decisiones a partir de ellas.
• Constancia en la búsqueda de soluciones a los problemas numéricos.
• Sensibilidad y gusto por la presentación ordenada y clara del proceso y del resultado seguido en problemas y cálculos numéricos.
• Perseverancia en la búsqueda de soluciones a los problemas.
COMPETENCIAS QUE SE TRABAJAN
• Identificar relaciones de proporcionalidad numérica (directa e inversa), y resolver problemas en los que se usan estas relaciones, haciendo hincapié en los problemas-tipo asociados a estas relaciones.
• Aplicar el razonamiento deductivo e inductivo en contextos numéricos y alfanuméricos.
• Valorar e integrarse en el trabajo en grupo para la realización de actividades de diversos tipos, como base del aprendizaje matemático, de la formación de la autoestima y de valores sociales.
3. CRITERIOS DE EVALUACIÓN
• Distinguir si dos razones forman proporción.
• Aplicar la propiedad fundamental de las proporciones en la resolución de diferentes problemas.
• Completar tablas de proporcionalidad y series de razones iguales.
• Distinguir si dos magnitudes son directa o inversamente proporcionales.
• Aplicar la regla de tres simple, tanto directa como inversa, en la resolución de problemas, estableciendo cuál debe aplicarse en cada caso.
• Utilizar los porcentajes para resolver distintos problemas.
4. SESIONES Y ACTIVIDADES DE APRENDIZAJE
Los contenidos de esta unidad didáctica se han impartido integramente en inglés utilizando el texto que se adjunta como Anexo 1, siguiendo el siguiente orden:
Sesión 1: Introduction, Direct proportionality. Examples.
Sesión 2: Direct Rule of Three. Examples.
Sesión 3: Inverse proportionality. Examples.
Sesión 4: Inverse rule of three.
Sesión 5: Percentages. Example.
Sesión 6 a 9:
Worksheet. Ejercicios del libro.
Sesión 10: Control.
Todos los contenidos se han impartido proyectando los materiales didácticos en cañón de video y produciendo las respuestas de los ejercicios conforme se escriben, escribiendo tanto la profesora como los alumnos.

INSTRUMENTOS DE EVALUACIÓN
Se realiza un examen en la última sesión. Se permite el uso de la calculadora. Todas las preguntas deben responderse en Inglés.
Se adjunta el examen.
RECURSOS DIDÁCTICOS
Se han utilizado los recursos didácticos que se anexan a continuación.
Anexo I
Definition . A ratio expresses the magnitude of quantities relative to each other.
A ratio is expressed algebraically as a quotient .
Example : For every spoon of sugar, you need 2 spoons of flour. The ratio is
1
2
.
In English we generally separate the two numbers in the ratio with a colon (:).
Suppose we want to write the ratio of 8 and 12. We can write this as 8:12 or as a
8 fraction
12
, and we say the ratio is eight to twelve . a
A proportion is an equality of two ratios. Example: b
 c x
Sometimes a change in one quantity causes a change in another quantity.
If these changes are related through equal factors , then the quantities are in direct proportion . Then the two quantities are directly proportional .
Example .
Suppose that you are buying cans of soup at the store. Let us imagine that they cost 50 cents each. Changing the number of cans that you buy will change the amount of money that you pay.
If you buy 4 cans, you will pay $2.00. But if you buy 8 cans. You will pay $4.00.
Both the number of cans and the cost changed by the same factor : they increase or decrease by the same factor.

If quantities A and B are in direct proportion and we have other measures of them as quantity A’ and quantity B’:
Quantity A

Quantity B
Quantity A’  Quantity B’
The following ratios are equal:
QuantityA
QuantityB

QuantityA '
QuantityB '
This ratio is called proportionality constant
Also:
QuantityA

QuantityB
QuantityA ' QuantityB '
For an equation of the form: a b
 c x where the variable to be worked out is x , the rule of three states that: b
 c x
 a
In this context, a is called the extreme of the proportion, and b and c are called the means . See these numbers in order: ( a ,b, c, x )
This rule is used to solve problems where two quantities are in direct proportion.
Example
Let’s say that we want to know how far a car will get in 7 hours, if we know that its speed is constant and that it already travelled 90 miles in the last 3 hours.
Converting the problem into ratios we get:
3 hours

90 miles
7 hours
 x miles
3 hours
90 miles
And also.

7 hours
 xmiles
3
90

7 x
90 miles
3 hours
 xmiles
7 hours

90
3
 x
7
The rule of three can be used to calculate x directly: x

90

7
= 210
3 miles.
The proportion factor is the constant
90
3
mph, it is the proportionality constant.

The formal definition of inverse proportion is:
Two quantities, A and B, are in inverse proportion if by whatever factor A changes, B changes by the reciprocal of that factor.
Example . Let us say that you are driving a car and you are going to travel 60 miles. Suppose that you spent 1 hour driving. Your average speed would be 60 mph.
If you spent 2 hours driving, your average speed would be 30 mph.
The two quantities, time and speed, changed by reciprocal factors: time changed by a factor of 2, whereas speed changed by a factor of ½.
That is, when two quantities change by reciprocal factors, they are inversely proportional or they are in inverse proportion.
This means that one quantity increases as much as the other decreases .
Then, the following is true
Quantity A

(60m)

Quantity
A’ 
(30m)

Quantity B
(1h)
Quantity
B’(2h)

60·1=30·2

'

'
QuantityA
And also:
QuantityB

QuantityB '
. 60/30=2/1
QuantityA '
This gives us a new rule of three: the inverse rule of three.
If two quantities A and B are inversely proportional:
Quantity A

Quantity B
Quantity A’  Quantity B’
We have the relationship:
QuantityA
QuantityA '

QuantityB '
QuantityB
, or
QuantityA

QuantityB
QuantityB '
QuantityA '

In problems, when the unknown x is to be calculated:

Quantities Numbers Numbers
A
B a
 c
 b x
If quantities A and B are inversely proportional, these numbers satisfy: b x
 c a
(Only quantities c
The inverse rule of three states that:
and a are swapped) to swap:
We work out: x
 a
 b c
Problem . Three people can build a fence in 7 hours. How long will it take five people to complete the same job?
Problem . A hose that delivers 220 gallons of water per hour can fill a pool in 16.5 hours. A second hose delivers 300 gallons of water per hour. o o
Using only the second hose, how long would it take to fill the pool?
Using both hoses together, how long would it take to fill the pool?
Hose: manguera
Only second hose:
Gph hours
220 --------- 16.5
300 --------- x INVERSE PROPORTION
Both hoses:
A percentage is the numerator of a fraction whose denominator is
100.
100% of any number is just the number unchanged. 200% of any number is twice the number.
A half can be written as: i) a percentage: 50%;

ii) a decimal: 0.5
1 iii) or a fraction:
2
.
Example 1 .
Calculate the 25% of 80.
25
25% means
100
(25 over 100), then:
25% of 80 =
25
100
of 80 =
25
100

80 =
25

80
100
= 20.
Notice that this can also be calculated with decimals as: 0 .
25

80

20 .
To calculate a percentage we use the corresponding fraction of
25% , that is
25
100
.
Or the product by the corresponding decimal number : 0.25
Percentages can also be calculated with the rule of three .
% :
Total Part
100

Number: 80

25 x x =
25

80
100
= 20

Example 2 .
In a students election, Sam got 66% of the votes polled.
If he got 363 votes, find the total number of voters.
Solution.
When we know the part and need to find the total , the rule of three is helpful:
Total Part
% 100

66 number x

363 x =
363

100
= 550 voters.
66
To calculate the total there is also a direct way: 363

100
66

550 voters.
To calculate the total we use the reciprocal fraction of 66%, that is
100
66 or the rule of three.

Example 3. Reductions.
A skateboard is reduced 25% in price in a sale. The old price was $120. Find the new price.
(We need to do a reduction)
25% of 120 is 30 ( 0 .
25

120

30 ). The reduction is $30.
120

30 = 90. The final price of the skateboard in the sale is $90.
The general way to get reductions, directly , is: 100

25 = 75.
75 % of 120 = 0 .
75

120

90 .
REDUCING A QUANTITY BY A PERCENTAGE
To make a reduction of an a % to a quantity:
1. Calculate b = 100
 a.
b
2. Multiply the quantity by
100
. (this fraction is less than 1)
Example 4 .
Increases.
A certain computer is advertised on the Web at a price of $1099, excluding VAT. If VAT is added at 17.5%, what price will you actually have to pay?
Solution. VAT: value added tax
= IVA
(We have to add the tax to the price)
17.5% of 1099 = 0 .
175

1099

192 .
352 is the tax.
The final price will be: 1099 + 192.325 = $ 1291.325.
The general way to get increases, directly , is: 100 + 17.5 =117.5.
117.5 % of 1099 = 1 .
175

1099

1291 .
325 .
INCREASING A QUANTITY BY A PERCENTAGE
To make an increase of an a % to a quantity:
3. Calculate b = 100 + a.
b
4. Multiply the quantity by
100
. (this fraction is greater than 1)
Example
Increase 40 € by a 15%
We are going to pay 100+15 =115%
We will pay: 1.15 · 40 = 46€

Anexo II
PROPORTIONALITY WORKSHEET
DIRECT PROPORTIONALITY
1. If two pencils cost $1.50, how many pencils can you buy with
2.
$9.00?
Jane ran 100 meters in 15 seconds. How long did she take to
3. run 1 meter? if 4/7 of a tank can be filled in 2 minutes, how many minutes will it take to fill the whole tank?
4. A car travels 125 miles in 3 hours. How far would it travel in 5 hours?
INVERSE PROPORTIONALITY
5. It takes 4 men 6 hours to repair a road. How long will it take 7
6. men to do the job if they work at the same rate?
3 workers build a wall in 12 hours. How long would it have
7.
8. taken for 6 equally productive workers?
A farmer has enough grain to feed 50 cattle for 10 days. He sells 10 cattle. For how many days will the grain last now?
In an army camp, there is food for 8 weeks for 1200 people.
After 3 weeks, 300 more soldiers joined the camp. For how many more weeks will the food last?
The groceries in a home of 4 members are enough for 30 9. days. If a guest comes and stays with them, how many days will the groceries last?
10. At the rate of 28 lines per page, a book has 300 pages. If the book has to contain only 280 pages, how many lines should a page contain?
11. 24 people can construct a house in 15 days. But the owner would like to finish the work in 12 days. How many workers should he employ?
PERCENTAGES
12. Solve: a. b. c.
87% of 41 is what?
What percent of 137 is 86?
What percent of 126 is 22? d. e. f.
17% of what is 156?
46 is what percent of 107?
120% of 118 is what?
What percent of 88.6 is 70? g. h. a.
9 is what percent of 84?
13. Find each percent change.
From 71 to 22. b. From 84 to 4.

c. d. e.
From $109 to $98.
From 33 to 47.
From 58 to 53.
From 79 to 94. f. g. h.
From 55 grams to 70 grams.
From 43 minutes to 28 minutes. i. From 54 m to 154 m.
MARKUP, DISCOUNT, AND TAX
14. Find the selling price of each item. a. b. a. b.
Cost of a comic book: $3.95 Markup: 20%.
Cost of a CD: $14.50 Markup: 30%. c. Cost of a computer: $1,850.00. Markup: 7%.
15. Find the selling price of each item.
Original price of concert tickets: $100.00. Discount: 21%
Original price of a book: $18.50. Discount: 45% c. Original price of a CD: $22.95. Discount: 10%
16. Find the selling price of each item. a. b. c.
Original price of a book: $49.95. Tax: 3%.
Original price of shorts: $19.99. Tax: 2%
Original price of an SUV: $42,000.00. Tax: 3%

Anexo III
PROPORTIONALITY WORKSHEET
DIRECT PROPORTIONALITY
1. If two pencils cost $1.50, how many pencils can you buy with
$9.00?
Pencils $
2-----------1.5 x------------ 9
2 x

1 .
5
9
, x

2

9
, x= 12 pencils
1 .
5
2. Jane ran 100 meters in 15 seconds. How long did she take to run 1 meter?
Metres seconds
100----------15
1 ---------- x
15 x

100
1
, x

15

1
100 x= 0.15 seconds
3. If 4/7 of a tank can be filled in 2 minutes, how many minutes will it take to fill the whole tank?
4. A car travels 125 miles in 3 hours. How far would it travel in 5 hours?
Miles hours Direct proportionality
125 -----------3
x ---------- 5
125 x

3
5
, x

125

5
3 x= 208 miles

INVERSE PROPORTIONALITY
5. It takes 4 men 6 hours to repair a road. How long will it take 7 men to do the job if they work at the same rate?
Men(+) hours(-)
4 ----------6
INVERSE PROPORTION
7 ---------- x
6

7
4
, x

6

4
SWAP : INTERCAMBIAR, dar la vuelta.
, x= 24/7=3.42 hours < 3.5 hours
7 x
0.42 hours is not 42 minutes. 3.42 hours is not 3h 42 min
1 hour -------60 minutes
3.42 h ------- x x=3.42·60/1 = 205.2 minutes
205.2 -180 =25.2
Final solution: 3 hours 25.2 min
6. 3 workers build a wall in 12 hours. How long would it have taken for 6 equally productive workers? workers(+) hours(-) INVERSE PROPORTION
3 ---------- 12
6 ---------- x
12 x

6
3 x= 6 hours
7. A farmer has enough grain to feed 50 cattle for 10 days. He sells 10 cattle. For how many days will the grain last now?
8. In an army camp, there is food for 8 weeks for 1200 people.
After 3 weeks, 300 more soldiers joined the camp. For how many more weeks will the food last?
Weeks (-) people (+) supplies: provisiones
5 ---------------- 1200 +300 people join the camp x ----------------- 1500
5 x

1500
1200
,
5 x

15
12
,
5 x

5
4
, x= 4 weeks

After the first 3 weeks with 1200 soldiers, the food will last for 4 more weeks with 1500 soldiers. 3 - 4= 7 weeks in total.
9. The groceries in a home of 4 members are enough for 30 days. If a guest comes and stays with them, how many days will the groceries last?
10. At the rate of 28 lines per page, a book has 300 pages. If the book has to contain only 280 pages, how many lines should a page contain?
11. 24 people can construct a house in 15 days. But the owner would like to finish the work in 12 days. How many workers should he employ?

PERCENTAGES
12. Solve: a. b.
86?
87% of 41 is what?
What percent of 137 is c.
22?
What percent of 126 is d. 17% of what is 156?
13. Find each percent change.
a. b. c.
From 71 to 22.
From 84 to 4.
From $109 to $98. final

98
109
=0.8990= initial
%
100
98 is the 89.9% of 109
Sol: 100 – 89.9= 10.1% reduction
Another way:
Difference in dollars:
109 – 98 =$11 is the discount,
11
109
10.09%
%
=
100
=0.1009, sol: d. e.
From 33 to 47.
From 58 to 53.
53
58
=0.9137-----91.37% we pay
Discount: 100 -91.37= 8.63% f. From 79 to 94. e.
107?
46 is what percent of f. g.
70?
120% of 118 is what?
What percent of 88.6 is h. 9 is what percent of 84? g. From 55 grams to 70 grams.
55
= 0.7857----78.57%
70
100 - 78.57= 21.43%
Another way: 70-55=15
15 ------ 70 x-------- 100 x= 21.43% h. From 43 minutes to 28 minutes.
28
=0.6511-----65.11%
43
Reduction: 100- 65.11=
34.89%
1- 0.6511=0.3489
28 ------43 razón en términos absolutos
0.65 --- 1 tanto por uno
65 ------ 100 tanto por ciento i. From 54 m to 154 m .
154

2 .
8518 =285.18%
54
100→285.18, increase:
185.18%

MARKUP, DISCOUNT, AND TAX
14. Find the selling price of each item. a. Cost of a comic book: $3.95 Markup: 20%.
3.95·1.2=….. b. Cost of a CD: $14.50 Markup: 30%.
130% of 14.5= 14.5· 130/100=14.5 · 1.3
=$18.85 final price c. Cost of a computer: $1,850.00. Markup: 7%. (Tax or VAT)
1850· 107/100=1850 · 1.07
= $1979.5
15. Find the selling price of each item. a. Original price of concert tickets: $100.00. Discount: 21%
Solution:
After the discount: 100% - 21% = 79% is what we are going to pay.
Then, the final price is:
$100 · 0.79 = $ 79 b. Original price of a book: $18.50. Discount: 45%
Solution
100 – 45 = 55 % we are going to pay.
55% of 18.5= 0.55 · 18.5 = 10.175,
Rounding to the nearest cent:
Final price: $10.18 c. Original price of a CD: $22.95. Discount: 10%
Solution
How much will we pay in percent? 100 – 10 = 90% = 90/100
= 0.9
90% of 22.95= 0.9 · 22.95 = 20.655.
$20.66
16. Find the selling price of each item. a. Original price of a book: $49.95. Tax: 3%.
49.95· 1.03 = 51.44
Find the tax:
49.95--------- 100
x-------- 3 x=$ 1.4985 (tax)
49.95 + 1.4985 =$51.44 b. Original price of shorts: $19.99. Tax: 2%

19.99 · 1.02 = c. Original price of an SUV: $42,000.00. Tax: 3%
42000 · 1.03 =
Textbook.97.
Atletismo: 25% de 1200= 0.25·1200=300 alumnos
Baloncesto : 15% de 1200 =0.15 · 1200=180 alumnos
Fútbol: 40% de 1200 = 0.4· 1200= 480 alumnos
Nada: 1200
– (300+180+480)= 240 alumnos
Nada: 25+15+40 =80% de 1200= 960 alumnos.
1200- 960 =240 alumnos
.

ANEXO IV. CONTROL
PROPORTIONALITY
Calculator is allowed. For every problem, what kind of proportionality do you use?
1. Suppose it takes 48 chicken fingers to feed a 4th grade class of 20 students. How many chicken fingers would be needed for 30 students?
2. It takes 5 men 8 hours to repair a road. How long will it take
4 men to do the job if they work at the same rate?
3. The groceries in a home of 5 members are enough for 30 days. If a guest comes and stays with them, how many days will the groceries last?
4. Jane ran 100 metres in 15 seconds. How long did she take to run 1 metre?
5. A wheel makes 1200 revolutions in 15 minutes. How many revolutions will it make in 43 minutes?
6. A tab that delivers 15 litres per minute fills a deposit in 20 minutes. A second tab delivers 25 litres per minute, a) Using only the second tab, how long will it take to fill the same deposit? b) Using both tabs together, how long will it take to fill it?
7. a) 87% of 41 is what? b) What percent of 126 is 22? c) 120% of 118 is what?
8. a) An item has cost 352€ to a shop owner. Mark it up so that there is a 12% benefit (get the selling price). b) A book was 28.50€, and it was sold for only 23.50€, what is the percent discount?
9. After a 15% discount, an appliance has cost 126€, what was its initial price?
Appliance: electrodoméstico.
10. Find the percent change from 58 to 53 (what percent
58 is decreased to get 53).