U3. Proportionality

Introductory activities

Quizizz:

1. What is 30% of 130?

2. A student gets 6 out of 8 in an exam. What mark he deserves out of 10?

3. There is a discount of 21% in a shop. If you buy a t-shirt whose price is 20 € and, after the discount, you have to pay 21% of VAT? How much is the final price, higher, smaller or equal to 20 €?

4. What percentaje of these emojis has a heart? 

Situación de aprendizaje: Elecciones.

Directly proportional magnitudes

Something you should remember:

Ratio (razón): A way to compare two numbers, the division of both numbers. 4:3 "ratio four to three", as in the scale of a map, 1:50000

Proportion (proporción): two equal ratios 4:3 = 1600:1200, usually expressed as fractions.

Directly proportional magnitudes: one rise as the other rise, their corresponding values form a proportion. The ratio is called, in this context, constant of proportionality. 

Examples of different magnitudes (not all are proportional):

• Price in euros and kilograms of oranges.

• Time in minutes to do a task and people doing it.

• Shoe size and height of people in cm.

• Weight of a jar with marbles and number of marbles in it.

Algorithm on solving many word problems on proportionality

As in: Antonio buy 4 workbooks by 6 euros. Paula is going to buy 3 workbooks. How much will they cost? Rubén has 15 euros. How many workbooks can he buy?

1. Use a table with two rows for the magnitudes. 

2. Wonder if the magnitudes are directly proportional or not (double the first goes with double the second).

3. Express the relations in the problem as a proportion. 

4. Solve for x (cross productos must be the same).

Percentages

Increasing and decreasing percentajes as products

Increasing a quantity, for example 23 by 5% means that you will add 5% to the initial 100%. The final result will be 105% of the initial quantity. So you can calculate that amount by multiplying 23 · 105% = 23 · 1.05 = 24.15.

For the same reason, decreasing, let's say, 39 by 12% can be calculated as 88% of 39, since 88% = 100% - 12%. Doing the maths, 39 · 88% = 39 · 0.88 = 34.32

1. Increase each of the following quantities by the given percentage, usign the multiplication method: 

   • 60 € by 4%; 

   • 12 kg by 8%; 

   • 450 g by 5%; 

   • 545 m by 10% 

2. Decrease each of the following quantities by the given percentage, using the multiplication method:

   • 34 € by 12%; 

   • 75 € by 20%; 

   • 340 kg by 15%; 

   • 670 cm by 23%

Interest

Inverse proportionality. Sharings.

Inverse proportion

How to solve inverse proportion problems (remember: a · b = a' · b')

Sharings 

Proportions with three variables

Three cats hunt three mice in three hours. How many cats will hunt a hundred mice in a hundred hours?

Word problems

1. A car travels 100 km with 4.5 litres of fuel. How far will it travel with 7 litres?

2. It take 8 students 9 hours to make a class work. How long would it take 6 students to make the same work?

3. It takes 12 people 15 days to harvest a crop of raspberries. How long would it take 18 people?

4. £1 is worth 1.21 €. How many euros is £9 worth?

5. A motorist drives at a steady speed and goes 232 km in 3 hours:

   1. What distance will she cover in 5 hours?

   2. How long will it take her to cover 193 km?

6. 5 litres of paint covers an area of 60 m². 

   1. What area will 15 litres of paint cover?

   2. How much paint is needed to cover an area of 240 m²?

7. A field provides grazing for 18 sheep for 8 days. How many days grazing would it provide for 24 sheep?

8. It takes an hour to mow a lawn using a mower with blades 14 inches wide. How long would it take using a mower with blades 12 inches wide?

9. 3 cm³ of aluminium weigh 8.1 g. Work out the weight of 10 cm³.

Criterios de evaluación y saberes básicos implicados en la unidad

Criterios de Evaluación

MAT.1.1.1.Iniciarse en la interpretación de problemas matemáticos sencillos, reconociendo los datos dados, estableciendo, de manera básica, las relaciones entre ellos y comprendiendo las preguntas formuladas.

MAT.1.1.2. Aplicar, en problemas de contextos cercanos de la vida cotidiana, herramientas y estrategias apropiadas, como pueden ser la descomposición en problemas más sencillos, el tanteo, el ensayo y error o la búsqueda de patrones, que contribuyan a la  resolución de problemas de su entorno más cercano.

MAT.1.1.3.Obtener las soluciones matemáticas en problemas de contextos cercanos de la vida cotidiana, activando los conocimientos necesarios, aceptando el error como parte del proceso.

MAT.1.2.1. Comprobar, de forma razonada la corrección de las soluciones de un problema, usando herramientas digitales como calculadoras, hojas de cálculo o programas específicos.

MAT.1.2.2. Comprobar, mediante la lectura comprensiva, la validez de las soluciones obtenidas en un problema comprobando su coherencia en el contexto planteado y evaluando el alcance y repercusión de estas soluciones desde diferentes perspectivas: igualdad de género, sostenibilidad, consumo responsable, equidad o no discriminación.

Saberes Básicos

MAT.3.A.3.1, MAT.3.B.1.2, MAT.3.D.4.2, MAT.3.E.2.3, MAT.3.A.1.1, MAT.3.A.4.4, MAT.3.D.6.2, MAT.3.D.6.3, MAT.3.C.4.1, MAT.3.D.1.1, MAT.3.D.2.1, MAT.3.A.2.4, MAT.3.A.4.2, MAT.3.E.1.2, MAT.3.E.1.3

Preguntas del examen

Problema 1. Problema de proporcionalidad directa.

Problema 2. Problema de repartos.

Problema 3. Problema de porcentajes.

Problema 4. Problema de interés compuesto.

Problema 5. Problema de proporcionalidad inversa.