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.

Exercise: Types of Proportionality

For each pair of quantities below, decide whether they are: directly proportional, Inversely proportional, or not proportional. Explain your choice briefly in each case.

A. The number of hours you work — and the amount of money you earn (at a fixed hourly rate).

B. The speed of a car — and the time it takes to travel a fixed distance.

C. The number of slices you cut a pizza into — and the size of each slice.

D. The number of students in a class — and the average grade of the class.

E. The length of one side of a square — and its area.

F. The distance you walk — and the time you spend walking (at a constant speed).

G. The price of one notebook — and the total price of 5 identical notebooks.

H. The temperature in °C — and the temperature in °F.

Can you invent two more pairs of quantities of every type?

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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 products must be the same).

Solve the following easy problems:

1. If 4 hours of work earn you €36, how much will you earn if you work 7 hours at the same rate?

2. A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?

3. 5 notebooks cost €12.50. How much will 8 notebooks cost?

4. To make pancakes for 4 people, you need 200 g of flour. How much flour is needed for 10 people?

5. A car uses 8 liters of petrol to travel 120 km. How many liters are needed for 300 km?

6. Downloading 15 songs takes 6 minutes. How long will it take to download 40 songs at the same rate?

7. A painter covers 12 m² in 30 minutes. How many square meters can he paint in 2 hours?

8. A phone company charges €9 for 3 GB of data. How much would 10 GB cost?

9. If you save €45 in 3 weeks, how much will you save in 8 weeks at the same rate?

10. A printer produces 120 pages in 8 minutes. How many pages will it print in 15 minutes?

Solutions (there is one wrong): €63, 300 km, €20, 500 g, 20 L, 16 min, 48 m², €30, €130, 225 pages

NGO Challenge: Math for reconstruction

In this gymkhana, your team will help an NGO working in Palestine. Each challenge represents a real problem they face during reconstruction. Two are traps! Solve carefully to unlock the final challenge and earn your sticker.

Inverse proportion word problems

a) Set the magnitudes in a table, x, y.

b) Identify that the relationship is inverse (one goes up, the other goes down).

c) Use the constant product rule: x1· y1 = x2 · y2

Substitute the known values and find the missing one.

Solve the following easy problems:

1. Six workers can build a wall in 10 days. How long will it take 15 workers to build the same wall?

2. A car takes 5 hours to travel a certain distance at 80 km/h. How long would it take if it goes at 100 km/h?

3. Four machines can fill 200 bottles in 3 hours. How long would it take six machines to fill the same number of bottles?

4. 12 students can clean the school yard in 45 minutes. How long will it take 9 students to clean it?

5. One pipe can fill a swimming pool in 6 hours. How long will it take three pipes working together?

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.