2. Fractions and decimals.

Fracciones portada

Which is larger, the pink or the blue area?

What fraction of the square is coloured in green?

Safari fotográfico

Busca fracciones fuera del instituto y hazles una foto. Publícala en tu carpeta de Classdojo. Al final de esta página podrás ver el álbum de fotos que hemos recogido.

Problema

Contenidos

  1. Fracciones equivalentes. / Equivalent fractions.
  2. Comparación y ordenación de fracciones. Suma y resta de fracciones. Multiplicación y división. Op. Combinadas. Potencias (exponentes negativos) / Comparing fractions. Addition and subtraction. Multiplication and division. Mixed operations. Powers (negative exponents).
  3. Notación científica. / Scientific notation.
  4. Formas decimal y fraccionaria de un número / Decimal and fractional expression of a number.
  5. Aproximaciones de un número decimal. Truncamiento y redondeo. / Approximating a decimal number. Rounding and truncation.
  6. Cálculo mental, cálculo escrito y calculadoras. / Mental calculation, written calculation and calculators.

Get equivalent fractions is useful to compare them. For example, to know who's bigger, 3/5 or 2/3, we can find equivalent fractions with the same denominator. That denominator can be the LCM of the denominators or any common multiple of them:

3/5 and 2/3 are equivalents to 9/15 and 10/15 respectively. As 10/15 > 9/15, 2/3 > 3/5.

Later we will use the same procedure of getting a common denominator to add or subtract fractions.

  1. Find equivalent fractions to the following: 12/30, 75/60, 20/18. (*) One with bigger numbers and another one with smaller numbers than the given ones.
    1. Which two of these fractions are equivalent to 1/4? 2/8, 5/16, 6/24, 11/40.
    2. Cancel each of these fractions to their lowest terms: 15/35, 24/54, 50/200, 36/48, 14/16, 7/21, 10/20, 42/60, 4/16, 18/27, 18/72, 84/140, 63/84, 35/90.
  2. Fill the gaps: 14/16 = ◊/24, 15/35 = 6/◊.
  3. Which of these pairs of fractions are equivalent?
  • 7/9, 28/18
  • 4/5, 2/3
  • 8/9, 16/27
  • 8/12, 2/3

4. Put the following fractions in order with the smallest first:

  • 17/24, 15/24
  • 9/20, 8/15
  • 5/6, 2/3, 3/4
  • 3/4, 1/2, 5/8
  • 1/3, 2/5, 3/10
  • 7/4, 9/8, 5/12, 11/18
  • 3/35, 6/25, 2/50, 5/6
  • 2/21, −3/14, 5/6, 7/12

5. Halla la fracción irreducible de 520/156 y 45/75


Operations with fractions

🔄Flipped classroom

Review with the help of the following links addition, subtraction, multiplication and division of fractions. You can find also combined operations and have to apply BEDMAS rule.

  1. Calculate each of the following:
    • 1/4 + 2/4 =
    • 2/9 + 5/9 =
    • 4/7 + 1/7 =
    • 7/8 - 4/8 =
    • 6/7 + 8/7 + 3/7 =
    • 4/9 + 7/9 - 8/9 =
    • 5/8 - 7/8 =
  2. Calculate each of the following:
    • 1/4 + 1/2 =
    • 3/4 - 1/2 =
    • 1/2 + 1/10 =
    • 3/5 + 2/10 =
    • 1/3 + 1/8 =
    • 5/8 + 1/6 =
    • 7/10 + 1/4 =
    • 2 − 1/3 =
    • 3/21 + 5/9 =
    • 4 − 28/5 =
    • 2/5 + 7/12 - 1/6 =
    • 8/9 − 1/2 + 3 =
    • 3 − 1/8 + 7/2 - 5/4 =
  3. Calculate and reduce the result to its lowest terms:
    • 1/3 · 9/4 =
    • 2/3 · 1/2 =
    • 1/4 · 2/7 =
    • −1/2 · 3/5 =
    • −1/5 · −1/5 =
    • 2/9 : 1/3 =
    • 1/5 : −2/5 =
    • 3/2 : 1/6 =
    • 3/4 · −5/9 =
  4. Copy, calculate and reduce:
    • 1/4 + 3/2 · 2/3 =
    • 5/6 · 4/15 − 3/5 · 20/18 =
    • 3/8 : 18/24 − 5/6 =
    • (3/5 + 1/10) : −14/15 =
    • −4/5 · (7/3 − 5/4) =
    • (1/2 − 3/4) : 5/6 =
    • 3/4 · (−5/9 + 2/3)² =
    • 1/2 + (−3/2)³ : 1/4 =
    • 5/4 , −4/9, −1/3 , −3/4, −13/15, −3/10
  5. Operations with fractions contest: make up a big operation with fractions whose result is, finally, 1. Rules:
    1. It must contain every operation (BEDMAS).
    2. You can repeat only twice every number. Notice that number is not the same than digits.
    3. Prices: a positive for the best three operations.

Powers

They are the same than with integers, for example (2/3)³ = (2/3) · (2/3) · (2/3) = (2·2·2)/(3·3·3) = 2³/3³ = 8/27.

Powers of negative exponents

Scientific notation

Planetary Suite by Steve Gildea

Foto: Planetary Suite by Steve Gildea

El peso en kg de los planetas de nuestro sistema solar es el siguiente: Mercurio (330200000000000000000000 kg), Venus (4869000000000000000000000 kg), Tierra (5974000000000000000000000 kg), Marte (641910000000000000000000 kg), Júpiter (1899000000000000000000000000 kg), Saturno (568800000000000000000000000 kg), Urano (86860000000000000000000000 kg) y Neptuno (102400000000000000000000000 kg). Ordena los planetas desde el más ligero al más pesado.

El peso de los planetas en kg expresado en notación científica es: Mercurio 3.302×10^23; Venus 4.869 × 10^24; Tierra 5.9736×10^24; Marte 6.4185 × 10^23; Júpiter 1.899×10^27; Saturno 5.688·10^26; Urano 8.686×10^25; Neptuno 1.024×10^26; Por lo tanto, ordenados, de menor a mayor, quedan así:

Mercurio < Marte < Venus < Tierra < Urano < Neptuno < Saturno < Júpiter

  1. Express the following quantities in scientific notation:
    1. Average distance from Earth to Sun: 150 000 000 Km.
    2. Weighest animal in the world (blue whale): 190 000 000 g
    3. Possible scrambles of a Rubik's cube: 43 252 003 274 489 856 000
    4. One trillion, one quatrillion, one quintillion.
    5. Mass of the Earth: 5974000000000000000000000 kg
    6. Possible combinations in Bonoloto: 13 983 816
  2. Expresa en notación decimal las siguientes cantidades:
    1. Número de células nerviosas del ser humano: 1.3·10¹⁰
    2. Número de respiraciones en el transcurso de una vida media: 5·10⁸
    3. Cantidad de moléculas en una cucharada de agua (6 cl):2,00738043×10²³
    4. Distancia de la Tierra a la estrella más cercana fuera del Sistema Solar: 4.13 × 10¹³ km.
    5. La edad, en días, de la Tierra:
  3. Write the following numbers in scientific notation:
    1. 0.13592
    2. -0.0038
    3. 0.00000013
    4. -0.567
    5. 0.00361
  4. Write in decimal notation:
    1. 1.03 x 10⁻²
    2. 8.862 x 10⁻¹
    3. 9.512 x 10⁻⁸
    4. -6.5 x 10⁻³
  5. What is bigger, the mass of the Sun or the mass of all the planets together?
  6. Express the following small numbers in scientific notation:
    1. Mass of a mosquito: 0.01 g
    2. Mass of the proton: 0.000 000 000 000 000 000 000 000 001 673 g
    3. Bohr radius: 0.000 000 000 053 m
    4. One Angstrom: 0.000 000 000 1 m
  7. Sort from lowest to highest: 2.7 · 10¹², 1.09 · 10⁶, 2.34 · 10⁸, 3.5 · 10⁶
  8. Light travels at c=3 · 10⁸ metres per second. How fast in this in km per hour? How many times faster is this than a car?
  9. Escribe el producto interior bruto de los 10 países más ricos del mundo en notación científica y ordénalos de mayor a menor. ¿Cuántas veces es superior el primero que el de España? ¿Cuantas veces es superior el de España que el de uno de los países más pobres?

Scientific notation in two calculators, where E+19 or e19 means ·10¹⁹

Decimal and fractional expressions of a number

Place value: the number 57.483 means 5 tens + 7 units + 4 tenths + 8 hundredths + 3 thousandths, that is, 5 · 10 + 7 · 1 + 4 · 1/10 + 8 · 1/100 + 3 · 1/1000 = 57 483/1 000

You can also convert a fraction into a decimal by dividing its terms, for example, 5/4 = 1.25 because 5 ÷ 4 = 1.25.

  1. Change each of these decimals into fractions, cancelling where possible: 0.7, 0.4, 0.5, 0.03, 0.06, 0.13, 0.25, 0.38, 0.55, 0.64.
  2. Change each of these fractions to decimals. Where necessary, give your answer correct to three decimal places: 1/2, 3/4, 3/5, 9/10, 1/3, 5/8, 2/3, 7/20, 7/11, 4/9.
  3. Put each of the following sets of numbers in order, with the smallest first.
    1. 0.6, 0.3, 1/2
    2. 2/5, 0.8, 0.3
    3. 0.35, 1/4, 0.15
    4. 7/10, 0.72, 0.71
    5. 0.8, 3/4, 0.7
    6. 0.08, 0.1, 1/20
    7. 0.55, 1/2, 0.4
    8. 5/4, 1.2, 1.23

There are some special numbers that cannot be expressed as fractions. Some examples are:

π (pi number) = 3.141592653589793238462643383279502884197169399375105820974...

√2 = 1.414213562373095048801688724209698078569671875376948073176...

Special non-periodical decimals: 1.2345678910111213141516171819202122...

They are called irrational numbers (números irracionales).

Approximating a decimal number: rounding and truncation

Have a look a this video to see how to truncate and round a number.

  1. Round the following numbers to the nearest unit:1.5, 0.72, 73.5, 91, 37.2, 746.1, 36.47, 69.65, 17, 7.56, 2.67, 73.27, 80.35, 9.56, 9.59, 9.50, 77.44, 64.38, 2.31
  2. Round the following numbers to the nearest tenth: 7.28, 13.94, 6.01, 27.60, 8.48, 23.87, 4.08, 6.74, 24.65, 43.37, 5.38, 2.48, 7.91, 7.79, 44.51, 2.35, 89.56, 6.77, 4.37, 3.91.
  3. Truncate these numbers to two digits: 13.5, 1.99, 0.777, 1.328, 0.1731, 87.1872, 196.5, 23.48, 456.2513, 7.2577, 4675.2579, 3.258, 396.1, 42.49, 0.5661, 99.6600, 7.105, 861.1, 8775, 9654.591
  4. Round:
    1. 43,507 to the hundredths;
    2. 5678,3215 to the thousandths;
    3. 39,85 to the units;
    4. 99,994 to the hundredths.

En la vida cotidiana usamos estos procedimientos según nos convenga. El redondeo es más preciso, pero exige un pequeño esfuerzo. El truncamiento es más rápido y suficiente como aproximación en muchas ocasiones. Las tiendas saben que somos perezosos y solemos truncar incluso, inconscientemente y aprovechan esa circunstancia para la publicidad.

5. Observa la siguiente foto de un catálogo de unos conocidos grandes almacenes. ¿Cuánto valen el patinete y la diana juntos, si redondeamos a las decenas? ¿Cuánto parece que valen si truncamos en el mismo sitio?

Approximations of calculations

Significant figures: those digits that carry meaning contributing to the precision of a number. Look how these numbers are rounded to one or two significant figures:

One SF:

33 → 30; 0.42 → 0.40; 5 470 → 5 000; 89.45 → 90.00;

998 → 1000; 1.89 → 2; 7.837 → 8.000; 0.000 056 → 0.000 060

Two SF:

123 → 120; 5.67008 → 5.70000; 34 699 → 35 000; 0.005 → 0.005

  1. Round each of these numbers to 1 significant figure: 46 313, 57 123, 30 569, 94 558, 85 299, 54.26, 85.18, 27.09, 96.432, 167.77, 0.5388, 0.2823, 0.005 84, 0.047 85, 0.000 876, 9.9, 89.5, 90.78, 199, 999.99
  2. Round the numbers of the last exercise 3 to two significant figures.
  3. Make a mental calculation of the following operations by rounding first to one significant figures. Use your calculator to check your answers:
    1. 5435 × 7.31
    2. 5280 × 3.211
    3. 63.24 × 3.514 × 4.2
    4. 3508 × 2.79
    5. 72.1 × 3.225 × 5.23
    6. 470 × 7.85 × 0.99
    7. 354 ÷ 49.8
    8. 36.8 ÷ 1.876
    9. 5974 ÷ 5.29
    10. 406.9 ÷ 0.783
Imagen de http://imgur.com/gallery/GeK1gAP

Word problems

  1. De un queso que pesa 1800 g nos hemos comido 7/9. ¿Cuántos gramos de queso nos hemos comido?

  3. Observa en el siguiente vídeo cómo se fabrica un compuesto homeopático.
  4. Sabiendo que ...

Links

Neper

John Neper, quien popularizó el uso del punto decimal en el siglo XVI, aunque se dice que fue inventado por Pitiscus.

Activities

Before starting the new unit, lets play a Math bingo.

Equivalent fractions. Sorting fractions

Who's bigger, 3/5 or 2/3?

Two fractions are equivalent if they have the same value, but with different expressions. For example, 1/2 and 2/4 are equivalent.

To quickly check if two fractions are equivalent, we can do crossing multiplication:


What fraction of the star on the right is shaded yellow?