Unit 2. Day 10.

Please get out paper for today’s lesson Name Date Period -------------------------------------------------------- Topic: Irrational numbers 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2)

Real Number System Real Irrational π, e, 𝟐 , 𝟕 , … Imaginary Rational -3.24, 2/3, 6.71, -5/2, … Irrational Integers -1, -2, -3, -4, … π, e, 𝟐 , 𝟕 , … Whole Natural Imaginary 1, 2, 3, 4, … −𝟐 , −𝟕 , −𝟏𝟑 , …

Imaginary Real Number System 1 2 9 2 − 81 −10 −2.7 10 16 𝜋 0. 6 7 5 𝑒 Rational Irrational

We have spent the past nine lessons working with RATIONAL numbers We have spent the past nine lessons working with RATIONAL numbers. We’ve converted, compared, ordered, added, subtracted, multiplied, and divided RATIONAL numbers. But there are also IRRATIONAL numbers! We want to understand IRRATIONAL numbers.

𝑊𝑒 𝑚𝑢𝑠𝑡 𝑓𝑖𝑟𝑠𝑡 𝑟𝑒𝑣𝑖𝑒𝑤 𝑠𝑜𝑙𝑣𝑖𝑛𝑔 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑊𝑒 𝑗𝑢𝑠𝑡 𝑡𝑎𝑙𝑘𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑖𝑠! 8.NS.A.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. We’ve done this 3 4 0.75000000… 0.75 2 3 We will now learn this 0.6666666666… 0.6666666666… 1.414213562… 𝑊𝑒 𝑚𝑢𝑠𝑡 𝑓𝑖𝑟𝑠𝑡 𝑟𝑒𝑣𝑖𝑒𝑤 𝑠𝑜𝑙𝑣𝑖𝑛𝑔 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑊𝑒 𝑗𝑢𝑠𝑡 𝑡𝑎𝑙𝑘𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑖𝑠!

Solving Equations Review 6th Grade Solving Equations Review

𝑥+4 = 7 𝑥+4 = 7 𝑥 = 3 𝑥 𝑥+4 = 7 = + −4 −4 Example A: 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥. + + Check: +4=7 3 𝑥+4 = 7 𝑥+4 = 7 𝑥 = + 3 7=7 −4 −4 𝑥 𝑥+4 = 7 = + + + + + + + + + + + − − − − − − − −

𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑦+2 =8 Example B*: 𝑚+4 =−1 Example C*:

𝑦 𝑦+2 =8 𝑦+2 =8 = 6 𝑦 𝑦+2 =8 = + Example B*: 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦. −2 −2 + + + Check: +2=8 6 𝑦 𝑦+2 =8 𝑦+2 =8 = + 6 8=8 −2 −2 𝑦 𝑦+2 =8 = + + + + + + + + + + − − − −

𝑚 𝑚+4 =−1 𝑚+4 =−1 = −5 𝑚+4 =−1 𝑚 = + −4 −4 Example C*: 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑚. + Check: +4=−1 −5 𝑚 𝑚+4 =−1 𝑚+4 =−1 = + −5 −1=−1 −4 −4 𝑚+4 =−1 𝑚 = + + + + − − − − − − − − −

Example D: 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 x. 1 3𝑥=15 𝑥 = 5 ∙ 3 3 Check: 3 =15 5 15=15

Example D: 𝑥+3 =15 𝑥 + = 12 1 3𝑥=15 𝑥 = 5 ∙ −3 −3 3 3

𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 Example E*: 4𝑤=24 Example F*: 5𝑒=30

Example E*: 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 w. 1 4𝑤=24 𝑤 = 6 ∙ 4 4 Check 4 =24 6 24=24

Example F*: 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑒 . 5𝑒=30 𝑒 = 6 ∙ 5 5 Check 5 =30 6 30=30

Converting Decimals into Fractions

Example --: Write the decimal as a fraction in simplest form. 0.81 81 100 3∙3∙3∙3 2∙2∙5∙5

Example G: Write the decimal as a fraction in simplest form. 0. 81 111111111111… 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 81 100

Example G: Write the decimal as a fraction in simplest form. 𝑛= 0. 81 100 100 𝑛 = 0.818181818181818181… 100𝑛 = 81.81818181818181818181… . 818181818181818181… 100𝑛 81.81818181818181818181… − − 1𝑛 0.818181818181818181… 99𝑛 81 99𝑛=81 99 99 𝑛= 81 99 = 9∙9 9∙11 9 11 =

Example H: Write the decimal as a fraction in simplest form. 𝑛= 0. 6 10 10 𝑛=0.666666666666666… 10𝑛 = 6.6666666666666666… . 666666666666666… 10𝑛 6.66666666666666666… − − 1𝑛 0.6666666666666666666… 9𝑛 6 9𝑛=6 9 9 𝑛= 6 9 = 2∙3 3∙3 2 3 =

0. 4 0. 45 Write the decimal as a fraction in simplest form. Example I*: 0. 45 Example J*:

Example I*: Write the decimal as a fraction in simplest form. 𝑛= 0. 4 10 10 𝑛=0.444444444444444444… 10𝑛 = 4.4444444444444444… . 444444444444444… 10𝑛 4.4444444444444444444… − − 1𝑛 0.4444444444444444444… 9𝑛 4 9𝑛=4 9 9 𝑛= 4 9 = 2∙2 3∙3 4 9 =

Example J*: Write the decimal as a fraction in simplest form. 𝑛= 0. 45 100 100 𝑛 = 0.454545454545454545… 100𝑛 = 45.454545454545454545… . 45454545454545454545… 100𝑛 45.454545454545454545… − − 1𝑛 0.454545454545454545… 99𝑛 45 99𝑛=45 99 99 𝑛= 45 99 = 3∙3∙5 3∙3∙11 5 11 =

Example K: Write the decimal as a fraction in simplest form. 𝑛= 0.5 3 10 10 𝑛=0.533333333333333333… 10𝑛 = 5.3333333333333333… . 53333333333333333… 4 10𝑛 5.3333333333333333333… 1 − − 1𝑛 0.5333333333333333333… 9𝑛 4 . 8 9𝑛=4.8 9 9 𝑛= 4.8 9 = 48 90 2∙2∙2∙2∙3 2∙3∙3∙5 8 15 = =

Example K: Write the decimal as a fraction in simplest form. 𝑛= 0.5 3 100 100 𝑛 = 0.533333333333333333… 100𝑛 = 53.33333333333333333… . 533333333333333333 2 100𝑛 53.333333333333333333… 1 − − 1𝑛 0.533333333333333333… 99𝑛 5 2 . 8 8 15 = 99𝑛=52.8 99 99 𝑛= 52.8 99 528 990 2∙2∙2∙2∙3∙11 2∙3∙3∙5∙11 = =

Example L: Write the decimal as a fraction in simplest form. 𝑛= 0. 1234 10000 10000 𝑛=0.1234123412341234… 10000𝑛 = 1234.123412341234… . 1234123412341234… 10000𝑛 1234.123412341234… − − 1𝑛 0.123412341234… 9999𝑛 1234 9999𝑛=1234 9999 9999 𝑛= 1234 9999

Example M: Write the decimal as a fraction in simplest form. 𝐿𝑒𝑡: 𝑛= 3.141592653589793… ? ? 𝑛=3.141592653589793…