Download What is an Irrational Number? π ≈ 3.1415926535 .... 1. Use your

Wh at i s an Irrati o n al Nu mb er?
π ≈ 3. 1415926535 . . . .
1. Use your calculator to find a fraction equal to π, to the nearest....tenth, thousandth, ten-­‐thousandth.
You should have had some difficulty in finding a fraction equal to pi (π) to the nearest ten-­‐thousandth. Here is pi expressed to a thousand places after the decimal point: 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513823066470938446095
5058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847568233786783165271201909145648566923
4603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519
4151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308
6021394946395224737190702179860943702770539217172931767523846748184676694051320005681271452635608277857713427577896091736371787214
6844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105
9731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428
755468731159562863882353787593751957781857780532 171226806613001927876611195909216420198 .... (this is rounded to the nearest tenth power of a googolth)
Imaging finding a fration equal to this!
T a r g e t : W ha t is a n Ir r a t iona l N um be r a nd how c a n you t e ll w he n a num be r is ir r a t iona l or not ?
Informa;on you need
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•
•
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The integers are the whole numbers and their opposites. The absolute value of a number is the distance that number is from zero. (Ex.: +3 and –3 have the same absolute value of 3).
A rational number is any number that can re-­‐written as a fraction using integers. There are irrational numbers. Some of them are pi, srt 2, etc.
2. Do these in the space provided
✴On a number line, place a dot showing the loca[on of the eight integers having lowest absolute value.
✴On another number line, place a dot showing the smallest integer.
✴Circle which of these you think are ra[onal numbers? Be prepared to discuss your reasons. 0.3 ½ 3 ½ –4 –1.2 ⅓ 0.333... ∏
3. Summarize
Write down how you think is the best way to tell if a number is rational or not.
Rational & Irrationa Numbers (8NS1, 8NS2) !
Lesson 1
To develop a beBer u nderstanding of ra;onal numbers, you should be preBy good at division. The two main methods for division by hand are long and short division. You should know both. You will see that short division is a much easier way to divide by hand when the divisor is easy to think with.
4. Do these on notebook paper
✴With your teacher, divide these quotients twice, once using long division and then again using short division: 5/8, 52/8 , 525/82
✴Use short division to find decimals for the halves, thirds, fourths, fifths, sixths, sevenths, eights, ninths, tenths, elevenths. Do this only for those fractions less than one.
✴Be prepard to talk with your class about any new ideas you have about how you think is the best way to tell if a number is rational or not
5. Prac;ce
✴Draw one number line, scale it by ones from –5 to +5. Place each of these values on the number line by drawing a dot at the approximate locations for the values and also labeling the dot with the number value. Circle any integer. Box any rational number. Triangle any irrational number. (Hint: If you’re not sure about a number, use your calculator to look at it’s decimal expansion. You may need help locating the correct key for 2 etc.) –3⅓ 2.9 120% 0.1200 –33⅓% 2 – 3 ∏
∏/4
✴Write about what you think is the main difference between rational and irrational numbers and how you can always determine if a number is rational.
✴Study the approximation for pi (above) for any patterns. Be ready to share what you have found.
Rational & Irrationa Numbers (8NS1, 8NS2) !
Lesson 1
Quiz
Explain the main difference between rational and irrational numbers and how you can always determine if a number is rational.
If you were asked to prove that pi wasn’t rational, how would you go about doing that?
Draw one number line, scale it by ones from –5 to +5. Place each of these values on the number line by drawing a dot at the approximate locations for the values and also labeling the dot with the number value. Circle any integer. Box any rational number. Triangle any irrational number. (Hint: If you’re not sure about a number, use your calculator to look at it’s decimal expansion. You may need help locating the correct key for 2 etc.) 0.11 –80% ∏
–0.1 266⅔% 2 ∏/2
–⅓
Explain why each given example is either a good one or poor one for each statement.
Irrational numbers have decimal expansions that never end and never repeat with a regular pattern. Example: 2/3
Rational numbers have decimal expansions that either terminate or repeat with a regular pattern. Example: ∏
Any number can be located on a number line by approximating. Example: ∏
Rational & Irrationa Numbers (8NS1, 8NS2) !
Lesson 1
T a r g e t : W ha t is a n Ir r a t iona l N um be r and how c an you te ll whe n a num be r is ir r ational or not?
Basic Ideas
• The integers are the whole numbers and their opposites. • The absolute value of a number is the distance that number is from zero. (Ex.: +3 and –3 have the same absolute value of 3).
• A rational number is any number that can re-­‐written as a fraction using integers. • There are irrational numbers. Some of them are pi, srt 2, etc.
• Short division
Conclusions
• Irrational numbers have decimal expansions that never end and never repeat with a regular pattern. • Rational numbers have decimal expansions that either terminate or repeat with a regular pattern.
• Any number can be located on a number line by approximating.
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Activities Checklist
1. Exploring fractions that approximate pi to the nearest ten-­‐thousandth
2. Placing integers, rational numbers and irrational numbers on a number line
3. Summary
4. Division to find decimal expansions for common fractions
5. Practice
_____ This grade is based upon the quiz. It is for showing progress toward meeting the overall standards, 8.NS.1&2 (see below) which will be evaluated later in the term.
8.NS.1 Know that numbers that are not ra[onal are called irra[onal. Understand informally that every number has a decimal expansion; for ra[onal numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a ra[onal number. 8.NS.2 Use ra[onal approxima[ons of irra[onal numbers to compare the size of irra[onal numbers, locate them approximately on a number line diagram, and es[mate the value of expressions (e.g., 2). For example, by trunca[ng the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to con[nue on to get bener approxima[ons.
Rational & Irrationa Numbers (8NS1, 8NS2) !
Lesson 1