Unit 1: Real Numbers Student Tracking Sheet Math 10 Common Name: Block: What I can do for this unit: After Practice After Review How I Did 1-1 I can sort a set of numbers into irrationals and rationals, including their subsets (natural, whole, integer, rational, irrational), convert rational numbers, and determine an approximate value of a given irrational number. 1-2 I can determine the GCF or LCM of a set of numbers and explain why the numbers 0 and 1 have no prime factors. 1-3 I can identify perfect squares and cubes and evaluate using factoring techniques. 1-4 I can convert between mixed radicals and entire radicals and express the meaning of the index of a radical. 1-5 I can solve word problems involving Real Numbers. Code Value Description N Not Yet Meeting Expectations I just don t get it. MM Minimally Meeting Expectations Barely got it, I need some prompting to help solve the question. M Meeting Expectations Got it, I understand the concept without help or prompting. F Fully Meeting Expectations Strong understanding, I understand the concept without help or prompting. Perhaps small mistakes or difficulty communicating methods used. E Exceeding Expectations Wow, nailed it! I can use this concept to solve problems I may have not seen in practice. I also get little details that may not be directly related to this target correct.
Unit 1: Real Numbers Math 10 Common 1 The Real Number System (B2.1-2,4, 2.) Identifying and Classifying : Rational vs. Irrational, Natural, Whole, and Integers Converting Real # s to decimals and vice-versa BEDMAS & Fraction Review 2 Factoring, GCF, and LCM (B1.1-1.3) What is a prime and composite number (0 and 1 also) Identifying factors of a number Finding prime factors of a number using factor trees Identifying Greatest Common Factor (GCF) of 2 or more numbers Identifying Least Common Multiple (LCM) of 2 or more numbers 3 Radicals (B2.7) Radical Sign, Index, and Radicand Intro to Square and Cube roots Estimating square roots and cube roots Determine if a number is a perfect square or cube and evaluate if it is Quiz 4 Multiplying Radicals, Entire versus Mixed (B2.5-2.6) Multiplying radicals with coefficients together By finding largest perfect square By factoring Mixed versus Entire Radicals 5 Simplifying Radicals (B2.5-2.6) Simplifying radicals Further multiplication 6 Review Quiz 7 Practice Exam Exam
Unit 1: Real Numbers Day 1 Math 10 Common 1-1 I can sort a set of numbers into irrationals and rationals, including their subsets (natural, whole, integer, rational, irrational), convert rational numbers, and determine an approximate value of a given irrational number. Real numbers include all numbers that can be placed on the number line. Real Numbers Rational Numbers: Numbers that can be written as a fraction of two integers. Written as a decimal, these numbers all terminate or repeat. ie. 9,12,, 5,.131313... They can be divided into: Natural: {1, 2, 3,...} Whole: {0, 1, 2, 3,...} Integers: {... 2, 1,0,1,2,3,...} Irrational Numbers: These numbers cannot be written as a fraction of two integers. Written as a decimal, these numbers will neither terminate nor repeat. ie. 7, 14,,5.139316269.... Identify each number as rational or irrational: 1) 9 2) 11 3) 4) 5.151151115... 5) 6) 7) 7.4261 ) 16 Name all sets to which each number belongs. 9) 10) 0 11) 12) 9 13) 14) 16 15) 12 16) 6.13113113...
17) 0.25 1) 25 19)! ) 20) 27 The following is a Venn diagram of the various subsets of real numbers. Fill in the appropriate numbers. 23 24 21 22 25 26 21) 22) 23) 24) 25) 26) To convert a non-repeating decimal to a fraction, put the decimal over the appropriate denominator, depending on place value, then reduce. ie. 5.295= 5 = 5 ### ## = # ## To convert a repeating decimal, use the Rule of 9. Determine the number of repeating digits and put the same number of 9 s in the denominator, then reduce. ie. 3.111...= 3 = 3 =
Identify each as rational or irrational. If rational, convert each of the following to an integer or fraction in lowest terms. 27) 20 2) 0.2 29) $0.4 30) 121 31) 144 32) 0. 06 33) 0.27 34) 0.2121121112 35) 0.06 36) 27 37) &2 3) 225 Review: BEDMAS and Fractions: Remember your order of operations: Brackets, Exponents, Division and Multiplication, Addition and Subtraction! Simplify all fractions. 39) 5 3! 7) 40)! 5)+7! 3) 41)!()) * )
42) # 43) # 44) +1! 4) 45) 5 # 46) 47) #! 0) 4) 20 49) 50) - 51) # 52) ) #! 5) 53) (!() 4 ( 54) 3 - + 55) 5 56)! )
Unit 1: Real Numbers Day 2 Math 10 Common 1-2 I can determine the GCF or LCM of a set of numbers and explain why the numbers 0 and 1 have no prime factors. Review: Identify each as rational or irrational. If rational, convert each of the following to an integer or fraction in lowest terms. 1) 2) 5.1 3) 125 Name all sets to which each number belongs. 4) 144 5) 6) 0 7) 21 BEDMAS and Fractions: Remember your order of operations: Brackets, Exponents, Division and Multiplication, Addition and Subtraction! Simplify all fractions. ) 1 ) 9) 10) Factors of a number are very useful for several math operations. For example, we need to find common factors to reduce fractions to lowest terms, and for simplifying when multiplying or dividing fractions. They are also very helpful to find the Lowest Common Denominator (LCD) for adding and subtracting fractions. Factors are numbers that divide evenly into another number. ex. Factors of 12: {1, 2, 3, 4, 6, 12} A prime number is any number that can only be divided by 1 and itself. Composite numbers are numbers that can be divided by numbers other than 1 and itself. Note that 0 and 1 are neither prime nor composite.
Divisibility Tests: 2: A whole number is divisible by 2 (even) if ends in a 0, 2, 4, 6, or. 3: A whole number is divisible by 3 if the sum of its digits are divisible by 3. 4: A whole number is divisible by 4 if last two digits are divisible by 4. 5: A whole number is divisible by 5 if ends in a 0 or 5. 6: A whole number is divisible by 6 if it is even and divisible by 3. 9: A whole number is divisible by 9 if the sum of its digits are divisible by 9. 10: A whole number is divisible by 10 if ends in a 0. All numbers can be broken down into a unique product of prime numbers. Factor trees are very useful for this exercise. 24 36 4 6 6 6 2 2 2 3 24 =2 2 2 3 = 2 3 2 3 2 3 36 = 2 2 3 3 =2 3 Write each of the following as product of prime numbers. 11) 72 12) 90 13) 312 14) 512 15) 2160 16) 4725 The largest number that divides evenly into two or more numbers is known as the Greatest Common Factor. This is extremely useful in Mathematics, such as when simplifying fractions or
factoring. To find the GCF, write each number as a product of primes, then circle all the numbers they have in common. Take all the common prime factors and multiply them together to get the GCF. ie. To find the GCF of 36 and 126: Find the GCF of each set of numbers. 36 =2 2 3 3 126 =2 3 3 7 GCF is: 2 3 3 = 1 17) 12,2 1) 54, 66 19) 4,136 20) 65,169 21) 1,10 22) 30,45,60 23) 1,36,72 24) 12,15,42 25) 2,42,4 The Least Common Multiple of a number is the smallest common non-zero multiple of two or more whole numbers. It is extremely useful for adding and dividing fractions (The LCD is the LCM of the denominators). To find the LCD: Method 1: Take multiples of the largest number until you find one that all the numbers divide into evenly. Method 2: Alternatively, find the prime factor of each number, select the primes that occur the greatest number of times in any one factor, then multiply those primes together.
ie. Find the LCD of 1 and 24. Method 1: Multiples of 24: 24, 4, 72 (72 is the smallest multiple of 24 that 1 goes into, therefore the LCM of 1 and 24 is 72. Method 2 : This method only works for finding the LCM of two numbers. 1 =2 3 3 24 =2 2 2 3 LCM = 2 2 2 3 3 = 72 Find the Lowest Common Multiple of each number. 26) 25, 50 27) 1, 45 2) 20, 55 29) 36, 4 30) 21, 30, 36 31) 12, 1, 24 Simplify the following fractions. Use your division properties when finding common factors. 32) 33) 34) $ 35) 36) 37) $ 3) Pencils come in packages of 10. Erasers come in packages of 12. Michelle wants to purchase the smallest number of pencils and erasers so that she will have exactly 1 eraser per pencil. How many packages of pencils and erasers should she buy 39) Shannon is making identical balloon arrangements for a party. She has 32 maroon balloons, 24 white balloons, and 16 orange balloons. She wants each arrangement to have the same number of each color. What is the greatest number of arrangements that she can make if every balloon is used
Unit 1: Real Numbers Day 3 Math 10 Common 1-3 I can identify perfect squares and cubes and evaluate them using factoring techniques. Review: Circle all the prime numbers and put a square box around the composite numbers. 1) 0, 1, 2, 3, 1, 25, 71, 117, 25 Name all sets to which each number belongs. 2) 144 3) 4) 0 5) 21 6) Express 56 as a product of prime factors. 7) Express 3024 as a product of prime factors. ) Find the GCF of 15 and 21. 9) Find the LCM of 24 and 54. 10) Find the GCF of 36, 12, and 252. 11) Find the LCM of 6,, and 20. To square a number means to raise it to the second power, or to multiply it by itself. 5 * +5 5+25 All real numbers can be written as a product of two other identical real numbers. Numbers that are perfect squares will have rational square roots. Numbers that are not perfect squares have irrational square roots.. Square roots can be written as - or -. For square roots we don t typically write the little 2, known as the index. Since 5 * +25 we know that 25+5. You should be familiar with the perfect squares from 0 * up to 15 *. 0,1,4,9,16,25,36,49,64,1,100,121,144,169,196,225 List the square root of each of the following if the square root is rational. If the square root is irrational, write irrational.
12) 196 13) 27 14) / 0 11 15) 0.25 16) / **2 34 17) / *2 1) 0 19) 45 20) 45 21) 121 22) 50.16 23) / 2 *1 24) 25) / 07 **2 26) 50.46 27) 125 To cube a number means to raise it to the third power. 4 +4 4 4+64. Thus the cube root of 64 is 4. We write this as 64+4. The little 3 is known as the index. It tells you what root you are finding (ie. the square root or the cube root). If there is no index shown, we assume it is a 2, meaning the square root. You should be familiar with the perfect cubes from 0 up to 5. 0, 1,, 27, 64, 125 We can estimate the square root of a number by seeing what perfect squares it is between. For example, 30 is between the perfect squares 25 and 36, therefore it between 5 and 6. To estimate a cube root we see what perfect cubes the number is between. 20 is between the perfect cubes and 27 so it is between 2 and 3. Radicals are the name given to finding a root of any degree. In this unit we focus on square and cube roots.
List the square or cube root of each of the following if the root is rational. If the root is irrational, write what two integers it is between. 2) 64 29) 71 30) / *2 31) 0.125 * 32) / 33) 15 34) 102 73 35) 125 We can determine if a number is a perfect square (or cube) by factoring it and looking for pairs (or triplets) of prime factors. If any prime factors exist that aren t pairs (or triplets) then the number is not a perfect square (or cube). We can then determine the number s square root (or cube root) by taking one prime factor from each pair (or triplet) and multiplying together. ie. Find the square root of 3600: 3600+2 3 3 * 5 * 3600+2 * 3 5+60 ie. Find the square root of 352: 352+2 3 * 7 * Note that there is an odd number of 2 s, 352 is irrational. ie. Find the cube root of 216: 216+2 3 216+2 3+6 Find the square root or cube root of each number as indicated. If the root is irrational, write irrational. 36) 576 37) 172 3) 27000 39) 72900 40) 3375 41) 100000 42) 172 43) 290000 44) 0.125 45) 529 46) 1000 47) 0.0004 Note that it is impossible to take the square root of a negative number. This is because the square of a positive is a positive (ie.5 5+25) and the square of a negative number is also positive (ie.5 5+25). It is impossible to multiply any real number by itself and get a negative number. Thus 25 is undefined and the square root of any negative number is undefined.
However, we can take the cube root of both positive and negative numbers ie.5 5 5+125 so 125+5 and (5) (5) (5)+125 so 125+5. Therefore the cube root of negative numbers is defined. Determine what values of - would make the following expressions undefined. 4) - 49) - 50) -1 51) -2 52) 5-53) -+3 54) 2-5 55) 5-+4 56) Calculate the surface area of a cube with side length 15 cm. 57) Find the volume of a cube with side length 21 cm. 5) Calculate the surface area of a cube with volume 216 cm. 59) Find the volume of a cube with surface area 46 cm *.
Unit 1: Real Numbers Day 4 Math 10 Common 1-4 I can convert mixed radicals into entire radicals. 1) Circle the irrational numbers: 1.353535,π,, 27, 172 #, $,5.121121112 % Evaluate each of the following: 2) 5 ( 15) ( ) 3) 16+ 74 4) 91125 % # 5) 1225 6) Express 5 as a product of prime factors. 7) Find the GCF of 12, 2, and 0 ) Find the LCM of 12, 24, and 40. When multiplying radicals, 9= 9. Thus 3 5= 15. This also works in reverse. 9= 9. Thus 5 20= 100=10. 5 4 3 9=5 2 3 3=90 3 4 2 24 5 3=1 5 6 3 ; 4 ;=12; (Note that ; ;= ; < =;) Thus 29 29=29 (No need to evaluate 29 < ). Evaluate each of the following. Evaluate square roots but do not simplify further. 9) 2 10) 7 9 11) 5 7 12) 5 6 13) 3 1 2 12 14) 5 3 20 15) ( > 3)( % 6) 16) ( < 2)( < @ 2)
2 3 is an example of a mixed radical (a rational multiplied by a radical). 12 is an example of an entire radical (there is no rational coefficient). We notice that 2 3= 4 3= 12. 2 3= 12. Every mixed radical can be written as an entire radical. Convert the following mixed radicals to entire radicals, then estimate the approximate value as a decimal. 17) 3 5 1) 5 2 19) 4 3 20) 6 2 21) 2 5 22) 4 2 23) 3 2 24) 9 3 Pythagorean Theorem: B 9 In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). In other words, < +9 < =B <. Find the length of the indicated side. Express your answer as a radical unless indicated otherwise. 25) 26) 7 ; 14 ; 15
27) A circle of diameter is inscribed in a square. Find the area of the square not covered by the circle (the shaded region). 2) Find the exact area of a rectangle that is 5 by 9 14. 29) Calculate the exact area of a triangle that has a base of 3 6 and a height of 4 5. 30) A rectangle has an area of 20 42. List two possible pairs of side lengths for this rectangle. 31) Find the sum of the areas of each triangle below. Round your answer to the nearest tenth. 32) Find the distance between the two points on the grid below. Hint: make a right angled triangle using the endpoints, then find the hypotenuse. 3 cm 4 cm 13 cm
Unit 1: Real Numbers Day 5 Math 10 Common 1-5 I can convert entire radicals into mixed radicals. 1) Circle the rational numbers: 5.171171117, 0.0016,,! 729, #,5.114 &, π $ 2) Convert.45 &&&&! to a fraction. 3) List all the prime factors of 96. 4) Evaluate 9261. Evaluate each of the following: 5) 19 19 6) 5 5 3 45 7) / 5 20 ) 0/ 1 920 3 4 642 9) A ladder is 4 m long and is leaning against a house. If the ladder is 1 m from the base of the house, how high is it up the house 10) A triangle has a base of 6 2 and a height of 3 15. What is its area 11) Express 650 as a product of prime factors. 12) Find the GCF of 20, 32, and 36 13) Find the LCM of 6, 10, and 24. Last lesson we learned how to write mixed radicals as entire radicals. Some entire square root radicals can be written as mixed radicals. We do this by pulling any perfect squares in the radicand out. 20= 4 5= 4 5=2 5 When all perfect squares are pulled out of the radicand, the radical is simplified. Thus 20 is not simplified but 2 5 is. We can use a similar process for cube roots. For cube roots, we pull any perfect cubes out.!! 24 = 3 = 3 =2 3!!!
Another way to simplify radicals is to write each radicand as a product of its primes, then looking for any factors that are present twice for square roots (three times for cube roots), then pulling out each prime once for every pair (or triplet). Factors without pairs (or triplets) remain in the radicand. 20= 2 2 5 =2 5 Simplify the following: 14) 50 15) 12 16) 1 17) 1) 75 19) 4 20) 300 21) 000 22) 3 0 23) 3 3 5 6 24) (4 5)( 3 15) 25) 1 4 54 26) 4! 10 27) 54 1! 2) 24! 29) 4 32!! 30) 0.25 2 31) 2 24 3 2!! 32) 4 3 2 1!! 33) 12 3 6 34) Order the following from greatest to least and place them on a number line: 5 6,14,6 5, 140 35) Order the following from least to greatest and place them on a number line:!!! 3 2,2 7, 55,4 36) Arrange the following in ascending order and place them on a number line. 2 5,4,3 2, 21 37) Arrange the following in descending order and place them on a number line.!!! 4 2,3 4, 120,5
Unit 1: Real Numbers Day 6 Math 10 Common Review Consider the following list of numbers: 7, 1,, 3, 0, 3.616616661, 5. 3,. List all: 1) Natural Numbers 2) Whole Numbers 3) Integers 4) Rational Numbers 5) Irrational Numbers 6) Real Numbers Consider the following list of numbers: 0, 1, 2, 3, 4, 6, 9, 11, 15, 17. List all: 7) Prime Numbers ) Composite Numbers Write each composite number as a product of its prime factors: 9) 3500 10) 1512 11) 9072 12) 14 Write the GCF for each set of numbers. 13) 126, 5 14) 4, 312 15) 16, 30
16) 216, 100 17) 2, 77, 4 1) 150, 600, 2250 Find the LCM for each set of numbers. 19) 1, 24 20) 24, 40 21) 45, 55 22) 2, 35 23) 90, 135, 225 24) 1, 27, 45 Determine if each number is rational or irrational. If rational, write as an integer or fraction. 25) +0. 4 26), -. //0 27). 1 1 2) 729 1 29) 512 30) 0. 6 1 31) 1600 32) 532 33) 160 Write each as an entire radical. 34) 5 2 35) 4 3 36) 2 2 1 37) 3 6 3) 3 2 1 39) 2 5
Simplify each radical. 40) 32 41) 4 42) 2 75 1 43) 3 24 44) 24 1 45) 3 250 Evaluate and simplify. 46) 3 6 2 6 47) 12 4) 3 10 5 5 1 1 49) 4 15 6 50) 2 7 3 14 51) 3 2 4 12 52) A Costco pie with a surface area of 1π cm / just fits inside its box. What is the area of the bottom of the box that is not covered by the pie 53) The volume of a cubic box is 21952 cm >. What is the surface area of the box 54) Calculate the exact area of a triangle with a base of 24 cm and a height of 24 cm. 55) Calculate the exact area of a rectangle with a length of 5 6 cm and a height of 3 30 cm. 56) Arrange the following in ascending order and place on a number line:, 2 5, 21, 3 2, 4 57) Arrange the following in descending order and place on a number line: 1 1 1 2 3, 3 2, 25, 3, 3
Unit 1: Real Numbers Key Math 10 Common Day 1: 1 20 are all real as well! 1) R 2) I 3) R 4) I 5) R 6) I 7) R ) I 9) I 10) R,I,W 11) R 12) R,I,W,N 13) R 14) R,I,W,N 15) R,I 16) I 17) R 1) R,I 19) R,I,W,N 20) I 21) Real 22) Irr 23) Rat 24) Integers 25) Whole 26) Naturals 27) I 2) R 2 9 29) R2 3 30) R 11 31) R 12 32) R 2 33 33)R 3 11 34) I 35)R 3 50 36) I 37)R 3 2 3) R 15 39) 26 40) 61 41) 9 42) 1 16 43) 7 15 44) 20 45)5 3 46) 17 36 49) 5 50) 2 51) 27 52) 25 53) 13 54) 73 55) 25 56) 64 12 2 2 30 47) 72 4) 25 2 Day 2: 1) 3 25 2)64 3) Irr. 4) Rat,Int 5) Rat 6) Rat,Int,Whole 7) Irr )23 11 1 9)12 35 10) 7 120 11) 2 " 3 $ 12) 2 3 $ 5 13) 2 " 3 13 14) 2 % 15) 2 & 3 " 5 16) 3 " 5 $ 7 17) 4 1) 6 19) 20) 13 21) 27 22) 15 23) 1 24) 3 25) 14 26) 50 27) 90 2) 220 29) 144 30) 1260 31) 72 32) ' ( 33) ') % 34) $& $* 35) " ) 36) ) % 37) + ) 3) 6 pack pencils, 5 pack erasers 39) Day 3: 0) Prime:2,3,71 Composite:1,25,117,25 1) Int, Rat, Real 2) Rat, Real 3) Rat, Int, Whole, Real 4) Irr 5) 2 3 7 6) 2 4 3 3 7 7) 3 ) 216 9) 4 10) 120 11) 14 12) 566 13) " '7 14) ' $ 15) Irr 16) 1 5 17) 0 1) Irr 19) Irr 20) 11 21)1 3 22)1 2 26) Irr 27) 4 2) Irr to 9 29) 5 3 30)1 2 31)1 4 23) Irr 24)14 15 25) 2 3 32) Irr 3 to 4 33) Irr 4 to 5 34) Irr 11 to 12 35) Irr 36) Irr 37) 30 3) 270 39) 15 40) 1000 41) 12 42) 1700 43) 1 2 44) 23 45) 10 46) 1 50 47) <0 4) none 49) <1 50) <2 51) >5 52) < 3 53) none 54) < 4 5 55) 1350 cm $ 56) 9261 cm " 57) 216 cm $ 5) 729 cm " Day 4: 1) ;, 27,5.12112112 2) 5 3) 12 4) 45 5) 35 6) 2 $ 3 7 $ 7) 4 ) 120 9) 4 10) 63 11) 35 12) 30 13) 6 216 14) 30 15) 4 1 16) 24 17) 45,6.6 1) 50,7.1 15 19) 4,6.9 20) 72,.5 21) 20, 4.4 22) 32,5.6 23) 1 4.2 24) 24315.7 25) 274 26) 132 27) 2; 2) 72 70 29) 6 30 30) 5 6 and 4 7,20 and 42,etc 31) 36 cm $ 32) 10 Day 5: 1) 0.0016, 729, 19 6,5.114 @ 2) 93 3) 2,3 4) 21 5) 19 6) 225 7) 1 ) 70 9) 15 10) 9 30 11 11) 2 5 $ 13 12) 4 13) 120 14) 5 2 15) 2 3 16) 3 2 17) 2 2 1) 5 3 19) 4 3 20) 10 3 21) 40 5 22) 12 5 23) 45 2 24) 60 3 25) 4 6 26) 9 3 2 27) 3 2 2) 2 3
29) 4 30) 7 31) 12 6 32) 24 2 2 36) 4,3 2,2 5, 21 37) 4 2,5, 120,3 4 Review: 33) 6 9 34) 14,6 5,5 6, 140 35) 3 2, 55,2 7,4 1) 1 2) 0,1 3) 3,0,1 4) 3,0, 9,1,5.3 @ 5) 7,;,3.616616661 6) All of them 7) 2, 3, 11, 17 ) 4, 6, 9, 15 9) 2 $ 5 " 7 10) 2 " 3 " 7 11) 2 & 3 & 7 12) 2 " 3 7 11 13) 42 14) 24 15) 2 16) 216 17) 7 1) 150 19) 72 20) 120 21) 495 22) 140 23) 1350 24) 270 25) $ 2) 9 29) 30) $ 31) 40 32) 1 33) Irr. " " 34) 50 35) 4 36) 37) 54 3) 54 39) 40 40) 4 2 41) 4 3 42) 10 3 43) 6 6 44) 2 3 45) 15 2 26) '& '* 27) %) '' 46) 36 47) 4 6 4) 75 2 49) 12 10 50) 42 2 51) 24 3 52) 324 1; cm $ 53) 4704 cm $ 54) 12 cm $ 55) 90 5 cm $ 56) π,4,3 2,2 5, 21 57) 2 3, 25,3,3 2,3 3