Math 6 Notes Unit One: Whole Numbers and Patterns

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1 Math 6 Notes Unit One: Whole Numbers and Patterns Whole Numbers The following is the set of numbers on the number line called whole numbers: {0, 1,, 3, 4, } All numbers in base 10 are made up of 10 digits: 0, 1,, 3, 4, 5, 6, 7, 8, and 9. The value of a digit depends on its placement within a number, also known as place value. In base 10, we have columns: 100,000 10,000, 1000, 100, 10, ,000's 10,000's 1000's 100's 10's 1's The digit s value is determined by the specific column in which it is located. The 5 in the following two examples have different values. In 53, the 5 has the value of 5 tens or 50. In 1,548, the 5 has the value of 5 hundreds or 500. Writing a Number in Expanded Notation To write a number in expanded notation, you write each digit as a product of that digit and its place value and find their sum. Once exponents are taught student should begin to see and use them as shown below. Examples: Write 73 in expanded notation Write 543 in expanded notation Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 1 of 14 Revised 013- CCSS

2 Write 1,07 in expanded notation Writing a Number in Word Form Examples: Write 73 in word form: Seventy-three Write 705 in word form: Seven hundred five Write 356 in word form: Two thousand three hundred fifty-six Writing in Standard Form Write five hundred sixteen thousand, one hundred fifty-four in standard form. 516,154 Write (1000) + 6(100) in standard form.,600 Ordering Whole Numbers To order whole numbers, you can compare them using place value. Then write them in order, usually from least to greatest. You can also graph the number on a number line; as you read the numbers from left to right, they will be in order from least to greatest. Order the numbers from least to greatest: 914, 954, Start at the left and compare the digits in the same place value position; look for the first place that the values differ. We are now comparing 10, 50 and 40. Since 10 is the smallest value and 50 the largest, we would order the numbers: 914, 94, 954 OR Use the number line. Graph the numbers on the number line and read them from left to right (for least to greatest) Reading from left to right: 914, 94, 954 Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page of 14 Revised 013- CCSS

3 NOTE: Be sure students understand the generalization that moving to the left on the number line means the values of the numbers are decreasing; left means less. Be sure students understand the generalization that moving to the right on the number line means the values of the numbers are increasing; right means greater. This generalization will serve them well with comparing and ordering integers and then again with comparing and ordering rationals. Rounding Whole Numbers To round a number, look at the digit immediately to the right of the place you are rounding. If that digit is 5 or more, round up. If that digit is less than 5, round down. Round 4,31 to the nearest hundred. hundred s place 4, 3 1 This digit is less than 5. Round down. 4,31 rounded to the nearest hundred is 4,300. Round 58,786 to the nearest , s place This digit is more than 5. Round up. 58,786 rounded to the nearest 1000 is 59,000. Use Rounding to Estimate Syllabus Objectives: (1.) The student will estimate by rounding to a given place value. (1.5) The student will use a variety of methods to estimate. Rounding is an effective way to estimate the sum or difference of a group of numbers. When estimating, you can round numbers to compatible numbers. Compatible numbers are numbers that are close to the given numbers that make estimation (or mental calculation) easier. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 3 of 14 Revised 013- CCSS

4 Estimate the sum to the nearest hundreds = 41 rounded to the nearest hundred is rounded to the nearest hundred is rounded to the nearest hundred is 100. Add the rounded values: = Sam is collecting aluminum cans to recycle. The following shows the number of cans he collected last week. About how many cans did Sam collect last week? Monday Tuesday Wednesday Thursday Friday 81 cans 48 cans 1 cans 75 cans 13 cans First choose the best place value to round each number. Since most of the numbers are under 100, round each number to the nearest tens. 81 rounds to rounds to 50 1 rounds to rounds to rounds to 130 Add the rounded values: Sam collected approximately 350 cans last week. Each of Margaret s energy bars weighs ounces. What is the approximate net weight of a box containing 68 bars? First, round 68 to Solution: About 770 Then, round to 11 x 11 ounces 770 Note: Since both numbers were rounded up, the estimated product will be an overestimate. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 4 of 14 Revised 013- CCSS

5 If each energy bar costs $.9, approximately how much will 68 bars cost? First, round $.9 to $.00 $.00 Solution: and round 68 to 70 x 70 About $ $ Note: Since one number was rounded down and one number was rounded up, initially you can t be sure if you have an overestimate or underestimate. Of course if you investigate a little deeper you can determine you have an underestimate. Ryan can run a mile in 8 minutes. If he runs for 34 minutes, about how many miles will he run? You can estimate 34 8 by finding compatible numbers = 4 Solution: About 4 miles Check for Understanding: Is this an overestimate or underestimate? Explain. In 000, the population in Tokyo was 8,11,54. Tokyo s area is 3 square miles. About how many people were there for each square mile in 000? You can estimate 8,11,54 3 by finding compatible numbers close to 8,11,54 and 3. 8,11,54 3 8,000, = 40,000 Remembering basic facts like 8 = 4 can help students find compatible numbers. Solution: About 40,000 people per square mile. Check for Understanding: Is this an overestimate or underestimate? Explain. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 5 of 14 Revised 013- CCSS

6 NEW 6.EE.1 Write and evaluate numerical expressions involving whole number exponents. Exponents An exponent is the superscript which tells how many times the base is used as a factor. exponent 3 base In the number 3, read to the third power or cubed, the is called the base and the 3 is called the exponent. Examples: To write an exponential in standard form, compute the products. i.e Special Case Examples: , ,000 Check for Understanding: What is the value of 5 10? 6 10? Etc? What pattern allows you to find the value of an exponential with base 10 quickly? Answer: The number of zeroes is equal to the exponent! Caution: If a number does not have an exponent, it is understood to have an exponent of ONE! Writing Numbers in Exponential Form Examples: Write 81 with a base of , 81 = 9x9 3333, therefore 81 3 Write 15 with a base of 5.? 4 Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 6 of 14 Revised 013- CCSS

7 ? , , therefore 15 5 Let us look at a pattern that will allow you to determine the values of exponential expressions with exponents of 1 or ? 1 1 3? ? ? ? ? 4 1 Any number to the power of 1 is equal to the number. That is, n 1 = n. Any number to the power of 0 is equal to one. That is, n 0 = 1. Again remind students that if there is no exponent, the exponent is always 1. Order of Operations Syllabus Objective: (1.3) The student will use the order of operations to evaluate expressions with whole numbers. The Order of Operations is just an agreement to compute problems the same way so everyone gets the same result, like wearing a wedding ring on the left ring finger or driving on the right side of the road. Order of Operations (PEMDAS or Please excuse my dear Aunt Sally s loud radio)* 1. Do all work inside the Parentheses and/or grouping symbols.. Evaluate Exponents. 3. Multiply/Divide from left to right.* 4. Add/Subtract from left to right.* *Emphasize that it is NOT always multiply-then divide, but rather which ever operation occurs first (going from left to right). Likewise, it is NOT always addthen subtract, but which of the two operations occurs first when looking from left to right. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 7 of 14 Revised 013- CCSS

8 Simplify the following expression Underline the first step. Simplify underlined step, & underline next step. Repeat simplify and underline process until finished. Note: each line is simpler than the line above it. Simplify the following expressions. (a) 35 Work: (b) Work: (c) 8 (13) 5 8 Work: 8 (13) Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 8 of 14 Revised 013- CCSS

9 Evaluate Evaluate Evaluate ,000 5, Evaluate Which is the greatest? A. 5 B. 4 3 C. 3 4 D A. 5 B C. So the answer is B D Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 9 of 14 Revised 013- CCSS

10 Properties of Real Numbers The properties of real numbers are rules used to simplify expressions and compute numbers more easily. Property Operation Algebra Numbers Key Words Commutative (change order) + a bb a Change Order Commutative a b b a 5 5 Change Order Associative + ( a b) c a( b c) (4 9) 6 4 (9 6) Associative ( a b) ca( b c) 9 (5) (95) Distributive ab ( c) aba c 5(3) = 5(0) + 5(3) Change Grouping Change Grouping Distribute Over ( ) Commutative Property of Addition a bb a Commutative Property of Multiplication a b b a Order does not matter!! Examples: = = 7 10 Associative Property of Addition ( a b) ca( b c) Associative Property of Multiplication ( a b) ca( b c) Examples: (7 + 8) + = 7 + (8 + ) (13 5) 4 = 13 (5 4) Distributive Property ab ( c) aba c (distribute over add/sub) 5 3 = 5 (0 + 3) = = = = 5 (10 + ) = = = 300 Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 10 of 14 Revised 013- CCSS

11 Arithmetic Sequence Syllabus Objective: (.5) The student will find missing terms in a sequence. A sequence is a set of numbers in a particular order. Each number is called a term of the sequence. {1, 3, 6, 10, 15} is a sequence with 5 terms. An arithmetic sequence is a sequence in which the terms change by the same amount each time. We are often asked to find the next term(s) in the sequence or identify the pattern. To find the next term in an arithmetic sequence, subtract any two consecutive terms, then add that difference to the last term to find the next term. (The CRT will refer to finding a pattern in lieu of using the word sequence.) Find the next term in the following arithmetic sequence: {5, 10, 15, 0, } Students know this pattern by heart so it is probably a good place to begin. They know they are skip counting by 5, or increasing by 5 each time so the next term in the sequence is 5. Find the next term in the following arithmetic sequence: {8, 19, 30, 41, } Now this one requires more thought. Choose two consecutive terms (8, 19) or (19, 30) and find their difference: = 11. Add the difference (11) to the last term = 5. 5 is the next term in the sequence. Using a table is a useful way to see the pattern and identify the next 3 terms. Position Value of the Term ? The next three terms would be 5, 63 and 74 because = 5, = 63, and = 74. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 11 of 14 Revised 013- CCSS

12 Identify the pattern and find the next 3 terms: {4, 10, 8, 14, 1, } Position Value of the Term ??? The 6 th term in the sequence would be 18 (which we obtain by adding 6 to the 1). The 7 th term in the sequence would be 16 (which we obtain by subtracting from 18). The 8 th term in the sequence would be (which we obtain by adding 6 to the 16). Identify the pattern and find the next term: {, 5, 8, 11, 14, } Position n th Value of the Term ??? The next term in the sequence would be 17 (which we obtain by adding 3 to the 14). Syllabus Objective: (.4) The student will generalize relationships from charts and tables with and without technology. (1.4) The student will use tables or charts to extend a pattern in order to describe a rule. Identify the pattern and find the next term: {5, 10, 15, 0, } Using a table: This arithmetic sequence has an infinite number of terms. The terms are obtained by adding 5 to the previous term. The next term would be 5. Position n th Value of the Term ? The pattern can be determined by looking for a relationship between the position and the value of the term. We can see that in each case the position is multiplied by 5 to obtain the value; hence, the n th term would be 5 times n, or 5n. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 1 of 14 Revised 013- CCSS

13 Make a table or chart to display the data. Write an expression for the rule. The height of a painting is 4 times its width. Create and complete a table. Width n th Height???????? Width n th Height ? This pattern is simple the height is four times the width so the rule is 4 width, 4 n or 4n. Write an expression for the rule for the following data. x y n th In this pattern both the x and y terms are increasing by one each time. Upon further examination we see the x increase by 7 each time to equal the y value or the difference between each pair of x and y terms shows an increase of 7 each time so the rule is add 7 or n + 7. Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 13 of 14 Revised 013- CCSS

14 BEWARE: NOT all sequences follow this simple format for finding the rule. Identify the pattern and find the next term: {, 5, 8, 11, 14, } Position n th Value of the Term ??? The next term in the sequence would be 17 (which we obtain by adding 3 to the 14). The pattern for this one is not so easily determined simple addition or multiplication does not work. We would have to try some combination of multiplication and addition/subtraction. In this case, the n th term would be 3n 1. (That is, take term 1, multiply by 3 and subtract 1 and you arrive at the value. Take term 4 and test it: multiply 4 by 3 and then subtract 1 and you arrive at the value 11 ). Holt, Chapter 1 Math 6, Unit 01: Whole Numbers and Patterns Page 14 of 14 Revised 013- CCSS