Math 7 Notes Unit Three: Applying Rational Numbers

Transcription

1 Math 7 Notes Unit Three: Applying Rational Numbers Strategy note to teachers: Typically students need more practice doing computations with fractions. You may want to consider teaching the sections on fractions first, and then introduce decimals as special fractions (rational numbers equivalent to fractions). This would allow you to reinforce the rules for fractions while teaching the sections on decimals. For instance, you could define a decimal as a special fraction and link all the rules introduced in fractions to decimals. That is when you line up decimal points and fill in zeros when adding or subtracting decimals, that s the same as finding a common denominator and making equivalent fractions when adding or subtracting fractions, etc. Estimating with Decimals Syllabus Objective: (.8) The student will estimate using a variety of methods. An estimation strategy for adding and subtracting decimals is to round each number to the nearest integer and then perform the operation. Examples: would be our estimate would be our estimate would be our estimate An estimation strategy for multiplying and dividing decimals: When multiplying, round numbers to the nearest non-zero number or to numbers that are easy to multiply. When dividing, round to numbers that divide evenly, leaving no remainders. Remember, the goal of estimating is to create a problem that can easily be done mentally. Examples: = would be our estimate 8 would be our estimate Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

2 Estimating can be used as a test-taking strategy. Use estimating to calculate your answer first, so you can eliminate any obviously wrong answers. A shopper buys 3 items weighing. ounces, 7.89 ounces and 3.5 ounces. What is the total weight? A B C..0 D. 5.5 Round the individual values: The only two reasonable answers are C) and D). Eliminating unreasonable answers can help students to avoid making careless errors. Adding and Subtracting Decimals Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Remember, since decimals are special fractions, the rules for computing with fractions also work for decimals. Note: Choose numbers carefully that are not reducible as fractions, so as to eliminate that distraction. To add.7 and.7, follow the algorithm for adding fractions: Algorithm for Addition of Fractions. Find the common denominator.. Make equivalent fractions. 3. Add the numerators.. Bring down the denominator. 5. Simplify = 00, =,000, , Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

3 . The denominator for.7 is 00. The denominator for.7 is,000. The least common denominator is,000.. Add a zero after.7 so it will have a common denominator of, Add the numerators by adding 70 and 7 = Bring down the common denominator of,000. This means there must be three digits to the right of the decimal point. Placing a decimal point gives.837 as the answer. After doing numerous decimal addition problems we can see a pattern which leads to a simpler algorithm that can be used to add/subtract decimals. Algorithm for Addition / Subtraction of Decimals. Rewrite the problems vertically, lining up the decimal points.. Fill in spaces with zeros. 3. Add or subtract the numbers.. Bring the decimal point straight down. ***When comparing the algorithm for adding/subtracting fractions and the algorithm for adding/subtracting decimals, we note that:. Lining up the decimal point is the same as finding the common denominator in fractions.. Filling in spaces with zeros is the same as making equivalent fractions. 3. Adding or subtracting the numbers is the same as adding the numerators.. Bringing the decimal point straight down is the same as bringing down the denominator Rewrite vertically, lining up the decimal points Fill in with zero to find the common denominator and make equivalent fractions Add and bring the decimal straight down Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 3 of

4 Multiplying Decimals Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Just as the algorithm for adding and subtracting decimals is related to the addition and subtraction of fractions, the algorithm for the multiplication of decimals also comes directly from the multiplication algorithm for fractions. Before we look at the fraction algorithm, let s first look at the algorithm for decimal multiplication. Algorithm for Multiplication of Decimals. Rewrite the numbers vertically.. Multiply normally, ignoring the decimal point. 3. Count the total number of digits to the right of the decimal points in the factors.. Count and place the decimal point that same number of places from right to left in the product (answer) Rewrite the problem vertically.. Multiply normally, ignoring the decimal point. 3. Count the number of digits to the right of the decimal points. There are two places to the right in the multiplicand (number on the top) and one place to the right in the multiplier (number on the bottom). + = Count and place the decimal point the same number of places (3) from right to left in the answer. Before going on, let s think of how this is related to the algorithm for multiplying fractions. The algorithm for multiplying fractions is to multiply the numerators, multiply the denominators, and then simplify. Multiplying the numbers while ignoring the decimal points is the same as multiplying the numerators. Counting the number of decimal places is the same as multiplying the denominators. In this problem, the denominators are 00 and = 000 which Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

5 shows three zeros, which means 3 decimal places. The point is the algorithm for multiplying decimals comes from the algorithm from multiplication of fractions. That should almost be expected since decimals are special fractions. Remember to point out to students that if a number has no decimal point, the decimal point is understood to go after the number So, when multiplying this number, it has NO decimal places after the decimal point. Multiplying by Positive Powers of 0 Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Examining the patterns discovered when multiplying by positive powers of 0 (0, 00,,000 ) we can do many multiplication problems mentally. What pattern do you see from these problems? 0 (.3) = (567.3) = 5673.,000 (.56) = (0.056) = (5.7) = 570,000 (0.356) = 35.6 We can generalize a rule from the observed pattern: When multiplying by positive powers of 0, move the decimal point to the right, the same number of places as there are zeros. 0 (3.75) 0 (3.75) = (5.37) 00 (5.37) = 53.7 One zero in 0, so move the decimal point one place to the right. Two zeros in 00, move the decimal point two places to the right. 000 (6.) Three zeros in,000; move the decimal point three places to the right. 000 (6.) = 6,00 Notice in this problem we had to fill in a couple of placeholders to move it three places to the right. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 5 of

6 Dividing Decimals by Integers Syllabus Objective: (.)The student will solve problems using operations on positive and negative numbers, including rationals. Algorithm for Dividing Decimals by Integers. Bring the decimal point straight up into the quotient.. Divide in the normal way and determine the sign Or use short division: ) Place decimal point in quotient. ) 7 divides into 39, 5 times with left. 3) Write remainder of, before. ) 7 divides into, 6 times evenly. Remember, when using short division, writing the remainder(s) in the dividend is showing work. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 6 of

7 Dividing Decimals and Integers by Decimals Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Algorithm for Dividing Decimals. In the divisor, move the decimal point as far to the right as possible.. In the dividend, move the decimal point the same number places to the right. 3. Bring the decimal point straight up into the quotient.. Divide in the normal way Move the decimal point places to the right in the divisor. Move the decimal point places to the right in the dividend. Bring up the decimal point and divide normally. By moving the decimal the same number of places to the right in the divisor and the dividend, we are essentially multiplying our original expression by one. We are making equivalent fractions by multiplying the numerator and denominator by the same number. If we move the decimal point one place, we are multiplying the numerator and denominator by 0. By moving it two places, we are multiplying the numerator and denominator by 00, etc =. 00 Dividing by Powers of 0 Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Examining the patterns discovered when dividing by powers of 0 (0, 00,,000 ) we can do many division problems mentally. What pattern do you see from these problems? = = = = = =.008 Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 7 of

8 We can generalize a rule from the above observed pattern: when dividing by positive powers of 0, move the decimal point to the left, the same number of places as there are zeros Since there is one zero in 0, move the decimal point one place to the left = Since there are two zeros in 00, move the decimal point two places to the left = There are three zeros in,000, so move the decimal point three places to the left =.009 Notice in this problem we had to fill in a couple of placeholders to move it three places to the left. Computing with Decimals and Signed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. The rules for adding, subtracting, multiplying and dividing decimals with signed numbers are the same as before, the only difference is you integrate the rules for integers. Compute: (.3) 6.70 Line up decimal points, fill in zeros, and add. +.3 Add negative numbers, the sum is negative Compute: ( 7.) Since subtract means add the opposite, we add 7. Line up decimal points, fill in zeros, and add. Adding numbers with different signs, we subtract the absolute values and take the sign of the greater absolute value so our sum is negative. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 8 of

9 Compute: ( 5.6)( 8) Multiply, count one decimal place in problem and place one decimal place in answer. Multiply negatives, the product is positive. Compute: = Move decimal point two places to the right in the divisor to make it a whole number. Move decimal point two places to the right in the dividend. Bring decimal point straight up into the quotient. Divide as usual Determine the sign. Solving Equations Containing Decimals Syllabus Objective: (3.5) The student will solve equations and inequalities in one variable with integer solutions. Strategy for Solving Equations: To solve linear equations, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use opposite (inverse) operations. x +.5 = Undo adding.5 by subtracting.5 from both sides. x = 8.8 Simplify and determine the sign. by 8..8 = 8n.8 8n = = n Undo multiplying by 8 by dividing both sides Simplify and determine the sign. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 9 of

10 y = y ( 5) = ( 5)( 0.7) 5 y =+ 3.5 Undo dividing by 5, by multiplying both sides by 5. Simplify and determine the sign. Estimation with Fractions Syllabus Objective: (.8) The student will estimate using a variety of methods. Round to 0 if the numerator is much smaller than the denominator. 3 7 Examples:,, Benchmarks for Rounding Fractions Round to ½ if the numerator is about half the denominator. Examples: 5 7,, Round to if the numerator is nearly equal to the denominator. Examples: ,, Examples: Round to 0, ½ or Estimating Sums and Differences Round each fraction or mixed number to the nearest half, and then simplify using the rules for signed numbers. Estimate is our estimate Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 0 of

11 Estimate is our estimate Estimating Products and Quotients Round each mixed number to the nearest integer, and then simplify = 88 is our estimate 5 7 ( ) 8 ( 3) 6 5 = 6 is our estimate Adding and Subtracting Fractions Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Common Denominators Let s say we have two cakes, one chocolate and the other vanilla. The chocolate cake was cut into fourths, the vanilla cake into thirds as shown below. You take one piece of each, as shown. Since you had pieces of cake, can you say you had of a cake? 7 Remember our definition of a fraction: the numerator indicates the number of equal size pieces you have Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

12 while the denominator indicates how many equal pieces make one whole cake Since your pieces are not equal, we can t say you have of a cake. 7 And clearly 7 pieces does not make one whole cake. We can conclude: The key is to cut the cakes into equally sized pieces. We ll cut the first cake (already in fourths) the same way the second cake was cut. And we ll cut the second cake (already in thirds) the same way the first cake was cut. So each cake ends up being cut into twelve equally sized pieces. Cutting the cake into EQUAL size pieces illustrates the idea of common denominator. Let s look at several different methods of finding a common denominator. A common denominator is a denominator that all other denominators will divide into evenly. Methods of Finding a Common Denominator. Multiply the denominators. List multiples of each denominator, use a common multiple. 3. Find the prime factorization of the denominators, and find the Least Common Multiple 3. Use the Simplifying/Reducing Method, especially for larger denominators. In our cake illustration, the common denominator is the number of pieces tin which the cakes can be cut so that everyone has the same size piece. Method : To find a common denominator of 3 and, multiply the denominators, 3 =. This technique is very useful when the denominators do not share any common factors other than. Method : To find a common denominator of 5 6 and 3, list the multiples of each denominator. Multiples of 6: 6,, 8, Mulitples of :, 8,, Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

13 Since is on each list of multiples, it is a common denominator. This technique is very useful when the greater denominator is a multiple of the smaller one. Such as, find the common denominator for 3 and 5. 6 is a common 6 denominator since it is a multiple of. Method 3: To find a common denominator of 3 8 and 7, find the prime factorization of each denominator. Factoring, 8= = 3 Multiply the prime factors, using overlapping factors only once, we get 3 =. This technique can be tedious and may be the least desirable method; although it is good practice working with prime factors. Method : To find the common denominator of 8 and 5 using the Simplifying Method, create a fraction using the two denominators: 8, and then simplify: 8 6 = 3. 6 For 8 = 3, cross multiply: 8 = 7 or 3 = 7. The common denominator is 7. Note: It does not matter if you use 8 or. 8 This is an especially good way of finding common denominators for fractions that have large denominators or fractions whose denominators are not that familiar to you. Adding and Subtracting Fractions with Like Denominators To add or subtract fractions, you must have equal pieces. If a cake is cut into 8 equal pieces and you eat three pieces today, and then eat four pieces tomorrow, you would have eaten a total of 7 pieces of cake, or 7 8 of the cake = The numerators are added to indicate the number of equal pieces that were eaten. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 3 of

14 The denominators are NOT added, because the denominator indicates the number of equal pieces in the cake. If we added them, we would get 6, but there are only 8 pieces of cake. Examples: 3 + = = Adding and Subtracting Fractions with Unlike Denominators Let s add to 3. Do we get 7? What does the picture tell us? + 3 Note that when creating your visuals, be sure to cut one figure vertically and the other horizontally. The pieces are NOT EQUAL, so we cannot add them. We need to divide each shape into equally sized pieces. Do this by using the horizontal cuts from the second visual on the first. Then use the vertical cuts from the first visual on the second. (Both will now be divided into twelfths.) 3 = = 3 7 Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

15 From the picture we can see that 3 is the same as and has the same value as 3. Adding the numerators, a total of 7 equally sized pieces are shaded and pieces make one unit. If you did a number of these problems, you would be able to find a way of adding and subtracting fractions without drawing the picture. Algorithm for Adding / Subtracting Fractions. Find a common denominator.. Make equivalent fractions. 3. Add/subtract the numerators.. Bring down the denominator. 5. Simplify To find the common denominator, multiply the denominators, since they are relatively prime (have no common factors greater than ). (3 5 = 5). Make equivalent fractions. (shown) 3. Add the numerators. (3 + 0 = 3). Bring down the denominator. (5) 5. Simplify. (not necessary in this problem) 3 = = Examples: = = = = Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 5 of

16 Adding and Subtracting Mixed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Algorithm for Adding/Subtracting Mixed Numbers. Find a common denominator.. Make equivalent fractions. 3. Add/subtract the whole numbers.. Add/subtract the numerators. 5. Bring down the denominator. 6. Simplify the fraction answer part, combining it with the whole number part = = = To find the common denominator, multiply the denominators, since they are relatively prime (have no common factors greater than ). 5 = 0 3 = = = Borrowing From a Whole Number Subtracting fractions and borrowing is as easy as getting change for your money. You have 8 one dollar bills and you have to give your friend $3.5. How much money would you have left? Since you don t have any coins, you would have to change one of the dollars into quarters. Why not ten dimes? Because you have to give your friend a quarter, so you get the change in terms of what you are working with quarters. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 6 of

17 8 dollars 7 dollars quarters 3 dollars quarter 3 dollars quarter Subtract to get: dollars 3 quarters Redoing this problem using fractions: In the last problem, we borrowed quarters because we were working with quarters. Now we will borrow ths for the same reason. We change to 5 because we are 5 working with fifths. Borrowing from Mixed Numbers For this problem you have 6 dollars and quarter in your pocket, and you have to give your brother $ dollars quarter 5 dollars 5 quarters dollars 3 quarters dollars 3 quarters Subtract to get: 3 dollars quarters Since we don t have enough quarters, we change dollar for quarters, adding that to the quarter we already had, that gives you 5 quarters = = 3 = 3 We change 6 to when we simplify. 5 and add the to make 5 5. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 7 of

18 Algorithm for Borrowing with Mixed Numbers. Find a common denominator.. Make equivalent fractions. 3. Borrow, if necessary. 3. Subtract the whole numbers.. Subtract the numerators. 5. Bring down the denominator. 6. Simplify if needed. This example has unlike denominators. 3 6 = 6 8 = = = 7 3 Multiplying Fractions and Mixed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Before we learn how to multiply fractions, let s revisit the concept of multiplication using whole numbers. When I have an example like 3 = we can model that in several ways. Since it can be read three groups of we can use We can show a rectangular array Show 3 groups of the addition model Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 8 of

19 Each representation shows a total of 6. Mathematically, we say3 = 6. Multiplication is defined as repeated addition. That won t change because we are using a different number set. In other words, to multiply fractions, I could also do repeated addition = = 3 A visual representation of multiplication of fractions would look like the following. = 3 In this example, we want half of one third. So the visual begins with the one third we have. Since we want only half of it, we cut the visual in half and shade. Now we see part double shaded and 6 total parts or 6. = 3 In this example, we want to take one third of. So the visual begins with the one fourth. We want one third of it. Now we visually see part doubly shaded, and total pieces. So =. 3 Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 9 of

20 Algorithm for Multiplying Fractions and Mixed Numbers. Make sure you have proper or improper fractions.. Cancel, if possible. 3. Multiply numerators.. Multiply denominators. 5. Simplify. 3 i i = i Since 5 5 it to 7. 3 is not a fraction, we convert 7 i can be written as 7 i 5 i 5 Using the commutative property, we can rewrite this as 7 i. 5i Using the associative property, we can rewrite this as 7 i. Simplify =. 5 7 Then multiply and simplify, as a mixed number. i = = Rather than going through all those steps, we could take a shortcut and cancel. 3 3 i i Make sure you have proper or improper fractions i Cancel 8 and 9 by common factor of Cancel 0 and 5 by common factor of 5. 8 i = = 8 Multiply numerators, multiply denominators, simplify. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 0 of

21 Dividing Fractions and Mixed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Before we learn how to divide fractions, let s revisit the concept of division using whole numbers. When I ask, how many s are there in 8, I can write that mathematically three ways To find out how many s there are in 8, we will use the subtraction model: 8 Now, how many times did we subtract? Count them: there are subtractions. So there are twos in eight. 6 Mathematically, we say 8 =. Division is defined as repeated subtraction. That won t change because we are using a different number set. In other words, to divide fractions, I could also do repeated subtraction. 0 Another way to look at this problem is using your experiences with money. How many quarters are there in $.50? Using repeated subtraction we have: = = 3 0 How many times did we subtract? Six. But this took a lot of time and space. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

22 A visual representation of division of fractions would look like the following. = 8 We have. Representing that would be Since the question we need to answer is how many ' 8 s are there in, we need to cut this entire diagram into eighths. Then count each of the shaded one eighths. As you can see there are four. So =. 8 5 = 6 3 We have 5. Representing that would be 6 Since the question we need to answer is how many ' 3 s are there in 5, we need to use the cuts 6 for thirds only. Then count each of the one thirds. As you can see that are. So 5 =. 6 3 Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of

23 Be careful to choose division examples that are easy to represent in visual form. Well, because some enjoy playing with numbers, they found a quick way of dividing fractions. They did this by looking at fractions that were to be divided and they noticed a pattern. And here is what they noticed. Algorithm for Dividing Fractions and Mixed Numbers. Make sure you have proper or improper fractions.. Invert the divisor ( nd number). 3. Cancel, if possible.. Multiply numerators. 5. Multiply denominators. 6. Determine your sign and simplify. The very simple reason we tip the divisor upside-down (use the reciprocal), then multiply for division of fractions is because it works. And it works faster than if we did repeated subtractions, not to mention it takes less time and less space i 5 (Invert the divisor.) 5 7 = 8 8 Multiply numerators and denominators, and simplify Make sure you have proper or improper fractions i Invert the divisor i Cancel 0 and by, and cancel 9 and 3 by i = Multiply numerators and denominators. 5 = 7 Simplify. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page 3 of

24 Computing with Fractions and Signed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. The rules for adding, subtracting, multiplying and dividing fractions with signed numbers are the same as before, the only difference is you integrate the rules for integers. 3 7 Invert divisor 3 7 i 5 = 8 8 Multiply numerators and denominators, determine sign and simplify. 9 5 i i i = + = 8 8 Solving Equations Containing Fractions Syllabus Objective: (3.5) The student will solve equations and inequalities in one variable with integer solutions. Strategy for Solving Equations: You solve equations containing fractions and signed numbers the same as you do with whole numbers; the strategy does not change. To solve linear equations, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use opposite (inverse) operations. Solve: Solve: x + = x + = x = 5 x = 5 3 x = x 5 5 = 3 0 x = or Undo adding one-third by subtracting onethird from both sides of the equation; make equivalent fractions with a common denominator of 5. Undo dividing by 5 by multiplying both sides by 5. Cancel. Multiply numerators and denominators, determine sign and simplify. Math 7, Unit 03: Applying Rational Numbers Holt: Chapter 3 Page of