Rational Numbers Module 3 pg. 57-102

3.1Rational Numbers and Decimals What is a rational number? Rational Numbers a number that can be written as a ratio of two integers a and b, where b is not ZERO. When divided it either terminates or repeats. How can you convert a rational number to a decimal?

Complete page 61 with a partner

Review division 1 2 that means 1 divided by 2 0. 5 2 1.0 10 -----

Writing Rational Numbers as Decimals read page 62 and do Terminating decimal a decimal that comes to an end when divided Repeating decimal a decimal when one or more digits repeat Together 1 5 1 6 Your Turn 15 20 2 3

Let’s Practice 2 8 3 18 7 12 4 7

Let’s Practice 2 8 =.25 3 18 = 0.16 7 12 = 0.583 4 7 = 0.571428

More Your Turn 5 16 13 33

Writing Mixed Numbers as Decimals read page 63 and do You rewrite the fractional part of the number as a decimal. Example: 8 9 20 = 8 + ( 9 20 ) = 8 + .45 = 8.45 Together 5 1 4 2 2 4

Practice 12 1 4 9 3 4 7 2 3 9 12 24

Guided Practice– CHECK Your Turn check- page 63 #4-8- CHECK Guided Practice– CHECK Complete 3.1 Practice and Problem Solving A/B worksheet

Fraction Review What is a fraction? Why do we use fractions? The parts of the fraction

Terminology NUMERATOR __________________ DENOMINATOR Simplest form Mixed number Improper fraction Equivalent fractions

Simplest form? Example: 4 16

Making fractions into simplest form/reducing Find a factor that both terms have in common and reduce. Example: 4 16 they both can be divided by 4 4 16 ÷ 4 = 1 4

Your Turn 4 24 8 20 2 6 3 15

4 24 = 1 6 8 20 = 2 5 2 6 = 1 3 3 15 = 1 5

Equivalent Fractions 3 8 = ? 16 1 2 = ? 12 10 25 = ? 5

3 8 = 6 16 1 2 = 6 12 10 25 = 2 5

Mixed Number to fraction 3 1 7 This means you have 3 wholes and 1 7 . One way is think 1 whole equals 7 7 . If we have 3 of these, we have 21 7 + 1 7 = 22 7 Or you have 3 1 7 = 22 7 + x

Your Turn 2 4 7 = 1 2 9 4 3 8 =

2 4 7 = 18 7 1 2 9 = 11 9 4 3 8 = 35 8

Fraction to Mixed Number You divide to get a fraction to a mixed number 1= whole # 11 7 = 11÷ 7 = 7 11 - 7 ---------- denominator 4(numerator) = 1 4 7 7

Your Turn 12 7 = 24 6 = 18 4 =

12 7 = 1 5 7 24 6 = 4 18 4 = 4 1 2

3.2 Adding Rational Numbers Layered Book creation Take 2 sheets of paper and fold them Staple and label them Title: Rational Numbers Adding Subtracting Multiplying Dividing

3.2 Adding Rational Numbers Adding Rational Numbers with the Same Sign Apply the rules for adding integers REMINDER: Same sign: find the absolute value, ADD, keep the same sign

Using a number line Using a number line is very helpful- 1. Find the scale This scale is by 1’s. You could cut them in half and use 0.5

Practice Add 2.5 + 1.5 First determine what scale would work and then use the number line just like we know how. Begin at 2.5 and move 1.5 to the right.

Adding Decimals LINE UP THE DECIMAL POINT ADD 7.5 + 2.13 7.5 + 2.13 + 2.13 --------------- 9.63

Same with fractions First determine what scale would work? Then use the number line just like we know how. 3+ 1 5 =

Determine the scale 3 1 7 + 3 7 = -1 3 4 + (-1 2 4 )= 4 5 + 2 5 =

Adding fractions Same denominator If the denominator is the same. ADD the numerator. Reduce to simplest form in needed 1 4 + 2 4 = 2 5 + 1 5 =

Practice 1 3 + 2 3 = - 1 4 + (- 2 4 )= 2 3 + 2 3 =

3 1 3 + 2 3 =4 -4 1 4 + (-2 2 4 )=- 3 3 4 2 3 + 2 3 = 1 1 3

Different denominator 3 8 + 3 4 = Find a common denominator = 8 Change to equivalent fractions 3 8 = 3 8 + 3 4 = 6 8 so 3 8 + 6 8 = 9 8 =1 1 8

What if they have different signs? Opposites signs--- you SUBTRACT using the same skills you learned for adding! 4.5 + (-6.5) = - 1 4 + 3 4 = 9 + (-15) =

What if they have different signs? Opposites signs--- you SUBTRACT using the same skills you learned for adding! 4.5 + (-6.5) =-2.5 - 1 4 + 3 4 = 2 4 9 + (-15) =-6

Additive Inverse The opposite or additive inverse, of a number is the same distance from 0 on a number line as the original number but on the other side of 0. What is the additive inverse of …. -3.5 = 4 1 4 = -8=

Additive Inverse The opposite or additive inverse, of a number is the same distance from 0 on a number line as the original number but on the other side of 0. What is the additive inverse of …. -3.5 = 3.5 4 1 4 = -4 1 4 -8= 8

Adding three or more rational numbers When adding you can use the Associative property and group all the positives and all the negatives. Or you can just go from right to left. Either way you get the same answer.

-5.25 + 3.75 +4.50 = - 1 4 + 3 4 + (- 2 4 ) = -8 + (-9) + 23 =

-5.25 + 3.75 +4.50 =3 - 1 4 + 3 4 + (- 2 4 ) = 0 -8 + (-9) + 23 = 6

3.3 Subtracting Rational Numbers What are rational numbers? Rational Numbers a number that can be written as a ratio of two integers a and b, where b is not ZERO. When divided it either terminates or repeats. When you subtract rational numbers, you use the same rules as subtracting integers!

What is the rule for subtracting integers?

Then follow the addition rules ADD THE OPPOSITE Then follow the addition rules

Video http://my.hrw.com/content/hmof/math/common/tools/videoplayer/player.html?contentSrc=7289/7289.xml Complete page 75-76

Partner Brainstorming Let’s explore! Work with your partner on page 77 and 78! Check points: #9 & #11

Practice time 6- (-7) = −3 3 4 - 4 3 4 = -8 – 3 = -0.75 – 4.75 =

6- (-7) = 6 + 7 = 13 −3 3 4 - 4 3 4 = −3 3 4 + (- 4 3 4 ) = −7 6 4 = -8 1 2 -8 – 3 = -8 + (-3) = -11 -0.75 – 4.75 = -0.75 + (-4.75)= -5.5

Guided Practice Work through page 79 with the following check points #4 #10 #14

3.4 Multiplying Rational Numbers The rules for the signs for rational numbers is the same as the rules for integers

3.4 Multiplying Rational Numbers The rules for the signs for rational numbers is the same as the rules for integers Same signs = positive product Opposite signs= negative product Signs of factor Sign of factor Sign of product + -

Let’s practice Take out your white boards and lets do the questions one at a time. -6(-5) = 9 x 3 = (-10)(3) = -2 (-2)(-3)= 4 (-2)(3) =

Decimal Review 3.6 x 3= 1 3.6 (1 space to move the decimal) X 3 --------- 10 8 1 space to move the decimal) Multiply and then look at the placement of the decimal. Move the decimal to the end of the number and count the spaces. In the product move the decimal the same number of spaces to the left. PRODUCT= 10.8 Check to make sure the answer is practical by rounding 3.6 is about 4 and 3 = 12. It is close so it makes sense.

Let’s Practice- Together -3 x -1.75 = 0.54(6) = -2.5 (2)= 1.8 x 5 =

Let’s Practice -3 x -1.75 = 5.25 0.54(6) = 3.24 -2.5 (2)= -5

Independent Practice -3 (-1.25)= 2(-3.5)= 1.2 ( 2.4) =

Independent Practice -3 (-1.25)= 3.75 2(-3.5)= -7 1.2 ( 2.4) = 2.88

What happens with 3 numbers If you have three numbers, take two at a time. Example: -1.5 (-2) x -4.2 3 x -4.2 -12.6 Worksheet practice- Individual Check points 4, 8, 12

Fraction Review ( 3 4 ) ( 1 2 )= 3 8 Multiply the numerator and the denominator. Simplify if needed ( 5 8 ) ( 2 10 )= 10 80 = 1 8

Or you can cross cancel When you look at the two fractions you will be multiplying, you can look to see if you can reduce before multiplying. 2 You can reduce either way. Cross cancel or simplify at the end. 1 ( 5 8 ) ( 3 10 )= 3 16

Let’s Practice 2 2 4 x 8 = 1 8 x 2 3 x 4 6 = - 1 5 x - 10 15 =

Let’s Practice 2 2 4 x 8 = 20 1 8 x 2 3 x 4 6 = 1 18 - 1 5 x - 10 15 = 2 15 -7 x - 1 5 = 1 2 5

3.5 Dividing Rational Numbers How do you divide rational numbers?

Lets Explore Select a partner and work to explore on page 89 and 90

Rules Same signs = positive quotient Opposite signs= negative quotient Sign of dividend Sign of divisor Sign of quotient + - Same signs = positive quotient Opposite signs= negative quotient

Dividing decimal review 4 ÷ 20 = 0.5 ÷ 20 = 4 ÷ 5.5= 24.24 ÷ 0.06=

Dividing decimal review 4 ÷ 20 =0.2 0.5 ÷ 20 = 0.025 4 ÷ 5.5= 0.72….. 24.24 ÷ 0.06= 404

Dividing Fraction Review Take one whole and divide by ½- how many pieces do you have? Take 4 and divide by ¼- how many pieces do you have? What conjecture can you form about dividing fractions?

When you divide fractions you multiply by the reciprocal. Practice 3 16 ÷ 1 8 = 7 ÷ 2 14 = 7 12 ÷ 2 9 =

Practice 3 16 ÷ 1 8 = 3 2 or 1 1 2 7 ÷ 2 14 = 49 7 12 ÷ 2 9 = 2 5 8

3.6 Applying Rational Number Operations How do you use different forms of rational numbers and strategically choose tools to solve problems? This is our goal: what does this mean? Different forms? Tools?

Picture Problem I want to hang a new picture of my family on wall. I want to be sure it is centered horizontally on the wall. How far from each edge of the wall should I place the picture?

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Holiday Baking Time Eila uses flour for each batch of cookies she makes. How many batches can Eila make if she uses all the flour? How much does the flour cost for one batch?

Eila uses flour for each batch of cookies she makes Eila uses flour for each batch of cookies she makes. How many batches can Eila make if she uses all the flour? How much does the flour cost for one batch? Requested information: Eila uses 1 1 4 cup for each batch She bought a 5 pound bag for $4.49 5 pounds contains seventy-six 1 4 - cup servings

The depth of Golden Trout Lake has been decreasing in recent years The depth of Golden Trout Lake has been decreasing in recent years. What is the depth today?

The depth of Golden Trout Lake has been decreasing in recent years The depth of Golden Trout Lake has been decreasing in recent years. What is the depth today? Requested information: Two years ago, the depth of the lake was 186.73 meters. The depth has changed at an average rate of -1 3 4 % per year.