Fractions 02/12/112lntaylor ©
Table of Contents Learning Objectives/Previous Knowledge Basic rules of fractions Adding fractions Adding and subtracting more than two fractions Ipsative Choice Multiplying fractions Cross Cancellation Dividing fractions Simplify polynomials with fractions /12/112lntaylor ©
Learning Objectives LO 1 LO 2 Understand what a fraction represents Perform basic operations with fractions LO 3Simplify expressions and solve equations with fractions TOC 02/12/112lntaylor ©
Definitions Definition 1A fraction is a way of expressing part of a whole number TOC Definition 2 A fraction is also called a ratio and is part of the rational number set Definition 3 A fraction consists of a numerator (the top) which represents the pieces Definition 4A fraction consists of a denominator (the bottom) which represents how many pieces in the whole number 02/12/112lntaylor ©
Previous knowledge PK 1 PK 2 Basic Operations and Properties Combine Like Terms PK 3Exponent Rules TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Adding and subtracting fractions requires cross multiplication Multiplying fractions requires straight across multiplication Rule 3Dividing requires flipping a fraction and multiplying straight across Rule 4Learn to “get rid” of fractions by turning expressions into equations Basic Rules of Fractions TOC 02/12/112lntaylor ©
Adding Fractions TOC 02/12/112lntaylor ©
Step 1 Step 2 Construct matrix with numerators on top and denominators on side Blank out boxes diagonally Step 3Multiply matrix Step 4Add the results; this becomes the numerator = 29 5 x 7 = Step 5Multiply left side numbers (denominators); this becomes the denominator 35 Step 6Reduce fraction if possible TOC 02/12/112lntaylor ©
Now you try TOC 02/12/112lntaylor ©
Step 1 Step 2 Construct matrix with numerators on top and denominators on side Blank out boxes diagonally Step 3Multiply matrix Step 4Add the results; this becomes the numerator = 41 4 x 7 = Step 5Multiply left side numbers (denominators); this becomes the denominator 28 Step 6Reduce fraction if possible TOC 02/12/112lntaylor ©
Now you try 3 ─ TOC 02/12/112lntaylor ©
Step 1 Step 2 Construct matrix with numerators on top and denominators on side Blank out boxes diagonally Step 3Multiply matrix Step 4Add the results; this becomes the numerator 3 ─ = 1 4 x 7 = Step 5Multiply left side numbers (denominators); this becomes the denominator 28 Step 6Reduce fraction if possible TOC 02/12/112lntaylor ©
Adding/Subtracting more than 2 fractions TOC 02/12/112lntaylor ©
Step 1 Step 2 Construct cascading matrix Blank out boxes diagonally Step 3Multiply matrix; you can only multiply by the box above! Step 4Add the results; this becomes the numerator Step 5Multiply left side numbers (denominators); this becomes the denominator Step 6Reduce fraction if possible ─ = 67 4 x 7 x 3 = 8484 TOC 02/12/112lntaylor ©
Now you try TOC 02/12/112lntaylor ©
Step 1 Step 2 Construct cascading matrix Blank out boxes diagonally Step 3Multiply matrix; you can only multiply by the box above! Step 4Add the results; this becomes the numerator Step 5Multiply left side numbers (denominators); this becomes the denominator Step 6Reduce fraction if possible ─ = x 7 x 6 = = TOC 02/12/112lntaylor ©
Is there another method? TOC 02/12/112lntaylor ©
Rooftop Method TOC 02/12/112lntaylor ©
Step 1 Step 2 Build a rooftop Add the results; this is your numerator Step 3Multiply the denominators; this is your denominator Step Reduce fraction if possible ─ x 7 x 6 = x 5 x 6 = x 7 (-1) = x 7 x 6 = = TOC 02/12/112lntaylor ©
Now you try! TOC 02/12/112lntaylor ©
Step 1 Step 2 Build a rooftop Add the results; this is your numerator Step 3Multiply the denominators; this is your denominator Step Reduce fraction if possible x 7 x 3 = x 5 x 3 = x 7 x 1 = x 7 x 3 = TOC 02/12/112lntaylor ©
Ipsative Choice Decide which method you will master Matrix or Rooftop? TOC 02/12/112lntaylor ©
Define Decide Ipsative choice means “forced choices” Either choice works – matrix or rooftop method MasterDo all problems the same way until you have mastered the method Ipsative Choice TOC 02/12/112lntaylor ©
Multiplying Fractions TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Adding and subtracting fractions requires cross multiplication Multiplying fractions requires straight across multiplication Rule 3Dividing requires flipping a fraction and multiplying straight across Rule 4Learn to “get rid” of fractions by turning expressions into equations Basic Rules of Fractions TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Multiply numerators; this becomes the new numerator Multiply denominators; this becomes the new denominator Rule 3Reduce fraction if possible (5)(5)= TOC 02/12/112lntaylor ©
Now you try! TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Multiply numerators; this becomes the new numerator Multiply denominators; this becomes the new denominator Rule 3Reduce fraction if possible (3)(3)= TOC 02/12/112lntaylor ©
Cross Cancellation TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Numerators can be moved anytime YOU want Reduce fraction Rule 3Multiply straight across (7)(7) (4) x 7 = 7 2 x 4 = 8 Rule 4Reduce fraction if possible TOC 02/12/112lntaylor ©
Now you try! TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Numerators can be moved anytime YOU want Reduce fraction Rule 3Multiply straight across (5)(5) (4) x 5 = 5 3 x 4 = 12 Rule 4Reduce fraction if possible TOC 02/12/112lntaylor ©
Dividing Fractions TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Adding and subtracting fractions requires cross multiplication Multiplying fractions requires straight across multiplication Rule 3Dividing requires flipping a fraction and multiplying straight across Rule 4Learn to “get rid” of fractions by turning expressions into equations Basic Rules of Fractions TOC 02/12/112lntaylor ©
Divide / TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Write top fraction Flip bottom fraction Rule 3Check for cross cancellation; you can here but we will skip it Rule 4Multiply straight across ─ x 5 = 15 4 x 9 = 36 Rule 5Reduce fraction if possible 5 12 TOC 02/12/112lntaylor ©
Now you try! / TOC 02/12/112lntaylor ©
Rule 1 Rule 2 Write top fraction Flip bottom fraction Rule 3Check for cross cancellation; none here Rule 4Multiply straight across ─ x 7 = 21 5 x 4 = 40 Rule 5Reduce fraction if possible TOC 02/12/112lntaylor ©
Simplify Expressions with Fractions TOC 02/12/112lntaylor ©
Simplify 2x 2 + 4x – 10x 3 5 TOC 02/12/112lntaylor ©
Step 4 Step 5 Combine like terms if necessary Divide by the y coefficient Step 6Simplify if possible Step 7You can erase the “= y ” if you want Step 2 Step 1 Turn the expression into an equation by introducing “ = y” Every term gets a denominator Step 3 Multiply every term’s numerator with every other denominator (Roof top method) 2x² 3 + 4x– 10x 1 5 =y (5)(1) 2x²+ 4x (3)(1) – 10x (3)(5)(1)(3)(5) = y 10x²+ 12x– 150x= 15y 10x² – 138x = 15y 10x² – 138x = y 15 x (10x – 138) 15 TOC 02/12/112lntaylor ©
Now you try! 2x 2 + 3x – 10x 7 5 TOC 02/12/112lntaylor ©
Step 4 Step 5 Combine like terms if necessary Divide by the y coefficient Step 6Simplify if possible Step 7You can erase the “= y ” if you want Step 2 Step 1 Turn the expression into an equation by introducing “ = y” Every term gets a denominator Step 3 Multiply every term’s numerator with every other denominator (Roof top method) 2x² 7 + 3x– 10x 1 5 =y (5)(1) 2x²+ 3x (7)(1) – 10x (7)(5)(1)(7)(5) = y 10x²+ 21x– 350x= 35y 10x² – 329x = 35y 10x² – 329x = y 35 x (10x – 329) 35 TOC 02/12/112lntaylor ©
End Fractions TOC 02/12/112lntaylor ©