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4.13: Solve Equations with Fractions (Part 2) - Mathematics


Solve Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to (1). For example, in the equation:

[dfrac{3}{4}x = 24 onumber ]

The coefficient of (x) is (dfrac{3}{4}). To solve for (x), we need its coefficient to be (1). Since the product of a number and its reciprocal is (1), our strategy here will be to isolate (x) by multiplying by the reciprocal of (dfrac{3}{4}). We will do this in Example (PageIndex{1}).

Example (PageIndex{8}): solve

Solve: (dfrac{3}{4}x = 24).

Solution

Multiply both sides by the reciprocal of the coefficient.( extcolor{red}{dfrac{4}{3}} cdot dfrac{3}{4} x = extcolor{red}{dfrac{4}{3}} cdot 24 )
Simplify.(1x = dfrac{4}{3} cdot dfrac{24}{1} )
Multiply.(x = 32 )

Check:

Substitute x = 32.(dfrac{3}{4} cdot 32 stackrel{?}{=} 24 )
Rewrite 32 as a fraction.(dfrac{3}{4} cdot dfrac{32}{1} stackrel{?}{=} 24 )
Multiply. The equation is true.(24 = 24 ; checkmark)

Notice that in the equation (dfrac{3}{4} x = 24), we could have divided both sides by (dfrac{3}{4}) to get (x) by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

Exercise (PageIndex{15})

Solve: (dfrac{2}{5}n = 14).

Answer

(35)

Exercise (PageIndex{16})

Solve: (dfrac{5}{6}y = 15).

Answer

(18)

Example (PageIndex{9}): solve

Solve: (− dfrac{3}{8}w = 72).

Solution

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.

Multiply both sides by the reciprocal of (− dfrac{3}{8}).( extcolor{red}{- dfrac{8}{3}} left(- dfrac{3}{8} w ight) = left( extcolor{red}{- dfrac{8}{3}} ight) 72 )
Simplify; reciprocals multiply to one.(1w = - dfrac{8}{3} cdot dfrac{72}{1} )
Multiply.(w = -192)

Check:

Let w = −192.(- dfrac{3}{8} (-192) stackrel{?}{=} 72 )
Multiply. It checks.(72 = 72 ; checkmark )

Exercise (PageIndex{17})

Solve: (− dfrac{4}{7}a = 52).

Answer

(-91)

Exercise (PageIndex{18})

Solve: (− dfrac{7}{9}w = 84).

Answer

(-108)

Translate Sentences to Equations and Solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Table (PageIndex{2})
Subtraction Property of Equality: For any real numbers a, b, and c, if a = b, then a − c = b − c.Addition Property of Equality: For any real numbers a, b, and c, if a = b, then a + c = b + c.
Division Property of Equality: For any numbers a, b, and c, where c ≠ 0 if a = b, then (dfrac{a}{c} = dfrac{b}{c}).Multiplication Property of Equality: For any real numbers a, b, and c if a = b, then ac = bc.

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality. In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

Example (PageIndex{10}): solve

Translate and solve: (n) divided by (6) is (−24).

Solution

Translate.
Multiply both sides by 6.( extcolor{red}{6} cdot dfrac{n}{6} = extcolor{red}{6} (-24))
Simplify.(n = -144 )
Check:Is −144 divided by 6 equal to −24?
Translate.(dfrac{-144}{6} stackrel{?}{=} -24)
Simplify. It checks.(-24 = -24 ; checkmark )

Exercise (PageIndex{19})

Translate and solve: (n) divided by (7) is equal to (−21).

Answer

(dfrac{n}{7} = -21); (n=-147)

Exercise (PageIndex{20})

Translate and solve: (n) divided by (8) is equal to (−56).

Answer

(dfrac{n}{8} = -56); (n=-448)

Example (PageIndex{11}): solve

Translate and solve: The quotient of (q) and (−5) is (70).

Solution

Translate.
Multiply both sides by −5.( extcolor{red}{-5} left(dfrac{q}{-5} ight) = extcolor{red}{-5} (70) )
Simplify.(q = -350)
Check:Is the quotient of −350 and −5 equal to 70?
Translate.(dfrac{-350}{-5} stackrel{?}{=} 70 )
Simplify. It checks.(70 = 70 ; checkmark )

Exercise (PageIndex{21})

Translate and solve: The quotient of (q) and (−8) is (72).

Answer

(dfrac{q}{-8} = 72); (q=-576)

Exercise (PageIndex{22})

Translate and solve: The quotient of (p) and (−9) is (81).

Answer

(dfrac{p}{-9} = 81); (p=-729)

Example (PageIndex{12}): solve

Translate and solve: Two-thirds of (f) is (18).

Solution

Translate.
Multiply both sides by (dfrac{3}{2}).( extcolor{red}{dfrac{3}{2}} cdot dfrac{2}{3} f = extcolor{red}{dfrac{3}{2}} cdot 18 )
Simplify.(f = 27 )
Check:Is two-thirds of 27 equal to 18?
Translate.(dfrac{2}{3} (27) stackrel{?}{=} 18)
Simplify. It checks.(18 = 18 ; checkmark )

Exercise (PageIndex{23})

Translate and solve: Two-fifths of (f) is (16).

Answer

(dfrac{2}{5}f = 16); (f=40)

Exercise (PageIndex{24})

Translate and solve: Three-fourths of (f) is (21).

Answer

(dfrac{3}{4}f = 21); (f=28)

Example (PageIndex{13}): solve

Translate and solve: The quotient of (m) and (dfrac{5}{6}) is (dfrac{3}{4}).

Solution

Translate.(dfrac{m}{dfrac{5}{6}} = dfrac{3}{4} )
Multiply both sides by (frac{5}{6}) to isolate m.(dfrac{5}{6} left(dfrac{m}{dfrac{5}{6}} ight) = dfrac{5}{6} left(dfrac{3}{4} ight) )
Simplify.(m = dfrac{5 cdot 3}{6 cdot 4})
Remove common factors and multiply.(m = dfrac{5}{8} )

Check:

Is the quotient of (dfrac{5}{8}) and (dfrac{5}{6}) equal to (dfrac{3}{4})?(dfrac{dfrac{5}{8}}{dfrac{5}{6}} stackrel{?}{=} dfrac{3}{4} )
Rewrite as division.(dfrac{5}{8} div dfrac{5}{6} stackrel{?}{=} dfrac{3}{4} )
Multiply the first fraction by the reciprocal of the second.(dfrac{5}{8} cdot dfrac{6}{5} stackrel{?}{=} dfrac{3}{4} )
Simplify.(dfrac{3}{4} = dfrac{3}{4} ; checkmark )

Our solution checks.

Exercise (PageIndex{25})

Translate and solve. The quotient of (n) and (dfrac{2}{3}) is (dfrac{5}{12}).

Answer

(dfrac{n}{dfrac{2}{3}} = dfrac{5}{12}); (n = dfrac{5}{18})

Exercise (PageIndex{26})

Translate and solve. The quotient of (c) and (dfrac{3}{8}) is (dfrac{4}{9}).

Answer

(dfrac{c}{dfrac{3}{8}} = dfrac{4}{9}); (c = dfrac{1}{6})

Example (PageIndex{14}): solve

Translate and solve: The sum of three-eighths and (x) is three and one-half.

Solution

Translate.
Use the Subtraction Property of Equality to subtract (dfrac{3}{8}) from both sides.(dfrac{3}{8} + x - dfrac{3}{8} = 3 dfrac{1}{2} - dfrac{3}{8} )
Combine like terms on the left side.(x = 3 dfrac{1}{2} - dfrac{3}{8} )
Convert mixed number to improper fraction.(x = 3 dfrac{1}{2} - dfrac{3}{8} )
Convert to equivalent fractions with LCD of 8.(x = dfrac{7}{2} - dfrac{3}{8} )
Subtract.(x = dfrac{25}{8} )
Write as a mixed number.(x = 3 dfrac{1}{8} )

We write the answer as a mixed number because the original problem used a mixed number. Check: Is the sum of three-eighths and (3 dfrac{1}{8}) equal to three and one-half?

Add.(3 dfrac{4}{8} stackrel{?}{=} 3 dfrac{1}{2} )
Simplify.(3 dfrac{1}{2} = 3 dfrac{1}{2} )

The solution checks.

Exercise (PageIndex{27})

Translate and solve: The sum of five-eighths and (x) is one-fourth.

Answer

(dfrac{5}{8}+x = dfrac{1}{4}); (x = -dfrac{3}{8})

Exercise (PageIndex{28})

Translate and solve: The difference of one-and-three-fourths and (x) is five-sixths.

Answer

(1dfrac{3}{4} - x = dfrac{5}{6}); (x = dfrac{11}{12})

Practice Makes Perfect

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

  1. x − (dfrac{2}{5}) = (dfrac{1}{10}):
    1. x = 1
    2. x = (dfrac{1}{2})
    3. x = (− dfrac{1}{2})
  2. y − (dfrac{1}{2}) = (dfrac{5}{12}):
    1. y = 1
    2. y = (dfrac{3}{4})
    3. y = (- dfrac{3}{4})
  3. h + (dfrac{3}{4}) = (dfrac{2}{5}):
    1. h = 1
    2. h = (dfrac{7}{20})
    3. h = (- dfrac{7}{20})
  4. k + (dfrac{2}{5}) = (dfrac{5}{6}):
    1. k = 1
    2. k = (dfrac{13}{30})
    3. k = (- dfrac{13}{30})

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

  1. y + (dfrac{1}{3}) = (dfrac{4}{3})
  2. m + (dfrac{3}{8}) = (dfrac{7}{8})
  3. f + (dfrac{9}{10}) = (dfrac{2}{5})
  4. h + (dfrac{5}{6}) = (dfrac{1}{6})
  5. a − (dfrac{5}{8}) = (- dfrac{7}{8})
  6. c − (dfrac{1}{4}) = (- dfrac{5}{4})
  7. x − (left(- dfrac{3}{20} ight)) = (- dfrac{11}{20})
  8. z − (left(- dfrac{5}{12} ight)) = (- dfrac{7}{12})
  9. n − (dfrac{1}{6}) = (dfrac{3}{4})
  10. p − (dfrac{3}{10}) = (dfrac{5}{8})
  11. s + (left(- dfrac{1}{2} ight)) = (- dfrac{8}{9})
  12. k + (left(- dfrac{1}{3} ight)) = (- dfrac{4}{5})
  13. 5j = 17
  14. 7k = 18
  15. −4w = 26
  16. −9v = 33

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

  1. (dfrac{f}{4}) = −20
  2. (dfrac{b}{3}) = −9
  3. (dfrac{y}{7}) = −21
  4. (dfrac{x}{8}) = −32
  5. (dfrac{p}{-5}) = −40
  6. (dfrac{q}{-4}) = −40
  7. (dfrac{r}{-12}) = −6
  8. (dfrac{s}{-15}) = −3
  9. −x = 23
  10. −y = 42
  11. −h = (− dfrac{5}{12})
  12. −k = (− dfrac{17}{20})
  13. (dfrac{4}{5})n = 20
  14. (dfrac{3}{10})p = 30
  15. (dfrac{3}{8})q = −48
  16. (dfrac{5}{2})m = −40
  17. (- dfrac{2}{9})a = 16
  18. (- dfrac{3}{7})b = 9
  19. (- dfrac{6}{11})u = −24
  20. (- dfrac{5}{12})v = −15

Mixed Practice

In the following exercises, solve.

  1. 3x = 0
  2. 8y = 0
  3. 4f = (dfrac{4}{5})
  4. 7g = (dfrac{7}{9})
  5. p + (dfrac{2}{3}) = (dfrac{1}{12})
  6. q + (dfrac{5}{6}) = (dfrac{1}{12})
  7. (dfrac{7}{8})m = (dfrac{1}{10})
  8. (dfrac{1}{4})n = (dfrac{7}{10})
  9. (- dfrac{2}{5}) = x + (dfrac{3}{4})
  10. (- dfrac{2}{3}) = y + (dfrac{3}{8})
  11. (dfrac{11}{20}) = −f
  12. (dfrac{8}{15}) = −d

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

  1. n divided by eight is −16.
  2. n divided by six is −24.
  3. m divided by −9 is −7.
  4. m divided by −7 is −8.
  5. The quotient of f and −3 is −18.
  6. The quotient of f and −4 is −20.
  7. The quotient of g and twelve is 8.
  8. The quotient of g and nine is 14.
  9. Three-fourths of q is 12.
  10. Two-fifths of q is 20.
  11. Seven-tenths of p is −63.
  12. Four-ninths of p is −28.
  13. m divided by 4 equals negative 6.
  14. The quotient of h and 2 is 43.
  15. Three-fourths of z is the same as 15.
  16. The quotient of a and (dfrac{2}{3}) is (dfrac{3}{4}).
  17. The sum of five-sixths and x is (dfrac{1}{2}).
  18. The sum of three-fourths and x is (dfrac{1}{8}).
  19. The difference of y and one-fourth is (- dfrac{1}{8}).
  20. The difference of y and one-third is (- dfrac{1}{6}).

Everyday Math

  1. Shopping Teresa bought a pair of shoes on sale for $48. The sale price was (dfrac{2}{3}) of the regular price. Find the regular price of the shoes by solving the equation (dfrac{2}{3})p = 48
  2. Playhouse The table in a child’s playhouse is (dfrac{3}{5}) of an adult-size table. The playhouse table is 18 inches high. Find the height of an adult-size table by solving the equation (dfrac{3}{5})h = 18.

Writing Exercises

  1. Example 4.100 describes three methods to solve the equation −y = 15. Which method do you prefer? Why?
  2. Richard thinks the solution to the equation (dfrac{3}{4})x = 24 is 16. Explain why Richard is wrong.

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


Whole numbers and properties

Integers and evaluating expressions

  • 16 Divisibility Rules
  • 17 Factors and Primes
  • 18 Prime Factorization
  • 19 Multiples and Least Common Multiple
  • 20 Greatest Common Factor
  • 21 Introduction to Fractions
  • 22 Equivalent Fractions (Part I)
  • 23 Reducing Fractions to Lowest Terms
  • 24 Equivalent Fractions (Part II)
  • 25 Improper Fractions and Mixed Numbers
  • 26 Comparing Proper Fractions
  • 27 Comparing Fractions Word Problems
  • 28 Adding and Subtracting Like Fractions
  • 29 Adding and Subtracting Unlike Fractions
  • 30 Adding Mixed Numbers
  • 31 Subtracting Mixed Numbers
  • 32 Multiplying Fractions
  • 33 Multiplying Mixed Numbers
  • 34 Dividing Fractions
  • 35 Dividing Mixed Numbers
  • 36 Adding and Subtracting Fractions
  • 37 Multiplying and Dividing Fractions
  • 38 Understanding Decimals
  • 39 Converting Decimals to Fractions
  • 40 Converting Fractions to Decimals
  • 41 Comparing Decimals
  • 42 Rounding Decimals
  • 43 Estimating Sums and Differences of Decimals
  • 44 Adding and Subtracting Decimals Word Problems
  • 45 Multiplying and Dividing Decimals Word Problems
  • 46 Powers of 10
  • 47 Converting from Scientific to Standard Notation
  • 48 Converting from Standard to Scientific Notation
  • 49 Terminating and Repeating Decimals
  • 50 Determining if a Number is Rational or Irrational

Simplifying expressions & solving equations

  • 51 Combining Like Terms
  • 52 Distributive Property
  • 53 Distributive Property and Combining Like Terms
  • 54 One-Step Equations
  • 55 Two-Step Equations
  • 56 Equations with Fractions
  • 57 Equations Involving Distributive
  • 58 Equations with Variable on Both Sides
  • 59 Equations with Variable on Both Sides and Fractions
  • 60 Equations with Variable on Both Sides and Distributive
  • 61 Equations with Decimals
  • 62 Equations with Decimals and Decimal Solutions
  • 63 Equations with Fraction Solutions
  • 64 Literal Equations
  • 65 Advanced Literal Equations

Ratio, proportion, & percent

  • 66 Introduction to Ratios
  • 67 Equal Ratios
  • 68 Unit Rate
  • 69 Introduction to Proportion
  • 70 Solving Proportions
  • 71 Proportion Word Problems
  • 72 Understanding Percents
  • 73 Fractions and Percents
  • 74 Decimals and Percents
  • 75 Percent Word Problems
  • 76 Percent Increase or Decrease
  • 77 Discount
  • 78 Sales Tax
  • 79 Interest

Inequalities and absolute value

Graphing and transformations

Linear equations and functions

  • 106 Recognizing Patterns
  • 107 Word Problems and Table Building
  • 108 Slope as a Rate of Change
  • 109 Slope of a Line
  • 110 Using Slope to Graph a Line
  • 111 Slope Formula
  • 112 Slope-Intercept Form
  • 113 Converting to Slope-Intercept Form and Graphing
  • 114 Interpreting Graphs
  • 115 Linear Parent Graph and Transformations
  • 116 Writing Equations of Lines
  • 117 Writing Equations of Lines Using Tables
  • 118 Direct Variation
  • 119 Applications of Direct Variation and Linear Functions
  • 120 Writing Equations of Lines in Standard Form
  • 121 Writing Equations of Lines Using the Point-Slope Formula
  • 122 Writing Equations of Lines Given Two Points
  • 123 Writing Equations of Parallel and Perpendicular Lines

Systems of linear equations

Exponents and polynomials

  • 136 Greatest Common Factor
  • 137 Factoring out the Greatest Common Factor
  • 138 Factoring Trinomials with Positive Constants
  • 139 Factoring Trinomials with Negative Constants
  • 140 Difference of Two Squares
  • 141 Factoring Trinomials with Lead Coefficients and Positive Constants
  • 142 Factoring Trinomials with Lead Coefficients and Negative Constants
  • 143 Factoring Completely
  • 144 Factoring by Grouping
  • 145 Beginning Polynomial Equations
  • 146 Intermediate Polynomial Equations

Rational expressions and equations

Radical expressions and equations

Imaginary and complex numbers

Quadratic functions and graphing

Angle pairs and perpendicular lines

Parallel lines and polygons

  • 228 Customary Unit Conversions
  • 229 Metric Unit Conversions
  • 230 Units of Measurement
  • 231 Area of Rectangles and Squares
  • 232 Advanced Area of Rectangles and Squares
  • 233 Area of Parallelograms
  • 234 Area of Triangles
  • 235 Area of Rhombuses
  • 236 Area of Trapezoids
  • 237 Area of Regular Polygons
  • 238 Area and Circumference of Circles
  • 247 Bar Graphs
  • 248 Line Graphs
  • 249 Circle Graphs
  • 250 Stem-and-Leaf Plots and Frequency Charts
  • 251 Histograms
  • 252 Scatterplots and Trends
  • 253 Range, Median, and Mode
  • 254 Box-and-Whisker Plots
  • 255 Mean
  • 256 Central Tendency Word Problems
  • 257 Simple Probability
  • 258 Experimental Probability
  • 259 Probability of Independent Events
  • 260 Probability of Dependent Events
  • 261 Simulations
  • 262 Tree Diagrams and the Counting Principle
  • 263 Permutations
  • 264 Combinations

Negative & rational exponents

Composite functions & inverses

Logarithms & exponential functions

  • 280 Evaluating Logarithms and Logarithmic vs. Exponential Form
  • 281 Solving Logarithmic Equations
  • 282 Logarithm Rules and Properties
  • 283 Evaluating Logarithms by Condensing or Expanding
  • 284 Solving Advanced Logarithmic Equations
  • 285 Logarithm Calculator Problems
  • 286 Exponential Equations and Change of Base Formula
  • 287 Exponential Growth and Decay
  • 288 Half Life and Doubling Time Formulas
  • 289 Natural Logarithms
  • 290 Solving Natural Logarithm Equations with ln and e

4.13: Solve Equations with Fractions (Part 2) - Mathematics

This unit will focus on understanding and recognizing the behaviour of the different families of functions, their domain and range, and what changes when functions are combined. The Topic we will be discussing are:

6. Partial fractions and other rational functions

7. Absolute value functions and equations

9. Classification of functions according to their mapping

10. Classification of functions according to their parity

11. Operations with functions

13. Transformations of functions

14. Properties of exponents

15. Exponential functions and equations

16. Properties of Logarithms

17. Logarithmic functions and equations

18. Trigonometric functions

19. Applications and modelling

In this lesson you will learn: (This is usually assumed known for HL)

- the concepts of domain and range.

- how to distinguish a functions from a relation using the vertical line test.

- to identify the graphs, end behaviour, domain and range of Linear functions.

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 72 - 85

- Watch: Video lessons part I on domain and range.

- Watch: Video Lesson part II on the Vertical Line Test

- Investigation: Please work on this investigation regarding the linear function. Solutions

- If you have any questions at any point, please ask them at this time.

- part III On linear and quadratic functions

Homework: Create a free account at inThinking at the link for the course Mathematics AA HL. Go to MAA then functions, then Equation of a straight line and do the two quizzes and the four ESQ.

In this lesson you will learn: (This is usually assumed known for HL)

- to understand the difference between graph and sketch.

- to identify important features on the graph of a function, such as x-intercepts, y-intercept, maxima, minima, asymptotes.

- to identify the graphs, end behaviour, domain and range of linear, quadratic, exponential, logarithmic, rational, radical and trigonometric functions

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 72 - 85

- part III On linear and quadratic functions

- Investigation: Please work on this investigation regarding the graphs of functions and the quadratic function.

- Submit: Look at the picture on Investigation 5, on page 83. Use it as inspiration to create your own original character. Please use Geogebra or Desmos to create your own character, playing with quadratics, and lines. Submit your picture here.

- If you have any questions at any point, please ask them at this time.

Task 3 prompt: Analysis of population rates of decrease and increase for various endangered species. How long will it take to bring them back to healthy levels?

Task 4: Choose your own topic.

Finish the exercises on quadratic functions.

In this lesson you will learn:

- determine the domain and range of rational and radical functions.

- to find the vertical and horizontal asymptotes of rational functions.

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 86 - 90.

- Watch: Video lessons part I, part II, Examples (optional) part III.

- Do: Mathematics Analysis and Approaches HL by Oxford, Exercise 2C (page 89) and Exercise 2D (page 90).

3 rational functions. How did you do?

In this lesson you will learn:

- determine the domain and range of rational and radical functions.

- interpret and graph piece-wise functions manually and using technology.

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 86 - 90.

- Homework check: Last lesson you were asked to complete the questions about application of rational functions. How did you do? See the solutions here.

- Investigation Piece-wise Functions. You will have only one lesson to complete this investigation.

In this lesson you will learn:

- to classify functions according to their mapping. (Part I)

- to classify functions according to their parity. (Part II)

- Read: Mathematics analysis and approaches by Oxford pages 102 - 108.

- Do: Mathematics Analysis and approaches by oxford, exercise 2K, page 105.

- Do: Mathematics Analysis and approaches by oxford, exercise 2L, page 108.

- Submit: Consider the information that you have learnt up to this point about functions. Please submit the answers to the exercises included in the following form.

In this lesson you will learn:

- how to add, subtract, multiply and divide functions. (Part I)

- to identify the domain and range of a resulting functions. (Part I)

- to do the composition of functions (Part II)

- to identify the domain and range of a composition. (Part II)

- Read: Mathematics analysis and approaches by Oxford pages 108 - 111.

- Watch: Video lessons Part I, Part II, Part III (examples 1, 2), Part IV (Examples 3, 4, 5)

- Do: Mathematics Analysis and approaches by oxford, exercise 2M, page 111.

- Do: Write a comment in the stream for this lesson giving your own example of a function that is the composition of three functions, f(g(h(x))) and state the individual functions f, g, and h. (example: f(g(h(x)))=1-(x+1)^2, where f(x)=1-x, g(x)=x^2, and h(x)=x+1)

In this lesson you will learn:

- The concept of identity function. (Part II)

- the concept of inverse function and how to find them. (Part I)

- the concept of self inverse. (Part II)

- how to find the domain and range of an inverse functions. (Part III)

- Read: Mathematics analysis and approaches by Oxford pages 112 - 116.

- Do: Mathematics Analysis and approaches by oxford, exercise 2N, page 116.

- Submit: Please submit the answers to the exercises included in the following form.

In this lesson you will learn:

- to transform functions using vertical and horizontal displacements. (Part I)

- to transform functions using magnifications. (Part II)

- to transform functions using reflections. (Part II)

- Read: Mathematics analysis and approaches by Oxford pages 117 - 127.

- Do: Mathematics Analysis and approaches by oxford, exercise 2O, page 118, questions 3, 5, and 6.

- Do: Mathematics Analysis and approaches by oxford, exercise 2P, page 120, question 2.

- Do: Mathematics Analysis and approaches by oxford, exercise 2Q, page 126, questions 1 and 3.

- Do: play this kahoot! (Link expires May 15th) Game PIN: 05886179

This is a practice lesson. In this lesson you will continue to learn:

- How to graph functions with multiple transformations

- How to write a function's equation undergoing multiple transformations

- Read: Mathematics analysis and approaches by Oxford pages 127 - 139.

- Watch: Video lessons Part I (Further Examples).

- Do: Mathematics Analysis and approaches by oxford, exercise 2R, page 134, question 2.

- Do: Mathematics Analysis and approaches by oxford, exercise 2S, page 139.

- Submit: Please submit the answers to the exercises included in the following form.

For this lesson you need to review beforehand:

- Properties of exponents (Bonus video)

In this lesson you will learn:

- How to solve equations with exponentials algebraically. (Part I)

- How to solve equations with exponentials using technology. (Part II)

- Graphs of exponential functions. (Part I)

- Read: Mathematics Analysis and approaches HL by Oxford, pages 460 - 464

- Read: Mathematics analysis and approaches HL by Oxford, pages 473 - 475 (476* Read only up to the yellow information box and recommend doing investigation 11 (to understand number e), you do not need to be concerned about the text that follows this box yet AND investigation 12 question 6.)

- Watch: Video lessons part I, Part II, Bonus (optional - watch first)

- Do: Mathematics Analysis and approaches HL by Oxford, exercise 7C questions 2, 6, 8, page 464.
-Do: Mathematics Analysis and Approaches HL by Oxford, exercise 7E questions 3, 4, 5, page 481-482.

- Do: Answer the question in the classroom for the corresponding lesson 19.


Unit: Manipulating and calculating with fractions

In this lesson we will look at what happens when we multiply a unit fraction by an integer.

Multiplying non-unit fractions with integers

In this lesson we will look at what happens when we multiply a non-unit fraction by an integer

Using area models to multiply fractions

In this lesson we will look at how we can use area models to multiply fractions together

Applying decimals and percentages to area multiplication

In this lesson we will look at how we can use area models to multiply decimals

Dividing a fraction by an integer

In this lesson we will look at methods to divide a fraction by an integer

Modelling fractions by division (Part 1)

In this lesson we will look at how we can create a model to help us to divide fractions

Modelling fractions by division (Part 2)

In this lesson we will develop our models for dividing fractions to look for patterns and solve more difficult problems.

Dividing Fractions in Mixed Contexts

In this lesson we will look at different contexts for dividing fractions

Adding and Subtracting Fractions (Part 1)

In this lesson we introduce the concept of adding and subtracting fractions

Adding and Subtracting Fractions (Part 2)

In this lesson we start to develop our understanding of adding and subtracting fractions

Adding and Subtracting Fractions (Part 3)

In this lesson we look at adding and subtracting fractions with different denominators

Fractions and Distributivity

In this lesson we will look at how we can use the laws of distribution to solve problems with fractions


Contents

The first part of the Rhind papyrus consists of reference tables and a collection of 21 arithmetic and 20 algebraic problems. The problems start out with simple fractional expressions, followed by completion (sekem) problems and more involved linear equations (aha problems). [3]

This table is followed by a much smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. For instance the division of 7 by 10 is recorded as:

7 divided by 10 yields 2/3 + 1/30

After these two tables, the papyrus records 91 problems altogether, which have been designated by moderns as problems (or numbers) 1–87, including four other items which have been designated as problems 7B, 59B, 61B and 82B. Problems 1–7, 7B and 8–40 are concerned with arithmetic and elementary algebra.

Problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4, and 1 + 2/3 + 1/3 = 2 by different fractions. Problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘‘aha’’ problems these are linear equations. Problem 32 for instance corresponds (in modern notation) to solving x + 1/3 x + 1/4 x = 2 for x. Problems 35–38 involve divisions of the heqat, which is an ancient Egyptian unit of volume. Beginning at this point, assorted units of measurement become much more important throughout the remainder of the papyrus, and indeed a major consideration throughout the rest of the papyrus is dimensional analysis. Problems 39 and 40 compute the division of loaves and use arithmetic progressions. [2]

The second part of the Rhind papyrus, being problems 41–59, 59B and 60, consists of geometry problems. Peet referred to these problems as "mensuration problems". [3]

Volumes Edit

Problems 41–46 show how to find the volume of both cylindrical and rectangular granaries. In problem 41 Ahmes computes the volume of a cylindrical granary. Given the diameter d and the height h, the volume V is given by:

Problem 47 is a table with fractional equalities which represent the ten situations where the physical volume quantity of "100 quadruple heqats" is divided by each of the multiples of ten, from ten through one hundred. The quotients are expressed in terms of Horus eye fractions, sometimes also using a much smaller unit of volume known as a "quadruple ro". The quadruple heqat and the quadruple ro are units of volume derived from the simpler heqat and ro, such that these four units of volume satisfy the following relationships: 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro. Thus,

100/10 quadruple heqat = 10 quadruple heqat 100/20 quadruple heqat = 5 quadruple heqat 100/30 quadruple heqat = (3 + 1/4 + 1/16 + 1/64) quadruple heqat + (1 + 2/3) quadruple ro 100/40 quadruple heqat = (2 + 1/2) quadruple heqat 100/50 quadruple heqat = 2 quadruple heqat 100/60 quadruple heqat = (1 + 1/2 + 1/8 + 1/32) quadruple heqat + (3 + 1/3) quadruple ro 100/70 quadruple heqat = (1 + 1/4 + 1/8 + 1/32 + 1/64) quadruple heqat + (2 + 1/14 + 1/21 + 1/42) quadruple ro 100/80 quadruple heqat = (1 + 1/4) quadruple heqat 100/90 quadruple heqat = (1 + 1/16 + 1/32 + 1/64) quadruple heqat + (1/2 + 1/18) quadruple ro 100/100 quadruple heqat = 1 quadruple heqat [2]

Areas Edit

Problems 48–55 show how to compute an assortment of areas. Problem 48 is notable in that it succinctly computes the area of a circle by approximating π. Specifically, problem 48 explicitly reinforces the convention (used throughout the geometry section) that "a circle's area stands to that of its circumscribing square in the ratio 64/81." Equivalently, the papyrus approximates π as 256/81, as was already noted above in the explanation of problem 41.

Other problems show how to find the area of rectangles, triangles and trapezoids.

Pyramids Edit

The final six problems are related to the slopes of pyramids. A seked problem is reported by : [8]

If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity found for the seked is the cotangent of the angle to the base of the pyramid and its face. [8]

The third part of the Rhind papyrus consists of the remainder of the 91 problems, being 61, 61B, 62–82, 82B, 83–84, and "numbers" 85–87, which are items that are not mathematical in nature. This final section contains more complicated tables of data (which frequently involve Horus eye fractions), several pefsu problems which are elementary algebraic problems concerning food preparation, and even an amusing problem (79) which is suggestive of geometric progressions, geometric series, and certain later problems and riddles in history. Problem 79 explicitly cites, "seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16807 hekats." In particular problem 79 concerns a situation in which 7 houses each contain seven cats, which all eat seven mice, each of which would have eaten seven ears of grain, each of which would have produced seven measures of grain. The third part of the Rhind papyrus is therefore a kind of miscellany, building on what has already been presented. Problem 61 is concerned with multiplications of fractions. Problem 61B, meanwhile, gives a general expression for computing 2/3 of 1/n, where n is odd. In modern notation the formula given is

The technique given in 61B is closely related to the derivation of the 2/n table.

Problems 62–68 are general problems of an algebraic nature. Problems 69–78 are all pefsu problems in some form or another. They involve computations regarding the strength of bread and beer, with respect to certain raw materials used in their production. [2]

Problem 79 sums five terms in a geometric progression. Its language is strongly suggestive of the more modern riddle and nursery rhyme "As I was going to St Ives". [3] Problems 80 and 81 compute Horus eye fractions of hinu (or heqats). The last four mathematical items, problems 82, 82B and 83–84, compute the amount of feed necessary for various animals, such as fowl and oxen. [2] However, these problems, especially 84, are plagued by pervasive ambiguity, confusion, and simple inaccuracy.

The final three items on the Rhind papyrus are designated as "numbers" 85–87, as opposed to "problems", and they are scattered widely across the papyrus's back side, or verso. They are, respectively, a small phrase which ends the document (and has a few possibilities for translation, given below), a piece of scrap paper unrelated to the body of the document, used to hold it together (yet containing words and Egyptian fractions which are by now familiar to a reader of the document), and a small historical note which is thought to have been written some time after the completion of the body of the papyrus's writing. This note is thought to describe events during the "Hyksos domination", a period of external interruption in ancient Egyptian society which is closely related with its second intermediary period. With these non-mathematical yet historically and philologically intriguing errata, the papyrus's writing comes to an end.

Much of the Rhind Papyrus' material is concerned with Ancient Egyptian units of measurement and especially the dimensional analysis used to convert between them. A concordance of units of measurement used in the papyrus is given in the image.


4th Grade Common Core Unit 4 Math Test: Fractions Part 2

**This test is also included in a year-long math test bundle that includes 6 unit tests, covering all the 4th grade Common Core Math Standards** here: 4th Grade Math Test Bundle: ALL Common Core Standards-6 Units-Entire Year

This Multiple Choice Unit 4 Math Test is aligned with the Common Core Standards, specifically for the Common Core Georgia Performance Standards in Unit 4. You can download my curriculum map for specific standards per unit for free here: 4th Grade Common Core Curriculum Map

This Test is 30 multiple-choice questions that include word problems and common core vocabulary. Give each student the test at the beginning of the unit as the Pretest and then give it again at the end of the unit as the Posttest. This way you can track each student's growth. Each test is provided with a growth chart so students and parents can visually see their growth.

MCC4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

MCC4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/1001.

MCC4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100 describe a length as 0.62 meters locate 0.62 on a number line diagram.


The link below will take you to a video I made on solving quadratic equations. Remember this is used to find the x-intercept. Good luck

Hi year 10 students, the following links will have extra tutorials on the topics we will be covering in week 6 and the end of week 5.

The midpoint of a line segment – using formula

Review of the pythagorean Theorem

Finding the distance between two points

Parallel Lines

Perpendicular Lines


The bar model basics with Dr Yeap Ban Har

Maths mastery expert Dr Yeap Ban Har demonstrates how bar models are taught in Singapore-style maths lessons.

Bar modelling and the CPA approach

The bar model method draws on the the Concrete, Pictorial, Abstract (CPA) approach — an essential maths mastery concept. The process begins with pupils exploring problems via concrete objects. Pupils then progress to drawing pictorial diagrams, and then to abstract algorithms and notations (such as the +, -, x and / symbols).

The example below explains how bar modelling moves from concrete maths models to pictorial representations.

As shown, the bar method is primarily pictorial. Pupils will naturally develop from handling concrete objects, to drawing pictorial representations, to creating abstract rectangles to illustrate a problem. With time and practice, pupils will no longer need to draw individual boxes/units. Instead, they will label one long rectangle/bar with a number. At this stage, the bars will be somewhat proportional. So, in the example above, the purple bar representing 12 cookies is longer than the orange bar representing 8 cookies.

The lasting advantages of bar modelling

On one hand, the Singapore maths model method — bar modelling — provides pupils with a powerful tool for solving word problems. However, the lasting power of bar modelling is that once pupils master the approach, they can easily use bar models year after year across many maths topics. For example, bar modelling is an excellent technique (but not the only one!) for tackling ratio problems, volume problems, fractions, and more.

Importantly, bar modelling leads students down the path towards mathematical fluency and number sense. Maths models using concrete or pictorial rectangles allow pupils to understand complex formulas (for example, algebra) on an intuitive, conceptual level. Instead of simply following the steps of any given formula, students will possess a strong understanding of what is actually happening when applying or working with formulas.

The result? A stable, transferable, and solid mathematical framework for approaching abstract concepts. Combined with other essential maths mastery strategies and concepts, bar modelling sets students up for long-term maths success.


Robert L.

My name is Robert L. I have been in the field of education for almost 20 years. I first started out as a substitute teacher in York County. My first full time job was as a seventh grade teacher at Sanford Junior High School. After my first child was born, I decided to work nights at Sanford Community Adult Education and also part time in the field of special education for Sanford Schools. A couple years later I took a position as an adjunct instructor with Southern Maine.

My name is Robert L. I have been in the field of education for almost 20 years. I first started out as a substitute teacher in York County. My first full time job was as a seventh grade teacher at Sanford Junior High School. After my first child was born, I decided to work nights at Sanford Community Adult Education and also part time in the field of special education for Sanford Schools. A couple years later I took a position as an adjunct instructor with Southern Maine Community College. I now teach at York County Community College as well as Portland High School. I love working with both kids and adults. The main way I try to help students succeed is by working neatly and checking all the work by using opposite operations for basic math and substitution for equations. I try to make sure students know that if they can verify that an answer is correct they can build confidence. I hope to help you out in the same way. Thank You!


Getting to Know Fractions Games

Games are a great way to practice skills and have fun. I believe that so much, I made some fraction cards for you and a few ideas on how to use them. And it’s free. That’s the best price, right?

I included an instruction sheet but I wanted to give you a couple of examples of the games in the Freebie that maybe require a little more instruction than is available on one sheet.

(Scroll to the end of this post to grab the download)

Create a Story Using Fraction Cards:

This game is meant for 2-4 people, varying levels will not hinder play. One player draws a card and starts the story. Let’s say I drew 1/4.

I might start the story with “There once was an evil queen who lived in a tall evil tower. Everyday she gathered her subjects. And every day she required 1/4 of them to give up all of their possessions to her, because she is evil. Soon the villagers were poor and starving until one day…

And then another student draws a card and picks up the story putting their fraction in the story somewhere.

Students learn creativity and get a visual of different kinds of fractions, which is really important later when you want them to solve problems with them.

Also, it’s fun! It’s casual and multiple ages/abilities can contribute. Which makes it perfect for families.

Write down the story as it’s being told and read it at the dinner table. Math and language arts, done.

Create Number Sentences:

Several games involve number sentences. By that I mean this: 1/2 + 1/4 = 3/4.

You can limit the operation (addition only) or open it up to any operation, or even a combination of operations.

In the multiplayer game, players draw 5 cards. Their goal is to create a number sentence. I did not include operation signs.

The rest of the cards are put face down in a pile (like Go Fish). These are the mystery cards. Each player takes turns. On their turn a player can:

A. Discard a card and draw a new one (either a mystery card or one someone else has discarded).

B. Lay down a finished number sentence. When a player puts down a number sentence, they read it out loud and get approval from the group, proving it with paper or objects if necessary. The player then draws as many cards as they laid down and play resumes with the next player. (I lay down the cards 1/4, 1/16, 5/16. I would read 1/4 + 1/16 = 5/16 or 5/16 – 1/16 = 1/4).

Assessing Understanding:

With all of the games: encourage students to talk about how they got their answers. Not every time, that would be tedious. But often enough that you can gauge where they are in their understanding and they gain confidence explaining.

A note about the numbers I chose. I only put in the most common fractions (halves, thirds, fourths, eighths, and sixteenths) because those are the ones I find myself using in my baking and measuring. I put the sixteenths on their own sheet so that you wouldn’t have to use them if you didn’t want to.

The instructions have ideas on how to use the cards for the 1st and 3rd step in this article.

I hope you find the games helpful and fun! If all goes well, there will be a few less children (and parents) crying over their fractions homework.

Danielle is a homeschooling mamma of 5. She is committed to making life with young children easier and sharing her passion for math. If you would like to learn more about teaching math to multiple age groups visit Blessedly Busy or follow her on: Facebook, Instagram, Pinterest or Twitter.

Want more hands on fraction fun? Grab the ebook Hands On Fraction Activities to explore all sorts of fraction concepts with rulers and paper folding.

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Watch the video: Εισαγωγή εξισώσεων στο Word (December 2021).