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التمكن من تحليل الكسور

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التمكن من تحليل الكسور

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سيكون الطلاب قادرين على تحليل الكسور الجبرية إلى عواملها الأولية.

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اختزال الكسور الجبرية

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اختزال الكسور الجبرية

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سيتمكن الطلاب من اختزال الكسور الجبرية إلى أبسط صورة.

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كيف نقوم بتحليل واختزال الكسور الجبرية؟

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Fractions: Learn How To Simplify Algebraic Fractions In Easy Steps And Prove Your Answer Part 2

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شرح كيفية تحليل العدد وتحديد عواملها وكيفية اختزال الكسور الجبرية.

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عمل في مجموعات صغيرة لتحليل مسائل محددة.

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التدريب المستقل

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التدريب المستقل

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صفحة عمل تحتوي على مسائل للتحليل والاختزال للعمل بشكل مستقل.

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التعلم المتمايز

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التعلم المتمايز

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أنشطة إضافية للطلاب الذين يحتاجون لمساعدة، وفرص تحدي للطلاب المتقدمين.

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Math classrooms are mosaics of strengths and experiences. When we have students with diverse backgrounds-with various languages, achievements, and interests-in the same space, everyone learns from each other and broadens their world.

On the flip side, though, teaching math to a broad array of students can be challenging. Do you struggle to reach all of your students? Are you a newer teacher who is looking to improve your practice? The strategies for differentiated instruction provided here might help you out.

What Is Differentiated Math Instruction?

Differentiated math instruction refers to the collection of techniques, strategies, and adaptations you can use to reach your diverse group of learners and make mathematics accessible to every single one. Dr. Timothy Kanold, former president of the National Council of Supervisors of Mathematics (NCSM)-and HMH author- clarifies that differentiation in a math lesson is "differentiation on the entry points into the task for support or the exit point to advance student thinking."

By applying various tools and strategies, such as incorporating technology, assigning hands-on projects, and teaching in math small-group formats, you can help every student meet expectations. We know that there are different schools of thought regarding what differentiation means. When we use the term, we are talking about providing student choice, voice, and agency. Differentiating instruction isn't meant to add more work to your day. Quite the opposite, in fact; it's meant as a teaching approach that will help you to reach more students in terms of accessibility and equity, making your job both easier and more effective in the long run.

Why Is Differentiating Math Instruction Important?

Some people think that math, more than any other subject, is the best fit for differentiation. Even though a 2018 survey by Texas Instruments found that 46% of kids said they really liked math, there are hundreds of books, websites, and memes discussing the difficulty of the subject. From the anxiety caused by there being only one correct answer to the cultural buy-in to the myth of being-or not being-a "math person" to the fear of solving a word problem, many students struggle with math. In addition, many students and educators alike find it hard to make the connection between math and the real world, which only increases disillusionment with the subject. That's why it's especially important to be open to new ways of providing instruction.

The National Council of Teachers of Mathematics (NCTM) promotes differentiating math instruction for differences in learning as well as differences in achievement, interest, and confidence. NCTM advises that the need is greater in middle and high school, as higher-level math relies on more complex reasoning. When you differentiate your math instruction, you support all learners by targeting and addressing specific needs of groups and individual students.

Examples of Differentiated Instruction in Math

Do you need ideas for how to differentiate your teaching to be sure your math students are progressing? Below are seven differentiation strategies for math instruction, along with ways that you can use them in your math classroom. They serve as examples of differentiated instruction in math and may work better for some classrooms and math topics than others. Customize these ideas however you need to serve you and your students.

Strategy 1: Math Centers

For this, you'll need to come up with a few activities your students can rotate through (be sure to browse our library of free activities and resources!), such as watching a video, reading an article, or solving a word problem. We spoke with Kristy McFarlane, an instructional supervisor at Sandshore Elementary School in New Jersey, about differentiation. She says math teachers at her school spend about 10 minutes on a mini-lesson for the whole class and then students spend about 15 minutes at various math centers. "They might meet with the teacher in a small group for extra help, use math software, do a game or project at the hands-on station, or do seat work based on the day's mini-lesson," she says.

Math centers are a powerful way to facilitate independent and small group learning within your classroom. Our program, for example, is known for embedding resources and instructional time to math centers. If a select group of your students are all struggling to, say, add fractions, they may benefit from an activity that has them practice finding least common denominators. Think about ways to customize the groupings and centers so they're perfect for your students' strengths, misconceptions, and interests, and make use of tools that strategically group students and recommend activities for you.

Strategy 2: Activity Cards

Choice is an important part of differentiation, and letting students decide how they want to spend their time is a great way to appeal to various learning preferences. You'll need to come up with math problems, tasks, or questions. As much as possible, use or create cards that span several lessons and offer options to work independently, with a partner, or in a small group. Ask for feedback so you can adjust future learning accordingly. Many of HMH's math programs, including , , and include inquiry-based task and project cards that help teachers differentiate.

Strategy 3: Choice Boards

As we just mentioned, giving students the ability to make decisions about their learning is an important part of differentiation. A choice board is a graphic organizer that gives students activities to choose from. There are different types of choice boards, but they need to focus on specific learning needs, interests, and skills. Choice boards increase student ownership; students pace themselves and get to decide how to engage with information, along with how to demonstrate their learning. Some teachers create different versions of the same choice board; others will color-code options to signify topic, activity type, or expected level of challenge. Check out the choice board we developed for remote learning. This board covers all subjects but also includes a free template to get you started on a math-only version.

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7 Strategies for Differentiated Math Instruction

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These strategies provide differentiated math instruction to K-12 teachers looking for ways to personalize their mathematics teaching.

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التقييم التكويني

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التقييم التكويني

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ملاحظات خلال الصف ومراجعة لأعمال الطلاب للتحقق من الفهم.

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التقييم النهائي

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التقييم النهائي

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اختبار قصير يتضمن أسئلة عن تحليل واختزال الكسور الجبرية.

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مناقشة جماعية

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مناقشة جماعية

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تلخيص ما تعلمه الطلاب اليوم من خلال حلقة نقاش.

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بطاقة الخروج

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بطاقة الخروج

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اطلب من كل طالب أن يكتب مثال على تحليل كسر جبري ويشرح خطواته.

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تمارين إضافية

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تمارين إضافية

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مسائل إضافية لتحليل واختزال الكسور الجبرية.

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One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

Learn more

Here we will learn about simplifying algebraic fractions, including different powers of x, quadratics, and the difference of two squares.

There are also simplifying algebraic fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you're still stuck.

What is simplifying algebraic fractions?

Simplifying algebraic fractions is simplifying a fraction that contains algebra so that the numerator and the denominator do not contain any common factors.

To do this, we need to be able to find common factors between the numerator and the denominator which can be cancelled down.

In order to simplify algebraic fractions, you must be confident with calculating with fractions.

If we want to express \frac{a}{b}+\frac{c}{d} or \frac{a}{b} - \frac{c}{d} as a single fraction, we need to find a common denominator and then add or subtract the resultant fractions.

The general form when adding fractions is:

\frac{a}{b}+\frac{c}{d}=\frac{a\times{d}}{b\times{d}}+\frac{b\times{c}}{b\times{d}}=\frac{ad+bc}{bd}

The general form for subtracting fractions is:

\frac{a}{b}-\frac{c}{d}=\frac{a\times{d}}{b\times{d}}-\frac{b\times{c}}{b\times{d}}=\frac{ad-bc}{bd}

E.g.
Calculate \frac{2}{5}+\frac{1}{4}

So if a = 2, b = 5, c = 1 and d = 4 , we can therefore say that

\frac{2}{5}+\frac{1}{4}=\frac{8}{20}+\frac{5}{20}=\frac{8+5}{20}=\frac{13}{20}

Below is a visual representation of this problem,

By converting the fractions so that they have a common denominator (in this case 20 ), we divide the fractions into smaller parts, then total those parts, all by using knowledge of equivalent fractions.

We can use this to simplify algebraic expressions:

E.g.
Let's look at \frac{5x}{15}

Here, the terms 5x and 15 share a common factor of 5 .

If we wrote this out as two products, we would get \frac{5\times{x}}{5\times{3}} .

We can simplify this by cancelling the common factor of 5 from the numerator and the denominator of the fraction \frac{5\times{x}}{5\times{3}}=\frac{x}{3} .

So by finding the highest common factor of the numerator and the denominator, we can show that the fraction \frac{5x}{15} can be expressed as as the more simple algebraic fraction \frac{x}{3} .

We can use knowledge of simplifying algebraic fractions to solve equations that include algebraic fractions. For more information on this, see the lesson on algebraic fractions.

Step-by-step guide: Algebraic fractions

What is simplifying algebraic fractions?

Simplify algebraic products and quotients

Sometimes algebraic fractions can be simplified by using the index laws. Here is an example where a product and a quotient can be simplified.

E.g.

Simplify

\frac{4x^5 \times 3x^2}{2x^3}

This can be simplified by first considering the numerator can be simplified using the product law of indices.

\frac{4x^5 \times 3x^2}{2x^3}=\frac{12x^7}{2x^3}

Then the fraction can be further simplified by using the quotient law of indices.

\frac{12x^7}{2x^3}=6x^4

How to simplify algebraic fractions

In order to simplify algebraic fractions:

1. Complete any calculations required in the question (+ − \times\div) .

2. Factorise the numerator and the denominator.

3. Cancel the common factor and simplify further if required.

Explain how to simplify algebraic fractions

Simplifying algebraic fractions worksheet

Get your free simplifying algebraic fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Simplifying algebraic fractions worksheet

Get your free simplifying algebraic fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Simplifying algebraic fractions examples

Example 1: Multiplying algebraic fractions

Simplify the algebraic expression

\frac{5}{x+2}\times\frac{x+2}{10}

Complete any calculations required in the question (+−\times\div) .

\frac{5}{x+2}\times\frac{x+2}{10}=\frac{5x+10}{10x+20}

2 Factorise the numerator and the denominator.

\frac{5x+10}{10x+20}=\frac{5(x+2)}{10(x+2)}

3 Cancel the common factor and simplify further if required.

\frac{5(x+2)}{10(x+2)}=\frac{5}{10}=\frac{1}{2}

Common misconceptions

Ignoring the denominators

When given an equation including an algebraic fraction, the denominators are ignored so the question is answered incorrectly.

When adding two fractions, the denominator must be the same. A common misconception for adding two fractions is to add the numerators and the denominators together because this method is emphasised when looking at multiplying fractions.

Cancelling terms incorrectly

Seeing the same term on the numerator and denominator can result in the misconception that they can both be cancelled.

Expanding double brackets incorrectly

When expanding double brackets, each term in the bracket needs to be multiplied by each term in the other bracket. By not treating the numerator as a single expression, mistakes are made when multiplying brackets together.

Simple algebraic expressions such as 5x + 4 can be incorrectly simplified to get 9x as the x is ignored; the assumption is that 5x + 4 is the same as (5 + 4)x which is not correct.

Factorising into a single bracket incorrectly

When factorising into a single bracket, the incorrect method occurs when the bracket is simply placed between the number and the x of the first term, then around the last term in the expression.

E.g.

Factorising 3x + 12 incorrectly, we would get 3(x + 12) .

Here the bracket is placed between the 3 and the x , and the 12 has not been factorised at all. When factorising, the common factor is placed on the outside of the bracket, and the factor pair that generates the term is placed on the inside, so for the example given 3x + 12 = 3(x + 4) .

Practice simplifying algebraic fractions questions

Simplifying algebraic fractions GCSE questions

1. Simplify the algebraic fraction

\frac{x^{2}-9x+20}{5x-20}

Write your solution in the form \frac{ax+b}{c} where a, b, and c are integers.
(3 marks)

Numerator is factorised: (x-5)(x-4)
(1)
Denominator is factorised: 5(x-4)
(1)
Fraction is simplified: \frac{x-5}{5}
(1)

2. (a) Justin is simplifying the algebraic fraction \frac{x^{2}-4}{x+2}

His work is written below.

Is Justin correct?

Explain your answer.

(b) Show that the fraction \frac{9x^{2}-16}{3x-4} can be simplified to 3x+4 .
(5 marks)

(a)

Justin has the correct answer but he has not shown the correct understanding
(1)
He should have factorised the numerator to get (x+2)(x-2)
(1)
He can then cancel the common factor of (x+2) , leaving the solution x-2
(1)
(b)

9x^2-16=(3x+4)(3x-4)
(1)
\frac{(3x+4)(3x-4)}{(3x-4)}=3x+4
(1)

3. Simplify fully

\frac{2x+1}{2x+10}\div{\frac{3+6x}{x+5}}
(4 marks)

\frac{2x+1}{2x+10}\div{\frac{3+6x}{x+5}}=\frac{2x+1}{2x+10}\times{\frac{x+5}{3+6x}}
(1)
=\frac{2x+1}{2(x+5)}\times{\frac{x+5}{3(1+2x)}}
(1)
=\frac{(2x+1)(x+5)}{6(x+5)(2x+1)}
(1)
=\frac{1}{6}
(1)

Learning checklist

You have now learned how to:

Simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:

Factorising quadratic expressions of the form x² + bx + c, including the difference of 2 squares; {factorising quadratic expressions of the form ax² + bx + c}

Simplifying expressions involving sums, products and powers, including the laws of indices

Still stuck?

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Free simplifying algebraic fractions GCSE maths revision guide, including step by step examples, exam questions and free worksheet.

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قراءة إضافية

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قراءة إضافية

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قراءة عن تطبيقات الحياة الحقيقية للكسور الجبرية.

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Have you ever wondered how Algebra may be applied to solve real-life problems?

We regularly see people using Algebra in many parts of everyday life; for instance, it is utilized in our morning schedule each day to measure the time you will spend in the shower, making breakfast, or driving to work.

The absence of "X" or "Y" doesn't imply that algebra is not around us; algebra's actual occurrences are uncountable. This exact and compact numerical language works wonderfully with practically all different subjects and everyday life.

In this article, we will grasp instances in real life where applications of algebra are needed and examples of applications of algebra in real life.
Also read:

What is Algebra?

Algebra is a part of mathematics that deals with symbols and the standards for controlling those symbols. The more basic parts of algebra are called elementary algebra, and the more abstract types are called Abstract Algebra or modern algebra.

Algebraic Expression

Let us consider the pattern below. It has been created using marbles. Here we see that the first column has 2 marbles, the second column has 3 marbles, the third column has 4 marbles and so on.

Thus we observe that every new column increases by 1 marble. We can write the representation as

The number of marbles used in a column =

position of the column + 1

or as

the number of marbles used in a column = n + 1.

Here n represents the position of the column. So 'n' is an example for a variable that can take any value 1,2,3... so on. Thus n + 1 is the algebraic expression formed with n as variable and constant 1.

Variable

A variable is a number that does not have a fixed value. The picture and the list below show some real-life examples, where the value of a variable changes with the change in place and time.

The temperature in different places also change.

The height of a growing child changes with time.

The speed of a car changes with time.

The age of people keeps on increasing year by year.

Constant

The value remains fixed for specific numbers that represent quantities or ideas that will not change. For example, the date of birth of a particular person, the normal human body temperature and capacity of a given container.

Framing algebraic expressions with given conditions

Now we will see how to frame an algebraic expression. The rules are that variables are to be represented with alphabetic letters, say lower case a-z and constants in numeric form.

Examples

1. Amanda has 10 storybooks more than Alex. Express the number of storybooks Amanda has in terms of the number of storybooks Alex has.

Let the number of storybooks Alex has = y

Therefore the number of storybooks Amanda has = y + 10

2. Sweets from a big box are equally distributed in 10 small boxes. Express the number of sweets in one small box in terms of the total number of sweets.

Let x be the total number of sweets.

Number of sweet boxes = 10

Therefore, the number of sweets in one box is = x/10

Solving Equations

Let us see how practical applications of algebra can be used to solve equations. You will often see equations like 3x + 4 = 5, where you want to find x.

Consider a situation from our daily life.

The cost of a book is £5 more than the cost of a pen. Let us take the cost of the pen as £x. Then the cost of the book is £ (x + 5) . If the cost of the book is £20, what is the cost of the pen?

We know that the book's cost is x + 5 and it is given that x + 5 = 20. This is an equation in the variable x.

A table is prepared as shown below for various values of x:

x

8

9

10

11

12

13

14

15

16

x + 5

13

14

14

16

17

18

19

20

21

It is clear from the table that x + 5 = 20 only for x = 15. So, the cost of the pen is £15.

In general we say that x = 15 is the solution of the equation x + 5 = 20. This is the trial and error method where we substitute different values for the variable that satisfies the given equation.

An equation has two parts which are connected by an equal to sign. The two parts or sides of an equation are denoted as LHS (Left Hand Side) and RHS (Right Hand Side). If LHS = RHS we get an equation. 2x = 6 is an

algebraic equation, whereas 3x > 10 or 4x < 12 are not equations.

Solving an equation using the Principle of Balances

Consider the balance given in the figure.

Four circles balance one square and a circle on the other side. The idea is we have to find out how many circles will balance a square. If we remove the circle from the left pan, we have only the square there. Since we removed a circle from the left pan, we have to remove the circle from the right pan also. Then there will be three circles in the right pan.
Example:
Now the balance looks like the one shown on the right. This is called the principle of balances. Using balancing equations, we can solve equations in a systematic way.

Solve using the principle of balances:

Benjamin's mother is three times as old as Benjamin. If Benjamin's mother is 39 years old, find Benjamin's age.

Solution:

Let Benjamin's age be x.

Benjamin's mother's age 3x = 39

3x/3 = 39/3 {Dividing by 3 on both the sides }

So, Benjamin's age = 13.

The same quantity can be added or subtracted to both sides of the equation. If the same amount is multiplied or divided on both the sides of an equation, it remains the same.

Forming an equation to find the unknown

Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems. So let's see another real-life example in the form of a puzzle.

Detailed Solution:

Our first supposition is that Uma buys at least one ball of each kind. Now let's say she buys x footballs, y cricket balls, and z table-tennis balls.

The question requires x + y + z = 100 [ 1 ]

It also requires 15 x + 1y + z/4 = 100 [ 2 ]Since we have 3 variables but only 2 equations we'll have to use the trial and error method to get at the solution.

Let's vary x the number of footballs and see what we get:

Suppose x = 1, then

y + z =99 and y + z/ 4 = 83z/4 = 14 3z = 56 z = 56/3 which is not a whole number.

Trying for x = 2 also fails and now

If x = 3, then z = 97 and y + z / 4 = 55 3z/4 =42 3z= 168 z = 168/3 = 56 which is a whole number!

And if z = 56 then y = 97 - z = 97 - 56 = 41

So the set of balls Uma buys is { 3 footballs, 41 cricket balls and 56 table tennis balls }

Algebra in Geometry

In Algebraic Geometry we study geometric objects and their assortment that are characterized by polynomial equations.

Examples of algebraic varieties' most studied classes are plane algebraic curves, including lines, circles, parabolas, ellipses, hyperbolas. There are also cubic curves like elliptic curves and quartic curves like lemniscates and Cassini ovals.

In real life, algebraic geometry can be used to study the dynamics properties of robotics mechanisms.

Source: Pinterest

A robot can move in continuous space with an infinite set of possible actions and states. When the robot has arms and legs that must also be controlled and the search space becomes many-dimensional. Robot's kinematics can be formulated as a polynomial equation system that can be solved using algebraic geometry tools.

Algebraic geometry is also widely used in statistics, control theory, and geometric modelling. There are also connections to string theory, game theory, graph matchings and integer programming.

Algebra in Computer Programming

The mathematical languages unite fields such as science, technology, and engineering into itself. That is why an individual intrigued by the field of computer programming and coding should figure out how to comprehend and control mathematical logic.

Strong comprehension of algebra incorporates characterizing the connections between objects, critical thinking with restricted factors, and analytical skill development to help execute decision making.

One such use of Algebra can be seen in Inference procedures used in Knowledge engineering. Variables and constant symbols are used as terms representing objects in real life.

The knowledge engineer adds a set of facts and specifies what is true, and the inference procedure figures out how to turn the facts into a solution to the problem.

Besides, because a fact is true regardless of what task one is trying to solve, knowledge bases can be reused for various tasks without modification.

Example for the task of inference

Take a sentence,

Everyone likes ice cream.

It is represented in First-order logic as

x Likes ( x, ice cream )

where x is the variable and is the universal quantifier that generalizes to all persons liking icecream.

If another sentence found in the knowledge base is as follows:

John likes ice cream

It is represented as Likes( John, ice cream)

The inference procedure will reason out from x Likes ( x, ice cream ) with the substitution {x/John} and infers Likes(John, ice cream) and concludes that John likes icecreams.

Biostatistics University of Florida

Other uses of algebra in programming are Ontology, error correction algorithms, Natural language processing, Neural networks, designing artificial intelligence programming languages such as LISP and PROLOG and theorem provers such as OTTER.

Algebra in Real Life

In real life there are a plethora of instances where Algebra is being used. It's utility is being universally quantified in all walks of our lives. For instance, take a shopping domain where we need to be budgeted with the cart items and some algebraic formulation is applied.

The economy of every country is analysed with the help of economists taking the help of algebra to solve the problems related to debts or loans.

Tom Evans's ANALOGY program (1968) solved geometric analogy problems that appear in IQ tests such as the one shown below.

Source: Artificial Intelligence by Stuart Russell

The use of algebra is multipurpose, and it goes handy in every sphere of our lives. It isn't just mathematicians, however, even most academicians, educationists, researchers, and experts from all different backgrounds collectively

agree with the adaptability of algebra.

Real-World Applications of Linear Algebra

What is Linear Algebra?

Linear algebra is the branch of mathematics concerning linear equations such as linear maps and their representations in vector spaces and matrices.

Example

The concept of classification can be simulated with the help of neural network structures that use a linear regression model. Here the training set is compared with the test data so that the learning algorithms generate outcomes to predict data related to decision making, medical diagnosis, statistical inferences, etc.

Applications

The most generally utilized use of linear algebra is certainly optimization, and the most broadly utilized sort of advancement is linear algebra. You can upgrade spending plans, your eating regimen, and your course to work

utilizing linear algebra, and this uniqueness starts to expose a lot of applications.

Other real-world applications of linear algebra include ranking in search engines, decision tree induction, testing software code in software engineering, graphics, facial recognition, prediction and so on.

Summary

In real life, algebra can be compared to a universally handy device or a sorcery wand that can help manage regular issues of life. Whenever life throws a maths problem at you, for example when you have to solve an equation or work out a geometrical problem, algebra is usually the best way to attack it.
Written by Jesy Margaret, Cuemath teacher
About Cuemath

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Algebra in real life | Applications of algebra

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Have you ever wondered how Algebra may be applied to solve real-life problems? We regularly see people using Algebra in many parts of everyday life; for instance, it is utilized in our morning schedule each day to measure the time you will spend in the shower, making breakfast, or driving to work.