Rational expressions and polynomials – Grade 11 math lesson
Lesson 7 of 23 in the Grade 11 math course
US version. US grades, US spelling and dollars ($). · UK Maths Department
A complete 45-minute lesson for grade 11 (ages 16–17) that runs on this page – at school or at home. Every answer is graded right away, with a worked explanation.
- Time
- 45 minutes + 20 min homework
- Grade
- Grade 11 (ages 16–17)
- You need
- Paper, a pencil and this page
- Graded
- Instantly, with explanations
For parents: how to use this lesson
You don't need to know this topic yourself. The lesson explains each step on screen, one tidy step at a time. Every answer is graded straight away with a worked explanation, so you never have to grade anything.
- Read what your child will learn and the big ideas together.
- Press Start the lesson below and sit with your child for the teaching and the worked examples (about 19 minutes).
- Let them do the practice on their own. They choose Getting started, On track or Challenge – start on On track and move up after a few right in a row.
- Finish with the exit ticket and the talk question: “Why can you not cancel the x in (x + 3)/x?”
- Later that day or the next day: the homework (20 minutes), then a practice test.
Watch out for
- Cancelling terms that are added ((x + 3)/x = 3). Only cancel factors: whole parentheses or terms that multiply.
- Adding the denominators. Find a common denominator first, as with numbers.
Tip for home: Check algebraic answers by substituting a number for x.
What your child will learn
Learning goal: simplify, add, subtract, multiply and divide algebraic fractions.
Success looks like
- I can simplify algebraic fractions.
- I can add and subtract them.
- I can multiply and divide them.
Key words
- algebraic fraction
- a fraction with letters
- factor
- write as a product
- common denominator
- a shared bottom
- cancel
- divide top and bottom by a common factor
What to have ready
- Paper and a pencil
- This page on a computer, tablet or phone
- For the hands-on part: Do the same calculation with numbers and with letters side by side to see the rules match.
The big ideas (for the adult)
Algebraic fractions follow the same rules as number fractions.
To simplify, factorise the top and bottom fully, then cancel common factors – never cancel single terms.
To add or subtract, use a common denominator; to multiply, multiply tops and bottoms; to divide, multiply by the reciprocal.
The lesson, step by step
| Part | Time | How it is checked |
|---|---|---|
| Starter: quick recall | 5 min | Graded by Classfern |
| Teaching input | 10 min | Not assessed, self-checked |
| Worked examples and guided practice | 9 min | Not assessed, graded by Classfern |
| Independent practice | 15 min | Graded by Classfern |
| Closure and exit ticket | 6 min | Graded by Classfern, self-checked |
Starter, "your turn", practice and exit ticket are graded by Classfern. Worked examples and teaching are not assessed. The hands-on task and the reflection are self-assessed. Explanations in talk are judged by the parent and not recorded.
The lesson
It runs here on the page. No sign-up and no names – progress is kept only in this browser.
Homework and practice at home
20 minutes: 10 practice questions, 3 multiple-choice questions and 2 to remember from earlier work, then a short reflection. New questions every time, all graded instantly.
More practice
More Grade 11 math lessons
- Complex numbers
- Radicals
- Vectors
- Laws of exponents
- Square roots and cube roots
- Systems of equations
- Arithmetic and geometric sequences
- Iteration
- Functions
- Graphing functions
- Exponential functions
- Trigonometric functions
- Law of sines and law of cosines
- Coordinate geometry
- Counting outcomes
- Experimental probability
- Compound events
- Conditional probability
- Expected value
- Box plots and interquartile range
- Scatter plots and lines of fit
- Sampling and making inferences
Curriculum
Common Core State Standards HSA.APR.D.6: “Rewrite simple rational expressions in different forms; write ^(a(x))/_(b(x)) in the form q(x) + ^(r(x))/_(b(x)), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.”
Common Core State Standards HSA.APR.B.2: “Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).”
Our lessons and questions are our own. They are not written or endorsed by any government department, testing agency or exam board.