Gr 9 Maths Worksheets (Ages 14-15) | Level M22: Real Numbers & Polynomials

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Help your child build absolute confidence and mastery in mathematics. This comprehensive workbook from MathSmith takes a highly structured, "one-rung-at-a-time" approach to Real Numbers and Polynomials, ensuring students fully understand concepts before moving forward.

Instead of overwhelming learners with random equations, this workbook uses a carefully scaffolded, predictable learning ladder. It is perfect for homework, structured independent study, or mastery practice.

📊 What Is Included?

1 Complete Level (Level M22)

27 Structured Sub-levels for gradual progression

108 Printable Worksheets

1,084 Total Problems for thorough practice

Full Answer Key online, unlocked by the order ID on your Payhip receipt, for quick, stress-free marking

🗺️ Curriculum Breakdown

This collection focuses deeply on mastering Real Numbers and Polynomials through a carefully curated system:

Which numbers there are (3 Sub-levels): Three rungs about what a number IS, before anything is done with one. A gap between two fractions has infinitely many fractions in it, and the way to see that is to write both over a much bigger denominator and count. A number that cannot be written as one whole number over another is irrational - and the two beliefs in the way are that such a number is one nobody has worked out yet, and that a decimal running on for ever is enough to make it one. A third of one runs on for ever. The strand ends with the families sitting inside one another, and with the claim that between them they use up every point on the line.

Writing a number that never finishes (3 Sub-levels): The level's first notation, and the level's first number with no digits at all. Dividing a fraction out can do only two things - stop, or start repeating - because there are only so many remainders to be had, and the bar says exactly which digits come back: a bar over one digit of nought point one six gives a sixth and a bar over both gives sixteen ninety-ninths, told apart by where it starts. Undoing the bar is nines for what is under it and noughts for what is in front of it. Then the number that no bar will reach, and the construction that finds it anyway: a semicircle, a perpendicular, and a length brought down onto the line that is exactly the root of a whole number, with nothing measured anywhere.

A root carried through arithmetic (7 Sub-levels): M18 asked what a root came to. These seven rungs ask what happens when it does not come to anything - when the root sign is not an instruction but the number's name. Simplifying first, because two surds can only be collected when what is under the bars is the same, and the root of 12 and the root of 27 do not look alike until they are. Then the pair of rules that look identical and are not: what is under the bars multiplies, and does not add. Then the index that is a fraction, which is what a root has been all along, and which lets every law of exponents work on one unchanged. The strand ends by moving a root out from under a bar - twice, because two terms underneath need the difference of two squares to shift them.

What a polynomial is worth at a number (8 Sub-levels): M19 multiplied brackets out and factorised what came back. These eight rungs ask a different question about the same objects: what the expression comes to when the letter is a number, and what that one answer tells you. A zero is a number that makes it nought. The remainder theorem says that the remainder after dividing by x minus a IS what it comes to at a, so one substitution replaces a long division - and the factor theorem is the same sentence with the remainder nought. Together they turn a cubic into a quadratic, which the splitting rung then finishes.

Saying what a product comes to without multiplying it (6 Sub-levels): Six rungs of identities, each of them a shortcut that is also an argument. The square of three terms has six terms in it, because a square cut across and down at two places has nine pieces and three matching pairs. A cube has four, and the middle two carry a three for the same kind of reason. Two cubes factorise into a small bracket and a big one that never goes further, and three cubes have a short cut that only works when the three numbers add to nought - which is why the condition is checked first every time. Then the identities turned on ordinary numbers, and a last rung where nothing says which method to use, because choosing is the skill.

🎯 Who Is This For?

Parents looking to provide structured math support at home.

Homeschoolers searching for a sequential curriculum to guarantee zero gaps in math skills.

Teachers & Tutors looking for bite-sized, scaffolded printables for classroom morning work or targeted intervention.

📖 How To Use This Workbook

Print the workbook, read each rung's method page, and let your child work inside the printed frame. Then check the answers your way: in the printable answer-key PDF included in this download, or online, one page at a time, with the order ID on your Payhip receipt (valid for six months). Please keep the answer key for the adult marking the work and do not share it with your child: the workbook only works if they have worked each answer out for themselves.

Step-by-step manual and video guide: https://ebooks.thecodecrafters.net/manual/

Check the answers: https://ebooks.thecodecrafters.net/answers

⚙️ Digital Delivery Details

Format: four instant-download PDFs (free sample, sample answers, workbook, full answer key). The answer key is also available online.

Print Ready: Easily printable on home or school printers.

Usage: Licensed for individual family or single-classroom use.