Collect like terms to simplify an algebraic expression. Part of HELM Book 1.3

A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).

Students seem to not realise that $\frac{a}{b}\times c=c\times\frac{a}{b}=\frac{a\times c}{b}=\frac{c\times a}{b}=a\times c \div b=a\div b\times c=c\div b \times a \ne c \div (b\times a)\ldots $ etc. This question is my attempt to help rectify this.

Simplifying algebraic expressions

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. SP = Statistics & Probability strand. Students read a percentage from the chart, and calculate that percentage of the number of children (randomly generated).

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are given the number of people that are fed with 200g of rice (randomised), and the total number of people to be fed (randomised). They need to calculate the weight of rice using proportional reasoning.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are given the number of pages read (randomised) and number of pages remaining (randomised), and asked to write the proportion read as a simple fraction.

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Find the remainder when dividing two polynomials, by algebraic long division.

Express $\displaystyle \frac{a}{x + b} \pm  \frac{c}{x + d}$ as an algebraic single fraction over a common denominator. 

Basic Algebra question, expand one set of brackets, coefficients may be positive or negative, pronumerals randomised.

Simple substitution into an algebraic expression. Includes powers, division, mulitplication, brackets.  Includes subscripts.

several statements are given regarding trig identities. student is to select which are true and which are false

These question are designed to be completed without using a calculator.

Fractions already have a common denominator. Addition and subtraction 50:50 split. Students shouldn't have to worry about reducing fractions by design.

Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression, and finally evaluate it at a given point.

The word problem is about the costs of sweets in a sweet shop.

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. Students are shown an infographic, and need to calculate a percentage from the information supplied. There are 7 different versions of this question.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are asked to find a fraction of a fraction. There are 9 different versions of this question.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression. The word problem is about the costs of sweets in a sweet shop. The quantity of each type of sweet is randomised.

This question was modified from a Newcastle University question.

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. Students must do a simple interest calculation (geven principal, rate and months, all of which are randomly generated) then subtract their answer from the supplied compound interest amount to find the difference.

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. Students are given the number of years worked, and the price paid. They use the table to look up the percentage discount, then calculate the full price. The number of years and price paid are randomised.

Used for LANTITE preparation (Australia). NA = Number & Algebra strand. Students are given sales and years of service. They read the retainer and commission from two tables, then calculate before-tax income. The sales and years of service are randomly generated.

(a) Equation given, and five coordinates. Student should select those coordinates which lie on the line. (b) Gradient and a point on line is given. Student is to calculate coordinates of other points on the line.

Extra practice on some basic algebra skills, including solving linear equations. You can try as many times as you like and also generate new versions of the questions for extra practice.

Manipulation of algebraic fractions

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Example of an explore mode question. Student is given a 2x2 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

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