Materials included

Algebra  Intermediate Phase, Grades 4 to 6

Fractions  Intermediate & Senior Phase, Grades 4 to 9

Rate and Ratio  Intermediate Phase, Grades 4 to 6

Derive the Natural Log  FET, grade 12+
Algebra
Elementary algebra involves the ideas of
 variable quantities,
 covariation (function), and
 equivalence of different formulas (expressions) that produce the same answers, as a consequence of the properties of the basic operations.
Apart from these ideas, elementary algebra also involves algebraic symbolism, and specifically:
 specifying a series of calculations with respect to any value of a variable (or variables) in a single expression, utilising specific syntactical conventions with respect to order of operations, and
 the use of letter symbols as placeholders for variables.
The Ukuqonda Institute Algebra booklet guides learners on a trajectory towards developing the different Algebraic concepts.
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Fractions
Learners would read something like ^{5}/_{8} simply as “five over eight” without recognising it as five eighths, which is what it actually means. It is critical that learners learn to say the fraction names, because without that there is no guarantee that they have a sound understanding of fractions. It is useful to distinguish the following three phases in the development of the concept of equivalent fractions in learners’ minds:
 Awareness that the same part of a whole, collection, quantity or unit of measurement can be described with different fractions.
 The ability to specify equivalent fractions.
 Producing equivalent fractions with a formula.
The Ukuqonda Institute Fraction booklet guides learners on this trajectory towards developing the Fraction concept.
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Rate and Ratio
It is better to withhold the words ratio and rate till the concepts have been experienced in at least two or more contexts. Postponing procedural based calculation of ratios and rates will be ideal and rather first allow learners to make sense of the situation. Be sure to encourage them to work sensibly with the information. Learners could use a range of different methods to find solutions.
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Derive the Natural Log
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What is the meaning and purpose of e and the natural logarithm?