Algebraic Reasoning Flashcards

All 7 cards in this deck

What is the difference between an equation and an identity?

An equation is true only for particular value(s) of the unknown; an identity is true for all values and is written with $\equiv$.
e.g. $3(x+2) \equiv 3x+6$

What are the steps to show algebraically that two expressions are equivalent?
e.g. show $2(x+3)+4 \equiv 2x+10$

1. Expand any brackets: $2x+6+4$
2. Collect like terms: $2x+10$
3. State this is the same as the other expression, so $2(x+3)+4 \equiv 2x+10$

What are the steps to use algebra to show a result about odd or even numbers always holds?
e.g. show the sum of two odd numbers is always even

1. Write general forms: $2m+1$ and $2n+1$
2. Add and simplify: $2m+2n+2$
3. Factorise and state the property: $2(m+n+1)$, a multiple of $2$, so even

What are the steps to use a shape with algebraic sides to show a given algebraic result?
e.g. a square of side $x+3$ has area $10\ \text{cm}^2$; show $x^2+6x=1$

1. Write the area (or perimeter) using the algebraic sides: $(x+3)(x+3)$
2. Expand and include the given value: $x^2+3x+3x+9=10$
3. Rearrange to the given result: $x^2+6x=1$

True or false? Showing that $n^2+n$ is even when $n=3$ proves that $n^2+n$ is even for every value of $n$.

False. One example only tests the claim; an algebraic argument is needed to show it always holds.

When an algebraic expression gives the length of a side of a shape, which values of the unknown must be rejected?

Any value that makes the expression zero or negative, because a length must be greater than $0$.
e.g. a side $4x-2$ cannot have $x=0.5$, as the side would be $0$ cm.

True or false? To explain why someone's suggested value of $x$ cannot be correct, it is enough to say that it does not look right.

False. You must substitute the value into the expression and state the impossible consequence, e.g. the side would then be $0$ cm.