Example: Consider trying to find out how many hours you spend watching TV in a week. Instead of adding up each day separately, you could use a formula where the letter $h$ represents the hours you watch each day. If you watch the same amount of TV each day, you could use the formula:

Here, $h$ is the unknown (the number of hours you watch each day), and this formula gives you a quick way to calculate the total.

• If we don't know what something is, we call it a letter. This helps us set up equations to solve problems where we can eventually figure out what the letter stands for.

Examples:

• $m+m=2m$
• $3p+2p=5p$ → 3 lots of something, plus 2 lots of something, gives you 5 lots of something.
• $16t²−4t²=12t²$ → 16 lots of something, minus 4 lots of something, gives you 12 lots of something.
• $10pq−7pq=3pq$

Example 1: Simple Simplification Problem: Simplify $4m+2p−m+6p$.

Solution:

• Step 1: Identify like terms. $4m$ and $−m$are like terms. Similarly, $2p$ and $6p$ are like terms.
• Step 2: Combine the like terms.

• Final Answer:

Example 2: Tricky Simplification Problem: Simplify $4t²−5t−2t+3t².

Solution:

• Step 1: Identify like terms. $4t²$ and $3t²$ are like terms. Similarly, $−5t$ and $−2t$ are like terms.
• Step 2: Combine the like terms.

• Final Answer:

Example 1: Simple Multiplication Problem: Simplify $5b×2c×3a$.

Solution:

• Step 1: Each term here is different, but that's fine because we're multiplying.
• Step 2: Multiply the numbers (coefficients) together first.

• Step 3: Now deal with the letters. Write them in alphabetical order and leave out the multiplication sign.

• Step 4: Put the numbers and letters together to form the final answer.

Example 2: Multiplying with Negative Numbers Problem: Simplify $4r×−3p×3r×q$.

Solution:

• Step 1: No problem with the different terms. We can still multiply them.
• Step 2: Multiply the numbers (coefficients) together first. Be very careful with the negatives!

Note: There was no number in front of $q$, so it's just $1$.

• Step 3: Now deal with the letters. Multiply them and arrange in alphabetical order.

Remember, if you multiply something by itself (like $r×r$), it just means you are squaring it.

• Step 4: Put the numbers and letters together to form the final answer.

Example 1: Simple Division Problem: Simplify→

Solution:

• Step 1: Different terms are no problem, just like in multiplication.
• Step 2: Divide the numbers (coefficients) first.

• Step 3: Now deal with the letters. Here, $z$ in the numerator and denominator can cancel each other out.

What happened here? When you divide the $z$ on the top by the z on the bottom, you're left with 1 (since anything divided by itself is 1). Multiplying by 1 doesn't change the value, so the $z$effectively disappears.

Example 2: Division with Fractions Problem: Simplify,

Solution:

• Step 1: Different terms are no problem.
• Step 2: Divide the numbers (coefficients) first.

Note: When you don't get a whole number, you need to use fractions!

• Step 3: Now deal with the letters. This requires some knowledge of indices (powers)
• The a on the bottom cancels out one a on the top, leaving a in the numerator.
• The $b³$ on the bottom cancels out $b$ on the top, leaving $b²$ in the denominator.

Example 1: Using a Diagram to Form Expressions Let's use a simple diagram to create expressions:

• Imagine we have a diagram where different blocks are labelled with letters representing different variables.
• We can form expressions by combining these letters according to their arrangements.

Diagram and Expressions:

• $b=2r$

• $b=4y$

• $2g=3r$

• $b+r=6y$

• $3r−3y=g$ Explanation:

• These expressions come from observing the relationships between the blocks and how many of each block correspond to others. For instance, if two red blocks ($r$) equal one blue block ($b$), then $b=2r$.

Example 2: Forming an Expression from a Story Problem: Phillip is going on a shopping trip. He needs to calculate the total cost of his items. He needs:

• 5 energy drinks

• 2 packs of coffee beans

• 1 box of chocolates He doesn't know the prices, so he uses letters:

• $e$ for the price of an energy drink

• $b$ for the price of a pack of coffee beans

• $g$ for the price of a box of chocolates

Step 1: Write the initial expression for the total cost:

Step 2: Adjust the expression based on new information. For example, Phillip decides to buy two more tins of beans and one less pear. Update the expression:

Step 3: Consider another scenario where his friend Bruno is coming over to study, so he needs twice as much of everything:

Final Expression:

Example 1: Simple Substitution Problem: Evaluate the expression $3ab$ given $a=2$ and $b=5$.

Solution:

• Step 1: Identify the values: $a=2$ and $b=5$.
• Step 2: Substitute the values into the expression:

• Step 3: Multiply the numbers together:

• Final Answer:

Example 2: Substitution with Multiple Variables Problem: Evaluate the expression $2ac−acd$ given $a=2$, $c=−3$, and $d=−10$.

Solution:

• Step 1: Identify the values: $a=2$, $c=−3$, and $d=−10$.
• Step 2: Substitute the values into the expression:

• Step 3: Combine the results:

• Final Answer:

Example 3: Substitution Involving a Square Problem: Evaluate the expression $5ad²$ given $a=2$ and $d=−10$.

Solution:

• Step 1: Identify the values: $a=2$ and $d=−10$.
• Step 2: Substitute the values into the expression, remembering to square $d$ first:

• Step 3: Multiply the numbers together:

• Final Answer: