Give and Take - Mental Arithmetic

Adding by Breaking Down: To add numbers mentally, break them into tens and ones. For example, to add $23 + 45$, think of $45$ as $40 + 5$. First add $23 + 40 = 63$, then add $63 + 5 = 68$. Visualize this as base-10 blocks where you first group all the long bars (tens) and then the small cubes (ones).

The Jump Strategy: Imagine a horizontal number line. To solve $34 + 20$, start at $34$ and take two big 'jumps' of $10$ units each to land on $44$ and then $54$. To subtract, simply 'hop' backward on the line.

Patterns on the 100-Chart: A 100-chart is a grid with 10 rows and 10 columns. Moving one step down adds $10$, and moving one step up subtracts $10$. Moving one step right adds $1$, and one step left subtracts $1$. Visualizing this grid helps in adding or subtracting multiples of $10$ instantly.

Subtraction by Counting On: Instead of taking away, you can find the difference by counting 'up' from the smaller number to the bigger number. For $60 - 55$, start at $55$ and count how many steps to reach $60$. This is like seeing two points on a path and measuring the distance between them.

The Compensation Method (Smart Addition): If you need to add a number ending in $9$ (like $19$), add $20$ instead and then subtract $1$. For $45 + 19$, think $45 + 20 = 65$, then $65 - 1 = 64$. This is like over-filling a container and then pouring a little bit back out to reach the exact level.

Fact Families and Inverse Relationships: Addition and subtraction are opposites. If you know $30 + 20 = 50$, you automatically know $50 - 20 = 30$ and $50 - 30 = 20$. Imagine a triangle with $50$ at the top and $30$ and $20$ at the bottom corners; the relationships flow between these three numbers.

Adding and Subtracting Hundreds: When dealing with larger numbers like $200 + 300$, focus on the hundreds digit. $2 + 3 = 5$, so $200 + 300 = 500$. Visualize this as large bundles of $100$ sticks being moved from one pile to another.