Within 20 (make ten, doubles, fact families) Knowledge-Based Study

Study notes

1. When adding two single-digit numbers whose sum is greater than 10, a fundamental mental math strategy is to make ten first.

   Because ten is the foundation of our place-value system, combining numbers to reach 10 makes mental calculation straightforward.

   Follow these steps to apply the strategy:

   1. Identify the larger addend and calculate how many more units are needed to reach 10.
   2. Decompose (split) the second addend into that needed amount and whatever remains.
   3. Add the needed amount to make 10, then add the remaining amount.

   Example: Calculate $8 + 5$.

   • Start with 8. We need 2 more to make 10.
   • Decompose 5 into $2 + 3$.
   • Add to make 10: $8 + 2 = 10$.
   • Add the remaining part: $10 + 3 = 13$.
   • Therefore, $8 + 5 = 13$.

   A doubles fact occurs when a number is added to itself. Knowing doubles by memory allows you to solve adjacent facts quickly.

   Core doubles facts within 20 include:

   • $6 + 6 = 12$
   • $7 + 7 = 14$
   • $8 + 8 = 16$
   • $9 + 9 = 18$

   Near doubles are addition problems where the addends are consecutive numbers (one more or one less than each other). You can solve them using a related double.

   Doubles Plus One:

   • To solve $7 + 8$, recognize that 8 is $7 + 1$.
   • Think of the double: $7 + 7 = 14$.
   • Add 1: $14 + 1 = 15$.
   • Thus, $7 + 8 = 15$.

   Doubles Minus One:

   • To solve $8 + 7$, you can also use the larger double: $8 + 8 = 16$.
   • Subtract 1: $16 - 1 = 15$.

   Addition and subtraction are inverse operations, meaning they undo each other. A fact family is a collection of related addition and subtraction facts that use the exact same three numbers.

   When the two addends are different, a fact family consists of four related equations: two addition facts and two subtraction facts.

   Example: Consider the numbers 7, 9, and 16.

   • $7 + 9 = 16$
   • $9 + 7 = 16$ (commutative property of addition)
   • $16 - 9 = 7$
   • $16 - 7 = 9$

   Note: If the two addends are identical (a doubles fact such as $8 + 8 = 16$), the family has only two distinct equations:

   • $8 + 8 = 16$
   • $16 - 8 = 8$

   Understanding fact families helps you subtract efficiently within 20 without counting backward one by one.

   Strategy 1: Think Addition

   To solve a subtraction problem, think of the missing addend in the related addition fact.

   • Problem: $15 - 8 = \text{?}$
   • Think: What number added to 8 gives 15?
   • Since $8 + 7 = 15$, it follows that $15 - 8 = 7$.

   Strategy 2: Bridge Through Ten to Subtract

   You can also use 10 as an intermediate step when subtracting across ten.

   Problem: $14 - 6 = \text{?}$

   1. Subtract just enough from 14 to reach 10: $14 - 4 = 10$.
   2. Since the subtrahend is 6, split 6 into $4 + 2$.
   3. Subtract the remaining 2 from 10: $10 - 2 = 8$.
   4. Therefore, $14 - 6 = 8$.