Skip-counting and multiples are patterns made by a rule. A rule like "start at 3, add 4" makes a whole list of numbers, and the list often has features the rule never mentions.
Key idea
To follow a rule, apply it again and again: start at 3 and add 4 to get 3, 7, 11, 15, 19, … To find the 10th number, count the jumps: there are 9 jumps of 4 after the first number, so it is $3 + 9 \times 4 = 39$.
Why it works
Every number in "start at 3, add 4" is odd. The rule does not say so, but you can explain it: 3 is odd, and adding an even number (4) never changes odd to even. If the rule added an odd number, the list would switch odd, even, odd, even.
Worked example: The next numbers
Start at 5 and add 10. Write the first five numbers.
- Add 10 each time: 5, 15, 25, 35, 45.
- Notice: every number ends in 5, because adding 10 never changes the ones digit.
Answer: 5, 15, 25, 35, 45
Worked example: A number far along
Start at 4 and add 3. What is the 8th number?
- From the 1st number to the 8th there are 7 jumps of 3.
- $4 + 7 \times 3 = 4 + 21 = 25$.
- Check by listing: 4, 7, 10, 13, 16, 19, 22, 25.
Answer: 25
Common mistake
Counting 8 jumps for the 8th number. The first number is where you start, so the 8th number is only 7 jumps away. List the first few to check.