How do we know if a number is even? What does it mean?
Definition
Some number of dots is even if I can divide the dots into pairs, and every dot has a partner.
Some number of dots is odd if, when I try to pair up the dots, I always have a single dot left over with no partner.
The number of dots is either even or odd. It’s a property of the quantity and is doesn’t change when you represent that quantity in different bases.
Problem 13
Which of these numbers represent an even number of dots? Explain how you decide.
\[22_{ten} \quad 319_{ten} \quad 133_{five} \quad 222_{five} \quad 11_{seven} \quad 11_{four}\]
Think / Pair / Share
Compare your answers to problem 13 with a partner. Then try these together:
 Count by twos to \(20_{ten}\).
 Count by twos to \(30_{four}\).
 Count by twos to \(51_{seven}\).
You know that you can tell if a base ten number is even just by looking at the ones place. But why is that true? That’s not the definition of an even number. There are a few key ideas behind this handy trick:
 In base ten, every number looks like $$\text{(some multiple of ten) + (ones digit)}$$ $$\begin{split} 53 &= 50 + 3 \\ 492 &= 490 + 2 \\ 45637289108 &= 45637289100 + 8 \end{split}$$
 Every multiple of ten is an even number, since $$10n = 2(5n),$$and two times a whole number is always even.
 Your whole number looks like this: $$\text{(some multiple of ten) + (ones digit)}$$ $$\text{(even number) + (ones digit),}$$
 Even plus even is even, and even plus odd is odd, so your whole number is even when the ones digit is even, and it’s odd when the ones digit is odd.
Think / Pair / Share
 Make sure you understand the explanation above. Does each piece make sense to you?
 In particular: Use the definition of even and odd above to explain the last step. Why is it true that even + even = even and even + odd = odd?
 What about odd + odd? Is that odd or even? Justify what you say.
Problem 14
 Write the numbers zero through fifteen in base seven:
base ten 
base seven 
0 
\(0_{seven}\) 
1 
\(1_{seven}\) 
2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

 Circle all of the even numbers in your list. How do you know they are even?
 Find a rule: how can you tell if a number is even when it’s written in base seven?
Problem 15
 Write the numbers zero through fifteen in base four:
base ten 
base four 
0 
\(0_{four}\) 
1 
\(1_{four}\) 
2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

 Circle all of the even numbers in your list. How do you know they are even?
 Find a rule: how can you tell if a number is even when it’s written in base four?
Think / Pair / Share
 Why are the rules for recognizing even numbers different in different bases?
 For either your base four rule or your base seven rule, can you explain why it works that way?