To solve the puzzle, we need to determine which class each child is in and who their teacher is, using the clues provided. Let’s break down the clues one by one:

1. **Miss Board is teaching Squirrel class this year.**

   - This means **Miss Board** is the teacher for the **Squirrel** class.

2. **Basil has a new teacher called Mr. Book.**

   - **Basil** is taught by **Mr. Book**.

3. **Hedgehog class is taught by Mrs. Marks.**

   - **Mrs. Marks** teaches the **Hedgehog** class.

4. **Basil is not in Squirrel class.**

   - Since **Basil** is not in **Squirrel** class, he cannot have **Miss Board** as his teacher. He is taught by **Mr. Book**, so he must be in a different class.

5. **Mrs. Marks is Jorge’s new teacher.**

   - **Jorge** is taught by **Mrs. Marks** and therefore is in the **Hedgehog** class.

6. **Zoe is in a class named after a bird.**

   - The bird classes are **Owl** and **Robin**. Therefore, **Zoe** is either in **Owl** or **Robin** class.

7. **Francis is in Squirrel class and is taught by Miss Board.**

   - **Francis** is in **Squirrel** class with **Miss Board** as his teacher.

8. **Rabbit class is taught by Mr. Book. Jorge is not in this class.**

   - **Rabbit** class is taught by **Mr. Book**. However, **Jorge** is not in this class, so **Basil** must be in **Rabbit** class.

Let’s summarize:

- **Francis** is in **Squirrel** class with **Miss Board**.

- **Jorge** is in **Hedgehog** class with **Mrs. Marks**.

- **Basil** is in **Rabbit** class with **Mr. Book**.

- **Zoe** must be in **Owl** class (as it’s a bird name) and is therefore taught by **Mr. Chalk**.

So, the solution is:

- **Basil**: **Rabbit** class, **Mr. Book**.

- **Francis**: **Squirrel** class, **Miss Board**.

- **Jorge**: **Hedgehog** class, **Mrs. Marks**.

- **Zoe**: **Owl** class, **Mr. Chalk**.