Most people tend to only understand math when they can translate it to the real world, something they can understand. You've given several examples of theoretical concepts that look like magic to someone who can't translate it into something meaningful.
But most of these (still somewhat elementary) concepts can easily be shown using real world examples.
She has insufficient understanding of fractions: she would understand what e.g. 3/4 would mean graphically, but does not understand that it's the same number as 6/8
Take a pizza, cut it into 4 pieces. Ask her how many slices she wants (let's say she says 2). Then continue cutting the pizza in 8 slices, and give her two of those slices. Wait for her to complain, or if she doesn't, ask her if she received as much pizza as she asked her for. Try to get her to explain to you that you didn't give her what she asked for.
The core idea to convey is that 2 slices when there were only 4 slices is exactly as much pizza as 4 slices are when there are 8 slices. Therefore, 2/4 = 4/8.
While a pizza may be impractical for a tutoring session, the same applies to a small cake or a chocolate bar.
and can not do operations with fractions
My elementary school teacher used water and measuring jugs (each containing a liter, but each jug was marked in 4 parts, or 5 parts, or ...) to first prove to us that e.g. 2/4 + 1/5 = 7/10. At this point, it was no longer a wild magical claim, but it was provably the case.
At that point, it became easier to try and understand (using unintuitive mathematical tricks) how we could've figured out that this was the case. You can do this in steps:
- First, using two jugs that are divided into 4 parts, prove that 2/4 + 2/4 = 4/4. She should be able to predict that when she sees it done with the jugs.
- Then, using a 4-part-jug and a 2-part-jug, prove to her that 2/4 = 1/2. Pour the liquid from one jug to the other to prove the point.
- Then ask her what 2/4 + 1/2 is going to be. Try to avoid asking her for a fraction, instead try to get her to state whether it's going to fill the jug or not.
To this day, I still use those jugs when doing fractions in my head. It really stuck with me because it's a visual and tangible representation of something that is otherwise just some arbitrary number magic.
Similarly, you can use the pizza/cake/chocolate bar to come up with similar real life examples.
She can not solve simple worded exercises like "A new phone costs double as an old one. If the new costs 200$, how much does the old one cost?"
Try to come up with simpler and simpler worded exercises, to the point of balantly easy maths. Try to figure out the point where she starts getting stuck.
- A new phone costs double as an old one. If the new one costs \$200, how much does the old one cost?
- A new phone costs \$10 more than an old one. If the new one costs \$200, how much does the old one cost?
- A new phone costs \$10 more than an old one. If the old costs \$200, how much does the new one cost? (this is slightly easier as the two sentences map more clearly to each other)
- You intended to buy a \$150 phone, but you ended up buying the fancier \$200 phone. How much did you overspend?
- You're going to buy a \$100 phone, but you want to also buy the \$20 phone case that goes with it. How much money should you bring to the shop?
If she really struggles with visualizing this, she may be more comfortable talking about an actual purchase she made instead of a theoretical one. Ask her about something she's bought or is wanting to buy. You can get her to do the same puzzle solving depending on the situation)
- If her parents paid for some of it, but not all, ask her how much it cost, how much her parents paid, and how much she paid for herself.
- If she's still saving up to buy it, ask her how much it costs and how much she has already saved. Then ask her how much she still needs to save.
Whatever you end up talking about, then follow up with a theoretical example that exactly mirrors the real life situation she already understands. This will very much telegraph to her that these two cases (the one she knows and the one she doesn't) are effectively the same but with different numbers/words.
Doing this repeatedly will eventually teach her to translate an unknown situation into a known one.
Stick to what they know
It's a cliché example, but in The Office there is a character called Kevin Malone, whose main characteristic is being allround dumb. It's established at one point that he can't count money, but he can count food. If ask him to count pies instead of dollars (or salads), the question becomes answerable to him.
Here's the scene in question
It's a really stupid joke, but it actually gets at the heart of how people do math: visualization. People are intuitively able to visualize that which they understand, and no one is able to visualize math without having been taught how to translate the abstract into the real (and equal), or vice versa. That is your job as a math educator.
Your task is to find the visualization that works for your student. That's going to be a highly personal process. Relate it to her hobbies and past experiences.
Ask her things the already (subconsciously) understands, no matter how trivial, and then present more theoretical examples that are blatantly the same situation as the one she already understands.
The goal is to get her to realize that these seemingly different situations are the same. The more blatant the similarity, the faster she'll figure it out.