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Can I say,

(i) Last Sunny morning, Nicky and his father went fishing . They went to a pond near their house. They also brought a rod and a pail to put the fish in it.

(ii) He was glad that he can eat the fish he caught.
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(i) Last Sunday morning, Nicky and his father went fishing . They went to a pond near their house. They took a rod and a pail to put the fish in.

(ii) He was glad that he could eat the fish he caught.
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Vincent Teo (i) Last Sunny Sunday morning, Nicky and his father went fishing . They went to a pond near their house. They also brought a rod, and a pail to put the fish in it.

(ii) He was glad that he can could eat the fish he caught.
Edit. MrM's is better, with "They took a rod . . " Brought is probably incorrect.
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I thought "took =brought".

because "brought" is the same meaning with "took", right?
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It's a grey area. Many people use them interchangeably.

Take this with you when you go.

Bring that with you when you come.

It depends on where you are when you make the statement / give the command, in relation to the destination.

I took my cell phone with me when I went to the movies. When the movie was over, I brought it back home with me.

You're at home when you make this statement. I think it would be incorrect to swap these verbs in this sentence.

To most people, I expect it would make no difference.

Phone conversations may be different. I haven't thought much about it.

Are you coming to my party?

Of course. I'm going to bring the beer!

Who wound dare argue that this is not correct?

It's really the same deal as with "come" and "go."

I'm planning to come / go to your house for Thanksgiving.

Go figure.

Joe lives at the lake. He's talking to Sam, who is visiting him:

"Last week Nicky and his father came to go fishing. They brought a pail." (No way would he say, "They took a pail.")

"They took the pail out in the boat with them."

"To go fishing" is a fixed expression. Could Joe say, "Last week Nicky and his father came fishing." ?? It's possible.