Apostle Paul used phrases such as "I would not have you ignorant" (or variation thereof) and "know ye not" to accomplish his goals regarding ignorance. He utilized the first phrase to impart information where there was a lack of understanding regarding a subject. He used the second phrase to remind his readers about concepts they understood but which needed reinforcing.
Below are Paul's uses of the two phrases in question and the topics they covered.
Jesus' role in the Exodus
Moreover, brethren, I would not that ye should be ignorant, how that all our fathers were under the cloud, and all passed through the sea . . . And did all drink the same spiritual drink: for they drank of that spiritual Rock that followed them: and that Rock was Christ" (1Corinthians 10:1, 4).
God's plan of salvation
For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in (Romans 11:25).
Symbolism of baptism
Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? (Romans 6:3).
Now concerning spiritual gifts, brethren, I would not have you ignorant . . . Now there are diversities of gifts, but the same Spirit. And there are differences of administrations, but the same Lord (1Corinthians 12:1, 4 - 5).
Necessity of living a holy life
Know ye not, that to whom ye yield yourselves servants to obey, his servants ye are to whom ye obey; whether of sin unto death, or of obedience unto righteousness? (Romans 6:16).
Know ye not that ye are the temple of God, and that the Spirit of God dwelleth in you? If any man defile the temple of God, him shall God destroy; for the temple of God is holy, which temple ye are (1Corinthians 3:16 - 17).
Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump? (1Corinthians 5:6).
Know ye not that the unrighteous shall not inherit the kingdom of God? Be not deceived: neither fornicators, nor idolaters, nor adulterers, nor effeminate, nor abusers of themselves with mankind, nor thieves, nor covetous, nor drunkards, nor revilers, nor extortioners, shall inherit the kingdom of God (1Corinthians 6:9 - 10).
Know ye not that your bodies are the members of Christ? shall I then take the members of Christ, and make them the members of an harlot? God forbid. What? know ye not that he which is joined to an harlot is one body? for two, saith he, shall be one flesh. But he that is joined unto the Lord is one spirit (1Corinthians 6:15 - 17).
What? know ye not that your body is the temple of the Holy Ghost which is in you, which ye have of God, and ye are not your own? (1Corinthians 6:19).
Know ye not that they which run in a race run all, but one receiveth the prize? So run, that ye may obtain (1Corinthians 9:24).
Examine yourselves, whether ye be in the faith; prove your own selves. Know ye not your own selves, how that Jesus Christ is in you, except ye be reprobates? (2Corinthians 13:5).
Though who serve should be supported
Do ye not know that they which minister about holy things live of the things of the temple? and they which wait at the altar are partakers with the altar? Even so hath the Lord ordained that they which preach the gospel should live of the gospel (1Corinthians 9:13 - 14).
Evangelism brings trials
For we would not, brethren, have you ignorant of our trouble which came to us in Asia, that we were pressed out of measure, above strength, insomuch that we despaired even of life (2Corinthians 1:8).
The resurrection of the dead
But I would not have you to be ignorant, brethren, concerning them which are asleep, that ye sorrow not, even as others which have no hope (1Thessalonians 4:13).
Future glory of Christians
Do ye not know that the saints shall judge the world and if the world shall be judged by you, are ye unworthy to judge the smallest matters? Know ye not that we shall judge angels? How much more things that pertain to this life? (1Corinthians 6:2 - 3).