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Read Genesis 15:13-16 & consider the fact that God spoke these words several hundred years before the Exodus.  Then consider Pharaoh’s enslavement of the Israelites, & even the slaughter of all the male Hebrew infants.  Was God on His throne during these horrific acts?  Why didn’t He do anything to prevent them?  What does it mean to say God was “sovereign” amid these events?

The words of Genesis 15:13-16 are incredibly prophetic, all the way down to the timeframe that they would be there, their enslavement, how they would leave after plundering the possessions of the Egyptians, & how they would wander in the wilderness for 40 years until the “iniquity of the Amorites” was complete.
However, the fact that God knew about these events also can be troublesome when you realize the massive loss of life that took place under Pharaoh.
The question “Was God on His throne during these horrific events” is, of course, rhetorical.  God is always enthroned in the heavens (Psalm 45:6; 47:8) & there is not one molecule in all of creation nor is there one moment in time when He is not in authority.
But this naturally begs the question, “Why didn’t He do anything to prevent them” is a difficult, but good question.  How can we answer this?

Everything God does is good & perfect.  He & all His works define “good” (Genesis 1:31; 50:20; 1 Chronicles 19:13). There is no evil in Him.  He is absolutely incorruptible (James 1:13).
Because of the above truths, we can have assurance that God is never unjust or susceptible to the accusation of evil.  But we do know this world is evil because of the presence of sin.  Human beings are the source of all evil in this world as all have inherited the propensity to sin from their parents.  Then they commit acts of sin in selfishness & vainglory.  This is the source of the evil in the book of Exodus.
The reason that God didn’t prevent them is twofold:

First, God did not prevent them because there is one destination for children who die in infancy: the presence of God.  God allowed this great evil because what Pharaoh intended for evil, God used for good (cf. Genesis 50:20).
Secondly, God knew the plans & the hardness of Pharaoh’s wicked heart (Exodus 3:19) & had already decided within Himself that He would use the destruction of Pharaoh to display His glory.  Paul gives us a glimpse of this in the New Testament (Romans 9:15-17).

Finally, how was God sovereign during this time?

Job gave us a great description of God’s sovereignty in Job 42:2 when he said, “I know that you can do all things, and that no purpose of yours can be thwarted.”
God’s sovereignty is on display in His creation & in how He turns the evil acts of His creatures to serve His glorious purposes. So far, this has been a consistent them of the Old Testament books we have heard.

Read the “riddle” contained in Exodus 34:5-7.  How does Christ resolve this riddle?

You have the LORD (Hebrew, Yahweh) STANDING beside Moses, proclaiming the name of the Lord.  The “riddle” is that God is spirit (John 4:24), which means that He is present everywhere at all times (Psalm 139:7-10).  So, how can God, who is spirit, be present in bodily form standing beside Moses?
Christ resolves this riddle because He is the Word made flesh (John 1:14).  But what many Christians neglect to understand is that even before Jesus took on human & was born as the child of Mary, there have been many times throughout the Old Testament where He took on bodily form. Whether it was God in bodily form or even some times where the “angel of the Lord” is acknowledge to be God in the flesh, we need to understand that as the pre-incarnate Jesus manifesting Himself among human beings (for further study, read this article from Answers in Genesis)

As we have considered, Moses was a murderer (just like other characters in the Bible such as King David & the Apostle Paul).  What do you think God calls such unlikely characters?

Because it gives God an opportunity to reveal His glory & His grace.  One of the goals of God is to display His grace as He works through imperfect & sinful human beings.
As we are tempted to play the game of comparison to these characters, we need to remember that if God only had the option of working with righteous & perfect human beings, then He would only have option: Jesus.  We are all sinful (Romans 3:23) & unworthy of being used by God.  This should free us from the guilt of unworthiness &, at the same time, give us hope that He would use us.

God saves His people from Egypt, hardens Pharaoh’s heart, & performs the many miraculous signs ultimately for His glory.  Does this make God vain or self-centered?  Why or why not?

Because knowing the glory of God is the reason for which we have been created.  When we seek our own glory, we believe & pass on an insufficient reality.  We have not been created to reflect our own glory & make a name for ourselves.  We are created to point people to the beauty of God’s character.
So, for God to do this (live for His glory) is His essential nature & mission (& it should be ours as well).

What significance does the Passover Lamb have in the story of Exodus?  What does Paul mean when He refers to Jesus as our Passover Lamb (1 Corinthians 5:7)?

It sets up an understanding of substitutionary atonement in the minds of the Hebrew people & in the minds of those reading the narrative.  The blood of a lamb would cause the wrath of God to pass over this particular family & child.  For this to happen the lamb had to die & the people had to follow God’s instruction as to how to appropriate this sacrifice for themselves.
Jesus is our Passover Lamb.  He has shed His blood for us & we appropriate the sacrifice through the means God has called us to: faith.