A Story of Trust and Provision
Hey Reader,
We are getting closer to the start of our school year over here, and I am in that in-between space where summer still feels present but the planning brain is starting to take over. I always have mixed feelings about this time of year. I love the fresh start that fall brings, and I also love the slower pace of summer, and the two things do not always want to coexist peacefully. If you are feeling that same tension, you are in good company!
Life
This week has been full of unexpected twists and turns, business pivots, and not-so-great news. But through it all, God has been at work, and his hand has been evident!
I want to tell you a story. Over the weekend, we had a garage sale. After the sale was over, we were counting the money, and we had $80 more than what we had written down. We couldn’t figure out where it had come from! We half-jokingly said that God had multiplied our money for some unknown reason, but we were thankful for it.
Fast forward to Sunday morning. We usually attend our church at it’s home campus, but this week we had actually gone to one of the other campuses for a special service. The sermon was exactly what we needed to hear, and we left feeling lighter than before. Then, as we were walking to our car, an older lady asked us if we could give her a ride down the road to the inn where she was currently living.
As my husband and the lady were talking in the front seat, she mentioned that she hoped the manager would be understanding today. When my husband asked what was going on, she said that it had been a rough week and she didn’t have enough money to keep her room.
Y’all. When Ian asked how much she needed, she said $80.
$80!
The exact amount of the mystery money.
God provided us with the exact amount this lady needed, and then guided her to us at just the right time so that we could bless her with what he had blessed us with.
God uses even the most mundane of circumstances to teach us and give us opportunities to bless others. ❤️
Homeschool + Math
Now, we have spent the last two weeks talking about fractions and integers and what they really mean and why understanding the concept matters more than knowing the procedure. Today I want to talk about a specific moment in fraction instruction that trips up more students than almost anything else in middle school math: dividing by a fraction.
You probably remember the rule. Keep the first fraction the same, change the division sign to multiplication, flip the second fraction upside down, and then multiply. Keep, change, flip. It works every time. But I want to ask you a question: do you know why it works?
If you had to explain to your child, without using the rule, why dividing by 1/2 gives you a larger number than what you started with, could you do it?
Most of us can’t. And that is not a personal failure; it is a curriculum failure. Division by a fraction is almost universally taught as a procedure, with the understanding assumed to follow from enough practice, but it rarely does.
Here is the conceptual key: division answers the question “how many of groups of that number fit into this number?” When you ask how many halves fit into 3, the answer is 6, because each whole contains two halves, and you have three wholes. Also, think about it: When you divide anything by two, you can think “what is half?” because multiplying by 1/2 is the same as dividing by two! That is true for all fractions. Dividing by 4 is the same as multiplying by 1/4, dividing by 6 is the same as multiplying by 1/6, and dividing by 4/5 is the same as multiplying 5/4. This is why multiplying by the reciprocal (keep, change, flip!) works. 😊
When a student understands division this way, as a question about how many groups fit and multiplying by the reciprocal, the behavior of dividing by a fraction stops being mysterious. Of course the answer will be larger when multiplying a whole number by a fraction, because you are asking how many small pieces fit into something, and small pieces fit more times, not fewer.
This is the kind of understanding that makes math makes sense rather than seem like a collection of arbitrary rules to memorize, and it is available to every student, at any level, once we stop rushing past the concept to get to the procedure.
If your student is in or approaching pre-algebra, this is exactly the kind of foundation we build in my Pre-Algebra Course. We don’t just learn how to perform operations, but also why they work and what they mean.
For classroom and co-op teachers: this concept is particularly powerful to teach with a visual model. Draw it out, use fraction bars, or use a number line. Showing students physically how many halves fit into 3 before introducing the algorithm gives the rule something to attach to.
Grace
What strikes me most about the fraction division concept is that the answer always seems counterintuitive at first. Dividing should make things smaller, and yet dividing by a fraction makes things larger. It feels wrong until you understand the why, and then it feels obvious, even almost inevitable.
Faith works like that sometimes too. The things God asks of us, like to release control, to trust in what we cannot see, to believe that surrender leads to freedom rather than loss, runs completely against our instincts. They feel backwards. And then we step into them and find that they are exactly as true as He said they were. It makes me think of the story that I shared at the beginning. I’ve also felt the challenge of trusting deep this last week when, once again, my carefully laid plans have all been turned on their head and once again I’m starting over, but I trust that God has a reason for all of it!
“Trust in the Lord with all your heart, and do not lean on your own understanding. In all your ways acknowledge him, and he will make straight your paths.” Proverbs 3:5–6 (ESV)
“For my thoughts are not your thoughts, neither are your ways my ways, declares the Lord. For as the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts.” Isaiah 55:8–9 (ESV)
We are not always going to understand the why. That’s not a flaw in God’s design, it’s an invitation to trust Someone whose understanding is bigger than ours. In math, we work toward the why, and it’s worth every bit of effort it takes to get there. In faith, the why is often given on the other side of obedience rather than before it, and sometimes, we won’t see the why till long after or even until heaven. Either way, the foundation holds, we just have to be willing to stand on it before we fully understand it.
See you soon!
- Mrs. Holman
