# WEST Math for the Middle Grades

Most of us learned to divide fractions.  We wrote the problem as it was presented in our math books:

3/4 ÷ 1/2

Then we were instructed to rewrite the problem changing the division to multiplication and flipping the second fraction:

3/4 x 2/1

At this point we could cancel, or we could multiply and reduce later.  Let’s multiply:

6/4

We can reduce to by dividing the numerator and denominator by 2:

3/2

And if necessary, we can change this to a mixed number:

1 1/2

Does this answer make sense?  It’s hard to tell.  We’ve followed a procedure, an algorithm, but do we understand what just happened?  For most of us, understanding the connection between the original division and the answer was missed.  Maybe your elementary teacher didn’t explain it.  Maybe he did and you didn’t understand or remember.

In Math for the Middle Grades we use a wonderful curriculum called Math Mammoth.  Math Mammoth is written by a Maria Miller, a homeschool mom who was born and educated in Finland and now lives in the U.S.  According to her website (click here), she was tutoring homeschooled children and saw that there was conceptual understanding missing from the children’s math education.  In her books she provides lessons in understanding math concepts, followed by lessons that teach the traditional algorithms for those concepts.

Students who join us for Math for the Middle Grades get a good start on understanding of operations with whole numbers, fractions, decimals, and percents.  This understanding provides some great advantages:

1. Greater ability to calculate mentally.  This is a great time saver when solving long, complicated algebra trigonometry equations.
2. Increased recognition of sensible answers.  When following an algorithm to divide decimal numbers or add fractions it’s helpful to have a rough estimate of what the answer should be.  That estimate can be made quickly by mental calculation when the operation is understood.
3. More easily connect real world problems with the math that represents them.  Solving real world problems can be difficult to solve when the process starts with picking the right algorithm.  Making a connection between the algorithm for multiplying decimal numbers and calculating the sales tax on a particular purchase seems arbitrary.  Understanding the concepts of decimal part of a number or percent of a number has a more direct connection to the problem.

Math for the Middle Grades help students in grades five and up with understanding the concepts described in this post and lots more.  In this worktext-based class we also study basic math facts, geometry, and graphs.  Most weeks also include a game that provides practice of math skills.

# Finishing Well

With only a few weeks left of classes at WEST, I’m thinking about how to finish well.  The first few weeks of a new school year seem strong, powerful, and effective.  A lot of energy is generated and used in starting a new thing.  I want the end of the year to feel like that.  The end of the year has the potential to be as productive as the beginning.  Here are a few things I’ve been thinking about:

The big picture, the goal, has to be in front of me.  I need to know what I’m working toward and how to know when I’ve finished. The finish line is not just a date.  It might not be a date at all.  The finish line is when I’m done with the work.

The goal is achieved by making many small choices. Sometimes there are many decisions in a day that could have a cumulative effect on whether or not I reach the finish line.

Patient endurance is key.  The initial excitement got me going, but patiently looking toward the goal and enduring through occasional, or sometimes frequent, discouragement will get me to the goal.

Sometimes I’ll need help from others.  It’s much better to ask for help than to miss the opportunity to achieve.

How can you finish well?  How can you keep the big picture in mind, make small choices, patiently endure, and ask for help today?

# Daylight Saving Time

We were so proud of ourselves!  We had managed to corral many children, get everyone ready to go, and arrive at church well before the service started.  After parking the car, we walked through the pedestrian-free parking lot thinking we were so early the previous service hadn’t ended.  The elevator ride ended with the doors opening onto the middle of the church service for which we thought we were early.  Before the doors completely opened, I pushed the button to return to the parking lot level.  As soon as the doors closed we whispered “Daylight Saving Time” and started laughing.  Having been defeated by the “spring forward” we went home.

This post is going to end up a lot like my last Monday post. Here’s an opportunity for research and learning:  Daylight Saving Time (DST) is an idea with a long and checkered past.  Follow the link below for an introduction to the history of DST, the pros and cons of DST, and the confusion it can cause.  It’s a great summary and could definitely be a jumping off point for further research.  Click here:

Brief History of Daylight Saving Time

# Risk Taker?

My Grama Raye gave each of her grandchildren the same Christmas gift each year.  We knew she would hand each of us an envelope containing a card and five dollars.   The ritual became a highlight of our family’s Christmas celebration.

One year Raye had a surprise for us.  We each had to choose between an envelope and an animal sculpted from aluminum foil.  The choice had to be made without the benefit of handling a foil animal.  I couldn’t imagine choosing one of the animals as my gift.  Some of my cousins, however, each gleefully received one of the lovely silver creatures.  I took an envelope, which I promptly opened.  Inside I found the expected card and five dollars.  My more adventurous cousins disassembled their foil sculptures and discovered each held five dollars in coins.

Looking back, this event showed me something about how I prefer to handle life, and specifically money.  I chose the sure thing, the low-risk option, the known.  Some of my cousins chose adventure and the fun of the moment.  Each of us felt comfortable with our choices.

As an adult, I have embraced my lack of comfort with risk. Knowing I prefer more security and certainty makes financial decisions easier. Debt?  No, that’s risky.  Six months no interest?  No, something might change.  Potentially high-yield investments?  No, they are generally higher risk.  And the older I get, the less risk I’m willing to take.

In Personal Finance class, we’ll be discussing how being aware of our risk tolerance can help us make financial decisions.  We’ll also learn ways to eliminate unnecessary risk by living without debt, saving, and investing wisely.

# Busy?

Last weekend I quilted and taught quilting at Camp Lebanon.  I quilted, helped others with their projects, slept, talked, listened, and ate.  Joy!  As the weekend was wrapping up I began planning time to quilt at home.  I chose projects to work on and deadlines for finishing a few things.  When I mentioned my plans, another quilter responded wistfully with something like, “I’ll be too busy with Christmas things to do any quilting at home.”  I nodded in understanding.  The activities of the Christmas season can be very time-consuming.

After hearing similar statements from a few people I began to question my initial empathy. What Christmas activities occupy so much of everyone’s time?  And why, when each mentioned being busy with Christmas activities, did each person seem less than joyful about the prospect?

Many people enjoy participating in many activities, keeping busy, and making Christmas fun.  Other don’t take much pleasure in the seasonally packed schedule but feel obligated to carry on holiday traditions.  Either way, the Christmas season can be exhausting, leaving many regretting their choices.  Maybe it’s time to look at things differently.

Many of us say “yes” to every invitation, carry on cooking and gift-giving traditions, and sometimes say “no” to something that seems to be just too much.  Maybe we should reverse that.  Instead of defaulting to “yes”, start with big picture goals.  Then decide on and schedule activities that fulfill the goals.  If an opportunity arises, ask how it will fulfill the goals set for the season.  If it helps fulfill a goal, do a quick pro and con list to be sure it’s worth the effort and cost. Maybe eliminate or change the traditions that aren’t serving the purpose or that are too costly.

Years ago, one of our Christmas traditions was Cookie Day.  My extended family would gather and make cookies and other Christmas treats, eat a delicious home-cooked lunch together, and pack the cookies on plates to give as gifts.  My mom did most of the planning and shopping.  The event was at my house.  By the end of the day, my mom and I were exhausted, and we would spend hours cleaning up afterward.  There were a couple of years that we continued this tradition because we felt obligated.  It was tradition! But it was too costly in effort and energy.  Eventuallly, we decided to fulfill our desire for family time in a way that everyone would find refreshing and fun.  Now we set aside a day for bowling and a meal at a restaurant just before or after Christmas.

This year my goals for Christmas are to focus on Jesus and spend time with family. In addition, I want to land at the end of the season with as little mess as possible in my home and in my relationships. My schedule will include more time for worship, prayer, and the Word, and I will be planning at least two events where my immediate family will gather, eat, and play.  At the same time, I’ll be keeping up routines at home so I can take part in other activities joyfully, without dreading the potentially disastrous shape my relationships and my house might be in if I’m too busy.

I’ve already turned down two Christmas activities.  One event would not have fulfilled any of my goals for Christmas.  A small part of the other event would have helped fulfill my goals, but only in tiny part. Participating in either of these events would have been counter-productive as far as my goal of maintaining my routines is concerned.

I think I’ve over-simplified a bit.  There will likely be more flexibility in what I do this Christmas, but starting with goals and plans will help me reduce regrets and exhaustion.

For the past five or six years, my son has cooked the turkey, mashed the potatoes, and prepared homemade gravy on Thanksgiving.  He started at age 10 or 11.  We’re not sure.  A couple of years ago, my husband thought it would be a good idea for someone else to learn to cook a turkey.  He and a different son shopped, prepared, and roasted the turkey.  Everyone else busily prepared side dishes:  cranberry sauce, sweet potatoes, scalloped corn, and of course mashed potatoes.  In all the busy-ness, no one noticed the lack of turkey aroma.

When the timer beeped, someone pulled the turkey out of the oven. The turkey looked beautiful.  Well, it looked beautiful, for a shrink-wrapped turkey.

We use roasting bags to cook our Thanksgiving bird.  Not much to clean up, and the turkey turns out perfect every time.  This year’s shoppers had spotted a box with a picture of a turkey on it and without reading the label they purchased and used a basting bag. This bag was not meant to cook in, but to marinate the bird in herbs and salt water before cooking.

About half of us opted to eat the turkey.  The other half concluded that, while the beautifully roasted poultry might not kill us that day, we have no idea what the long-term effects would be of eating turkey roasted in a basting bag.  We opted for a turkey-less Thanksgiving. It’s not like there was nothing else to eat!

This story came to mind in part because we’re approaching Thanksgiving, and in part, because I’ve scored lots of math tests and other math assignments since school started.  I will probably never cease to be surprised about the number of errors caused by not reading the directions.  Instead, students have looked at the picture, or the numbers, or the symbols and done what they thought should be done with them.  They saw the turkey on the box and roasted the turkey.

Students, please read the instructions.  Yes, you did some math, but if you didn’t answer the question you’ve likely missed the point of the problem.  And once you’ve solved something, go back and read again.  Sometimes we solve and we think we’re done, but the question has not been answered.  Pay attention to the details.  It will pay off in real life.  The turkey will be delicious!

# Pray

The sound of cars crashing could be heard above the sound of the radio and the babbling of my five month old baby in the back seat of my car. I looked toward the sound and saw a shower of glass falling on two cars across the busy street. Without thinking, I parked my car on the shoulder, locked it, and ran across the street.

In the driver’s seat of the car closer to me was a young woman with a gash on her forehead that matched the curve of the steering wheel. I looked around to see who else was there to help. A young man looking dazed and confused walked toward me. I asked him if he was involved in the accident. He said he was not involved, so I told him to call the police and ask for an ambulance. He looked confused. I pointed to the store across the street and told him they had a phone and he should go use it. I turned back to the driver, kept her talking, and prayed for her.

About then another man approached. He had a bloody forehead and was obviously in shock. The other driver. I told him to sit down and stay put.

A few minutes later emergency workers arrived. I left.

When I got back to my car my baby was happily cooing in his car seat. We left and went about our business.

Looking back, if I had thought about what was happening I would have driven by and not stopped to help. I had a baby with me. What was I thinking?

Many days later I was with some friends from church. One of them mentioned seeing me as she drove by the accident. Her response was to pray for the people involved and for me.  I credit my calmness in the situation, and the safety of my baby, to God acting on my behalf in part because of my friend praying.

Up to this point in my life, it had seemed obvious to me to pray for those in ugly situations.  The lesson for me was that I should respond to every situation with prayer and pray for everyone in the situation. Specifically, I should pray for the people involved, and also for the people providing immediate help, the people who will provide ongoing care, and the many people who will have opportunities to serve and offer grace to people in need.

# Pencil Snob

Pencils and paper, pens and a stapler . . .  These are a few of my favorite things!  The wall of pens and pencils at Office Max makes me smile.  The seasonal school supply department at Target makes me giddy.  Even the relatively small selection of office supplies at Hy-Vee is a reason for me to stop and gawk.  As much as I love a new pencil with a fresh eraser, or a three-ring binder full of graph paper, there are other things I don’t like.  Not all school supplies are of equal value. Some are wonderful, useful tools, and others just don’t do what they are supposed to do.

I love Dixon Ticonderoga pencils.  They write smoothly, erase easily, and sharpen to a beautiful point.  At Costco they come in a box of ninety-six.  That new box always feels like a lifetime supply.  I must be a cat.

There are many Dixon Ticonderoga enthusiasts.  We are sometimes referred to as pencil snobs and we wear the title with pride.  We love our Ticonderoga pencils, and we want to share the joy with everyone!

A student struggling with her pencil is an opportunity for conversion.  While working with a student her pencil lead broke.  I gave her a Ticonderoga and she continued writing.  A moment later she erased something.  Her mother, who had been quietly observing, exclaimed, “It erased everything so easily!”  Another convert.

When I find other pencils in my home I usually just throw them away.  While it seems wasteful, I believe it’s wasteful to provide inferior tools to our children, and ourselves, and expect excellent work.  Excellent tools make it easier to produce high-quality work. Excellent tools make it more likely we will enjoy our work and therefore invest more effort and more time.  When I find other supplies I love as much as my Dixon Ticonderoga pencils I’ll be sure to let you know.  What tools have you used that exceeded your expectations?

# Make Change

My son and I were waiting in the checkout line at the grocery store when I observed something that made me want to take action.  The customer in front of me got her total and handed the cashier a \$100 bill.  The cashier thought she pushed the right buttons, but the register did not display the amount of change to give the customer.  The cashier asked the bagger, who was also a cashier, what to do.

At that moment I had to turn around and compose myself.  I had to resist…

After a moment I was able to turn back in the direction of the cashiers.  The bagger grabbed a calculator and began entering numbers.

I had to turn away again.  I couldn’t watch.  Resisting the pull to act was so difficult!

The second cashier finished using the calculator.  The first cashier gave the customer her change.  I was still resisting.  All through my transaction I was biting my tongue.

I so badly wanted to teach that cashier how to count change.

Counting change from the purchase price to the amount given is relatively simple and involves little more than counting skills and familiarity with the values of coins.  The goal is to count up from the purchase price to the amount the customer presented using the fewest coins and bills possible.

Here’s an example:  the purchase total is \$2.89.  The cashier says “That will be \$2.89,” and as the customer hands the cashier a \$10 bill, “out of 10.”  The cashier places the \$10 bill on the register, not in it, counts the change from the drawer into her hand, then from her hand to the customer’s hand.

From the cash drawer to the cashier’s hand:

“The total is \$2.89

out of 10

(picks up a penny) \$2.90

(picks up a dime) \$3.00

(picks up a \$1 bill) \$4.00

(picks up a \$1 bill) \$5.00

(picks up a \$5.00 bill) \$10”

The cashier then counts the change into the customers hand in the same way.  There is no need to subtract.

When the changed is calculated by the cash register, counting the change back to the customer from the purchase price up to the amount given is a good way to check your work and demonstrate to the customer they are getting the correct change.

In Math for the Middle Grades at  WEST  we’ll be learning this skill and practicing it a few times this year.

# Are You an Mathlete or a Weekend Warrior?

A weekend warrior, according to Merriam-Webster, is a person who participates in a usually physically strenuous activity only on weekends or part-time.

Because of the sporadic participation in a sport, and the enthusiasm for the game, weekend warriors are prone to injury.  Medscape.com lists the following as common weekend warrior injuries:  Achilles tendon rupture, plantar fasciitis, tennis elbow, ankle sprain, shin splints.  Once injured, the weekend warrior is temporarily, or sometimes permanently, unable to participate in the sport she loves.

To improve athletic skills, increase endurance, and avoid injury, an athlete will practice almost every day, sometimes for hours each day.  Athletes will also take breaks between workouts to recover from the exertion.

Math is like a sport.  There is a muscle in our brains called the Math Muscle.  When it is exercised only on weekends, or only on the day before a weekly class meets, it is prone to exhaustion or injury.  Once exhausted or injured it is unavailable, at least temporarily, for further exercise.

Of course the Math Muscle is fictional.  But the results of being a weekend warrior math student are not fictional.  How can the Math Muscle be strengthened?  How can we keep it in shape and learn more?  How can we increase our mental endurance?

Math is a set of skills that develop best when practiced many times a week.  Math students benefit from a training schedule similar to that which is effective for athletes.  Focused work time nearly every day and plenty of time for other activities will yield greater math skills with less exhaustion and fewer injuries.

## 5 Common Weekend Warrior Injuries

Brian Hamzavi, MS, MD; David A Forsh, MD  |  July 13, 2015