Refreshing Rose – Wordless Wednesday

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Here is a refreshing rose from my yard. Enjoy your Wednesday!

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Snowballs in Late Spring – Wordless Wednesday

Yes, we have snowballs in late spring.

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I refer to this as a snowball tree, but it’s technically called Viburnum Opulus.

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I look forward to seeing it blossom every spring. It only stays in bloom for a short time.

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Enjoy your Wednesday.

Flowers for Mama – Wordless Wednesday

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When the Explorer, 6, returned from recess, he carried with him a surprise for me.  He presented me with a bouquet of fragrant wildflowers he’d picked from our yard.  😉

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“Gratitude is the fairest blossom which springs from the soul.” ~Henry Ward Beecher

Nutrition Nudge


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As the warm spring wind blows away winter’s cold, I’ve decided to make some changes around here.

Throughout our day we usually have two snack sessions: the first at about 10 a.m., and the second around 3 p.m.  As you can imagine, we get our share of processed foods.  For example, we usually have granola bars, pretzels, and any number of crackers plentiful in the cupboard.  Though I have no problem with our eating of snacks, I do think we eat too many processed foods during these times.

My children do like fruit. But too often, in the rush of our busy days, crackers or pretzels are what they go for. And the lonely fruit sits and rots, forgotten in the deep dark fridge.

I’ve come up with a possible solution: I’ve decided to make our 10 a.m. morning snack be a fruit snack.  I’m not talking about fruit juice or items that contain fruit ingredients, I’m talking about eating actual fruit – bananas, oranges, grapes, apples, etc.  Just simple, refreshing real fruit.

How am I going to get my children to go along with it?  Well, not in a extreme, pushy way.

In addition to suggesting a morning fruit snack, my plan is to set out freshly washed fruit around 9:45 a.m. With shiny fruit smiling at them in a bright, colorful bowl, what’s to stop them?

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If that doesn’t work, then I’ll move on to plan B – which will be to tell them they have to eat a piece of fruit sometime before they go to bed.  They may just take the fruit early, and get it over with.

For us, I think this is a change worth making.  Imagine how much of a positive impact it could have on their growth and development if they add in another piece of fruit each day over time.  My goal is to gently and welcomingly develop better eating habits in them.  Morning snack time seems like a great time to make sure fruit is included in their day. If I’m lucky, they may even carry the habit into adulthood.

 And do you know what I’ve noticed?  They often watch me to see what I’m snacking on.  Yes, I’ll be joining them in the morning fruit snack.  No excuses for me, either.  😉

How do you encourage healthier eating habits in your family?

Paper Tie – Wordless Wednesday

What’s a child to do when he finishes his lessons early?  Perhaps design a tie by folding, coloring, and taping the paper art to his shirt?

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And the reverse must be a different pattern…

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Created by the now ‘Sharp Dressed Explorer’, 6.  😉

Mapping the World with Art Review

South America drawn by the Artist, 13.
South America drawn by the Artist, 13.

One of our favorite curricula that we’ve used during the past year has been Ellen McHenry’s Mapping the World with Art (MtWwA).  I’ve had several readers email me curious about how we use it.  This week we finished our lessons and are working on the final project.  Here is my overview of how we used it and thoughts on the curriculum.

Overview:

I originally purchased MtWwA for Geography only, but soon realized that it’s a combination of geography, cartography, art and even a little history.  It works best for middle school and up (though my 8 year old enjoyed it when he was able to go at his own pace).  The book is divided into four parts: 30 Readings, 30 Map Drawings, 23 Activities, and a Final Project where students create a world map.

The Readings section begins with history of the first maps and guides the reader in how they’ve been developed through time – with earlier maps being less complex than later ones. The reader is taken on a journey to a better understanding of how and why maps developed from necessity and exploration from ancient Greece through modern times.  My children and I appreciated the perspective it gave us.  For example, we were impressed when comparing a map that was made when the Pilgrims landed in America in 1650 with a modern map on our wall – realizing how much more we know now and that 1650 was not that long ago!  The section finishes with a discussion of how GPS satellites have revealed the entire surface of the Earth to us.  Interesting.

Next is the Map Drawing section intended for students to draw maps of countries and continents of the world.  In this portion each lesson shows detailed step-by-step instructions.  We purchased the optional DVD instructional video in which the author leads you through the map drawings.  It is interesting to hear the author’s take on each map.  At the beginning of each DVD lesson, the tools needed for that particular lesson are displayed on the screen so you know what to have available for that day.  (Tools needed throughout the course include a ruler, compass, protractor, pencil with eraser, large eraser, and a black waterproof pen.)

The third part is an optional Activities section which provides projects and reinforcement for the earlier topics.  Some of the activities include extra drawing practice, arts and crafts, games, and review worksheets.

Finally, there is a final project. It gives students the opportunity to put all the information they’ve learned together by creating a world map.  Templates are provided to photocopy to guide them along.

How we used it:

Like many people we don’t have unlimited time, so I chose Thursdays to cover only the parts of the program that I thought would work best for us.  We covered about one lesson per week consisting of the Readings and Map Drawing sections.   During our morning journal time, I’d cover the Readings.   This would often spark fascinating, meaningful discussions about both history and the level of understanding of the geography of the world for those who lived in earlier times.

In the afternoons we’d watch the DVD’s and follow along with the author whose knowledge on the subject was impressive.  The DVDs give you the option to pause to wait for everyone to have the time they need, and I’d also have the book open as an additional help if anyone wanted to see the drawings each step of the way.

We did not use the Activities section.

For the final project there are several options, and I chose the option of photocopying the 6 page templates to give the general idea how the continents and oceans fit together.  I thought this would best fit our goals and would keep frustrations to a minimum.  The final project is important to help students visualize how all the pieces they’ve drawn fit together as a whole.

When done in the time frame I used, some of the lessons may take multiple weeks to cover.  For example, Map Drawing 29 has parts A-G.  It took us 3 Thursdays to get through it.  Overall, it took us about 1.5 years to get through the entire book (with taking the summer months off.)

My thoughts:

We did not try to rush to get through this, and I considered it a fun elective.

Drawing the maps and thinking through the shape of the land helped them remember the information. In addition, we learned some more in depth drawing skills with compasses and protractors while sketching the continents and countries.  The final project drawing at the end helped cement the locations and gave us an important big picture view.  These factors combined with the fascinating history in the readings made for a positive educational experience.  For these reasons, I highly recommend Mapping the World with Art.

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Cape Horn, by the Artist.
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Do you see the ‘boot’ of Italy?
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Australia, by the Investigator, 11.
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Siberia, by the Investigator.
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Hawaiian Islands, by the Architect, 8.
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Papua, New Guinea, by the Architect.
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Scary Monster, by the Explorer, 5.

Forming First Impressions: Kindergarten Math

The Explorer, 5, recently told me that his favorite subject is Math.  At this age he’s forming early impressions on the blank canvas that will become his Math Education.  Developing a strong base is important, but so is lighting a spark of interest.  Led partly by his fascination levels, I focus on three areas with my Kindergartener: math facts, new concepts, and problem solving.

1. Math facts

Mastery of the math facts will greatly reduce frustration for him down the road with things like muti-digit multiplication, long division, and many other areas.  It’s important that he learn the concept behind the idea.

I start him on skip-counting by introducing a hundred number chart.  After he’s learned to count to 100 (by ones), we learn the two’s, five’s and ten’s.  We then cover both odd and even numbers and what digits each type will end in.

We recently learned the three’s. For example:  I have him start at zero and count up three, and then cover the number with a rectangular counter stopping at 30 – for starters.  The idea is that he’ll begin to notice a pattern.  [He observed that the three’s have both odd and even numbers.]

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Hundred number chart, counting by 3’s starting at 0.

Then he removes the counter and writes down the numbers – beginning at zero then up to 30.  (If he gets frustrated writing the numbers down, I just have him read them while I write them for him.) What I like about this is that not only does he learn to count by three’s, but he also realizes that it starts at zero.  He reads them forward and backwards a few times.  Then we focus on memorizing.

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Recording the 3’s as he removes them

After he’s mastered the 3’s starting at zero, I then have him pick another starting number – say one.  We then count up by 3’s but notice that our numbers are now different depending on our starting point.

So far this year, we’ve covered the one’s (0 to 100); two’s (0 to 100); three’s (0 to 30); five’s (0 to 100); ten’s (0 to 100), the hundred’s (0 to 1000) and the odd numbers (1 to 21).  Next week we plan to start the four’s (0 to 40).  How many we’ll cover depends; I use his interest level to help guide me about how far to go.  It is not a race, though, and it’s far more important to me that he understands the idea.

2.  New Concepts

By this I refer mainly to the learning of the arithmetic operations:  addition, subtraction, multiplication, and division.  I often use a curriculum as my guide, and some of my favorites are Singapore Math, MEP Math, and Miquon Math.  However, I believe that any curriculum is a tool that should help and not hinder.  I often find myself filling in concepts that I think a curriculum should have spent more time on.  I use a Place Value Activity Kit to demonstrate all four operations.

We start by learning the plus and minus facts up to 10 (all the combinations), then we expand it up to 20.

Interestingly, we recently learned three different methods for subtraction. For example, consider:  15-8 = 7.

 Method 1: Start at 15 on our number line and count back 8 steps.  We stop at 7.

 Method 2: Think of 15 as a number bond of 10 and 5.  I ask my son if we can take 8 away from 5. (We’ve not learned negative numbers yet.)  He replies that we cannot, but we can take 8 away from 10.  So we calculate (10-8) and we still have our 5 from before so our answer is 2 + 5 = 7.

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Method 2 demonstrated with place value kit

 Method 3: Start at 15.  Think about where our nearest ten is when subtracting. Answer:10.  Think about how many units we would take away from 15 to get to our nearest 10.  Answer: 5.  Then subtract 5 from 15 and next subtract 3 more. (15-5-3=7)

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Method 3 explained.

Having learned three different methods hopefully causes him to analyze, compare, and develop a greater number sense regarding subtraction.

So far we’ve worked though the concepts of addition and subtraction mainly, and also a good start in multiplication.

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Multiplication demonstrated with place value kit, and noting that ‘x’ means ‘groups of’.
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More multiplication showing that multiplying can be thought of as repeated addition.

Having already introduced multiplication (with skip counting),  I then use examples like the one above with a place value kit to go into more depth by showing that multiplication is repeated addition and that ‘x’ can be thought of as ‘groups of’.  For division, I only introduce it with the place value kit and show how division is related to multiplication.

[As an aside: If a student is confused by a concept, I recommend trying another curriculum to explain it a different way.  Use the curricula that best helps your child understand OR simply explain it in your own words how you understand it.]

3.  Problem Solving

How I interpret it, problems solving means that a student has enough mastery of a concept that he can apply it to solve a problem.

Thus far for this type of activity, my favorite curricula are Singapore Math Intensive Practice and Challenging Word Problems workbooks.  What I like best about these are the way they encourage a child to visualize the problems in order to better understand and solve them.  Also, questions are asked from many different angles so that the child has to really think about what is being asked.

Recently we worked on a problem where he had to think about how a larger number can be made out of two smaller numbers to solve puzzle-like problems related to subtraction.  We pulled out our Cuisenaire rods to help visualize the combinations that make 10: (1,9); (9,1); (2,8); (8,2); (3,7); (7,3); (6,4); (4,6); and (5,5) – to remind us of our math facts. (Hopefully he sees the pattern that addition is commutative). Manipulatives can be a great addition when solving a complicated problem.

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Looking for a pattern and using C-rods to visualize.

Of course, problem solving can also bring feelings of frustration. During these trials of deep thought, he grows in knowledge and develops math endurance.  I’ve found that giving praise when he’s solved a problem works wonders, and it’ll likely light a fire and make him want to learn more.

Overall, our plan is that we cover math fact practice and new concepts daily.  Problem solving (puzzle-type questions) may only happen two or three days per week.  We touch on all three areas every week.

It’s my intention that Math time be fun for him at this age.  Demonstrating that problem solving can be both rewarding and enjoyable [hopefully] paints a positive first impression.

AoPS Intro to Algebra Review – An Active Approach

For a middle or high school student who is motivated to take a more active role in learning Algebra, check out Art of Problem Solving Introduction to Algebra, 2nd Edition, by Richard Rusczyk.  Intended for students in grades 6-10, I’ve found it to be effective in not only teaching Algebra, but also in developing general problem solving skills.

Appearance

The pages are uncluttered with mainly black and white print and a few boxes highlighted in a peaceful blue – making it easy to follow and not distracting.

Jump Right In

Building concepts in a logical, step-by-step manner, the textbook is thoughtfully written to the student. The method appears to be this: the author carefully introduces new topics and engages the student by posing simple, related example problems, and then gives her a chance to jump in and develop her own solutions before presenting the answer to her.  For example, the chapter on quadratic equations begins by defining quadratic terms and expressions.  Then it leads the student by posing five such questions – gradually increasing in difficulty – related to quadratics.  This starts the student in the process of thinking about how she will answer the questions.  But the solutions are not revealed just yet.  The idea seems to be to motivate the student to reach her own solution before the book explains it to her.  Eventually the solution is explained, but by that time the student will be more aware of it, since she should have at least begun to construct her own.  Contrary to many textbooks I’ve seen, more up front thinking is expected of the student rather than being spoon-fed the material.  This active involvement makes learning math concepts less dull, more enjoyable and easier to remember.

Breaking It Down

Throughout the book there is a parallel with how new material is presented and the fact that complicated problems can often be solved by breaking them into smaller parts or steps.  For a quick example, when imaginary numbers are introducedand defined, the student is asked to evaluate a few simple cases on his own (which he should be able to do by applying the definition).  Next he is asked to solve aquestion that is slightly more difficult, but building off the previous question.  Then another – more difficult, but similar.  Finally, he is to simplify a set of more complicated imaginary number questions.  But if he has been solving the little problems along the way, the difficult questions are more easily diffused because he can break the difficult problem into smaller, simpler parts.  It no longer looks scary or confusing, and he will probably be able to quickly calculate the answer.  As I said above, only after he has had time to develop his own solution are the explanations presented in the text.  This approach makes sense, and is a helpful model for problem solving in general.

Following the above-mentioned example problems, there are Exercises that evaluate the concepts for each section.

At the end of each chapter is a thorough Summary section which highlights definitions, concepts, and/or other important information.  Many of the these sections also contain Problem Solving Strategies, and it is worth your time to read, absorb and apply these.  Next are Review Problems which seem to help measure how well a student understands the chapter.  And finally there is a Challenge Problems section in which will be found the more difficult problems that are likely to really stretch the student’s understanding and help her master the material.

Competitions

Problems previously seen in MathCounts, the American Mathematics Competitions (AMC8, AMC10, and AMC12,) the American Invitational Mathematics Examination (AIME), the USA Mathematical Olympiad (USAMO) and others are included in some of the exercise, review, and challenge problems.  Working these problems not only helps with understanding the idea of the chapter, but also in preparing for future contests.  In our situation, this book greatly contributed to the preparation of my daughter in past competitions.

Additional Helps

AoPS’s website also provides Intro to Algebra Videos which give the student another visual aid if desired.  Though my daughter did not use these, they are there if you would like them – which I appreciate.

There is a Hints section located in the back of the book for selected problems.  Any problem with a hint will note that at the end of the problem.

I strongly recommend the Solutions Manual.  It provides well written explanations with step-by-step solutions.

Caveats

It is important to read the How to Use This Book section at the beginning of the book for clarification.  A student should attempt solving a tough problem several times before looking at any hints in the back of the book.  For example, a student may work on a difficult problem for half of an hour or more, but not solve it.  She may feel frustrated and that she is getting nowhere.  But she is learning where her weaknesses and strengths are.  At some point she may want to look at a hint or review the concept notes of the chapter.

If you find that the lessons are taking too long, another option is to pick and choose challenge problems.

Highly Recommended

Overall, we thought highly of this book because there were challenging problems that made you think. 

The book seems creatively written with the intention of not only teaching Algebra, but also developing robust problem solving skills in its reader.  From the perspective of a teacher and especially if a student uses the materials as intended, I think it accomplishes this goal.