# Critical Thinking and Sledding: How to Get the Most Fun Out of Snow.

On Mount Lemmon, the kids are understanding that there are different stages of snow, or forms of snow.  Lightly dusted snow can melt quickly once the sun comes up.  Snow that has been there for a while can be hard as a rock.  They must waste no time when the snow is just perfect because it might not be like that the next day, or even later in the day.

For this particular day, we were going to check out a favorite area to go sledding.  They had full faith that there was the perfect snow for sledding and they were right!  The rest of the afternoon was filled with pure blissful fun!  However, it was the next morning that would prove to be the challenge.

The next morning, the kids wanted to try sledding again, but there was not any “new” snow. It was the same patch that was used the previous day.  It was also windy the previous night which moved a lot of debris.  As usual, the kids went to the top of the path to sled down and it was a disappointment.  So I began asking them some questions to help in their thinking of what was potentially slowing them down.  Here is a list of things we came up with:

1.  Pinecones, pine needles, and rocks were on the path, which slowed them down.
2. A smoother surface would help them sled down faster
3. The layer of snow was not as thick as the previous day because it was pressed down from yesterday’s sledding and no new snow had fallen.
4. To sled down, someone had to push the person on the sled for a bit to get going, or having mom pull them down worked too!

The kids still had fun and they helped each other out with getting all the debris out of the way, taking turns pushing each other, and tried to lay flat on the snow board sometimes to go faster.  It was their second time sledding and I think they understand how to think critically to get the most fun out of snow.  I also did my own turn in sledding down the incline to show them what I can do to sled faster to see if that would work for them too!  So where is the math in all of this?

In Mathematics, critical thinking is really important.  When you come up with a problem that has a lot of “problems” in it, trying to look at what is keeping you from solving it out is important!  When you can critically look at a situation, without getting upset or feeling frustrated, to see what is slowing you down in solving the problem in the first place, you will become more successful.  It also takes practice.

Since sledding is not intimidating to the kids, it was easier for them to think critically and not give up so quickly.  The more times the kids can do this for things that are “no big deals” to them, the easier it will be to show them how to apply this for any sort of math problem.  It is also important to purposefully tell them that these are the same tools of thinking to apply in any math problem they encounter.

Here are a few questions to ask your child when they are having trouble with solving a math problem:

1. What is the word problem asking you to do?
2. Have you done this type of math problem before? If you have, how did you solve that one? (this is like thinking back to when the snow was perfect for sledding the day before)
3. What is slowing you down? Is there too much information given to you?  What do you really need to solve this problem? (this is like the pinecones and rocks in the path)
4. Would you like me to help you, or have your brother or sister help you? (this is like me showing them how I sled down the incline to show them what could work)
5. What if I showed you how to do a similar problem and then you can see if what I did would help you solve the next one?

In reality, the first time we directly apply critical thinking to a situation may not mean that they will be able to transfer that when working with math problems.  It takes time, practice, and directly referring back to a situation in context.  Be patient, everyone learns differently, and most of all, have fun!  Mathematics really should not be presented as something difficult to learn. It was not made to be that way.  Have fun with it!  Enjoy!

This particular batch of cookies are shortbread cookies and the filling is from using the last bit of jams that we had in the our kitchen. 🙂

The weather is finally getting cooler and we can begin using the oven for baking!  Baking is a great time to learn about mathematics!  The best part of it is making arrays with the cookies.   An array is an arrangement of objects, pictures, or numbers in columns and rows.  An array is especially easy to make when we place the cookie dough onto the cookie sheet.  The reason for me even doing this is because I am working with my oldest son on how to think about multiplication in several different ways.  One way to think about multiplication is by making arrays of columns by rows to find the total product.  The most important part of multiplication, for me, is also learning how to apply the multiplication to everyday things in his life.

Before baking the cookies, the instructions on the box/recipe will tell you how many cookies the mixture will make.  Ask the question out loud about how many dollops of cookie dough can fit onto one oven sheet.  For this particular batch, we were able to make 30 cookies.

After the cookies are baked, the next question I ask my oldest is how many rows do we have on the cookie and how many columns.  For this particular one, we have six columns and five rows.  How many cookies do we can by multiplying 6 and 5?  To get this answer he can count out the total cookies that are on the cookie sheet, or have a chance to figure it out in a different way.

Here comes another situation with having more than one child in the family, how to distribute/divide the cookies equally?  We have three kids in our household who have three different tastes.  One would like to have strawberry jam as a filling.  The second child does not like anything on the cookie.  The third child likes apple butter as a filling.  The conversation is now about division and distribution.  The kids then need to decide how many cookies each of them will have and if it needs to be the same number of cookies.  They need to figure out how to divide thirty cookies by three kids equally.  So they did!  The results are the pictures below. 🙂

With your child, you may want to do fewer columns and rows. You can point out small arrays within the bigger array of cookies.  For example, you can make ask how many cookies are in a 2 by 3 array.  How about a 3 by 3 array?  Look at the shapes of these arrays too!  When you make a 2 by 3 array, is this a rectangle, or a square?  When you make a 3 by 3 array, is this a rectangle, or a square?  The answer is that a 3 by 3 array will make a square and visually will make the square number of 9!  Check out a 4 by 4 array on the cookie sheet, is this also a square?  So is 16 a square number, which is 4 multiplied by 4?  How cool is that!

In any case, make this a fun and tasty mathematical exploration.  Always remember that this is not only a time to learn about math, but a time to get together and make something of love! Enjoy!

# Are You Taller Than a Cow Parsnip?: Measuring in Units of Nature

Cow Parsnip (Heracleum lanatum)

I am so grateful for such a wonderful paradise on Mount Lemmon, being so close to the desert, and being surrounded by a beautiful community.  What a better place to go learn mathematics because there is math everywhere on the mountain.  What a great time to be up on Mount Lemmon and go exploring!

One of my favorite plants on the mountain is the Cow Parsnip.  These are my favorite wonders of the area!  Cow Parsnips remind me of a child where they begin as small little ones and grow above and beyond 5 feet tall, or close to it.  So I guess the big question is, are you taller than a Cow Parsnip?

What is great about being outdoors is that plants do not know what inches, centimeters, or feet are.  In Nature, there is an abundance of resources use to measure and one of these can be the Cow Parsnip of course! So go for a wonderful walk and look for one of these beauties.  Stand up next to one and measure yourself.  Who is taller, you or the Cow Parsnip?

Take the kids out for a walk and see how they measure against these wonderful giants.  Start using the words of comparison like, “shorter than”, or “taller than”.  How many of you would take to be as tall as the Cow Parsnip?  What about the blossoms?  How many blossoms would it take to be the length of your hand?  How about the width of your hand?  Take a look at those leaves!  How many of your hands would it take to be the same length, or width of those leaves?

This is cute and all, but how does learning Mathematics really measure up to the “real” world?  Take a child, for instance, maybe that child is sad because he/she hasn’t grown as tall as everyone else, but take a look at the Cow Parsnip with how small the Cow Parsnip began.  The Cow Parsnip starts out tiny and then shoots up to be taller than any other flowering plant I know.

Success can be measured using blossoms and stalks of plants, or the number of tree rings a tree can have in a lifetime.  Nature has so many opportunities where we can learn and explore in Mathematics.  What I would like everyone to get out of these posts is that the world of Mathematics encompasses more than what we can imagine.  The earlier we start our kids and grandkids in interacting and learning from Nature, the more they will have more reason to protect what is treasured by the rest of us.

# Diversity Exists in Nature To Teach Us a Lesson: Counting, Pattern Finding, and More!

I could not resist in getting our home a bunch of beautiful red Tulips.  If you have read the other posts about looking at flowers in Nature to look for patterns, well this one is not any different.  Tulips taught me a lesson about diversity.

Fascinated about the black and yellow six-pointed star, I counted the points on the Stigma of the flower.  Three points on the Stigma, then six Stamen, then six red petals on the Tulip.

In Nature, there are flowers demonstrating these patterns of either doubling the original number from the Stigma, or simply repeating the original number from the Stigma.  However, this second Tulip (pictured below) surprised me because this Tulip demonstrated the same doubling pattern, but a different original number.

Looking at this particular Tulip, the Stigma has four points, not three points.  Then that quantity of four is doubled to eight Stamen and eight petals!  Wow!  I love Nature!!!! Yes, I realize that these Tulips were not cultivated in the wild, but this one grew to be much different then the other red Tulips.  So this hit me with a great example of showing kids and ourselves as to how Nature and Mathematics shows us that Diversity still exists even when we think we are all the same.

These two Tulips have the same color pattern, have the same green stem, and are both called Tulips.  However, both are still different and unique from each other, which is beautiful!

So how is this important with Mathematics?  Well, just by understanding how to count, how to compare and contrast, and how to find patterns, that can simply take you to a lesson of learning and appreciating Diversity.  Yes, Nature has an abundance of beautiful flowers of different shapes, sizes, and colors!  Each of them have their own purpose of making the world function and be beautiful! How exciting it is to further see that even when we think a bunch of flowers look the same, Diversity still exists!

This is why Mathematics is so very important and near and dear to my heart.  Mathematics is not just solely about numbers, but extends to our very own life and life lessons that we might encounter from day to day.  What a wonderful way to introduce this particular lesson to your child about how Nature can beautifully teach us about the importance of Diversity and celebrating the Diversity!  🙂

The world of Mathematics is in your home and outside, so go explore what new lessons are out there!  Always remember that it does not matter who crosses the finish line first, it is actually getting there that is more important!

If you would like to see the article describing number patterns on flowers, go to

https://learningmathwithmom.wordpress.com/2016/07/01/theres-math-in-them-there-mountains-spending-time-outside-exploring-math-and-building-vocabulary/

# Whether Freshly Collected in Your Basket, Hard-Boiled, Scrambled, or Colored, Eggs Teach Us About Equality: Making Division “Egg”citing!

We just colored these hard-boiled eggs that you see in the picture above.  Each child wants to make sure that he/she has the same number of eggs, so I took this opportunity to explore division and practice it too! Regardless if we were going to color the eggs, or eat them, everyone wants to have their equal share, so distributing the eggs equally was a good way to show how division works.  This goes for eating the eggs for breakfast, as a snack, or with any food item, the concept of equality is always there.

After boiling the eggs, we needed to let them cool off.  Afterwards, each child was handed their plate for their eggs they were going to color.  We started with 15 eggs and I asked the older ones how many eggs each of us should get if there are three of us.  Then I did the distribution of the eggs one at a time.  Each of the three plates received one egg at first, then two, then three, checking each time if each plate has the same number of eggs. We checked back to see how were left and if we could still distribute those eggs.  We did this until all eggs were equally distributed.

Each plate now had 5 eggs for each of the 3 plates.  This makes 15 eggs divided by 3 plates to equal 5 eggs each.  But what if we had one egg leftover, or two eggs leftover? We could divide each egg into three equal parts and distribute those equally, but we face the concept of context.

In this context of coloring eggs, it does not make sense to divide a hard-boiled egg into three equal parts because who wants to color a third of an egg?  This is a wonderful conversation to have with your child.

In this “egg”citing activity, this is about starting with a quantity of eggs that will result in everyone having an equal number of whole eggs.  If you scramble your eggs, then it might not matter about everyone each getting and egg and a third of an egg because the context is different.  Sometimes, mathematics is all about the context of it all.

Another important piece is talking about what other contexts hold division in distributing everything equally?  What cultures and communities share all their food equally with everyone?  How does this relate to using division and how important is it to divide equally?  What are some other examples where things need to be divided equally and why is that important?  Asking ourselves and our children these questions places the mathematics of division in a contextual situation, something more real to us in life than what is on paper.

Trust me, Mathematics is really “egg”citing!  So get to exploring and think back to how you practice Mathematics in your life!  Enjoy!

# Math, It’s What Soup is Made Of: Counting, Measuring, and Shapes.

It’s now beginning to look and feel like winter for us. My youngest and I made soup together and like so many times, I squeeze in a math moment.

Before we get to wash our vegetables, we count how many stalks of celery we have, how many carrots, potatoes, and zucchini we have. Then it is time for washing.

When we cut our vegetables, what shape does celery look like when we chop it this way? Crescents? How about the carrots? Circles? What about the zucchini? Don’t forget the garlic and onions too!

What about the potatoes? What types of shapes do you see? Why is it important to make sure we cut the potatoes the same way and size? Would it take longer or shorter to cook small pieces?

Then put the vegetables in a pot and start measuring. How many cups of water, or cups ofchicken/vegetable broth do we add to the soup? How long do you think it will take to cook? These are are great questions to ask. Even if your child says, “it will take a billion years to cook,” that might just be accurate for them because in their world, a few minutes feels like a million years anyway . 😊😊

Making soup is not a new concept, but it is one of those moments to ask them to count, or identify shapes, or measure out in cups. At least do one of those things and you have placed math in your child’s environment. Most importantly, you spent some time together, made a mess, and had fun. 😊

Enjoy!

# Stars in the Forest, Not Just in the Sky: Geometric Stars in Plant Life.

On Mount Lemmon, this is the first time I have seen a Scarlet Gilia.  It is so beautiful and unlike any flower that I have seen on the mountain.  This particular Scarlet Gilia was located on Turkey Run across from the community center.  After having the gift of encountering this, it encouraged me to share this out with the rest of you and your kids that Stars do exist in our world, not just in the sky.

For the little ones, stars just may be the easiest shape to learn because they are so special looking.  So why not begin with teaching them about things that are shaped like stars and point them out in the plant life and in food!

Up on Mount Lemmon, you have the beautiful Scarlet Gilia that is a five-pointed star.  The Cow Parsnips that bloom earlier in the summer look like spheres from far away, but look closer and you see little individual blossoms that are five-pointed stars!  As these bloom until October, Richardson’s geraniums on the mountain are these beautiful dark lavender blossoms that are also five pointed stars.  So go take a walk with the kids, or even for yourself and go find these beautiful stars during the day.

For other plants, it depends on where you live.  If you live in cooler areas, you may be able to look at the Lilies of the area and notice that their blooms are six-pointed stars.  Take the Hollyhock.  Look from within the bloom and you will see this beautiful green five-pointed star in the center.  In the desert, look for the Aloe Vera plants and look from above to see the star.

In food, we have stars that form on the top tomato plants (five-pointed stars), onion blossoms (six-pointed stars), and on the pomegranate fruit (six-pointed stars).  If you cut an apple horizontally, you get to see a star there as well (five pointed star)! So get the kids in the kitchen and explore which fruits and veggies have stars in them, or cut them into stars.

To learn about shapes and to teach them to your young child, it really is not difficult.  You either need to look at the food you eat, or look at the plants outside.  Begin with teaching them about stars because they are everywhere you are,  up in the sky or growing out of the ground.  It just takes a moment to open your eyes and the eyes of your children so that you get to see and enjoy the beautiful geometries, the beautiful shapes that are already around you.  🙂

Learning about geometry at such an early stage in life gives them the opportunity to see it everywhere.  Once a child, or any person of any age, is able to see the geometry, it becomes more tangible when it is time to learn more about geometry in the classroom.  Geometry then becomes more relevant and more important to them to learn.

Have a wonderful time searching for the stars!

# Taking a Closer Look at Nature’s Flower, the Carnation: Counting and Comparing

It was a very hot day of over 100 degrees outside and we needed something to do in the house.  After having carnations left over from a bouquet, I made the decision to use them to teach my children and my nephew about the parts of the flowers and counting.  For this, I gave each child a carnation, a piece of paper, tape, and glue sticks.

The question that I gave them was to find out if the number of petals for each flower were the same as the other carnations.  The other task I wanted them to answer was to count the number of stamen and compare it to the number of petals as well.

So the children began taking a part the flower petals and spreading them out.  All I have to say is how beautiful everything smelled in the room! As you look at the photos, some of them had 37 petals, while another had 36 petals.  The numbers were all within range of each other.  It was really interesting for them to work carefully in pulling apart the carnation flower from the stem and then finding out how difficult it was to pull the stamen and stigma apart from each other.

For the number of stamen and comparing this to the numbers of petals, I will have you figure this out with your own children at home.  This idea came from my time taking photos of the flowers on the mountain close to us.  Sometimes the comparison was equal, while other times it was double or half of the other.  The idea for doing this and observing is to acknowledge that there is a relationship here.

This activity took about 10-20 minutes to do.  Most of them could do this on their own, but sometimes needed help in taping, or glueing this onto the paper.  We did let the petals dry and we put the rest of the flowers into the compost.

On the photo above, you see the number 28 and then the number 36.  For this, I asked one of the them to guess the number of petals he had on his paper.  He guessed 28 and then we counted them together to get 36.  So his estimate was not too far off from what the actual count was.  Estimating is a wonderful skill to have, so even doing this is a great example of how to have your child practice estimation. When you go get blueberries, or raisins in a box, have the child estimate the number and then count them and eat them!

Remember, doing mathematics at home does not have to be fancy.  You can go outside and look at the flowers and count petals.  If you have cut flowers at home and would like to get some more use out of them, think about doing this activity.  You do not even need to have paper and glue handy.  Just have them take a part the flower and start counting.  🙂  Take a closer look around you and start counting and comparing.  🙂 Below are more examples of the work the children did that day!

# Radials in Nature: Plant Life That Reflects the Starburst, Sunburst, and Nature’s Fireworks.

There are so many beautiful teachable moments up on the mountain and everywhere else, when it comes to plant life.  As children, we are taught about circles and squares and triangles, but what about radials-starbursts and sunbursts?

Radials are something that begin in the center and bursts out from there, like fireworks.  They capture our attention.  They make us stop and look and admire the beautiful way that it grows.  Some could be easily described as growing in a circular pattern, or like a globe.  This is true, but it could also be described as a radial pattern.

Look at the photo above of the Wheeler Thistle.  This flower is blooming at Rose Canyon Lake near Mount Lemmon, AZ.  From afar, it looks more like a globe, or a sphere.  When you look a little closer, it looks like fireworks!  Most of the grasses on Mount Lemmon also grow from the center and branch out as well. It is really amazing just how many radial forms are out there.

When you are walking through the desert, look at the Aloe Veras and Agaves.  Take a moment to look from above and look down to the center. They all begin from the center and grow outwards in a fireworks pattern.

Is it enough for my small child to just know what a triangle is, a square, a circle, a rectangle, a heart?  The answer is yes.  If you can show more shapes at an earlier age, then you can open their eyes to see a lot  more than just what we are normally taught.  In some cases, things do not necessarily belong just with circles, or just with triangles.  Depending on how you see the plant for that moment, it could be both, or something entirely different.  This is how mathematics is, observing the behavior and seeing for what it is for that one moment.

Take the Blue Globe Glow Thistle.  Once the petals have fallen off and I look from above the blossom, I could say that this is a radial pattern, something that looks like fireworks.  However, when the petals are still there, as shown below, I could say that this looks like a sphere, or little stars on a sphere.  It just depends on how you look at this flower.  Nature shows us how many different names and answers we can come up with by the way we look at things.  In many cases, problems in mathematics may have more than one answer, just like this Blue Globe Glow Thistle. 🙂

More Examples of Radials in Nature:

So there you go, radials are another shape, or growth pattern in plants that we can teach our children about and ourselves.  I will continue to say that Mathematics is everywhere.  The Mathematics out in Nature is free and teaches us that there is more out there than we are taught in the classroom and there is always more than one answer to something, it just depends on how we observe for that moment in time.  If you do not have plants growing where you live, print these out, or point them out to your children.  Have fun with your kids outside as much as you can!

P.S

# Math on the Mountain: The Geometry of Streams, Bubbles, and Chocolate Milk

Mount Lemmon has been getting a lot of rain. With rain, comes running streams. Running streams bring a great lesson in geometry…..bubbles!

Take a walk and listen to the running water. Ask the kids to go find the water and see if there are any bubbles forming. Look at the bubbles and ask the kids to name the shapes they see. Do they see Spheres? Do they see Circles? What happens when the Spheres get pushed together, what other shapes do they form? Do they see Pentagons, or Hexagons (five-sided or six-sided shapes)? How would you be able to recreate these shapes at home? How about with milk?

If it raining outside, or you need to take a snack break, give the kids some milk….chocolate milk if you have it.  Get a straw and have them blow bubbles in the milk.  What happens?  Do Spheres take shape? What happens if they get pushed together?  Do you see the same shapes as from the stream?  Do you see a rounded triangle?  Well, that is called a curvilinear triangle!

Look Closer.  You will see a Curvilinear Triangle to the left of the center of the picture.  Copyright 2016 Christina Grossman All Rights Reserved

Take a look at the size of the shapes when looking at the bubbles in streams, or in the milk.  Are all the bubbles the same size?  What do you notice when big bubbles are pushed together?  Do you notice a pattern in the bubble formations?  These are great questions to ask, even if asking just one or two of these questions.  It creates another way of looking at something that we might have taken for granted, especially up on the mountain. For instance, blowing bubbles and why are they always spherical….this is something to think about.

The next time you have some bubble solution, bubble wands, and/or a bubble machine, ask this question:  Does the bubble always come out as a sphere regardless of the shape of the bubble wand?  Why do think that happens?

Here is a website that has a brief description about that:  http://bubbles.org/html/questions/round.htm

Remember, these small lessons are to help the child and ourselves to begin seeing and thinking about the world for what it is.  The mathematics is there, it is free for everyone to learn, and there is an abundance of math to explore inside your home and outside, on the mountain, on a desert, and everywhere else.  🙂

If you would like a recipe to make bubble solution, here is a link for you (I am not paid to show support for this site, as well as the one above regarding the spherical shape of bubbles).  Enjoy!

http://www.wikihow.com/Make-Bubble-Solution