Math with Mandalas: Perpendicular Bisector

We learned about bisecting angles and perpendicular bisectors through creating Mandalas.

Creating Mandalas with children is a fun way to teach math concepts that doesn't feel like work. Kids can learn about degrees, angles, bisecting, radius, diameter, circumference, perpendicular, parallel among other geometry concepts.

To begin, all that's needed is a good quality compass, paper and a straight edge (like a ruler). There is an endless possibility of mandalas that can be created. We have a book of simple geometric mandalas and find it very educational to try to recreate them. Interestingly enough, there usually ends up being more than one way each mandala can be created. In other words, sometimes steps can proceed in different orders, or alternative steps can be used which arrive at the same result. After walking kids step-by-step through the creation of several mandalas, I like to give them a challenge mandala and see if they can create one on their own.

Each mandala created requires some construction lines which end up being erased in the final version. Therefore, it's best to draw them lightly. Here are the steps we used to create the above mandala.

1. Draw a horizontal line on your paper with a straight edge. The line should be approximately the diameter of the outside circle. Set the compass radius to slightly greater than on half of the line. Place the pointer on the end of the line and create an arc above the line. Create another arc below the line. Repeat placing the compass point on the other end of the line.

2. Create a perpendicular bisector by using a straight edge to connect the two crossing arc points.

3. Set the compass radius to the desired radius for the center circle. Draw the center circle placing the compass point on the point where the two lines cross.

4. Bisect the perpendicular angles. Place the compass point on one point where the circle and straight line cross. Create an arc just outside the center circle. Repeat the process placing the compass point on the adjacent point where the circle and other straight line cross. The two small arcs should cross.

5. Bisect the angle connecting the center point of the circle and the crossing arc point with a straight edge.

6. Repeat the process to bisect the other 90 degree angle.

7. Find the center point for one of the eight surrounding circles. Referring back to the first picture, you can see that the center is difficult to locate, but a tangent point on the edge of each of the eight surrounding circles lies on the point where the 45 degree angle line and center circle cross. Therefore, set the compass radius to the same radius as the center circle. Place the compass point on the point where the 45 degree line and center circle cross. Create an arc to mark the center of the circle crossing the vertical line.

8. Create one of the eight surrounding circles. Place the compass point on the point where the vertical line and arc meet. Draw the circle.

9. Repeat the process creating three more circles.

10.  Using the same method, repeat the process to create the four arcs which are not quite complete circles.

11. Draw the outer circle. Open the compass so the radius is set for the size of the outer circle measuring the point with the sketched geometry.

12. Erase all construction and undesired lines.

13. Use a sharpie or black marker to define the desired lines.

14. Color the mandala any way you choose.

The Tuttle Twins

The Tuttle Twins is a new series of books to help children learn about complex political and economic concepts.

There are currently five books in the series targeted at children ages 6-13. Each 50 page book focuses on a government policy or concept that has real life consequences for all citizens.

The Tuttle Twins and the Creature from Jekyll Island
The Tuttle Twins Learn about the Law
The Tuttle Twins and the Miraculous Pencil
The Tuttle Twins and the Food Truck Fiasco
The Tuttle Twins and the Road to Surfdom

For example, the Food Truck Fiasco explores protectionist laws designed to prevent competition when the owner of a restaurant convinces the city to pass laws that make it difficult for food truck vendors to continue to compete. Eminent Domain, Free Market, and concepts of liberty are among those introduced.

Because these books are targeted at young children, they are brief and the language is simple. The story line does a good job of introducing complex topics but doesn't go into a lot of detail, and does little to address opposing view points. Overall, they are a descent introduction especially if the parents encourage further discussion.

An unrelated follow on series that continues to explore similar complex topics is Richard Maybury Uncle Eric Books. The Uncle Eric books are better for children a little older. (Ages 12-18)

* I did not receive any compensation for this recommendation. I'm just a homeschooling mom who has found many products that I like. If you're interested in the products I recommend on this blog I want to make it easy for you to find them. 
 ** I am an Amazon associate and receive a small portion of the sales on orders made after clicking in from this site, which I promptly spend on homeschooling books and supplies for my children.

The Importance of Skip-Counting

Skip-counting is a fundamental math skill which is often not covered by math curriculums.

What is skip-counting? Quite simply, skip-counting is counting by multiples of a number. The following list shows skip-counting by the numbers 2 through 9. In each example, the list could continue on indefinitely.

2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

If kids begin elementary math with a strong foundation in number sense, which involves skip-counting, it enables them to rapidly absorb higher level math concepts. It is not unreasonable to cover math that is typically taught in grades 3-6 in less than a year. This is because there are lots of math concepts with roots in skip-counting. It takes the majority of math time in grades 3-6 to cover these concepts. Once kids understand and can rapidly skip-count, they can fly through these numerous math concepts.

Not only does skip-counting have benefit because it enables children to rapidly progress through math curriculum. Skip-counting has value on its own. I often use skip-counting when knitting and crafting. Instead of counting stitches by 1's, counting by 3's or 4's is much more efficient. Many people use skip-counting during their work day. A store owner uses skip-counting to take inventory of products. It comes in handy when playing games and in estimating. Since skip-counting is so valuable and can be fun to learn, it shouldn't be skipped.


Math Concepts Based on Skip-Counting 

Lowest Common Factor
Greatest Common Factor
Adding and Subtracting Fractions
Reducing Fractions
Prime and Composite Numbers
Recognizing Number Patterns and Sequences

Multiplication and Division

Numerous math concepts are based on skip-counting. Multiplication, for example, is directly related to skip-counting. The numbers in the list above are the answers to the multiplication tables.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

Skip-counting by 4's, referring to the above list, 4x1=4, 4x2=8, 4x3=12, 4x4=16 and so on. The answers are the same as the list of skip-counting numbers. Therefore, if kids learn to skip-count, they have already memorized the answers to the multiplication tables, before being asked to multiply. In other words, if children understand what skip-counting is and how to do it, multiplication becomes second nature.

The concept works equally well with division. Again, referring to 4's as an example: 4÷4=1, 8÷4=2, 12÷4=3, 16÷4=4. In this case, they have memorized the problems, but can count up to the correct answer.

Greatest Common Factor and Lowest Common Multiple

Greatest common factor (GCF), Lowest common multiple (LCM) and factoring are concepts taught in conjunction with fractions. Likewise with multiplication and division, having a good handle on skip-counting eases understanding of these concepts.

For example, the greatest common factor of the numbers 12 and 16 is 4. Why? Because both 12 and 16 are part of skip-counting by 4's. They are also both part of skip-counting by 2's, but the "greatest" word in the question requires the greater of the possible numbers. This answer of 4 can be visually seen by scanning the above list of skip-counting numbers and finding the biggest number that contains both 12 and 16 in its list.

Lowest common multiple is related to the greatest common factor in a similar way that division is related to multiplication. For example, 21 is the lowest common multiple of 3 and 7 because when referring to the list of numbers above, 21 is the lowest number in both the list for 3's and 7's.

When to Begin Skip-Counting?

If a math curriculum does cover skip-counting, the lessons usually begin around 2nd or 3rd grade. In general, this makes sense because kids should have a good handle on counting the numbers up to 100, counting backwards, and have a basic understanding of addition and subtraction. Although counting to 100 should really be second nature before beginning skip-counting, the other concepts could be mastered in conjunction with skip-counting. So for many kids, 2nd or 3rd grade works just fine, but kids as young as 4 or 5 years old can begin to master skip-counting.

Skip-counting by the numbers 2 and 5 usually comes first, because they are the easiest. Next it's best to work the way through the numbers concentrating on one until it is mastered as kids can become confused by advancing too quickly.

What's Next?

Once the child can skip-count well with all the numbers, or with any one number, the concepts of multiplication and division can be introduced. By placing effort on skip-counting, and making it fun, the pain parents often feel when teaching multiplication can be greatly reduced.

When my oldest daughter was about 4 years old a neighbor and friend of mine asked me, "Do you know how important skip-counting is in learning to multiply?" No was my answer that day, but I quickly learned how right she was. I followed her advice and used this method with all three of my own children who each learn very differently. Despite their differences, they all had fun learning to skip-count and rapidly learned follow-on math concepts. If you have a young child who has not begun learning math, are struggling to teach multiplication, factors, or fractions, or have an older child that could use some review, I highly advocate for a little skip-counting practice.

Best International Historical Fiction

By reading my daughter and I have studied many historical topics. Although we often read historical fiction in connection with our history studies, it happens to be a favorite genre of both my daughter and myself. These books are great reads for upper elementary and middle school students, and are excellent for sparking interest in historical topics.

Snow Treasure tells the story of school children in the north who save the town's treasure from Nazi's during World War II occupation.

The Well of Sacrifice
Red Sand Blue Sky (Girls First!)

The Endless Steppe: Growing Up in Siberia
During World War II a family is set to Siberia. This is the story of how they adjust to this harsh land during hostile times.

Viking Adventure
The Vikings discovered America long before Columbus. In this story, a young Viking boy leaves home seeking adventure. Trained for a life of exploration, he is part of a voyage to Vineland.

The Apprentice

The Children of the New Forest
During the time the British kicked out the king, a family of soldier's children are orphaned and go into hiding. Their lives change dramatically as they learn to hunt, cook and care for themselves.

Trapped by the Ice!: Shackleton's Amazing Antarctic Adventure - Shackleton, an explorer, lead an expedition to the South Pole. During his journey the crew suffered many hardships but were able to overcome due to the outstanding leadership of Shackleton.

* I did not receive any compensation for this recommendation. I'm just a homeschooling mom who has found many products that I like. If you're interested in the products I recommend on this blog I want to make it easy for you to find them. 
** I am an Amazon associate and receive a small portion of the sales on orders made after clicking in from this site, which I promptly spend on homeschooling books and supplies for my children.


We are trying something new with science. With our microscope the kids are inspired to explore bugs, pond water, fish scales, hair, foods and more. The first time we got it out my son couldn't wait to look at his hair. Following the book The Ultimate Guide to Your Microscope by Shar Levine is helping us learn the basics of how to use the mircoscope and create different types of slides by doing actual activities.

I'm going to encourage the kids to get the microscope out at least once per week. How do you use your microscope?

 * I did not receive any compensation for this recommendation. I'm just a homeschooling mom who has found many products that I like. If you're interested in the products I recommend on this blog I want to make it easy for you to find them. 
 ** I am an Amazon associate and receive a small portion of the sales on orders made after clicking in from this site, which I promptly spend on homeschooling books and supplies for my children.
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