Category: Math

Inch by Inch: Introducing Measurement

By Mariah Bruehl,

Playful Learning: Introducing Measurement

Learning to measure our world is a welcome experience for young children. Children love being able to use “real tools” to measure things that they encounter on a daily basis. This experience introduces children to both standard and non-standard measurement and helps them to develop the important skill of estimation. For the younger mathematicians this activity is great for number formation and identification


  • One small Lego® (approx. 1 inch)
  • Ruler and/or Tape Measure
  • 12 by 2in. piece of paper
  • Lego measurement sheet (see below)
  • Body measurement sheet (see below)

The Process

  1. Start with a small Lego® (approximately one inch) and the Lego® measurement sheet (printable M14). Ask your child to hunt around the house and see how many items they can find that are approximately the same length as the Lego®. It is fun to you’re your child a clip board, which makes the search that much more “official.” Encourage your child to record her/his discoveries on the measurement sheet.
  2. When your child is finished, explain that grown-ups do the same thing when they want to know how long an item is—but instead of using Legos® they use rulers or measuring tapes. Next, take out the 12”x2” pieces of paper and invite your child to make her/his own ruler. Explain that he/she can use their Legos® to make the marks (starting at one edge, making a mark after each Lego® all they way to the end of the sheet). When she/he is finished making and numbering their rulers, take out one of your rulers and show your child how similar they are. Your child may notice that each sheet had enough space for 12 Legos®. You can mention to her/him that those spaces are called inches. You can also bring up the fact that 12 inches makes one foot.
  3. It is helpful to remember that for this experience it is not important that your child’s measurements are exact. What is important is that your child begins to create mental images of items that are approximately an inch long. This will help them tremendously when it come to estimating measurement as well as gives them internal mechanism for knowing if their results are in the right ball park.
  4. Once your child’s handmade rulers are complete you can encourage her/him to put her/his new tools to work. Print out the body measurement sheet (printable M15) and invite your child to measure the various parts of her/his bodies (it is nice to work with a partner for this part of the experience). Explain that she/he can start out by estimating how long an item will be, then have her/him measure the item and finish by trying to figure out how far off she/he was from the estimation.
  5. It is fun to compare and contrast the different sizes of nose, fingers, etc. among your family members

Book Love

More to Explore

After your child is comfortable measuring items by the inch, you can repeat the same process with the smallest Lego® piece (see bottom portion of printable M14) and introduce the centimeter as another unit of measurement.

Challenge your children to measure longer items with the use of a tape measure (keep an eye out for hurt fingers) or yardstick.

Maker sure to check out our Playful Picks for Introducing Measurement.


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Playful Learning: Introducing Measurement Playful Learning: Introducing Measurement Playful Learning: Number Line


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Playful Picks: Resources for Measuring

By Mariah Bruehl,


Learning to measure our world is a welcome experience for young children. Children love being able to use “real tools” to measure things that they encounter on a daily basis.

Here are our favorite picks for introducing children to measurement. It’s fun to mix up the tools children have available to measure in a variety of different circumstances. Beware, you may just find that your child wants to bring their tape measure with them wherever they go!

Maker sure to check out our fun measuring activity, Inch by Inch: Introducing Measurement.

1. Inchimals – Fun way to learn about measuring!

2. How Big is Foot? by Rolf Myller

3. Length by Henry Pluckrose

4. Wood Folding Ruler

5. Inch by Inch by Leo Leonni – One of our favorite authors!

6. Measuring Penny by Loreen Leedy

7. Fabric Measuring Tape

8. Mini Tape Measure Key Chain – A wonderful first measuring tape (ages 4+)

9. Classic Wooden Ruler – Every house needs one!

Measurement Activities

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Playful Learning: Introducing Measurement Playful Learning: Introducing Measurement Playful Learning: Number Line

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Math Literature: Infinity and Me

By Mariah Bruehl,


Literature with math concepts is often the perfect way to start a math lesson. Literature naturally engages students and often initiates meaningful, rich discussions. Some discussions may focus on the math concept being presented, other discussions might focus more on connections and experiences with the story line, while still other may center on the illustrations and artistic techniques. Literature draws us in and keeps us together around a central theme but also lets us enter based on our personal experiences and interests. Therefore, math literature is a powerful, all reaching tool when sharing math concepts with our students. Often, we can reach most, if not all students with a powerful story.

Authors of math literature often also do a great job of personalizing difficult concepts. Kate Hosford has done just this in her book Infinity and Me. Her words along with Gabi Swiatkowska’s gorgeous illustrations personalize the concept of infinity. Uma, the main character, questions her friends and family members on how each person imagines infinity. The answers are varied and imaginative but also sweetly personal. Young readers will love to engage with this delightful story about a difficult, but now accessible concept.

After reading and discussing this book with your child, try writing an infinity poem:

  1. Brainstorm ideas about infinity including: (1) feelings, (2) images, (3) activities one might want to do and (4) wonderings or questions.
  2. Write a poem about the concept of infinity together to model writing to complete a sentence starter. Ask for ideas and show how to write these in sentence form.
  3. Using the brainstorming list, have your child or students write a poem of their own.
  4. Share finished poems with pairs or the whole class.

More to Explore…

  1. Draw a picture to share one of the ideas you presented in your poem. Look closely at the illustrations by Gabi Swiakowska for ideas.
  2. Visit Kate Hosford’s website and download the curriculum pack for more engaging activities.



Playful Geometry: Turning Circles Into Squares

By Mariah Bruehl,

circlestosquares Have you tried the Playful Geometry activity, Turning Circles Into Triangles?  Would you ever guess that a smooth, curved shape could give rise to one of angles and straight lines?  It’s pretty neat, right?  Today we’ll expand on that activity and play around with turning circles in to squares.

Are you ready to give it a try?  We’ll look at two different ways of creating squares from circles.  In the first example, the circles and points are provided for you on our printable.  This would be the most appropriate method to use with younger children (ages 6-9 years).  The second method works well for older children (10+ years) who are practicing or are comfortable with their compass  skills.

Before starting, gather your supplies:

  • a copy of our printables or
  • graph paper
  • a ruler
  • a pencil
  • a compass with pencil

If you’d like, you can review the characteristics of a square with your child:

  • All four sides are equal in length
  • The opposite sides are parallel to each other
  • All angles are equal
  • If you were to divide a square in half diagonally, the dividing line would bisect its angles.

Method 1
Before you begin following the directions on the printable, notice with your child the arrangement of circles and dots.  How many circles are there?  Are they all the same size?  How are they arranged (touching, overlapping)?  How many points are there?  Where do the points lay?  Follow the directions to connect the points and create a square.  Why did that work?

Now move on to the second set of circles.  How many circles are there now?  Once you have created squares following the directions, you can take it a step further by connecting the blackpoints with diagonal lines.  Did that create any more squares?  How many total squares can you make with this set of five circles?

Method 2
1. Create a circle on a piece of graph paper by placing the point of the compass on a place where two lines intersect.  When the circle is complete, darken the center point with your pencil.
2. Now place the point of the compass on the circumference of the circle horizontally from the center point of the circle.  You can follow the line on the graph paper.  Draw a second circle and again darken the center point.  Each circle should cross through the center of the other.
3. Using a ruler, connect the two center dots with a horizontal line.  Now draw a vertical line to connect where the two circles intersect.
4. Place the point of the compass where the horizontal and vertical lines intersect.  Place the pencil of the compass on the center point of one of the circles and draw another circle.  With a ruler, connect the four points where this third circle crosses the horizontal and vertical lines.  It should look like this…


You can also follow the instructions in this Khan Academy video.

If you wanted to create even more squares, where would you add more circles?  Can you come up with other arrangements of circles that would allow you to draw squares?  Go ahead…  Have fun playing around with it and see what you can create.


Playful Geometry: Turning Circles into Triangles

By Mariah Bruehl,


Years ago I picked up the book A Beginners Guide to Constructing the Universe by Michael Schneider.  It was exactly what I had been looking for to enhance my children’s geometry lessons.  Today I am sharing one of the first lessons from the book and one of my children’s favorite – making triangles from circles.

You will need:

  • Graph paper
  • A ruler or other straight edge tool
  • A compass with a sharpened pencil.  Note – If manipulating a compass is a challenge for your child, circles can also be made using a strip of cardboard, a pushpin, and a pencil or pen as demonstrated in the photo above.

The lesson begins with your child drawing a circle around a center point  (created by the compass point).  A point on the graph paper is the perfect place to start.  Having the center of the circle marked is important for this activity.  You will see why shortly.  Once the circle is drawn I like to make a pencil dot on the center point to it can easily be seen.

Now place the point of the compass (or pin on the cardboard strip) on the circumference of the circle horizontally from the center point of the circle.  You can follow the line on the graph paper.  Draw a second circle and again darken the center point.  Each circle should cross through the center of the other.

Make a dot where the circles intersect at the top.  Using the ruler, draw a straight line from the top dot to each center point.  Then connect each center dot with a horizontal line.  This is the birth of the triangle!  How neat is that?


It doesn’t end there though.  This is a great time to ask “I wonder” questions and explore deeper by adding more points to connect.  Our printable will help get you started.

After completing the printable, you will see that all the triangles created are equilateral triangles.  If you connect other points does a different type of triangle emerge?  Do you notice other shapes?


Have fun and don’t be afraid to experiment or add your own flair!


Estimation Magic with Greg Tang

By Mariah Bruehl,


Greg Tang is a wonderful children’s book author who emphasizes number sense strategies for children. His books are colorful, poetic and encourage children to think through figuring out how many by using clever grouping strategies. Tang provides children with clues through poetry on how to group numbers. Children learn through these strategies that it is quicker to count by making groups than counting by ones with a larger group of objects.

Activity: Estimation Magic

This activity is best suited for children ranging in ages from five through eight.


  • Math for All Seasons by Greg Tang
  • Or the e-book version of this book available on the website
  • Pencils
  • Printable: Estimation Magic

Playful Learning: Estimation Magic

  1. Share the cover and title of Greg Tang’s book Math for All Seasons. Discuss how Greg Tang wants children to learn how to count quickly by making groups of numbers instead of counting by ones.
  2. Explain that you will be sharing a picture with a poem for about ten seconds, will then close the book and ask for estimations of how many.
  3. Share with children how the poem provides clues to how Greg Tang would group the objects to count quicker.
  4. Take children through the first picture together as a group. Discuss estimations, clues provided in the poem and the actual number of objects.
  5. The routine for reading the book could progress like this: read aloud the poem while sharing the picture, children listen and try to figure out how many, close the page, ask for estimations, then figure out the actual answer together using the strategies suggested in the poem.
  6. Using clipboards, pencils and the Estimation Magic printable children record estimations, actual answers and explain how they grouped the objects.
  7. While reading, continually discuss strategies for children to practice. Later, this activity makes for a fun buddy reading activity where children can quiz one another.

Extension Ideas:

  1. Take photographs of everyday objects, placing them in groups, and use these pictures to have children estimate with.
  2. Have children create pictures for others to estimate with. This provides children with a chance think about grouping that make sense for quick counting. Children might create groups of twos and threes to make groups of five for example.

Greg Tang’s Books…

  • The Grapes of Math by Greg Tang
  • The Best of Times by Greg Tang
  • Math for all Seasons by Greg Tang
  • MATH-terpieces by Greg Tang
  • Math Fables by Greg Tang
  • Math Potatoes by Greg Tang
  • Math Fables Too by Greg Tang
  • Math Appeal by Greg Tang

Greg Tang’s website is full of interactive math games, resources for teachers and interactive e-books of his original paperback books.

*This post contains Amazon affiliate links.

Mystery Bags for Older Children

By Mariah Bruehl,

Mystery Bag TitleMystery bags (bags in which you put items for a child to identify by sense of touch only) are commonly used with children in the three to eight year age range and are a fun, engaging way for children to sharpen their tactile perception.  It’s a classic activity that you perhaps have done many times with your child.  I certainly have with all three of mine and one thing I noticed is that no matter what their age, they can’t resist the intrigue of a mystery!  But how can you continue the fun while still providing a challenge for older children?

Here are some twists to give a try…


What’s in Common

(classification, problem-solving)

Choose items for the mystery bag that all have one thing in common.  It could be items with a similar purpose or that you might use for a particular activity (art, cooking, reading, getting reading for bed, swimming, etc), items that share the same beginning or ending sound, or anything else you can come up with.  Ask your child to visualize each item as he feels it and to say what it is.  Ask him to tell you as soon as he thinks he’s knows what each item has in common.

To make it a bit more challenging, choose items that can also be categorized into sub-sets.  For example, the objects might fall in the category of “toys” but can be further divided into animals, wooden toys, and vehicles.  This can be a fun way to introduce a new unit study or skill.



(critical thinking, reasoning)

Inference is the skill of reaching a conclusion using observation, prior knowledge, and reasoning.  It is considered a foundational skill that is used across the curriculum, especially in reading and science.
To prepare an inference mystery bag, gather items that tell a story about a person (interests, occupation, etc.), a place, or a situation.

For example, you might pretend that you are meeting your sister’s friend for the first time and, so that you can learn a little about her first, your sister has given you a bag of some of her belongings for clues.   What kinds of things might she do or like?  Does she have children or a pet?  Or maybe a neighbor was telling you about the great vacation he just got back from and he’s brought you a bag of souvenirs from the trip.   Where did he go and what types of things did he do?  You can really have fun getting creative with this version!

In this variation, the child pulls out the items one at a time and makes his inference as to what information the object gives.  Ask your child questions about what he’s basing his reasoning on and if there are any other conclusions that could be considered.

Mystery Bag for Older Children

Touch Drawing

(tactile perception, observation, visualization)

Choose one object with an interesting form and texture but not too complex (something from nature is our favorite type of object to use) and place it in a bag that your child cannot see through.  Ask your child to reach in and feel the object without peeking.  Prompt him to think about it’s texture, size, and shape.  Is it smooth or rough?  Ridged?  Scaly?  Do all sides have the same texture?  Is it thin, long, pointy?  Is it soft or hard?  Once your child has enough information to be able to visualize the object it’s time to put pencil to paper and try to capture the object’s image in a drawing.  Allow your child to feel the object as many times as he wants while he draws but still no peeking.  The level of details that your child notices will support the accuracy of the drawing.  Once the drawing is complete your child may take the object out and compare it to the drawing.  Discuss the similarities and differences.

Mystery Bags for Older Children

Have fun with these variations!  And if you have other clever mystery bags activities, please do share.


Outdoor Math Games

By Mariah Bruehl,

Outdoor Math GamesSummer is often when we all want to be outside to enjoy the sun, play with friends and explore nature. Fun in the sun can also coincide, playfully, with keeping up our math skills. The games and activities below can be played both outside and inside. The key is that they are portable and can be enjoyed in both environments.


Bracelet Fact Game:

Materials: pipe cleaners, colored tape or thin post-it notes

This activity helps children to remember basic facts in an area of your choice. Help your child or group of students decide on facts to practice (addition, subtraction, multiplication or division). Start with five to learn and practice. These should be facts that the child knows how to figure out but needs to gain in recall speed.


Activity: To play this as a game with two players, have each player play with the same facts on their bracelet. One player keeps the facts on the questions and the other player turns hers’ over to the answers. The first player asks a question or gives an answer and the other player locates it on her bracelet. For example, if player one calls out 5 + 5, the second player finds the answer 10 on her bracelet and calls it out. Starting with the answer is fun too, where the answer of 10 is called out and the other player locates the question 5 + 5. Children can also play on their own by quizzing themselves and checking answers.

Outdoor Math Games

Beach Ball Facts:

Materials: large beach ball, sharpie pen

This game can be played with as little as one player to a whole large group of players.  Before playing write math facts on the spaces of a beach ball. Choose facts based on your child’s needs. To play alone, the player simply tosses the ball up in the air, catches it, and answers the question closest to him on the beach ball. If playing with other children, the ball is tossed back and forth and each player answers the question closest to him when he catches the ball. This game is easy to take with you and can be used in all different subject areas by simply changing the questions.

Outdoor Math Games

I Have, Who Has Card Game

Materials: I Have, Who Has card set (click on photo above to download)

This is another game that can be taken anywhere but does require a group of children. To play, pass out each child one or two cards, depending on size of group, and have them figure out the answer to their questions. Have students form a circle and the player with the start card begins. Students play until the player with the end card plays. Encourage children to answer quickly so that it becomes a fast paced, quick recall, game.


Hopscotch Counting or Facts

Materials: sidewalk chalk, and flat small sticks

I just taught my own children the game of hopscotch and we played for hours. In this version, it is played the same way as the original with a little twist in the labeling of the boxes. Instead of labeling each square with numbers one through ten, have your child practice skip counting. We labeled boxes by 2’s and 3’s to practice but you could decide on something different. Another fun idea is to start the first number higher, maybe 22, and then count by 2’s. Your boxes would be labeled: 22, 24, 26, 28, 30, 32, 34, 36, 38, and 40. Facts can also be written in the boxes with or without the answer. You could make boxes labeled 1+1 = 2, 2+2 = 4, etc. and have your child say the facts as he/she moves through the box.




Have fun!


Snowflake Math

By Mariah Bruehl,

Snowflake Math

Anytime learning can be rooted in the world around them children thrive.  This is true for mathematics-based explorations, just as it is for science and history.  Right now snow, lots and lots of snow, surrounds my children.


There are many activities that can be done with snowflakes that enhance children’s mathematical understandings.

Snowflake Math

First, if you can, observe snowflakes outside; watch them fall, catch them on a gloved hand, or on black paper, and observe them with a magnifying glass or a camera zoom.  With older children you can try a snowflake fossil technique.  If the weather is not cooperating with you look at snowflake books or pictures of snowflakes.  When observing the snowflakes ask your children questions such as, “How many sides does your snowflake have?”  “Do all of the snowflakes we see have the same number of sides?”  “It’s said that no two snowflakes are exactly alike, but can you find any that are similar?”


Next make paper snowflakes, there are many methods for making paper snowflakes, but one of the simplest methods starts with round paper coffee filters and a pair of scissors, appropriately sized for your child.  First have your child fold the filter in half.  Then count the “valleys” in the filter.  Our coffee filter halves had 12 valleys.  We knew we wanted to fold this piece into thirds so we divided 12 by 3, arriving at 4.  This told us we needed to have each section have 4 “valleys”.  Now fold the half-moon filter into 3rds, using these “valleys” as a guide.  Finally fold the filter one last time.  You have essentially folded the coffee filter into 12 pieces, but after cutting you will be left with a six-sided snowflake.

Snowflake Math

If you are folding snowflakes with older students now might be the time to take out a protractor.

Snowflake Math

Instead of counting the “valleys” in the filter and using a division problem to decide where to fold you could instead try to figure out the angles you will need to create.  Present questions such as, “If the paper snowflake is a circle and therefore 360 degrees, how many degrees will the angle be when we divide the circle into sixths?”  “Into twelfths?”

Snowflake Math

Now cut out your design making sure not to cut all the way through the folded edge.

Snowflake Math

Snowflakes also lend themselves to discussions of fractions.  When you observe snowflakes in their natural setting it is very rare to find some that is completely intact.  They have taken quite a ride to get to the ground and often have broken branches.  You can observe what fractions of the branches are broken.

Snowflake Math

Symmetry is another mathematical concept that can be reviewed when observing snowflakes.  Leaving aside the damage that can happen to a snowflake while it falls, snowflakes form in a symmetrical manner.  Each snowflake starts as a hexagonal prism and since each arm of the flake forms under the same weather conditions each branch will be identical.  After creating their paper snowflakes children can find the lines of symmetry by reviewing their folds, or using a straight edge, or pencil, to mark them.


Math really is all around us, especially in those little flakes currently falling outside my house!

Building Place Value Understanding

By Mariah Bruehl,

Building Plave Value Understanding

The mathematical concept of place value is an essential one for young learners to grasp. Place value understanding will help children move on – and thrive – with more advanced math.


“Place value” is the phrase we use to talk about how numbers, when arranged in different places, show different amounts. For example, in the number 24, there are 2 groups of “tens” and 4 single “ones.” In the number 487, there are 4 “hundreds,” 8 “tens” and 7 “ones.” If we switch the number 4 with the number 8, we suddenly have 847 – a completely different number, with a different amount of “hundreds” and “tens.”


Place value is a concept that begins to be taught once children can count to 100, and know basic addition and subtraction facts. Kindergarten, 1st, and 2nd grades are when most place value fundamentals are learned. Most children need quite a lot of practice and exposure to place value, in a variety of approaches.


Given how important place value is to young mathematicians, how can you support your children in cementing this concept? Since place value understanding is a building block of math understanding, what better way to practice place value than with actual building blocks?


What kind of blocks do you have in your home? Wooden blocks? Legos or Duplos? Another kind? Anything will work! For children just beginning to work in place value, or who need extra support (typically ages 5-6), start with a smaller quantity of larger blocks. For children who are more advanced (typically ages 7-8), try large amounts of tiny blocks.


Pull out the blocks and challenge your child to build with them – this is the open-ended part! Ask your child to build something that resonates with their interests, or with content area being studied at home or in school. Maybe that means building a castle, a parking garage, a barn, a bookstore, a fairy path…the list goes on and on. For even more playfulness, try creating a story together about the creation being built.


Once the building is complete, ask your child to estimate how many blocks were used. Remember, use age-appropriate quantities – you can ask preschoolers this question if the number of blocks is somewhere between 5 and 20, and by the time a child is 7 or 8, as many as hundreds of blocks can be used.


Then it’s time to count the blocks. Sadly, this means that the building must come down. If you know your child will be particularly averse to deconstructing the building, take a photo, draw a picture of the building, or wait until the next day to take it apart. Photos of the creations could also be used to show your child his or her progress over time – once they’re building with hundreds of blocks, it will amaze them that there was a time when they were only building with a few groups of 10s.


As the blocks are being counted, support your child in putting them into groups of 10. If you used fewer than 10 blocks, point out how there are no groups of 10 at all. For most children 5 and older, beginning with more than 10 blocks is appropriate.


Stacking the blocks into groups of 10, or connecting them if possible, will help visualize the concept – if there are 40 blocks stacked into groups of 10, you can discuss how there are 40 blocks total, but that there are 4 groups of 10 blocks. If you are going above 100 blocks, put the blocks into groups of 10 first, then prompt your child to put those groups into groups of 100 (keep the groups of 10 connected, though, so you don’t end up with a pile of 100 loose blocks).


Use the printable place value mat below to continue. Ask your child to write in the number of blocks that were used for the building creation.

Building Place Value UnderstandingThen, ask how many groups of 100 blocks (if applicable) were used; your child will write in how many groups of 100 were used. If the total number of blocks was 132, your child will write in that there was 1 group of 100. Continue with groups of 10. How many groups of 10 blocks were used? Again, if the number was 56, your child will write that there were 5 groups of 10 used. Then, ask how many single blocks, not in a group of 10, were used. If the number was 32, then there were 3 groups of 10 blocks used, and 2 single blocks, not in a group of 10.


What can your child build and count? Leave a photo in the comments section to show what they are creating!