Using Folding to Teach Fractions

Equivalent fractions can be a difficult concept for kids to truly visualize and understand.  We can teach them to multiply the numerator and denominator by the same number, but they may not fully understand WHY that trick works.  Fortunately, there are some very fun ways to help your students understand this concept more deeply, so they can apply it and use it more flexibly later on.



Folding (or cutting) is one of the most concrete ways for students to visualize how equivalent fractions work. 

Here’s an example:
1.    Take a piece of paper.
2.    Fold it in half.
3.    Color one half and leave the other half blank.
4.    Fold it back in half.
5.    NOW, fold it AGAIN, to make fourths.
6.    Open it up!   Voila!  Now it’s clear that the same ½ that was colored is now actually 2 parts (because of the new fold).  BUT, where there used to only be 2 total pieces, there are now FOUR. 

This process makes it easier for kids to see the doubling that is happening when we make equivalent fractions.   The numerator (shaded part) is being doubled by folding the paper in half. AND, the denominator (totally pieces) is ALSO being doubled by folding the paper in half.   So ½ will become 2/4 because both the top and bottom are doubled.  And we didn’t change the are that was shaded, so we know the two fractions are equivalent!

PRO tip: Have the students write an equation to match each folding that they do.  For the above example, one equation could be: 1/2 x 2/2 = 2/4. 

This exercise can be repeated over and over until students start to see the how the equivalent fractions are related.  They can even refold papers that are already folded or fold papers in a sequence to make a string of equivalent fractions.  Start with doubling (folding in half), then move on to tripling or quadrupling.  Have different students work with different sized papers to help them generalize their understandings for different size wholes.

For ready-made, guided discovery folding equivalent fractions lessons, check out my best-selling Third Grade Fraction unit.



Happy (equivalent fractions) Teaching!

Christine Cadalzo

Why I Don't Teach Rounding Tricks (anymore!)



My very first year teaching, I taught a self-contained special education class of 16 students in a New York City public school.  Did you know you can do that with absolutely no experience or qualifications?  Yeah….

I had absolutely no idea what I was doing, but I sure as heck tried.  But, when it came time to teach rounding to my 3rd -5th graders who had little to no place value understanding, I thought I was so clever.   I got a yard stick and a hula hoop, gave each child a slate with a number on it, made a human-sized big number to round,  and taught them this trick:

1.    Circle the place you are rounding to.  (That person got in the hula hoop.)
2.    Underline the place to the right.  (Yep, that’s what the yard stick was for…)
3.    The underlined number (person with the stick) is the decider.  If they are strong enough (5 or higher), they can order the one in the hula hoop to go up to the next digit.  If not (0-4), then they cannot do anything. 
4.    The circled number (person in the hula hoop) does not like being bossed.  They get annoyed and turn everyone behind them (to the right) into zeros. 

Obviously, these steps work, and if students follow them, they they will be able to round.  But I wouldn’t say that they LEARNED to round or that they UNDERSTAND rounding… that’s a whole other thing.   But, in my naivety, I thought that, if the kids got the right answer, they understood it.  I kind of wish I could go back and super apologize to that class…

Here’s what I missed:
Teaching kids to round isn’t really about teaching them to get the answer (like most of math!).  It’s an uber important part of the puzzle in constructing place value/ base ten concepts.  When I taught then tricks instead of helping them truly understand rounding, I was robbing them of crucial time spent working with place value and understanding how our base ten system works.  (And believe me, I paid for this mistake later on, when I tried to teach estimation, computation, word problems, and every other concept that relies on a solid place value understanding.)


Here's a better idea: I should have helped my students discover rounding patterns so they could generalize and extend their understanding of how the base ten system works.   

I also should have helped them develop a stronger concept of what rounding actually is, why we use it, and how it works.




Yes, teaching “tricks” is fun and sometimes helps kids feel more successful.  If I could do it again, I would teach the rounding tricks AFTER students had a solid understanding of how to round without using the tricks.  That way, it’s like coming up with a shortcut for something you already know how to do, instead of skipping over the heart of the concept to get to the ‘trick.’


For more ideas on a better way to teach rounding, click here for my rounding unit.


Happy (rounding) Teaching!!

Christine Cadalzo

Writing ABC Books with Upper Elementary Students


I love using ABC books with my upper elementary students.   I like the organization of them and the familiar patterns.  Some of the letters are easier to find ideas, and some push the students to think more deeply and creatively.  But the thing I LOVE about using alphabet books with upper elementary students is just how incredibly flexible they are.  You can create an ABC book for just about anything.

Pro Tip: If you're substitute teaching in an upper elementary class, this is a great activity.  You can prepare the materials ahead of time, even without knowing what the students are learning about, and adapt the activity by changing the content at the last minute.  Super engaging and super flexible!!




There are two main options for organizing Alphabet books with upper elementary students:
1. Students can each create their own.  This will require more content and time on the part of the students.
2. Students can each create one page to make a class ABC book.  This is quicker, and will provide a reference for your students for the rest of the year.
Note: If you have more than 26 students, repeat some of the more popular letters, like "S" and "T," or assign some students to do the cover, editing, write an introduction, or create a digital copy of the book.

Here are just a few ways you can use ABC books for those chaotic beginning/ end of the school year times:
-For back to school: each student can create and "All About Me" book as a get to know you activity.  Or, they can create one for a partner to introduce that person to the class.
-At the beginning of the year, students can create alphabet books of positive examples for character education, and then add to it throughout the year.  (Think: I is for integrity, with an example of a time someone in the class or a role model demonstrated integrity.)
-At the end of the year, students can create alphabet books as scrapbooks or memory books of what they have learned throughout the year.
-At the end of the year, students can also create "The ABCs of Fourth Grade" books, where they give life advice to next year's fourth graders.  The incoming students LOVE reading these on the first day of school!

And even more (Common Core aligned) ideas for using ABC books with upper elementary students:

For ELA:
-keep an ABC book of character traits in each student's reading folder/ binder.  They can add to the book throughout the year, or they can start a new one with each read aloud.
-create alphabet books of lessons/ morals/ themes that students come across in literature throughout the year.
-as spelling references or personal dictionaries
-create ABC books of linking, temporal, or transition words and phrases for writing
-as a place to store "juicy" words that students come across in their reading and learning.  They can then use their words as a source of inspiration when they are writing.   They can also do this for nonliteral language- keeping an alphabet book of similies, metaphors, idioms, and other figurative language.
-to study informational text features: students can draw diagrams, write labels, and add captions as they create a page for each letter of the alphabet.  They can choose a topic or create one for a topic the class is studying.
-create ABC books of ELA vocabulary: narrator, opinion, theme, revise, illustrator, etc.
-alphabet books are also a great place to store new vocabulary words- almost like a personal dictionary of new words for the students to review and try using
-create alphabet books of parts of speech: an ABC book of adverbs, for example

For math:
-make digital ABC books by taking photos of things that look like each letter- students can learn about angles by labeling the different types of angles in each letter.
-draw each letter as a block letter and draw the lines of symmetry for each one.  Bonus points if you can add rotational symmetry!
-make an alphabet book of math vocabulary to use as a reference all year long.

For content areas:
-as an alternative way to publish a research project- students can make an ABC book about their topic
-there are ABC books available about most cities/ countries.  Have the students read as many of them as possible, and then create an alphabet book about their hometown.






Happy (alphabet book) Teaching!!

Christine Cadalzo




Geometry Sorts


What is a Geometry Sort?
A geometry sort is when students are classifying shapes into categories based on the geometric attributes of those shapes.

Why Geometry Sorts?
-Geometry sorts help students focus on the identifying geometric attributes (number of sides, parallel lines, angle sizes), as opposed to other attributes such as color, orientation, and shape size.
-Geometric attribute sorts help students look for and notice patterns among shapes with a common attribute. They may notice that all rectangles also have two sets of parallel sides, for example. -Sorts challenge and push student thinking about classifying shapes. Does a square go in the “rectangle” or “not a rectangle” category?  Challenging thinking like this helps students expand their understanding of what a rectangle is, and helps them see squares as a subset of rectangles, rather than a separate category. It helps them see that not all rectangles look a certain way.
-Once students have classified the shapes into categories, they can draw additional shapes that fit each category. This pushes their thinking about geometric attributes even further.


Types of Geometric Attribute Sorts:
(Listed in order from simplest to most complex.)

1. In/ Out (example/ nonexample):
Students classify shapes into two categories. The shape either has the geometric attribute or it doesn’t.
Example: polygon/ not polygons






2. Mutually Exclusive Groups:
Students classify shapes in to two or more mutually exclusive categories. Each shape can only fit into one category based on its geometric attributes.
Example: classifying triangles by number of equal sides





3. Side-by-Side Sorts:
Students classify the same set of shapes into two different sets of categories, so they can compare and contrast the two different classifications.
Example: sort triangles by angle size, and then sort them by side length





4. Venn Diagram:
Students classify shapes onto a Venn Diagram so they can see the geometric attributes they have in common.
Example: comparing and contrasting the geometric attributes of a rhombus and a rectangle





5. Overlapping Circles (Hierarchy):
Students classify shapes into overlapping circles. The largest circle represents the largest category of attributes (e.g., polygons), and the smallest circle represents the smallest/ most specific category of geometric attributes (e.g., squares). This way of classifying is more complex because students have to consider subgroups and the hierarchy of attributes.
Example: polygons/ quadrilaterals/ parallelograms/ rectangles/ squares






Discussion Questions about Classifying Shapes:
-What geometric attribute makes this a ____ ?
-How do you decide how to classify a shape?
-Why did you put this shape in this category?
-How could you change this shape so it would go in a different category?
-What geometric attribute do all the shapes in this category have in common? Do they also have anything else in common?
-What would you call this shape? Why? Could you call it anything else?
-What geometric attribute would a shape need to be in this category?
-Are there any geometric attributes that would keep a shape OUT of this category?
-How are ___ and ___ related?
-Do the shapes in these two different categories have any geometric attributes in common? Are these subcategories of a bigger category?






PS:  Looking for some pre-made, Common Core- aligned sorts, lessons, and activities?

Click here for my



Happy Teaching (and Sorting)!!

 Christine Cadalzo