Why Students Struggle with Adding and Subtracting Fractions with Unlike Denominators

If your students are struggling with adding and subtracting fractions with unlike denominators, the problem could actually be with:

-understanding of how to compose and decompose fractions.  Can they add and subtract fractions with LIKE denominators, first?  Do they show understanding and flexibility of this concept, or do they only know that they have to add the numerators and not why?

-fluency with multiples and the multiplication times tables.   Multiples are the basic patterns on which equivalent fractions work.   If students are struggling to skip count or to work with basic multiplication facts, they will inevitably get stuck on this one step of the process.  If they have to stop and think about think about a multiplicaion fact, they tend to lose the larger picture of the problem they are working on and are much more likely to make errors and lose their place completely.  The more familiar students are with multiples, the easier it will be for them to quickly recognize the lowest common multiple, or lowest common denominator, for any two fractions.

-understanding fraction equivalency.  In order to add or subtract fractions with unlike denominators, we have to be able to convert one or both of them to an equivalent fraction, and we have to understand why that works in order to fully apply it.  Students need a solid understanding of how equivalent fractions are formed and the patterns they create, so that they can fluently generate and recognize equivalent fractions.

-fluency generating equivalent fractions.  This is a step where students will get stuck if they cannot form equivalent fractions quickly and accurately.

-understanding of concepts of fractional size and being able to use benchmarks and estimation to judge whether or not their answer is reasonable.   This is important for making sure their answers makes sense and realizing when they’ve made a mistake, so they can correct it. 

If students are adding and subtracting mixed numbers with unlike denominators, they’ll also need strong:

-understanding of improper fractions, mixed numbers, and how they relate to each other.  Students should be able to move fluently between the two, and should have a strong understanding of how they are connected.  If students lack basic conceptual understandings for improper fractions and mixed numbers, manipulating them and working with them will be very difficult for them.

-fluency and understanding converting between improper fractions and mixed numbers.  Converting is not always necessary for addition or subtraction, but it offers students another option for how to solve the problem, and another strategy for working efficiently. 

If your students are struggling with adding and subtracting fractions with unlike denominators, it’s possible that the breakdown is in one or more of these areas.  Do a quick assessment to find out which one(s) and provide students with support to strengthen their conceptual weaknesses and practice in areas where they need to become more fluent.  Students who are strong in all of these areas will be able to add and subtract fractions with unlike denominators fairly easily.  It’s a difficult concept, but if you break it down, the pieces are much more manageable for our students.

Happy (fraction) Teaching!!

Christine Cadalzo

Using Assessments to Empower Students

Assessments can be a stressful event for students.  They can feel like punishment for not learning ‘enough’ or being ‘smart enough.’  Students can develop low self-esteem and feel unmotivated if they think that they “should’ be able to do everything perfectly, all the time.   And teachers can become stressed and overwhelmed trying to push students harder and calm their fears simultaneously. 

But assessments can also be a tool of empowerment for students.  When you know what you can already do and what you need to work on, you feel more in control of your learning.  If we teach kids to view assessments as a way to measure what they CAN do, instead of what they can’t, it will be enable them to feel proud of their accomplishments.  

Here are some tips for using assessments to empower:

-Use rubrics and other holistic grading systems for qualitative work.  Students may not always see the change in their overall grade, but when it’s broken down, they can see it more clearly.  They may score an overall level 3 on two consecutive essays, but a rubric will help them see that all of their work on editing has improved their punctuation mark tremendously.

-Track and celebrate progress.  The more often we can do this, the better.  Tracking and celebrating helps us stick to our goals and stay motivated to pursue new ones.

-Accept that being able to do things with a little (or a lot) of help is part of the process.  It’s a step up from not being able to do them at all.  Sometimes kids (and adults!) think that, if they can’t do something completely and totally on their own, then that is equal to not being able to do it.  In reality, it’s more of a progression, and good teachers know this and help their students understand it.  We can’t do something, then we can do it with a lot of help, then with a little help, then on our own. 

The self-assessment at the bottom of my one-page math assessments. 

-Break big goals into smaller goals so students don’t feel overwhelmed.  Sometimes standards contain multiple parts and concepts.  Students can master each part individually on their way to mastering the entire standard.  It doesn’t have to be ‘all or nothing.’

(a screenshot from my editable standards checklists)

-Know when perfection is needed and when accuracy is important, and when it’s not.  In other words, know when to focus on the product and when to focus on the process.   

-Keep students portfolios.  There is nothing better than watching kids, at the end of the school year, going back through their work from September.  They don’t always realize how much they are learning day-to-day, but when they look back over months, they can see how much progress they have made.  I love when kids say things like, “OH!  I can’t believe I used to think…!!”

-Remember that learning is not linear- especially for more complex tasks and concepts.  Kids’ brains are constantly growing and developing.  They have good days and bad days, just like us.  They may be able to do something brilliantly one day and struggle with it another day.  It’s a process, and that’s ok, and students should know that so they don’t become frustrated.   After all, no one goes off to college and still can’t tie their shoes… it always works out if you keep trying.   

Happy Teaching (and Empowering)!!

Christine Cadalzo

Place Value in Upper Elementary

By the time students get to 4th and 5th grades, they should already have strong conceptual understanding of the following place value concepts:
-counting by ones, fives, tens, hundreds, thousands, etc.
-each place value is 10 of the place value to the right.
-writing numbers in words and numerals.
-working with expanded form for smaller numbers.
-comparing and ordering smaller numbers.
-rounding whole numbers.
-using place value understanding when adding and subtracting.
-multiplying by ten.

Hopefully, the students have been taught to use these concepts, and not just to apply ‘tricks’ to get the answers.  If they’ve been using shortcuts (eg, ‘just add a 0’ for multiplying by 10, or ‘circle the place, look at the number to the right, and change everything to 0’ for rounding, or ‘just carry the 1’ when regrouping in addition), then helping students understand harder place value concepts in upper elementary is going to be a real struggle.  If they’ve truly developed the conceptual understanding for these place value concepts, then the more complex ideas in the upper grades should build naturally on what they have already learned.  (And the same going forward- what they will learn in middle and high school should build naturally on the place value concepts from upper elementary, making them even more important.)

In upper elementary, the focus shifts to larger/ smaller numbers, including decimals, and to generalizing and applying the base ten concepts.  Many of the concepts are the same ideas, just now applied to larger numbers and decimals.  In lower grades, students learn to compare numbers to the thousands place.  In upper grades, they extend this understanding to comparing larger, multidigit numbers and decimals.  In fourth and fifth grade, students work on place value concepts such as:
-understanding that the place value to the right is 1/10 of the one to the left, and the one to the left is 10 times the one to the right.
-use rounding and estimation fluently and apply them as a check on the reasonableness of a number or answer.
-comparing and ordering decimals and larger numbers.
-working with expanded form for decimals and larger numbers.
-apply place value understanding when performing arithmetic.
-applying patterns of zeros and ‘moving the decimal’ when multiplying or dividing by ten.
-beginning to work with exponents (an extension of multiplying by ten)

To help develop these concepts in upper elementary, try:
-relating numbers to money.  This works for both very large numbers (a billionaire is worth ONE HUNDRED millionaires), and for decimals (dollars and cents are a very concrete way for kids to understand ones, tenths, and hundredths.)
-working with calculation problems where it makes more sense to apply place value concepts than to use the standard algorithm.  1,000- 999 is a perfect example of this.  Many students will struggle trying to borrow from the 1 in the thousands place, across all those zeros, but students with strong place value understanding will see that 999 is only one away from one thousand. 
-do a lot of estimating… especially with money.  A great way to practice this is by estimating the total grocery bill for a list of items.
-continue working with number lines.  (They aren’t just for learning to count from 0-100!)  Making a decimal number line from 0.00 to 1.00, counting by tenths and hundredths is a great way for students to make sense of how tenths, hundredths, and ones connect and why the zeros at the end of decimals are not always necessary.  Try having each pair or group make a decimal number line between two different whole numbers (one group does 0.00-1.00, one does 1.00-2.00, etc.), so students can see how decimals connect to whole numbers, and will help rounding to the nearest whole make more sense.
-using real life examples of large numbers and decimals.  It can be hard to make these numbers concrete for kids, and finding real life examples can help.
-do math talks and strategy talks where students share ideas and strategies based on place value understanding.  If your students aren’t at that level yet, start with modeling it for them.   
-encourage students to use strategies other than the standard algorithms.  The standard algorithms are very practical, and students definitely need to be able to use them.  But we also have calculators that can replicate the standard algorithms for us.  What calculators cannot replicate is the more flexible thinking and understanding that comes from solving problems using a variety of strategies and the strong place value concepts that develop from working with numbers in this way.

Happy (Place Value) Teaching!!
Christine Cadalzo 

Fraction Sort

Sorting fractions is a great way to help students apply and solidify their understandings of fractions.  It's a fairly straightforward, simple activity that allows for a lot of really good, higher order thinking and reasoning.  

How to do a fraction sort:
1. Decide on the categories.  Start with ‘between zero and one half,’ ‘equal to one half,’ and ‘between one half and one whole.’  Add in ‘equal to zero,’ ‘equal to one,’ and ‘greater than one’ as needed.
(This is also a great opportunity for differentiation.  Stronger students can have more refined categories- dividing by fourths or even eighths, and going above one and including improper fractions and mixed numbers.
2. Make cards for each category. 
(To differentiate, make more or fewer cards, or make the cards more or less obvious- eg. 99/100 is more obviously close to 1 than is 77/99.)
3. Have the students sort the cards into the categories.  This is a great center or partner activity, and is a wonderful opportunity for rich discussion.

4. Share and discuss the sorts.
Misconceptions about fractions tend to be revealed here, and provide an opportunity for reteaching and discussion.  It’s also a good chance for students to share their strategies and verbalize their thinking.  (Hopefully, some minor ‘arguments’ will arise and allow students to ‘convince’ the other person!)
5. Have the students choose one card to write about and explain how and why they chose the category for that card.  Have them use visual models and justify their responses.   This works great as an assessment or portfolio piece- you can even assign the students a new (differentiated) fraction that hasn’t yet been sorted, to see how well they can individually apply what they have learned.

Why to do a fraction sort:
-helps with estimating and judging whether or not a response is reasonable
-helps solidify understandings about fraction benchmarks and how fractions are ordered from zero to one
-helps develop fraction on a number concepts, which are needed for measuring to the fraction of an inch
-provides practice and application with equivalent fractions, comparing fractions, and ordering fractions
-reveals misconceptions about basic understandings of fractions, their size, and how they are compared and ordered
-explaining and writing about their decisions helps students organize their thinking, and communicate using visual models and explanations

What to look for when students are sorting, sharing, and explaining:
-reasoning based on the relationship between the numerator and denominator
-reasoning based on not only how BIG the numerator is, but also based on the size of the missing piece needed to make a whole.  (eg., 9/10 is close to one because there’s only 1/10 missing, and 1/10 is a small piece.)
-identifying equivalent fractions and using that information to help make decisions.  (eg., 6/10 is equal to 3/5, so it’s between one half and one whole.)
-ability to clearly explain the thinking and strategy behind why they put a card in a certain category.
-reasoning based on using benchmarks. (eg., 5/10 is equal to half, so 6/10 is more than half)
-using critical thinking and asking good questions- making sure they agree with their partner’s or group’s sorting decisions
-drawing visual models to clarify, verify, or explain their thinking

Happy (fraction sorting &) Teaching!!
Christine Cadalzo

This lesson and materials are part of my Equivalent & Comparing Fractions math unit (4.NF.1-2). 

Coloring to Help with Measuring

Why kids struggle with measuring inches (and how coloring rulers can help):

1. Measuring in inches requires a different way of thinking about fractions. The most common way for kids to think about fractions is the area model- where a space is divided into equal parts.  When measuring inches, we have to think about fractions as a point on a line – the linear model- which can be confusing to students.  It’s hard for them to connect the idea of
½ of a cookie with a line on a ruler being ½ of an inch.  
Coloring inch measurements on rulers can help students make that connection between the area model and the linear model of fractions.  When they color in half of an inch on a ruler, they can SEE why that line is the halfway mark for measuring inches. 

2. Students need a strong understanding of equivalent fractions to understand the lines on the ruler when measuring inches.  The ruler lines count like this: whole number, 1/8,
¼, 3/8, ½, 5/8, ¾, 7/8, whole number.  That’s very confusing for kids who are just starting to understand both measurement and fractions. 
Coloring the inch measurements on a ruler can help students see the patterns and begin to make sense of this confusing order of fractions.  Color-coding the different fractions of an inch will help students see the progression from one fraction to the next, and will help them internalize the sequence. 

3. There are SO MANY LINES to look at when measuring inches on a ruler!  It can be really hard to know which one(s) to focus on and which ones mean which fractions. 
Coloring the measurement lines on a ruler will help students to make sense of all of those lines.  In order to color the ruler, students have to focus on the measurement lines, which will help them see the patterns.  They will begin to notice that some measurement lines are longer, and that these lines connect to the fractions with the smaller denominators (half inches and quarter inches).  They’ll be able to visually see that the bigger spaces/ bigger fractions have the longer measurement markings on the ruler, and the smaller spaces/ smaller fractions (eighths) have the smaller measurement markings on the ruler. 

Tips for Using Coloring to Teach Measuring Inches on a Ruler:

-Give students printed rulers with only the lines on them. Have the students color in the measurements (whole, half, quarter, or eighth of an inch) and label the lines of the ruler with the correct measurements. (To align to the Common Core, second graders should work with whole inches, third graders should work with halves and quarter inches, and fourth and fifth graders should work with eighths of an inch.)

-Start by using rulers marked with only whole inches, then add half inches, quarter inches, and eventually eighths or even sixteenths of an inch.

-Differentiate by writing in some of the measurements for students who need support and challenging students with completely blank rulers.

-Give the students paper rulers with only whole inches marked.  Have them draw the half lines and color each inch in halves.  Or, have them divide each inch into fourths or eighths, mark each line, and color each fractional piece.

-Use color-coding to help kids see the patterns: Color all the spaces from the whole to ¼ inch blue, the spaces from ¼ to ½ yellow, from ½ to ¾ green, and from ¾ to the whole red.  Students will begin to notice the pattern: whole, ¼, ½, ¾, whole, etc. The same can be done with ½ inches or eighths of an inch.

Happy Teaching (to the Nearest Quarter Inch)!
Christine Cadalzo


Here's a ready-made math unit for teaching students to measure to the nearest quarter inch!