Friday, December 29, 2017

Fractions in 2nd Grade

Last month, a reader asked me how to teach fractions to a 2nd grader.  I'm assuming that the reader is asking me to provide the magic formula for a child of a specific skill set (which I haven't measured) paired with a parent who has a specific skill set (which I haven't measured).  This is a tough problem that requires the Force.  I retreated to the island of Skellig Michael off the coast of Ireland so I could meditate.

This article is going to describe how to teach a math topic (in this case fractions).  The approach for fractions shares a lot with other topics.  Fractions is especially important in math because it is slightly abstract and always multi-step, follows 3 straight years of spoon-feeding one step problems in school, and therefore befuddles students and parents alike for want of cognitive skills.

The Parent

Before we launch into this topic as a parent/coach/Jedi Master, we need to take a step back to appreciate the importance of this math topic.  It is fascinating, alluring, captivating.  Without this appreciation, it's hard to pass our love of math to our children.  If your child is ready to take on fractions at a young age, you will convince no one by saying "Fractions are totally boring and pointless to me but I'm going to make you do them anyway".

What are our goals for fractions?  We want the child to know fractions so that they can look at a super hard problem and see the answer right away.  We want the child to emerge from these fraction studies with a formidable skill set that can be applied to other quantitative areas.   We want to present this child with something like exponents or algebra someday and the child will say "Leave me alone.  I can do this all by myself".

There are some years from K to 8 where you want your child to do 45 minutes of math a day and be really, really good at it.  In our case, there were 3 of these years.  I wouldn't do it every year because the child will learn to hate math, and it's not necessary anyway.  If this happens to be one of these years for your child, in addition to what I describe in this article, get them a decent workbook that includes fractions and make them do every single problem in the book no matter how long it takes.  Otherwise, just do what I recommend here.

Fraction Foundation
The first thing we need is a high level definition of fractions.   When you divide 20 by 4, you end up with 5.  This means splitting 20 into 4 groups gives us 5 in each group.  If you have 20 skittles, but I'm only going to let you eat 1/4 of them, you're only going to get 5.  These are two different concepts, but the exact same mathematical operation, namely 20 ÷ 4 = 5.

What does it mean to divide 7 by 2?  What does it mean to divide 1 by 3?

There are two times when we have these discussions.  The first time is when I think we're going to be studying fractions in a few months or next year.  I call this Power Bucketing.  This discussion will create a brand new bucket in the child's brain called 'Fractions', and when the child sees fractions in school, while the other kids are trying to come to terms with fractions, my child will already have an empty filing cabinet in their brain for fractions and will have a permanent head start in this area.

When I teach fractions, we spend the first week just asking what fractions are.  I will give the child 10 to 20 minutes for them to think through these simple problems, like 7 ÷ 2 = 3 1/2.  After we've exhausted the mental capacities of the child, I'll ask for a picture or show them how to diagram fractions.

If you look through 2nd, 3rd, and 4th grade math curriculum on the topic of fractions, fractions are introduced slowly.  I'm not going to speed through this process.  Please view this Ted Talk on J.J. Abrams from 2007.  Look at that box with a question mark (in the video).  As long as the box is sealed, your child's imagination is in play.  As soon as you open it and describe its contents, you've ruined your child.  Let the child figure out what is in the box on their own.

Fraction Lifestyle
At this point, you can introduce fractions into your conversation.  Think about a really smart parent with multiple PhD's who just talks their child into Stanford.  We want to be like this parent, only not as nerdy.  The two most obvious uses of fractions are time and baking.  Get your child a brownie mix and make them do all of the work.  Put post it notes on the refrigerator reminding yourself to talk about time only in fractions, as in 'it's 1/3 past 5, what time is it?'  By the way, my older child has been in charge of making desert for years thanks to fractions.

Fraction Overdrive
This page from IXL describes the basic fraction related skills expected of 4th graders.  You can also look at grades 5 and 6 because fractions is going to appear every year from now on.  I didn't read any of it because it's too boring.

Instead, like all topics in math before calculus, with the exception of geometry, we simply have to state the obvious.  How to you add, subtract, multiple, and divide fractions?  Throw in 2 more operators (greater than and less than) and transformations (aka equals) and that's pretty much our goal.  This is exactly 8 things to learn (transformations are 2 things - equivalent fractions and transformation to and from mixed fractions).

This little exercise is going to be repeated with rational numbers, exponents, complex numbers, and other pre-pre-algebra topics.  When this child is doing algebra for the first time at age 9, and is stuck while trying to reduce a really complicated algebraic expression, I say 'Dude, you've only got 4 possible operations - addition, subtraction, multiplication and division, just try all 4 of them to see which one works.'

When your child sees decimals and percentages some day, we'll have 2 additional transformations involving fractions.

Where did I get all of this material?  I spent a month thinking about it.  Your child's teacher does not have a month to spend on fractions because there are typically 6 to 8 topics each day, plus statistics.  With some math topics, I also wiki and read about Egyptian or Babylonian history.  Your child's teacher won't have time to do this either.  She has 8 subjects and 30 kids of cognitive profiles to teach.  You have 1 child and fractions.

The Student

Children are natural learners.  Once the parent is prepared (95% of the battle), the rest is easy.  Just give your child as long as it takes and don't help at all.  Ask the questions and expect your child to work things out mentally, when your child doesn't succeed, ask them to draw the picture.  Help as needed, but only after the child has exhausted their mental faculties.  I generally observe mental exhaustion takes place at about 20 or 25 minutes (because I always choose really hard material), and I'm prepared to sit there, sometimes silently, for 20 or 25 minutes.

If you hand a 4th grade book to your child, there will be gaps hidden in 2nd and 3rd grade material.  The child will get stuck on a problem, and the way forward is material that they either never had or never mastered.   Be prepared at all times to go back to 2nd or 3rd grade material as needed.  Suppose they get stuck on a really hard problem, and you can see that it involves transforming from mixed fractions or comparisons.  Take a few days off and do some problems involving mixed fractions or comparisons.  IXL is good for this.

Step 1 - Comparisons and Transformations
I'm not sure why a book would be needed at all.  The most important fractions are 1/2, 1/3, 1/4 ... 1/10.  If the denominator is greater than 2, then you've got 2/3, 2/4, 2/5 ...,  3/4, 3/5, 3/6... and so on.  Then you can multiple any fraction by 2/2, 3/3, 4/4 and you've got a set of un-reduced equivalent fractions.

Pick any 2, and ask for >, <  or  =.

If your child was adept at division and had a really strong number sense, I would not create flash cards to drill my child on fraction comparisons.  If your child did not have a strong number sense because they never had really great curriculum at age 4 or 5 that built number sense, I would not only create flash cards, but I would create spread sheets with 100's of problems from the fraction list and drill the child until their number sense was invincible.  In our case, we did this at age 4 with SSCC and never looked back.  Except when we did this again.  And one other time.

You can search the internet for "comparing fractions worksheet" and see thousands of examples.  If I never met your child and you only gave me 30 minutes to address "comparing fractions", I would print 3 of these: One with pictures, one with simple fractions, and one for harder fractions (involving primes versus composite numbers, like 12/13 versus 10/12) and I would find out quickly where they are.

This exercise requires transformations - like comparing 5/6 and 10/12.   This is a two step problem.  5/6 and 9/12 would be a more obvious two step problem.

Throw an integer in, like 2 1/3, and you've got the other transformation to get to 7/3.  We never go in the other direction, from 7/3 to 2 1/3.  When I see this in a book, I comment that this is lame.  In higher order math, we only work with 7/3, or 142/25, and never mixed fractions.  Also, as I mentioned before 6 ÷ 3 = 6/3 (this is impossible to write in a vertical line, but basically I'm writing division problems as fractions and never using ÷ again).

Step 2 - The Other Arithmatic Operators
Once we 'get' fractions and practice transformations, we have to tackle addition and subtraction, then multiplication and division.

Addition and subtraction involves transformation.  We can't add apples and oranges.  We have to transform one or both.  This is why transforming and comparing fractions is a prerequisite.

Pictures might help if you didn't spend any time doing step 1.  We usually just skip to the hard parts, but you need to read Step 4 below to see why.

Note that this is a 2 or 3 step problem.   These types of problems reward a child who works slowly and a parent who doesn't expect correct answers.  If the child is expected to do a lot of problems, expected to get them correct, and expect to do them quickly, the child will fail at multi-step problems.  Because of this, I have settled on one or 'a couple' of problems as our daily routine until the child builds speed.

If this child was 10 years old, I would expect the child to devise and explain a formula for adding fractions.  Before this age I never even hint that there is such a thing as a formula.  I want the child to go through the 5 or 6 substeps every time, using working memory, because amazing and surprising subskills will develop in that child's brain that will pay off in a big way later on.

For an 8 or 9 year old, I would want to see a picture and an explanation of what is happening.  I would also try out 1/2 + 1/3, 1/3 + 1/4 etc from the list I explained above.  But I would do this every time he was stuck on something like 5/11 + 2/3, because this age desperately needs intuition number sense and now's the time to develop it.  This is really going to slow down the topic, but if you do it right, you'll save many years later on not having to explain math topics.

Multiplication and division require starting all over again with this article, both parent and child section, with each operation.   What does it mean to divide 3 by 1/2 or 7 by 1/2?  What does it mean to divide 1 by 1/3?  How about dividing 4/5 by 2/3?  The same basic questions are asked about multiplying fractions.  What is 1/3 times 3/4?  Before algebra in about 6th or 7th grade, I would want this child to think through the meaning of these problems every time instead of just turning 4/5 x 2/3 into (4 x 2)/(5 x 3), because if the child skips thinking through these each time, they will get to algebra ready to calculate but unable to understand.  This approach precludes some problems and precludes lots of practice.  This approach involves a few problems over a much longer period.

Diagrams work really well in understanding multiplication and division.  These will be articles on their own so I'm not going to cover it here.  Have you ever read a history book that starts with the beginning of time, evolution, 40,000 years ago etc until it gets to the main topic, which might be 1972?  That's how I handle these topics.

How Bad Can It Be?
The biggest challenge with teaching your child math is coming to terms with how stupid your child is.  You're doing something that you just did the day before, and your child not only forgot what he learned the day before, he can't even add.  He does a single problem in 30 minutes and it's totally wrong.  There are 29 problems on the page that are not completed.  It's a disaster.

This is the make or break moment in your child's academic career.  You have the choice between a future surgeon with join doctorate degrees in Sumerian literature and Bioengineering, or a kid who drops out of community college to form a rock band.  The choice is yours.

I'm usually pretty pleased and announce that will pick up problem #2 the next day.  I can do this because in the futile mess I see cognitive skills developing.  Within a few months, my child is making adequate progress and I'm looking for books on Sumerian literature on Amazon.

Sometimes  I am discouraged and ask how he could possibly screw up such a simple problem.  After I say something like this, he will spend the next few weeks perfecting a base guitar riff.

How Good It Can Be
Once you've taken on a few topics like this once, each successive topic is easier and more fun.  The key is that 6 to 8 months of hard work pays off, and you can see that doing a single problem for 2 weeks and getting nowhere is normal and leads to ripping through pages down the road and eating math for lunch.  For a parent, it requires nothing short of faith to get through the first few weeks.

For those of you who took my advice to do EDM Grade 2 in Kindergarten, you already know this.  For those of you who do TPM, which is not all that mathy but is really thinky, you're ready to start.  Unfortunately, in both cases, nothing ever gets easier and you still have to go through the whole painful learning curve with new maths.  But doing Algebra II with a 9 year old and going through a painful learning curve is much more gratifying than doing decimals and going through no learning curve.

Last week, my child was struggling on a problem from his Algebra I final exam.  We stopped using math books altogether and just take tests, figuring things out on the spot.  Sometimes, we'll take a break and do some worksheets on a new topic.  Anyway, there were 4 maths involved in this topic, and he didn't know 3.  He didn't even know the formula for the area of a circle.  It took us over an hour to do a single problem, what with all the backtracking.

Then I realized I accidentally grabbed the Algebra II final.  When we went back to the Algebra I final, he had 6 questions of the form "What is 42% of 66?" and didn't know how to do them.  Arrrgggghhhh!

In each case, we took apart, figured out, and mastered new topics on the spot.  This is the skill set that I want.  This is the skill set behind the MAP test, for very important reasons.  If you can get this skill set down early on, say fractions, then it's just a matter of plowing through pre-algebra, functions, algebra, geometry, trigonometry, calculus (AB and BC), linear algebra, real analysis and series, and then statistics. 


  1. Where can I get a good list of nonfiction for advanced 2nd grader? I've been reading the handbook you recommended and that has mostly novels and picture books...etc. Thank you!

    1. There is a great reading list at the 4-6 grade level for this age group that is historical fiction, which is more appropriate than non-fiction, unless you give me a gender and a subject area. Give me some time to publish this list. I've been totally side lined by other activities.

    2. I'm still working on this. I've been travelling lately, and sitting on a plane pondering next steps. Getting there soon.

  2. My second grader will finish your test prep 3 book in about a month. It has been a slow pace, he's been doing only 1-2 problems on weekend but we finished both test prep 2 and 3 fairly quickly. Where do I go from there? Feel like running out of thinking math books again and no math plan and books lined up for the rest of the year and summer before his GT starts. He is not extremely advanced to do SAT questions or anything but I need challenging math that makes him think and struggle daily. NEED HELP PLEASE!!!

    1. How about doing Math Olympiad book? Would that be a good idea???

    2. Before you buy any books, download free released tests from competitive math. Try 2nd, 3rd, and 4th grade tests. Some times all three grades are on one test. If you do 1 or 2 questions. If this goes well, I'll know what could be next. The problem you are experiencing is that the challenging math after TPM is really hard, but it's math, so you may come across problems and not know which solution strategy to use. Of course, just tell me the problem and I'll help. If comp math goes really well, you qualify for Math Olympiad books. If not, Challenge Math for 4th grade isn't a bad choice for you. Very few kids really get into compititve math type work.

    3. Is that Edward Zaccaro's?