Category Archives: Math Fluency

Ours is to Wonder Why

It began almost accidentally. After fooling around with some interesting magic squares recently I decided to put a couple in front of my students. I wanted to step away from our work with division of fractions momentarily and ignite their curiosities. Below is the fist magic square I showed them.

ms1I won’t bore you by telling you about all of the amazing patterns inside this square – they are yours to discover. In class we shared things we noticed. One of my students noted something like “It goes 8 – 5 – 2 diagonally: each number three less than the one before?” (Notice the question mark?) I said “Are you asking me? You’ve got to make a statement! You’ve made a discovery! You’ve got to stand up and shout EUREKA! So from here on out the class made really exciting discoveries punctuated by shouts of “Eureka!”

Then we looked at the following magic square.

ms2What do you notice? We made one discovery after another – the class was reaching a fevered pitch: math was rocking the house, and the students’ excitement and engagement was running into the red. Time to change gears and use some of this fuel on division of fractions.

Division of fractions – something like 2/3 ÷ 1/4 – is an odd thing to get your head around. We’ve been working with visual models for a little while but it still takes some mental work to see this stuff. In the old days there used to be this saying in elementary math about dividing fractions: “Ours is not to wonder why, just invert and multiply.” Catchy, but this takes all of the understanding right out of the equation (math joke!). Maybe the problem looks like this:

grid1Here we see a rectangle that is cut into thirds vertically and fourths horizontally. Each column represents one-third and each row is one-fourth. So the question is: How many of the shaded areas will fit into the green area? If we rotate the one-fourth, we can fit two full fourths into the green area (below). In the remaining green area we can only fit 2/3 of a quarter strip.

grid-div2So the number of fourths that can fit is 2 and 2/3. You can think about it this way: How many groups of 1/4 can I make with 2/3 of a whole? Scoop, scoop, little scoop.

If we look at the equations we can make some sense of these as well. Notice that in the equation below, we just combine the fractions into one fraction and the result is another way of looking at the picture above! How many 3/4 can fit into 2 wholes? – except this time the whole is actually just one-quarter! Do you see it?

eq1Maybe you don’t like that one… OK, I can live with that. But what if we did something interesting like found a common denominator for the fractions. 2/3 = 8/12 and 1/4 = 3/12.

eq2In this equation, the quotient of the first expression gives us 1 in the denominator! So it effectively becomes 8 ÷ 3 or 8/3, which is 2 and 2/3. This technique just melts the fraction division away.

All of this happened in one class. And there were more problems than just this one and more voices than just mine. Students were explaining what they saw and why it made sense, and they were shouting “Eureka!” every time they understood something new. Many students were standing, and hands were waving all around the room. I can tell you that at times it got a little crazy. But this is how, at 12:20pm, after a long morning of classes culminating in a math lesson on the division of fractions, a group of 25 fifth-graders spontaneously erupted into applause. It certainly doesn’t happen every day.

Ours is definitely to wonder why.

Places and Bases

Recently in math class we have been working with decimals. This is one of the areas in math where a solid foundation in place value will take you a long way. If for instance you want to know the product of 15 and 3.2 it’s nice to know that 10 x 3.2 is simply a decimal move and 5 x 3.2 is just half again. I am always surprised to find that place value is not well understood by 5th grade. I guess I need to stop being surprised. But perhaps place value is just a little more complicated than we give it credit for, and maybe there is just not enough emphasis in the earlier grades.

Let’s look at the base-ten place value table:

Thousands Hundreds Tens Ones Tenths Hundredths
1000 100 10 1 1/10 1/100
103 102 101 100 10-1 10-2

The reason for the ease of computations with 10 and powers of 10 is that ten is the base. This is why when you multiply 10 times anything “you just add a zero.” I try to teach my students to use the more mathematically accurate phrasing of “you just move the decimal,” but it’s no more challenging. The ones, or units, are the center of the place value system – all other places are exponentially larger or smaller than the base to the zero power.

Exponent review:

  • All numbers to the zero power are equal to 1
  • Negative exponents are just powers of base (ten in this case) in the denominator.

To understand place value, I find that it is helpful to examine what life is like in other bases. Let’s look at a base-five place value table like the one above.

One hundred twenty-fives Twenty-fives Fives Ones Fifths Twenty-fifths
1000 100 10 1 1/10 1/100
53 52 51 50 5-1 5-2

In base 5 there is no numeral 6, or 5 for that matter. There are only 0, 1, 2, 3, and 4. Once you get to 5 the notation is 10 or more clearly written, 10five. The number 12ten is 22five, and the number 78ten is 303five.

When we want to multiply any number in base five times five all we have to do is move the decimal! 115 (which is 6 to us) times 105 (which is 5)  is 110five (which is 30 to us). The product is still true regardless of the base. Likewise 22.2five (which is 12 and two-fifths) times 5 is 222five (12.4 * 5 = 62).

The place value system is dense, but when we understand the places and bases of our number system everything makes a whole lot more sense (or perhaps cents or pence if you live on the other side of the pond fence).


90-10We were playing this game in math class today called “I have, You need” during which the number ninety-ten came to my attention.

The game is pretty simple – one person thinks of a number between 1 and 99 and the other person comes up with the number that will add to that number to make 100. Warming up we start with numbers like 30, 45, 75 etc. Then we move on to more difficult numbers like 37. Many students will think first that 73 is the matching pair. This is because they want to make 100 and they know that 30 and 70 go together. When I explained ninety-ten, the problem became much easier. They were now trying to make nine tens, so 63 was immediately identified.

Sometimes we need to think about numbers a little differently. From place value to working from left to right, there is much more freedom in math than we’ve been led to believe. Create something new!

YouCubed (and more on fluency)

youcubed-thumbJo Boaler, a math education researcher and professor at Stanford University, launched a new website this past year (YouCubed) to help teachers, students, and parents navigate math education. She recently published a short paper on math fluency. In it she discusses the problems with associating math fluency with speed or memorization.

Interestingly, the Common Core intends this de-emphasis on speed but the word “fluency” is often misunderstood by textbook publishers. The newly adopted Bridges Curriculum seems not fall into this category – their strategy for building fluency is based strongly on number sense. (Read an excerpt from Jo’s paper below)

This past September the Conservative education minister for England, a man with no education experience, insisted that all students in England memorize all their times tables up to 12 x 12 by the age of 9. This requirement has now been placed into the UK’s mathematics curriculum and will result, I predict, in rising levels of math anxiety and students turning away from mathematics in record numbers. The US is moving in the opposite direction, as the new Common Core State Standards (CCSS) de-emphasize the rote memorization of math facts. Unfortunately misinterpretations of the meaning of the word ‘fluency’ in the CCSS are commonplace and publishers continue to emphasize rote memorization, encouraging the persistence of damaging classroom practices across the United States.