# Multiply Before Divide

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Let’s ponder for a second.  How would you teach students how to solve this problem:

2.4 x 1.3

Would you multiply 24 x 13 and then count decimal places?  In years past, this is the method I taught students.  No reason to count decimal places other than it will give the correct placement of the decimal point in any problem.  I anticipated this strategy being taught to the group of rising 6th graders I’ve had the privilege to work with this summer.

In our previous session, we made sense of the standard algorithm.  I planned to transition next to dividing decimals using the standard algorithm.  As I consulted with Van de Walle on how to make this concept make sense for students, he gave this advice.

With this, I changed my plan to address multiplying decimals before dividing decimals.  In thinking forward, I knew I wanted the kiddos to apply the place value understanding we discussed with the standard algorithm as they divided decimals.  As I considered Van de Walle’s advice and what would be the most effective way to make these connections, I decided to focus on the relationship between whole numbers and decimals.  This might be easier for me to show you my thoughts rather than type it out:

Disregard the fact it says, “Multiplying Decimals in Context using the Standard Algorithm”. 🙂

I wanted the group to think about the relationship between the whole numbers and the decimals, so I begin with the Same, Different strategy.  They noticed the obvious like, same digits, some have decimals while others do not and 130 has a zero while 13 does not.  This was the perfect setup for what I wanted to highlight for them.

Let’s consider 13 x 17 = 221 and 13 x 1.7 and the relationship between 17 and 1.7.  1.7 is ten times smaller than 17, therefore the product of 13 and 1.7 will be ten times smaller.  What about 13 x 17 and 1.3 x 1.7?  1.3 is ten times smaller than 13 and 1.7 is ten times smaller than 17.  This makes the factors one hundred times smaller than the original whole number factors.  Which will in turn make the product one hundred times smaller than the original product of 221.

After conducting a think aloud, I assigned some parallel problems for practice to the group.  With the parallel problems, they worked in partner pairs answering 3 questions.  Each partner had a different problem, but both problems had the same answer.  This helped generate conversations around their thinking.

The biggest struggle was determining what the product would be when it is ten times, one hundred times or one thousand times smaller than the original.

In 4th grade, students learn as digits shift to the left, they become ten times larger.  Or as the standard states, the digit is ten times what it represented in the place to the right.  4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

In 5th grade, students learn as digits shift to the left, they become ten times larger; and as digits shift to the right the become ten times smaller.  5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

I have yet to find in the standards where it states the decimal point moves.  THAT’S BECAUSE THE DECIMAL NEVER MOVES!  IT IS ALWAYS BETWEEN THE ONES AND TENTHS PLACES.  Now remember, students’ understanding of multiplying decimals will support their understanding of dividing decimals.

The images above and below shows the shifting of digits as it becomes ten times smaller and one hundred times smaller.  Images similar to this helps to reinforce the two standards learned in 4th and 5th grade and debunks the idea of the decimal point moving. (Oh and I kind of told them if their teachers says the decimal point moves not to believe them.). I may have said it once or twice before, but if you are teaching students the decimal point moves, you are undermining the place value understanding emphasized 1st through 5th grade.

The group asked if I would give them a problem in which they could practice by first solving the multiplication problem of whole numbers and then determine the products of related problems.

In the long run, they will understand why counting decimal places of factors will match the decimal places in the product.  This will help them understand division problems such as 9.8 divided by 0.3, which I will share in my next post.

# Mom, The Decimal Point is Moving Again!

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I’ve had the pleasure of modeling a formative assessment lesson/classroom challenge in all regular ed 6th grade classes over the past two weeks. I must say, I absolutely love how these lessons are formulated to pull out the misconceptions lurking below the surface of students’ understanding. These times were no different.

Because this task focuses on converting decimals to percentages and percentages to decimals, I kind of figure I would have to tackle the notion of the decimal point moving. I thought this would be a perfect time to make the explicit connection to place value standards from 4th and 5th grade for teachers and students. Just to be clear I’m referring to 4.NBT.1 and 5.NBT.1. As we wrapped up the lesson, in every class, students discussed how they converted the decimal to a percentage by moving the decimal point two spaces to the right and converted a percentage to a decimal by moving the decimal point two spaces to the left. I had them right where they needed to be.

Research shows students’ misconceptions aren’t alleviated by the teacher telling them their thinking is off. Having multiple opportunities where their thinking is challenged through experiences and conversations with peers is what alleviates the misconceptions. So I wrote this on the board:

We entered the values of a decimal and its percentage equivalent. I asked students the location of the decimal point for each number. Ensuring students attend to precision, I asked guiding questions to get them to articulate the decimal point was between the ones and tenths place for each number. I asked students, if the decimal point moved. Many were able to conclude the decimal did not move. We had to determine what actually moved.

Students grappled with the idea that in fact the decimal did not move but it was the digits which shifted.  In one class, a student even articulated, when we multiplied by 100, the number become 100 times larger!  Yes!!

Anchor charts will serve as a reminder of what is actually happening to numbers when multiplying or dividing by 100.  My hope is as teachers continue the conversations about the concept, they will attend to precision and discussing the magnitude of numbers instead of stating something which is mathematically impossible.

Next stop, 8th grade classes studying scientific notation because, “Mom, the decimal point is moving again!”

# Never Wave My Flag

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Don’t get me wrong, procedural fluency is very necessary. However, if procedural fluency is your only means of understanding you have a huge problem. This problem extends pass performing well on a state assessment. When procedures fail you, what do you have to rely on but conceptual understanding. An Englishman put it this way, “The only place procedural understanding brings success with problem solving is in American classrooms. In all other cases, we must rely on our conceptual understanding.” (I hope I quoted you accurately Phil.).

So when I discovered that majority of my students were procedurally fluent with converting fractions to decimals and didn’t understand at all why the rules they applied so successfully even worked, I couldn’t turn a blind eye. In the midst of feeling like I was ready to give into the ways of rules and tricks without understanding, I took a stand. I would not give up on them and I would not give up on my beliefs.

This lesson was implemented.

I thought it would be the bridge to understanding for my kids. I figured if I started from the bottom, the basics, with modeling things would click. What I discovered was they had minimal experience with modeling. Most didn’t know where to begin with setting up a model to demonstrate division. They didn’t get the idea of the physical manipulation of the blocks showed the division. Instead they thought they needed to represent the dividend, divisor as well as the division symbol. I also discovered they were limited in their reasoning abilities due to desire, experience or cognition. Needless to say, of the papers I’ve reviewed, no one was able to make the connection between why the rule in which they are proficient works.

I’m not giving up though. My next steps will be more explicit. Not explicit in telling them why it works. But explicit in my questioning as we model the division of 1 divided by 4. “What can we do to physical put 1 whole into 4 groups?” Students should conclude the whole can be exchanged for ten tenths. Draw the model. “What number is represented by the model?” My goal is for students to see adding the decimal point and zero is the ten tenths needed to physical put 1 whole into 4 groups. So the problem doesn’t become ten divided by 4 but actually 10 tenths divided by 4. Which is still 1 divided by 4.

I know moving forward it will be important to spend time working with modeling with manipulatives . I’ll probably end up using some fifth grade tasks from the GA math frameworks to help activate prior knowledge through the use of modeling with manipulatives.

# Hand On the White Flag

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Fighting the “good fight” can be a rewarding thing. To know you are a part of molding thinkers and mathematicians brings great feelings of a job well done. You’re fighting for all the little guys and gals that others may have been unequipped to do so.

When you are a middle school math teacher you have two options as I see it. You can carry on the tradition of rules and tricks minus the understanding. Or you can provide bricks for a shaky foundation put in place by the most well meaning teachers. I chose the latter. And so far this year, my students have not been resistant to the remodeling of their math thinking. Until today…

After conducting a number talk yesterday to discuss changing a fraction to a decimal, we as a class identified two strategies we could apply for this. One being, find an equivalent fraction with a denominator of 10 or 100 and the other divide the numerator by the denominator. We attempted to apply those strategies today bringing to light the old issue of being fluent with a procedure but not understanding why the procedure works.

Student: to change 5/6 to a decimal, you…well you can’t divide 5 by 6 so you put a decimal and zero. And 50 divided by 6 is…
Me: actually, you can divide 5 by 6 and we are getting ready to do it.
Student: well just add a decimal and zero
Me: Let’s think about what number we can multiply 6 by to 5.
Random students: 1. 0. -1

I inquired why do we need to add a decimal and zero and the only response given was, “Because you can’t divide 5 by 6.” Twenty minutes we spend on looking at numbers that would give us less than or just about 5. Many students lack the concept of fractions and decimals being part of a whole. Some wouldn’t budge from their add a decimal and zero rule and in turn decided not to make sense of the quantities. I could tell there were many that had just checked out. Those bricks I was trying to help lay sat squarely on my chest. I put my hand on the white flag, grasped it tightly and had a private conversation with myself.

Was it worth the struggle the students were facing as their rules of old were challenged? Was it worth the terrible feeling of knowing your students, a large majority of them, just weren’t getting it? I wanted to say, “yes, we just add the decimal and zero” and follow the steps of long division as I waved my white flag.

I left work feeling defeated, conflicted, and lost on what to do. In deciding to go for a walk after having such a tough day, a song came to mind. “Started From the Bottom” by Drake which discussed his struggles coming up before making it big as a rapper. Then it hit me, start from the bottom, the basics and strategically work my way up to what number I could multiply 6 by to get 5. I’m working a bottom up lesson to implement tomorrow. I’m hoping this bottom up remediation approach will help students make the connection between the procedure and why it works.

Started from the bottom now we’re here
Started from the bottom now my whole team ________here
Started from the bottom now we’re here
Started from the bottom now the whole team here, _______
Started from the bottom now we’re here
Started from the bottom now my whole team here, _______
Started from the bottom now we’re here
Started from the bottom now the whole team ______ here

– Drake, “Started From the Bottom” lyrics