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.

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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:

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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.

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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.

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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. IMG_9430(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.

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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.

Thanks for reading.

Having Fun with Scientific Notation Part 2

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I can’t really say students traditionally struggle when using operations with scientific notation.  Some of my students were apprehensive about the concept as we transitioned from converting to computing.  Again the saving grace was beginning with the conceptual using base ten blocks.

Students were able to see the quantities being added having the same blocks like when we were adding (6 x 10^3) + (3 x 10^3) or related blocks like with (6 x 10^3) + (3 x 10^2).  The discussion of regrouping blocks to add unlike blocks went over well allowing a student to conclude this:

 

We used a combination of concrete manipulatives and virtual manipulatives.

By the end of the lesson a student stated, “Is it really that easy?  I thought scientific notation would be hard!”

Placing Place Value in Middle School

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*I know the title is corny, but the content below is great.”*

What happens to place value understanding between elementary and middle school?  From my observations, when tricks are introduced, the need for place value understanding is no longer necessary.  Why know they relative size of numbers when I can “add” zeros and move decimal points?

If you are a middle school teacher, there’s a chance you didn’t know that in 4th grade students 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. (4.NBT.1) and in 5th grade students 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 (5.NBT.1).  Why is that important for you to know?  In middle school, students begin to look at percentages and scientific notation, among other concept extensions, which are based in place value.  If we decide to make explicit connections between concepts students already know and understand to new information, we have essentially cut the amount to commit to memory down.  It makes me think of Van de Walle discussion on how understanding the commutative property cuts multiplication problems in half.  Well, if students know as a digit shifts left it is getting ten times larger and as it shifts right it is getting ten times smaller, it can be applied to solving percentage problems and scientific notation problems.  However, if that connections is not made, students now have to memorize in which direction the decimal point “moves” or how many zeros to “add” in addition to solving the problem.  All new information that does not make sense, nor can the reasonableness of the answer be determined.

Here are some activities I have used to help make the explicit connection of place value with scientific notation and percentages:

Exploring Powers Of 10

Scientific Notation Prezi

Powers of 10 Kahoot Session

Percentages- nzmaths

Translating Fractions, Decimals and Percents

 

Matching scientific notation expressions

  

The discussion is based on a conjecture made by a student after looking patterns connected to place value and scientific notation

Lesson on Deck! :-)

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If you ask any teacher what is their number one enemy, they would most likely say, “Time!”  I wanted to wrangle the moving decimal point in 8th grade, however, time did not permit.  But you better believe when the time comes for me to lasso that decimal point and park it in its proper place, between the ones and tenths places, I will use this lesson along with this activity.

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!”