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.

What Did You Say About Cheeseburgers

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Yes, it’s how we learned it.  Yes, most people will say, “It worked for me, so it will work for them!”  The standard algorithm has definitely been a bone of contention in many conversations.  On another, but connected note, while watching Spiderman into the Spider Verse, my youngest was singing along with one of the songs.  A friend of mine made the comment, “Kids can hear a song once and be able to sing it, but yet they can’t remember things in school.”  (With the voice of Carrie) I couldn’t help but wonder, “If concepts had more structure that is easily identified by students, would learning be more fluid”.  This reminded me of a talk I watched of William McCallum on The Story of Algebra.  In it he discussed finding structure within a poem to help memorize the poem.  He then connected it to the structure found within the Common Core Standards.

In 5th grade, students solve division of whole numbers using partial quotients.  A huge component of partial quotients is considering the entire quantity of the dividend and the application of place value understanding.  For example:

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When dividing 691, students should think how many 8s are in all of 691.  After working with multiple of 10s in prior opportunities, students will conclude there are about 80 eights in all of 691.  Fast forward to 6th grade when students are first introduced to the standard algorithm for division, it’s important to carry that same thinking with a heavy undertone of place value understanding.

In our first session as Great Girl Mathematicians, we reviewed division using partial quotients.  By the second session, a young rising 6th grade male joined the group, thus removing Girl from our group name.  The second session was spent connecting partial quotients to standard algorithm.

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We began our session with the partial quotients strategy, but instead of writing the quotients to the side, we staked them on top as they would see them within the standard algorithm.  They could also see the place value connection of the why behind the standard algorithm.  I paralleled this approach with the standard algorithm.  I explicitly modeled my thinking:

“I notice there are 69 tens in 691.  I know 8 x 8 is 64, but I need 64 tens.  Therefore, I can multiply 8 by 8 tens which is 64 tens.  Because it is 8 tens, I will record the 8 in the tens place of the quotient.”

As I did the subtract, I attended to precision by stating, “69 tens minus 64 tens is 5 tens”.  Speaking of it in this manner gives meaning to why the next number is ‘brought down’.  In this problem, the next number is in the ones place, resulting in 51.  Because 8 x 6 is 48, the 6 is placed in the ones place of the quotient.

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As the kids worked through their practice problems, some needed to use the staked partial quotients before rewriting it as the standard algorithm.  Others caught on quickly.   While others grappled with where to place the numbers in the quotients.  Using place value as our guiding thought helped them to make sense of why to record numbers in certain places.

This approach debukes the cheeseburger mnemonic device which causes students to not consider the magnitude of the dividend making the thought “does my answer make sense” seem like a distant after thought.  Being explicit about the place value understanding creates the structure to help students make connections between and among concepts.

 

It Started as Great Girl Mathematicians

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After hearing a friend say how he has been previewing math concepts with his daughter to provide a foundation he knew might be missed in class, I decided to support my daughter in the same manner.  My oldest daughter is entering middle school this upcoming school year.  As a building level coach and district specialist, I’ve observed the struggles students have with transitioning from 5th grade standards to 6th grade standards.  From my experience, this struggle is related to middle school teachers not having a complete understanding of the 5th grade standards and how they directly connect to their grade level standards.

With this knowledge, I decided to create a plan to preview the concepts my daughter will learn in Unit 1 of 6th grade math.  These concepts include:

  • Divide multi-digit whole numbers using the standard algorithm
  • Add, subtract, multiply and divide decimals fluently
  • Apply GCF and LCM
  • Interpret the quotient of fractions

My oldest hasn’t had the easiest time in math.  She has battled with being flexible in her mathematical thinking and gets bogged down with procedures.  In school she has learned many tricks without understanding.  I’ve found it to be difficult to counteract these tricks at home once they are the topic of discussion during class.  But finally, conceptual understanding will have the upper hand, or at least that’s the plan.

As I pondered the best way to approach this preview, I thought of what would be most helpful and effective for Jocelyn.  She’s one who needs to know what’s next and the order in which things will happen.  She’s a social butterfly who is empowered by forward thinking friends.  With all of this, I decided it would be best to open the opportunity up to some of her closest friends.  A group of girls, a mixture of those who have excelled and others who have struggled.

So here’s the plan:

  • Meet at the library twice a week for 30 minutes each time
  • Be explicit about the connection between 5th grade standards and 6th grade standards
  • Assign a “project” for the girls to do for the week while at home.  These projects consist of 3 act tasks, open middle problems and tasks from the Georgia Department of Education mathematics frameworks.
  • Provide the group with notes to use during the unit of study once school starts
  • Use the app GroupMe so the girls can converse about the projects.  I called the group Great Girl Mathematicians, mainly because I wanted to instill that this group of girls is great and are mathematicians no matter the struggles.

In subsequent posts, I’ll share the actual content we discuss.  With them, I hope it will help 6th grade teachers and parents of rising 6th graders to make the transition to middle school more empowering and less awkward.