Multiply Before Divide

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

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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.

 

Unit 5 Linear Functions with Desmos

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I’m sure it has been done before, but here’s my take on teaching an entire unit using Desmos.  Before I begin, I’d be remiss if I didn’t share these articles.  Being completely transparent, I had this Desmos unit idea before my friend Turtle shared these articles with me.  Human Contact is Now a Luxury Good and How Busy Hands Can Alter Our Brain Chemistry a quick synopsis of the articles, working with your hands promotes happy brains and an increase in brain activity.  Too much screen time creates changes in the brain such as thinning and decreases the thinking and has been linked to depression.  Please take some time to read both of these articles.

With that said, my Desmos unit does not include an absence of peer to peer dialogue or teacher student conversations.

Screenshot 2019-04-06 at 7.12.39 PM  In the state of Georgia, 8th grade unit 5 covers linear functions.

Unit Suggested Timeline: 8 – 10 days

Suggested Sequence of Instruction:

    1. Revisit graphs of proportional relationships. 8.EE.5 (To be taught concurrently with #2 and 3)
    2. Connecting representations of proportional relationships. 8.EE.5 (To be taught concurrently with #1 and 3) (1 day)
    3. Comparing features of different proportional relationships. Connect unit rate to slope through a context 8.EE.5 (1 day)
    4. Use slope triangles to derive change in y over change in x. 8.EE.6 (1 day)
    5. Derive the equation for slope intercept form, y =mx + b. 8.EE.6 (To be taught with #4)
    6. Determine slope from a graph, table or linear equation. 8.EE.6 (3-4 days) (To be taught with #5)
    7. Interpreting slope in context. 8.EE.6 (To be taught with #4-6)
    8. Compare and contrast linear and nonlinear functions using tables, graphs and equations. (Emphasize y=mx + b as equation of a straight line) 8.F.3 (2 days)
    9. Create examples and non-examples of linear equations. 8.F.3 (To be taught with #8)

Suggested Activities:

  1. Click Battle  8.EE.5
  2. Sugar Sugar  8.EE.5
  3. Polygraph: Lines, Part 2  8.EE.6
  4. Investigating Rate of Change  8.EE.6
  5. Points on a Line– (with paper overlaps to create the similar triangles)  8.EE.6
  6. Which is Steepest?  8.EE.6
  7. Land the Plane  8.EE.6
  8. Match My Line– (Slides 1 -7) 8.EE.6
  9. Graphing Calculator with Lesson 13 from Illustrative Mathematics Open Up Resources  8.EE.6
  10. Investigating T-Shirt Offers  8.EE.6
  11. Charge!  8.EE.6
  12. Graphing Calculator with Introduction to Linear Function from Illustrative Mathematics  8.F.3
  13. Card Sort: Linear or Nonlinear  8.F.3

I’m Selfish

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There they were, the proof in the pudding in the form of pancakes.  I had to admit, they didn’t taste half bad.  The Cocoa Puffs within them were a bit mushy, but the Fruity Pebbles added an interesting flavor.  The most important piece of this, is I realized how selfish I had been.

I sat on the couch at my friend Nickeva’s house discussing things as friends often do.  Her girls, free to be who they fully want to be, casually asked if they could make pancakes.  “Sure”, without skipping a bit their mother responded.

Thought 1: How many times had my girls asked to be crafty in the kitchen and I shut them down without giving it a second thought?

They moved about the kitchen with ease, this wasn’t their first time making major moves in there.  A 3rd grader and a kindergartner, real chefs in the kitchen as their mother continued to sit on the couch with me.  Thankfully, they invited my girls to join in on making the masterpiece.  There were laughs mixed in with directions.  I saw them pull out the bag of Fruity Pebbles.  What would they do with that?  Aren’t they making pancakes?

Thought 2: Look how free they are to create without my input.  How naturally creativity comes to kids when we allow them to explore and investigate.

While the pancakes cooked, I saw them pull out a plate, fill it with water and dish soap.  Then came the straw.

Thought 3: That’s wasting dish soap!!  My girls know better than to do that!  Ugh! Why am I like that?

That visit blessed me more than I realized in that moment.  My girls walked away inspired, with a feeling of accomplishment.  The next morning, they were all about making breakfast!  (Although I had to draw the line at cooking the bacon.)  I reflected and realized how my unbalanced focus on money stifled my children’s creativity.  How my need for control and order limited the opportunities they had to explore passions and investigate things that intrigue them.

This got me thinking about the students who enter our classrooms everyday.  How many of them have been stifled at home and are desperate for a moment of creativity?  How many of them crave to have a chance to investigate and explore? How are we contributing to the opportunity gap, by not allowing our children, our students to tap into their natural curiosity and innate abilities?

I’m certain if I were to inquire, Nickeva would tell me about the times she guided her daughters around the kitchen or how she sign them up for cooking classes.  She would mention the various opportunities her girls have had prior to the pancakes on that Saturday.  What does that mean for me?  What does that mean for you?  Time and repetition of opportunities.  Guidance without majority of the control.  Being the guide on the side.  And I think the most important part is understanding this life is bigger than you and me.  What are we doing with the time we have with our own kids, with our own students.  What do we communicate to them when we limit their opportunities?

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The Push Without Relationships

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“As much as he pushed me, I never doubted his love for me.” – Billy Donovan about Rick Pitino

As an educator, what does this quote mean to you?  In hearing this, I couldn’t help but think about the students I have encountered in my career.  Not the ones I tagged as a model student.  Not the ones I would say to their parents, “I wish I had a classroom full of them!”  No, I thought of the ones who had me in tears, who had me wanting to holler and throw up both my hands.  The ones who I whispered to myself about them, “you are not my enemy”.  The ones who when they were absent I secretly rejoiced because I knew the class would run smoothly.

I thought intently about would they feel about me the way Billy still feels about Rick.  I reflected on what would cause a player to say that about their coach.  What would cause a student to say that about their teacher?  Relationship.

I truly believe that educators desire to push their students to their max potential.  Okay, well most educators.  I’ve been in conversations with teachers who have the best ideas for instruction that will engage their students.  In collaborative planning, hopes are high and expectations are as well.  Walk into their classrooms and it appears the conversations had during planning were nothing more than lip service.  Students, aren’t engaged.  They’re actually calling out, calling the teacher by their first name and walking about freely.  They’re up opening the classroom door for no apparent reason.

How can there be such a disconnect?  Relationship.  You can have the best laid plans for your classroom, but without relationship those plans can easily go awry.  Relationship makes room for the necessary pushes needed to get students to want to persevere through the low floor, high ceiling tasks.  It’s relationship that encourages students to receive the push that helps them work in spite of the shaky foundation they may have.

All it takes, a simple “it’s good to see you” or “I’m glad you’re here” as students enter the classroom.  A smile (before December) when they pass by.  Attentively listening as they share their thoughts or perspective.  It takes a more intentional honoring of what your students say during instruction.  A high five, when they share their mathematical thinking.  Making turn and talk a pervasive practice in your classroom to show you value each students’ input.

Relationship is the important aspect of teaching that gets you through the tough moments.  It causes you to see past the misbehavior of students and see them as a person, a human.  Without it, your days are longer than you want them to be.  Without it, you students resent you for wanting and doing what’s best for them.  Without it, will they ever say, “I knew he/she loved me…”?

 

 

A Case of Teacher Envy

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Okay, before I begin, let me be clear.  I am grateful for the position in which God has blessed me.  I know He has placed me in it for a reason.  With that said, I must admit, I have a bad case of teacher envy!

When you have the opportunity to observe other teachers, you get to see a lot.  Some days can be really dark and you find yourself in need of something to lift your spirits.  For certain, I know of 2 classrooms in which I find a math sanctuary.  I know, at any given moment when I walk in, there’s going to be some goodness for my math soul.

But have you ever seen something in another teacher’s class that made you cry, “I wanna do that!!”  I kid you not, just about every time I walk in @tenaciousXpert 8th grade class, my math soul cries out, “I wanna do that!”  From small groups set up in the four quadrants of the coordinate plane created on her floor to the reinforcing of vocabulary when she calls on the groups, “someone from quadrant four answer…”.

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When you talk with her about her students, she holds them in high regards.  I have never once heard her utter the words, “My kids can’t…”  I’ve only ever heard her talk about what she’s planning to do to ensure her students reach the level of the standards.  She talks about the cognitive processes and executive functions.

You probably can’t imagine why I might be jealous of her.

Because of this:

While working on this standard:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

 

She intentionally built small group activities for students to see the connection between why the interior angles of a triangle  add up to 180 degrees, to finding the exterior angles of triangles to the angle of parallel lines cut by a transversal.  So methodical and purposeful.

When she told me about her plans for Pythagorean Theorem, I really wanted to hate on her, but that’s not a part of my nature.  I could only celebrate with her, as her students developed aha moments which carried them through accomplishing Taco Cart.  The class worked together to conceptually understand WHY a^2 + b^2 = c^2!

(Yes, she cut out foam pieces for this activity.)

Each time she sent me a picture, my math soul cried, “I wanna do that!”  While I thought I was doing big things in my class in this unit, she is setting up real world scenarios in her classroom.

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“LIKE, WHO DOES THIS!” my math soul shouted.  I want to be her right now, I remember thinking.  To see the kids make connections is so exciting to me, I always want to be there for it.  I want to be a part of the math goodness.

Then there’s this!

NEED I SAY MORE!  But I will.  With tasks such as In and Out Burger, she instills perseverance and math proud in each of her students.  They go home and work on math over the weekend because…wait for it…they are too excited about the math to wait until the next class period.

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I miss that, I envy that she has the opportunity to expose her students to their full potential each and everyday.  But I’m grateful I can witness her greatness.