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

Creating a Remediation They Want to Come to!

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At the end of every 1st semester, students who failed a course are invited to participate in Comet Academy, my schools grade recovery/remediation sessions.  This year, those attending Comet Academy come for 2 1/2 hours for 6 consecutive Saturday mornings.  During our first session on January 30th, students were given a 30 questions, multiple choice pretest covering 5 key standards from 1st semester.  I was assigned the 8th graders who failed to which I was very excited.  8th grade is the one grade level I haven’t been able to infiltrate with the use of manipulatives and building conceptual understanding.

Students completed the pretest in about an hours time, followed by checking the answers of a partner.  Stunned by the results, I knew immediately, 6 sessions, even at 150 minutes a pop, wouldn’t be enough to get the students to pass the post test.  The highest score was 11 out of 30 correct, ugh!

After proposing to the math AP about the need for a math boot camp for students who failed to stabilize their foundation for high school, I got busy planning.  8th graders who failed 1st semester, whether they attend Comet Academy or not, would be pulled twice a week for 30 minutes each time for 5 weeks.  This would occur during their Connections classes (electives, specials, not the core classes).  This boot camp would only offer additional support and would not be rewarded with a grade, extra credit or anything tangible outside of a better understanding of the content.  For students who are often motivated by outside factors, I needed to ensure the activities were enticing enough to get them to come week after week.

Meaningful Practice

The standards are not new for students, therefore a mix of meaningful practice and concept development is necessary.  In their regular classroom, students are subjected to worksheet after worksheet or textbook page after textbook page for practice.  In boot camp we used a Solving Equations Bingo game to practice solving equations.  Students filled in the answers: x=3, x=11.5, x= 1 1/2, x=5, x= 17, x=4, x=17 and -72 =x into the Bingo board.  Then I read off equations in which students needed to solve in order to cover the correct answer.

We also played a game of Knockout!  This idea was taken from the basketball game Knockout!  In the basketball version, participates line up behind the free throw line, the first two people in line have a basketball.  The goal is for the second person in line to make a shot before the person in front of them in order to knock the player out of the game.  In boot camp, students sat in a straight line and the second person tried to correctly answer the math problem before the person in front of them did in order to knock them out of the game.  Here’s the PowerPoint with the game: Boot Camp 2-9 and 2-11.

To encourage collaboration, students were grouped (they chose girls against boys) and given whiteboards to record the answers to problems.  The group representatives would hold up their whiteboard showing their answers.  In order to receive a point, teams had to get the correct answer, but also shot a tiny basketball into a toy hoop.

Concept Development

To help to continue to build student understanding of solving equations and integer rules, I used lessons from Hands-On Standards and incorporated color tiles and algeblocks with lessons like this Exponent Activity.

File_001   File_005

Formative Assessment

I have to know where they are from day to day because I don’t have a lot of time to cover the material.  Therefore, each session has a connected formative assessment.  This helps me to plan differentiated lessons even within the small group of students I see on the different days.  I’ve used a two question quiz, a portion of a FAL and a Ticket out the Door pictured below.  Students were able to choose what type of question they wanted to answer which is a formative assessment within a formative assessment :-).

 

Special Education Math Framework

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It’s been a while since I have sat down to collect my thoughts and as the Bible says, “write them down and make them plain”.  This may result in a series of multiple posts in the next coming days.

 

Coming back from winter break always creates the opportunity to visit every math classroom to see how teachers have jumped back into the swing of things.  I was encouraged by the many teachers who have started 2016 with a bang of differentiation, small groups and formative assessments.  I was also encouraged by the goals teachers set for themselves to improve during semester 2.  One teacher’s goal became the spring board for one of the many projects in which I am currently working.

The special ed teacher approach me to share she felt the interactive textbook wasn’t helping her students understand the concepts.  She desired to know more instructional strategies she could use to better meet their needs.  During a pre-conference, we discussed how using more hands on activities and manipulatives would help students in understanding the concepts.  This prompted the conversation of what resources would provide meaningful activities while incorporating manipulatives.  We looked at using Hands-On Standards (HOS) and resources from nzmaths.  The conference ended with the scheduling of a model lesson on implementing a HOS lesson involving functions.  During our post conference, she expressed excitement about the observations she made during the lesson.  Students were discussing the mathematics and making connections between previously taught concepts and functions.  Students were more engaged than when they worked from the textbook.  All this in the MILD special education classroom.

After seeing the lack of hands-on experiences in other special ed classrooms, the special ed AP and I discussed ways to increase this instructional strategy. What spawned from that conversation was the Special Education Math Framework for Grace Snell.

Balanced Numeracy Adapted from our district’s balanced numeracy document we developed a scope of a lesson for special ed teachers.  It was important for it to mesh with our school initiative of gradual release.  Beginning the lesson with an activity from HOS brought in the hands on experience while satisfying the whole group portion of gradual release.  As the lesson transitions to the work session where students are working independently, the same manipulatives from the HOS mini-lesson can be used and any knowledge or understanding gained can be applied.  So those teachers who find comfort in the pages of a textbook are able to stay comfortable for the time being.

Happening this Friday is an optional professional learning session on implementing HOS and using the manipulatives.  My goal is to get every special ed teacher comfortable with using manipulatives on a regular basis.  I predict when we get to that point, teachers will find the textbook does not fulfill the same need as once before.

Pythagoras Square

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During the summer I had the opportunity to work alongside my math friends implementing foundations of algebra professional learning sessions. Although I was the one conducting some sessions, I also put myself in the position of a learner. One of my takeaways was using the Pythagoras Square to help build conceptual understanding of square and cube numbers as well as square roots and cube roots. 

So Wednesday morning when I was asked to cover an 8th grade class (the teacher was out and there wasn’t a sub to cover) I gladly abliged. They are working on simplifying radical expressions and the primary means of introducing this idea was through the use of prime factorization. I wanted students (and teachers) to understand the why behind the simplified solution. 

When students came into class from connections they were asked to view this model and record what they noticed and what they wondered. 

  
Students’ notices included: the different colors, it looks like an arrow, the figures getting larger and the similarities of a multiplication table. A few students wondered if the model could be extended. So I asked a few to jump in to help extend the model. 

   
    
   
We didn’t complete the entire square due to time. However, we did take the time to focus in on the perfect squares and their lengths. We were able to make the connection between the area of the squares and the radicands and the lengths of the squares and the square roots. 

From there we jumped into simplifying radical expressions.  We discussed identifying factors of the radicands and determining which factor pair contained a perfect square. 

   
 We concluded the lesson by students completing 10 problems on a worksheet left by the teacher for the substitute. I looked over the papers afterwards and realized some students didn’t quite understand how to write the answer once they determined the square root. For example, they would record the square root of 64 as square root of 8, not just 8. Other students needed help making the connection of identifying the perfect square strategy to the prime factorization strategy. 

Anyone who has followed for a while could guess my next step, pulling small groups. Within the groups students built squares using color tiles as we discussed square numbers and square roots. For students who recorded answers inaccurately we looked at a square made of 9 color tiles. By determining the length students were able to see why that answer is recorded as 3, not square root of 3.  For students who needed to make a connection between identifying the perfect square and the prime factorization strategies, we looked at how the prime numbers can show the lengths of the perfect square. 

I’ll have to check with the teacher to see how students faired on the common formative assessment given on Friday. 

Thinking Out Loud

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One of the great things about coaching at a school where you previously taught is you get to see former students and talk with them about their new year. Recently, a former student approached me and said, “You know the difference between this year and last year is you taught us the why and how. This year, my teacher only teaches us how, not why and it confuses me because I don’t understand.”  This was powerful because he was a student who resisted my student-centered approach. He hated doing small groups and often asked if he could just do work out of the textbook. 

Within days of this encounter, I read an article from SEDL Insight titled, “Teaching Mathematics Conceptually“. While sitting and reading the article, I couldn’t help but think out loud. My thoughts can be categorized into 3 sections; “Aha!”, “Yeessss!” and “Amen!”

Aha!

  • To say you are “reducing” a fraction creates misconceptions, the fraction is not smaller. Use simplify instead. 
  • Mathematics is a precise discipline and teachers don’t realize the slightest deviation from true mathematic language can make the content incorrect. 
  • Exponents indicate how many times the base appears as a factor, not number of times a number is multiplied. 
  • -(-4) should be interpreted as the opposite of -4. 

Yeesss!

  • Algorithms and shortcuts are not bad, the key to using them to help rather than hinder understanding lies in the sequence of events. 
  • Relying only on algorithms and procedures and focusing on shortcuts result in teaching efficiency, not mathematics. 
  • As a result of naked math, teachers and students easily lose sight of the meaning of numbers and numerals. 

Amen!

  • Memorization of rules and mastery of computation are not the same as true knowledge of mathematical concepts and ideas. 
  • Shortcuts become the means to get answers with the unfortunate result of bypassing conceptual understanding.  This makes it difficult for students to understand more advance or complex mathematics. 
  • By not taking shortcuts and insisting on illustrating the full process (cra),  it’s possible to make explicit connections among concepts. 

CGI in Middle School 

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If I could do last year’s lessons on solving equations all over again I would begin with CGI. As a whole, 7th graders at my school struggled with solving equations last year. And as I’ve been in 8th grade classes and had conversations with 8th grade teachers I see the struggle continues. I’ve concluded that many students do not understand the real meaning of the equal sign.

When you ask students what the equal sign means the overwhelming answer is “equals”. When you ask if they can give another meaning the look of confusion covers their faces as they respond, “equal to?”

Starting with CGI, Cognitively Guided Instruction, helps students to make sense of a concept through making sense of a context. Through this process of thinking, students are able to conceptually understand a mathematical idea such as dividing fractions, addition and subtraction and even solving equations.

This past week I worked on a few CGI problems for middle school concepts. My plan is to do a couple of professional learning sessions where we can focus on this instructional strategy. 8th grade is my first group. An 8th grade teacher asked if Number Talks could be used to introduce systems of equations. What a perfect fit for CGI.

Here’s a problem I wrote. Students were instructed to only draw diagrams with labels to solve the problem.


Pictured above are two students beginning thoughts.


How two students served in a special education resource class thought about it. 
 More processing going on.



Students realized the answers needed to be 18, not more and not less. They had to use reasoning to determine how much of each fish was needed in order for it to be equivalent to 18. Within their explanations, students expressed they had to determine one unknown in order to determine the other unknown.  Imagine students being able to develop their understanding of systems of equations in this manner. Making the bridge to the abstract would make more sense.

Here are other CGI problems I have written and examples of more. If you have a source for more please let me know.

The Oath of a Math Geek

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My oath to my students:
I solemnly swear to take you through the CRA process of understanding.

I promise to allow you time to develop your understanding and not force you into the abstract world before you are ready.

I promise to never use tricks that do not conceptually make sense or rules without your understanding in order to move on to the next page, task, unit, concept or prepare you for a test.

I will do everything within my power to help you never ever have a story that begins, “I was never good at math”.