Math: Justify Your Answer

We'll use this problem first this morning. Then we'll use PARCC sample problems to lead our study.

MCAS tests led us to create a number of graphic organizers to help students learn and practice the skill of explaining their thinking and answering questions with depth and precision.

PARCC presents a somewhat new type of question and perhaps new expectations for best answers with regard to math problem base assessments.

I looked at the online information again to get a deeper idea of expectations.

How do we best help students explain what they know and how they solved a problem using concise mathematical language?

Throughout the year I've been working on this with some success, but today I'll dig deeper using the information we learned from the SRSD approach to writing persuasive essays.

As we work together in class on PARCC sample problems and others, we'll use this template to guide our share.

During the share, we'll discuss how we can concisely explain process and knowledge as we justify answers using mathematical language, numbers, and models. We'll also recognize that the transition words can be bullets, letters, or the words "first, second, third. . ."

As test taking efforts and expectations evolve and our efforts to coach students with regard to explaining their thinking and justifying answers deepens, I'm sure the graphic organizers we find and use will change and grow. I know the aim of this work is to deepen students' ability to understand, manipulate, express, and share their mathematical thinking, and that's a good aim for later math knowledge, skill, concept, and success.

Afterward
One issue that arises again and again with this kind of teaching is pacing. Some quickly grasp the content while others struggle. Differentiation works, but time is always a challenging factor given the depth and breadth of concepts to cover. This will be a challenge I continue to think about as we reach for deeper, richer math instruction and learning.

Flexibility with Fraction Computation

The young boy said, "It's hard to remember what to do when you add and when you multiply fractions."

"I know," I responded, "your brain has to keep switching. They call that flexibility."

When I teach I think about the brain and how it reacts to information. Earlier in the year students wrote essays that examined the similarities and differences when calculating with whole numbers. Today they'll make models and write explanations related to fraction computation. I believe that the time to think deeply about each fraction operation process will help students to differentiate the meaning and process.

Students will use the learning menu to guide their work in the computer lab.

I'm looking forward to coaching students as they dig in deep to learn about fraction computation. Let's see what they come up with.

Don't Underline Key Words? Why Not?

Many math education experts say, "Don't tell students to underline key words!"

I disagree.

Those math experts think of key words as words that determine a specific algorithm or thinking about a problem. I define key words differently. I think of key words as any important words in a problem or text.

As one who gets quickly swayed and confused by too much text, it's so helpful for me to read with a pencil underlining as I go along. I underline the words that help me solve a problem or decipher the text for deeper meaning.

The simple act of underlining seems to send a message to the brain that says, "This is important--don't forget it!" Recent research on hand-brain connection seems to affirm this.

Another good reason for underlining is that it slows you down and makes you focus on the text.

Once I started using my pen to read well, my reading speed and comprehension increased tenfold. Not only do I underline key words but I follow this common process:

• Focus on the question or problem. Understand what the job is.
• Read with care; underline and write notes that pertain to the job, goal, problem, or solution.
• Read over and organize the notes to inform your solution.
• Check over your work.

Hence, I disagree with those who tell students not to underline key words--underlining slows you down and helps you to find your way through complex text which better prepares you for finding the desired solution and/or information.  That's been my experience. 

Let me know if and why you disagree. 

More PARCC Practice

We've been using the PARCC practice test to practice for the upcoming math test.

Problem by problem we've been analyzing questions, making models, and practicing responses.

The explicit teaching is a bit dull, but productive, as we navigate each problem, trying out online tools, and thinking about ways students can use the scrap paper to organize their thinking and find answers. I continue to favor the depth the new common core standards create for learning at the fifth grade level, and I know that the teaching related to the tests will improve as we incorporate successful, blended units of study for each standards-base content area.

The practice problems vary in format, needed thinking, and math skill, knowledge, and concept.

If our State continues to use PARCC next year, I'll think about how we can employ more of these question types into our typical math units and formative assessments. I did a bit of that this year, but not enough as the problems were new.

It's also likely that some of these question types will be short-lived as some of the practice questions seem ill-fitted for young minds and mathematical understanding.  Time will tell in this regard.

The PARCC journey continues.

Next week we'll begin the tests. I'm not allowed to look at the questions or report on that process, but hopefully, sooner than later in the year, I'll be able to analyze the published results and look carefully at the published questions asked to help me prepare for next year's teaching/learning expectations and goals.

Math Path: March/April 2015

It's time to re-look at the math path as we enter the last three months of the year.  What's important?

PARCC Tests
No matter how one feels about standardized tests, the tests are here and students are expected to be prepared for the assessments. In two weeks students will take the PARCC Performance Based Assessment (PBA) tests and then in May they'll take the End of Year (EOY) PARCC test. That's four PARCC math tests.

The preparation will include the following:

• Taking the official math practice test.
• Reviewing the questions and format of the online practice test with students.
• Deepening students' understanding of the end-of-year standards such as fraction computation, line graphs, and polygon properties.
• Reviewing all content taught this year with a large number of online and offline learning activities.

Systemwide Math Assessment

Our fifth grade students also take a system-wide assessment that reviews the year's learning. To prepare students for doing their best on this test, we'll do a lot of practice and review of past concepts, skills, and knowledge.

Computation Proficiency

Both the system-wide test and PARCC require accuracy and precision with large number computation. The only way to assure students' skill in that area is to provide steady practice so we'll continue to practice large number computation in a number of ways.

Math Depth and Enjoyment

All this test prep and standards-base teaching has to be engaging in order to reach for depth and investment. Hence I'll look for ways to employ the standards of mathematical practice (SMPs) to reach for that depth and engagement in collaborative, empowering ways.

Specific Roll-Out

Core/Enrichment RTI Groups

RTI Tuesdays: Computation Practice and Order of Operations

RTI Wednesdays: Math TED Wednesdays

Test Prep

• Review of PARCC online practice test. 3/18-3/19

Content Depth

Review concept with PBA, discuss, branch out to practice with meaningful problems, practice. 

• Review of prime factorization. 3/23
• Review of simplifying fractions. 3/25
• Review of finding the common denominator. 3/30
• Review of adding/subtracting fractions using fraction bars, model making, computation. 3/30
• Review of area model, fraction multiplication and division. 3/31
• Review of improper, proper, and mixed fractions. 4/2
• Problem solving with fractions. 4/6
• Mixed fraction computation. 4/8
• Line plots with fractions 4/9
• Polygon review 4/13

Content Review

• Review of all content covered with home study and in-class review activities, online practice, and games. 4/20, 4/27, 4/10, 4/14

Marble Maze Days: 4/15-4/16 for Project Completion

PARCC Practice Test Reflections

Today I gave my twenty-two fifth graders the PARCC practice test.  We were all a bit anxious because we had not given or taken the test before.

Fortunately our school administration put in a lot of time upfront to make sure the technology was all set, test support was in place, teacher trainings were held, and multiple informal conversations at PLC also occurred.

Technology
Our computers worked well for the most part. There was one minor glitch on my computer which the tech staff is looking into, and there were a few steps in the process I had to learn better for test delivery ease and accuracy. Students had no problem logging in and getting started.

Test Support Materials and Classroom Set-Up
I stretched out the desks quite a bit, but I want to stretch them out even more to give students ample space to work and to give me ample space to get around and troubleshoot with any computer glitches that may occur.

I xeroxed and stapled a packet of scrap paper including graph paper, lined paper, and plain paper. That served the students well for the most part, however, a few students would have profited from some large square paper.

I want students to have their books ready to read in case they finish the test early, and I want to work on the timing aspect of the test with regard to everyone starting at the same time. I think I'll clip their test tickets to their scrap paper packet to support that.

Also, laptops need to be charged, pencils sharpened, headphones ready, and the classroom quiet and clean for best work.

Content Issues
There were some questions that many students had. Since it was the practice test, I could talk with them about content issues.

One issue was a coordinate grid where they had to infer what the x and y categories were. Many had trouble making that inference as they are used to using graphs where the axes are labeled.

Other issues related to whether the answer had to be simplified and do you consider a mixed number a fraction. I looked that up and found that a mixed number can be considered a mixed fraction. The question called for a fraction answer so that was confusing.  What do you think?

A word that confused students was "combined" as I haven't used that much.

Students also need more rehearsal with fraction computation, the area model, and volume.

Time
I gave students more time to complete the practice test than the test allows, yet only a few finished. I anticipate frustration by several students with regard to the time factor as many won't be able to finish. It takes young children a lot of time to calculate, read, problem solve, and check their work. Although they've had a lot of math this year, I still don't think they're ready to complete tests at this length with the time allotted, but perhaps I'll be surprised. Let's see.

Preparation for the Tests
The tests take up eight mornings. To prepare well I'll have to come in early to make sure the computers are ready, the room set up, and the materials at-hand.

I'll need to have sharpened pencils, extra scrap paper packets, large square scrap, erasers, and a few extra headphones on hand.

We'll also have to make extra sure that we clean up the room, take down all content posters, and relay a "do your best" strategy.

Challenge
With eight PARCC mornings the challenge is going to be to keep students' self confidence strong. These tests are not fun for many students, and for some the tests present a very difficult challenge. The other challenge is going to be the challenge of keeping the regular, learning program going with so many interruptions in the schedule due to shared teaching/tool resources and the fact that our usual routine of learning will be interrupted by lost time, time for the tests.

I'll meet that challenge in ways I've mentioned previously which are to complement the serious, quiet test taking with hands-on, engaging activities such as the marble maze simple machine project. My colleague who teaches ELA is doing the same thing by employing a student-centered, heart-ful ELA project.

It helps to take the practice test and assess student/teacher needs right afterwards so that teachers can prepare students well for the tests, and so that students can do the best possible on these tests.  If you have other reflections related to these tests and student confidence and success, please share.

TED Wednesdays: Math RTI

Our Wednesday math RTI time resources have been altered due to PARCC tests and a couple of other scheduling issues.

That leaves me with about 36 fifth graders to teach in an approximately 20-minute period without our usual access to technology.

What's a teacher to do?

I decided to make those Wednesday TED Wednesdays. On TED Wednesdays we'll watch a TED talk, and explore the mathematical idea relayed during that talk with discussion and hands-on activities.

Today we started with making origami cats and dogs and then we watched the Lang talk below.  Next week we'll focus on mathematical thinking, and then after that we'll examine math puzzles.

I had heard a long time ago that Gates favored a type of professional development for educators where experts educate many through the use of video and other online vehicles. Last week when I attended a Mahesh Sharma's math workshop, I thought that he'd be a good candidate for that kind of professional development. In a similar fashion, sometimes a TED Talk is a terrific vehicle for students learning.

There's never one way or medium for all share and growth, but online learning is definitely a positive path for some of the teaching/learning we want to do.

PARCC: Practice Test #1

Today, students in one of my math classes will take the official PARCC practice test.

My colleague gave the ELA practice test yesterday so the students and I will profit from that experience.

First, I'll organize the learning environment and materials in the following ways:

• spread out desks in two long rows so I can easily walk up and down the rows to troubleshoot when necessary.
• separate desks to give students room and prevent students from looking at each others' work.
• take down math posters.
• I'll make each child a packet of scrap paper that they can use to do their figuring on. They'll have a similar packet during the actual test. 

Next, I'll give students a pep talk. During the pep talk, I'll add the following comments:

• The goal of the test is to show off what you know and do your best.
• The goal of the practice test is become familiar with the test routine, tools, and structure.
• As you take the test, write down strategies that you think are helpful with regard to completing this test successfully. We'll share those strategies to help each other.
• During the practice test, you may ask any questions that you'd like, but during the actual test you may only ask questions related to the tools or process, but not the content--so ask away today.
• Remember this is the first year for the official test, and it's likely that there may be problems, so don't worry about that.
• Does anyone have any questions or comments?

After that, students will sign on. I'll read related directions and they'll get started. 

We may not have enough time to finish the test due to time constraints, but at least students will get a chance to become familiar with the tools and structure of the eight PARCC tests to come, five during March/April and three more in May.  

If you have any suggestions with regard to this process, please share. As their teacher, my aim is to support students when it comes to showing off their knowledge, and doing their best. 

Reflection: A Pause in the Math Teaching/Learning Path

Standardized tests offer us one window to student learning needs and
success. The key is to streamline those tests so that there is time for
worthy, interactive, rich learning endeavor which builds a love of learning
and depth of understanding and application. 

Yesterday our team received a collection of positive standardized test scores related to students' math knowledge, concept, and skill.

The scores which were very positive prompted reflection with regard to next steps in the math program. Our efforts have been successful and we want to continue down that teaching/learning path.

I analyzed the scores first and acknowledged that we had made some good decisions as a team with respect to Response To Intervention (RTI) grouping/focus and math interventions. I also realized that the depth of the new common core standards had impacted students' efforts since throughout the math teaching year we've stayed focused on those standards. Finally, I believe the blended approach to math teaching and learning, frequent formative assessment/reflection, and varied learning menus supported students' growth.

The scores also moved me to think deeply about the math days to come which take on a bit of a staccato pattern interrupted by testing days and special events. As I thought ahead, I decided that we'll meld explicit teaching with lots of review and practice.

The explicit teaching will focus on building depth. To do that I'll follow the lead of two experts I recently learned from, Michael LaFosse, the origami expert, and Mahesh Sharma, a math education expert. LaFosse and Sharma, experts with significant experience related to math and education, focused on precise language, concrete models, hard work, attention to detail, educators' foundation knowledge, and lots question-and-answer student-teacher interaction.

For practice and review we'll use That Quiz, a simple, straightforward math platform that lends itself to independent study menus and student practice and review.

Our goals include the following:

• Strengthening student ability to apply concept, skill, and knowledge to solve multi-step math problems in efficient, explicit ways.
• Review and deepening of all math concepts, skills, and knowledge taught.
• Review of PARCC test strategy, process, and math content.
• Re-look at math RTI groups and set the course for the next leg of the RTI efforts.
• Completion of Khan Academy math for grades 3,4, and 5. Completion of grade 6 for advanced students. 

The success of a math program depends on the following:

• Educators' deep knowledge and skill related to math content, concept, and skill. 
• A blended teaching/learning program that includes a multi-faceted platform of tools and resources.
• The use of concrete and visual models. 
• Lots of time for math talk and discussion.
• Regular formative assessment, review, reflection, and revision. 
• Time on task with students.
• Differentiation as one way to meet multiple student needs. 

This weekend I'm sure I'll have more to add to this post as I listen to two more math experts, Keith Devlin and Diane Briars, at the Teaching and Learning 2015 Conference. 

It's an exciting time to be a math teacher thanks to the wonderful tools and expertise available to lead the way. If you have any thoughts on this post, please write a comment. I look forward to learning more. 

Origami Artist: Michael LaFosse

Students and teachers were mesmerized today as Michael LaFosse taught math with origami. I recommend.

LaFosse's work has invigorated lots of discussion and follow-up math and origami at the school. He led us to watch the Lang Ted Talk below about origami and math.

He also invigorated our fraction bar work and exploration.

His visit to our school has been a catalyst for deep and meaningful learning.

Quality Work and Reflection

At the end of each unit of study, students should have the time to reflect, create, and share an illustration of the learning experience. The illustration could be a photo, drawing, writing piece, sculpture, film, comic strip or any display of their learning that is meaningful and valuable to the student.

Quality work and share like this serves to build students' investment in learning and metacognitive skills with regard to their learning journey and personal interests, needs, goals, and vision.

Too often we spend time on learning items that don't matter rather than building in enough time to create and complete quality work that does matter.

In thinking about this with regard to units we're working on now, students will create the following quality reflection and share:

Marble Maze Machines: photos and written reflections that will be placed in their portfolios.

Fractions: posters and an end unit creative project that can be shared with younger children as well as placed in their online or offline portfolios.

Science films and study: a collection of wonderfully illustrated and labeled diagrams that include written descriptions and reflections.

Math Review: Site bites and sound bites that helps a child review, share, and deepen concept memory and understanding.

Portfolio Days: Time set aside to update the portfolios and add to the reflections in those special books. Also, time for the creation of online portfolios as well.

RICE: Math Problem Solving Focus

Today we'll step back to focus on strategy more than content.

Using a number of fairly easy to solve math word problems, we'll use the acronym RICE to lead our study.

I'll first model the use of RICE with a focus on careful reading, visualization, strategy choice, calculation, and communication.

Then I'll ask students to take a risk and come up to teach the class the following problems.

In the afternoon, students will take the final part of a system-wide assessment that's focused on math problem solving.

As they take the test, I'll walk around to see who applies the strategies we reviewed.

Computation Practice and Test Taking

Today students will practice computation skill.

Prior to a test there's always that question related to how much practice is good, and what level of practice overdoes it.

I'll give students a chance to practice quite a bit this morning to warm them up for this afternoon's computation test.

Just before the test, I'll share the following test strategies--strategies that match this test:

• Take your time.
• Copy the figures correctly and neatly.
• Calculate accurately.
• Use graph paper if it makes it easier for you to line up the numbers and complete the problem correctly. 
• Check your work with the inverse operation and/or in other ways.
• Bubble accurately.
• Keep track of what problem you're on.
• If you don't know an answer, circle and move ahead as later questions may help you to remember how to answer that question.
• When done check over your bubbles, close the booklet, and quietly read.

In all, the most important directions for students are "Do Your Best" and "Show Off What You Know" as the test results will help us to teach you well.  Onward. 

Fraction Framework: Outstanding Presentation Snapshot

Mahesh Sharma provided a meaningful framework for fraction instruction today at a one-day conference. For years I've heard about Sharma's expertise with regard to math and math education, and today I was finally able to experience this gifted and knowledgeable educator's presentation. I recommend that every math teacher and program leader seek out Sharma's expertise if possible.

Now, after a great day of learning, the question is how will I apply all that I learned. I'll begin by stepping back a bit with my students and reviewing many fraction concepts I've covered, but this time I'll use the depth that Sharma modeled today.

General Teaching Tips

I'll apply the following teaching tips:

• Be mindful of the message of the nonverbal communication you use during math class.
• Encourage students to write it down for better retention.
• Don't let the class be "hijacked" by the four top performers.
• Let students know that math is hard work and it does take perseverance to learn it well.

Review Multiplication, Division, and Decimal Computation

We'll review the fact that multiplication can be thought of as arrays, equal groups, area and repeated addition. We'll also review that division is repeated subtraction, partitioning, arrays, and related to the dimensions of the area of a rectangle. Also show the meaning behind the decimal point with regard to aligning the decimal points while adding and subtracting and counting the places for decimal multiplication.

Review Prime Factorization, Lowest Common Multiple and Greatest Common Factor

• Lead students to the fact that all terminating decimals have prime factors of 2 and 5.
• Use the double method of finding prime factors.
  

Review Symmetry

Naming Fractions
With a focus on using his language, concept, procedure roll-out of content, we'll first revisit benchmark fractions by drawing bar and number line models of those fractions and naming the fractions in multiple ways. For example 3/4 can be named:

• 3/4
• three fourths (1/4 + 1/4 + 1/4 or 3(1/4)
• three fourth
• 75%
• .75 or 75/100
• 3 divided by 4 or 3/4
• 3 X 1/4 = 1/4 X 3
• 3 groups of 1/4
• 3:4 the ratio of 3 to 4
• three fourths of the whole

Make Fraction Bar Models By Folding Paper

Make a large number of one inch strips of paper. Pass out eight pieces to each student. Begin with a focus on one whole, then move onto halves, thirds, fourths, sixths, eighths, and tenths. Each time you focus on the paper, ask the following questions:

• What am I putting through the fraction "machine?" (one whole)
• What machine am I using? (halves, thirds, fourths. . . .)
• How many parts did I make?
• Are the parts equal?
• What is the name of each part?
• How many equal parts make a whole?
• What is the new name?

Develop Fraction Concepts

• Name fractions
• Locate fractions
• Relate fractions to multiplication and division, use variables.
• Compare the relationships within and amongst fractions.
• Prompt students to observe patterns, conjecture, and summarize.

Develop Concept and Procedural Knowledge of Fractions

• Apply congruency as you develop a concept with the concrete, pictorial and abstract. Use a multi-sensory approach.
• Manifest fractions:
• parts to whole
• comparing quantities: ration, scale
• comparison of quantities with a standard such as decimals or percent.
• comparison of comparison: proportion
• rational numbers.
• Procedure
• intuitive: mix, combine with conceptual schema
• concrete: exact number, efficient, elegant, generalize
• pictorial representation with numbers and models
• application: intra, interdisciplinary, extra curricular
• communication/teaching: mastery

Fraction Computation

• To add or subtract fractions they must have a "common experience" or common denominator.
• Study fraction size, denominators, relationships, different "experiences" or behaviors.
• Determine:
• What is the whole under discussion.
• How many parts is the whole divided into?
• Are the parts equal?
• Name each part?
• What is the new name?
• Use area models with cuisenaire rods, base ten blocks, folding paper, and pictoral models.

I highly recommend Mahesh Sharma's math workshops for your math teaching development. He focuses on deep understanding and mathematical thinking. The framework and pedagogy he presents helps one to meet the standards with depth and meaning. It was a great day of learning, one I look forward to sharing with my students. 

Math Workshop: Fraction Compare Posters

One page of one student's project.

We're coming to the end of the fraction compare poster project. It's been more arduous for students than I expected, but once we passed the initial struggle, student buzz began as children uncovered one tool or strategy after another to support each others' work and inform the project.

A first cut of any new project creates some discomfort and a lot of revision, but always at the end, I recognize the worth of the group share, edit, and revision.

The project has prompted students to think deeply about fractions. Today, I'll edit students work and send them back to make edits, enrich, and complete the posters. We'll hang the posters outside of the classroom to inform our future work and teach others too. I congratulate the students for their willingness to struggle with the project since it's that "just right" struggle that creates learning.

PARCC Performance Assessment Practice

During March, I'm going to focus the math study on PARCC Performance Assessment Practice.  I'll use the PARCC Practice test as a jumping off place for this practice, and craft learning experiences for students that mirror the test questions.

Is this teaching to the test? Perhaps, however for students to perform well on these tests, they need a lot of practice with process, attention to detail, model making, checking their work, reading carefully, and more. The multi-step nature of the problems require steady attention and focus. Plus the fact that the tests are online, changes the way they'll need to approach each problem.

So what pattern of study will I present to prepare students for the tests.

Problem Presentation
First, I'll present a problem on the white board, and ask students to solve the problem on their own giving them about five minutes time.

Dissect the Problem, Model, and Solve Together
Next, we'll dissect the problem together by highlighting key words, making models, using number lines and other tools, and checking our work. We'll discuss how to set up the work space for multi-step problems so the data doesn't get mixed up or lost. We'll also discuss ways that we can input data so that we're inputting the correct figures and/or other information.

Practice
After that I'll give students a number of similar problems online or offline to practice with a partner, small group, or teacher.

Assess
Then, about once a week, I'll give students an independent assessment to assess their skill with performance based tasks.

As much as possible I'll try to craft assessments that have interesting, meaningful data and problems. I'll use Tuvalabs.com, in part, for that data. I'll also match problems to other events and content we're focused on at the time. In fact, I hope I can weave in the science standards for the MCAS test into this daily practice.

What's positive about this approach?

• Students will have the chance to practice their "attention to detail" skill with regard to math problem solving.
• Students will practice math skill and content that they've already studied in various ways.
• Students will be well prepared for the tests which will alleviate stress and worry.
• Students will continue to build a strong math foundation and see math connected to real world, relevant, and meaningful events and problems. 

In the meantime, we'll continue to study the remaining standards of fractions, volume, line plots, and polygons as well as continued practice of computation skill and order of operations with whole numbers and fractions.

There's a lot of a catch-up in math since standards have changed for these fifth graders--lots to learn, and we'll approach this learning with a day-to-day practice and study approach.

Comparing Fractions: Lesson Language

Today students will study how to compare fractions with greater skill and depth.

First, I'll introduce the process with the lesson language below.

Then students will apply that process as they work on their fraction compare posters.

Later students and I will review the concept using this worksheet.

During the explicit teaching time, students will be seated at desks with notebooks. I will be at the front of the room leading the learning activity. My student teacher will walk around helping students follow the lesson.

Then during the project time, students will spread out in the room and next-door computer lab to continue working on their compare posters. At that time I'll coach individual and small groups of students as needed.

Take a look at the language below. Is this how you would teach this concept? How would you change the language or progression? What would you add?

Tomorrow, I'll work backwards a bit, and explore the concept and skill of simplifying fractions.

Learning Language

1. Let’s compare ⅞ and ⅚.

Take a few  minutes to draw and compare these fractions in your notebooks.

2. Let’s examine these fractions?

What do we know about these fractions?

• names?
• simplest form?
• closer to 0 or 1? How do you know?
• where do they fall on the number line?
• how far away from ½?
• what do they look like as a bar model?

3. Let’s compare the fractions using a number line.

Some fractions are easy to imagine and compare. We can easily place ½ and ¾ on a number line. ½ is less than ¾. Where do you think the fractions above belong on the number line?

4. Finding a common denominator.

• When fractions are not easy to compare, we can find a common denominator to compare those fractions. A common denominator is in a sense “splitting each fraction amount into the same number of parts.”
• A common denominator is a multiple that is common to each denominator.
• What is a common multiple for 6 and 8? 
• We can multiply the numbers by each other to find a common multiple. So 6 X 8 = 48, and that would be a common multiple, but that’s a big number and more difficult to work with. 
• Is there a multiple of less value that is a common multiple of 6 and 8? This number would be the Least Common Multiple (LCM) or Lowest Common Denominator (LCD)
• We can count up by the largest number and stop when we reach a number that is also a multiple of 6 like this 8, 16, 24! 
• 24 is a multiple of 8 and 6.
• We can change our images to fractions with 24 parts. That would look like this.

Even with a good tool, it’s difficult to work with only models when converting fractions to fractions with common denominators. There’s an easier numerical way to do this.

We know that when we multiply or divide any number by one, we get an equal number. The same is true for fractions when we multiply or divide any fraction by a fraction that equals 1 (1/1, 2/2/, 3/3, 4/4. . .), we will get an equal fraction.

Now with common denominators the fractions are easy to compare, add, and subtract.

What is the difference between these fractions?

If we add these fractions together, what is the total?

Teaching Math: Should We Personify Numbers?

When Professor Keith Devlin discussed the "behavior" of numbers during the Stanford Mathematical Thinking MOOC, it changed the way I thought about math and numbers.

When I used the notion of numerical "behavior" in class, I noticed that students began to think about numbers differently. They grasped the deeper understanding of operations and number relationships.

So now, at the start of the fraction unit, I'm going to personify numbers to help students understand fractions with greater depth. We'll essentially create benchmark fraction profiles like the one below? I can imagine that later students could turn these profiles into animations, short movies, skits, or stories to extend the learning.

Do you personify numbers when you teach math? If you do, what value does that process have? If you don't think this is a good idea, tell me why?

Sight Bites: Teaching Fractions

One reason I like fractions so much is that the study lends itself to lots of artistic creation as you problem solve and create models.

Currently students are creating fraction bars with Google table--it's a great way to start the study.

Later I'll have students make "sight bites" like the one I crafted below to demonstrate the "behavior" of the operations when it comes to fractions. They'll likely marry their "sight bites" to meaningful numbers, problems, and data.

There are so many avenues to travel when it comes to math learning today. I'm looking forward to this next chapter of the math study year with my students, a fraction action unit. Onward.

Fifth Grade Fraction Learning Path

Example of Granite School District Math Vocabulary Cards

Students will begin the fraction unit this week.

I created a draft of the learning path today on the class math website. I'll continue to work on the path this week and share with families and students on Friday.

In the meantime, students will make the bridge between the computation and decimal units to fractions with the following activities.

I want to give a shout-out to the Granite School District who has graciously shared their terrific math vocabulary cards with math teachers throughout the globe--a terrific resource.

Let me know if you have any great additions for the learning path ahead. In the meantime, feel free to use any of the resources listed.