Representing Fractions with Thinking Blocks

Many classrooms in my school are in the midst of reviewing fraction concepts. Throughout the school students are finding fractional pieces, converting fractions to decimals, and identifying fractions on number lines.  For the past week students in second grade have been identifying fractional parts.  Earlier in the week students completed the page below during a math station.  Students did well on the first two pages, but struggled a bit when identifying fractions on a number line.

Representing 3/4 in many different ways

Representing 3/4 in different ways

This was a challenge for some students as many are more familiar with identifying fractions within objects (in a circle/rectangle).  Moving from identifying fraction to placing them on a number line can be a stretch.  Many students have already started to decompose numbers and have completed “fraction-of” problems.  These types of activities have helped reinforce the number line and fraction connection.  Next week students will be assessed on the fraction unit and many classrooms move into geometry concepts.  Before focusing in on geometry, I wanted to give student an opportunity to visualize fractions and use them with more complex word problems.

As I was looking for supplemental material I came across a Tweet by Paula (@plnaugle). She referenced Thinking Blocks  as a resource that she uses with an interactive whiteboard. I looked into the site and thought that it might be useful for my grades 2-3 classes since the app allows students the opportunity to solve fraction problems visually.  Specifically, I downloaded the fraction app on the school iPads.   Yesterday a second and third grade class used this app in their classroom as a guided activity.  The app was introduced to the class and I modeled the different steps involved in solving the problems.

Modeling

Modeling

The students were then asked to find a comfy place in the room and complete a minimum of three exercises.  What’s nice is that the problems are picked at random, so students aren’t on the same problem at the same time.  There’s also a feedback box that assists in guiding students towards labeling the correct parts of the fractions.

Click to enlarge

Click to enlarge

I helped the students as needed, but many were able to use the virtual manipulatives and generated feedback to stay on track.  Some students completed three problems, while some went beyond and tried out five.  After about 12 minutes the class gathered and we reflected on the perseverance that was needed and celebrated successes. This activity gave students an opportunity to make mistakes and persevere.  I’ll be keeping this app in my repertoire for the future.

Using Comics in Math Class

Comics

Comics/cartoons have been used to communicate important issues for many years.  Education has even been part of the comic movement.  Susan O’hanian has demonstrated with her website how cartoons can communicate and start important conversations.  I’ve tinkered with comics in my math classroom this year.

I believe that humor has a place in the classroom.   Comics can bring in a humor aspect, as well as practicality and motivation that can engage students.  My students are starting their algebra unit this week and I’ve been looking for new ways to introduce combining like terms and solving for unknown variables.  In the past I’ve used Hands-on-Equations and different types of narratives that explain how like terms can be combined.  After a lot of searching,  I ended up using pages 5 – 8 in this pdf to help introduce the concept this year.  Students responded well to the comic and I believe it helped them complete the activity in the document below.  I used one of the practice sheets as a model and the second sheet was completed and shared in student groups.

Combining Like Terms Activity

When I look back at some of my favorite K-12 teachers, many of them were able to connect and build rapport with students quickly and use humor appropriately.  I’m going to explore how to use  comics a bit more in my classroom over the next few weeks.  I may even have students create their own through comic creator apps on the iPad.

How do you use comics in the classroom?

Coffee and Mathematical Understanding

Coffee and Math Connection

Coffee and Math Connection

I spent the majority of this past week visiting with friends and family.  My destination ended up being in snowy and icy northern Michigan.   The house that I was staying in lost power for five straight days.  Thankfully the house had an efficient fireplace and a small gas-powered generator that ignited a few space heaters to keep one of the rooms fairly warm as outside temperatures hovered around 20 degrees.

In normal circumstances, the first person that wakes up in the house starts to brew the coffee for the family.  Since we had no power the coffee machine wasn’t an option.  You see, my family definitely enjoys their coffee. Being official coffee addicts my family has a decent understanding of the integral parts of the coffee-making process: hot water, filter, coffee grounds and cup. The one missing ingredient in this process was the hot water.  One of the family members found a pan and began to boil water on top of the fireplace.

Coffee

The water was then used to complete the coffee making process.  Success!  All of the family was able to sit around the fire and drink our coffee.

coffee maker

As you can imagine or already know, they’re many ways to make coffee.  My family knows this and that understanding led us to a solution that was adequate.  We substituted a different process in the coffee making flowchart and arrived at decent tasting coffee. Regardless of the process used, the user ended up with the same solution.  In the end some type of hot coffee was served.  Understanding the key components of any process allows opportunities to substitute yet arrive at the same solution. Because my family knew the process in-depth we we’re able to substitute the missing item and still have the same result.  I feel like this type of thinking applies to the classroom.

Having a limited understanding of place value and number sense can limit opportunities for students.  Students need to be exposed to an array of methods to complete problems, not just shortcuts.  Only understanding the formula/shortcut doesn’t necessarily show mastery of a particular concept.

At the upper elementary level students are expected to find the product of  3+ digit numbers.  If students have been exposed to using only the traditional method to solve these types of problems they know how to multiply large numbers using one method.  Although that process might be effective for them, it doesn’t cement a deep understanding of multiplication.  Students often have problems when decimals are introduced when finding the product of these types of problems.  Having a more in-depth understanding of place value and multiplication can give students the tools to solve more complex math problems.

On the other hand, if a student has been given opportunities to use repeated addition, partial-products, lattice and traditional methods, students might have a better understanding of the role that place value has in the multiplication process.  Having that understanding of place value will help students when they approach decimal computation and throughout their academic career.  Having multiple tools/strategies also encourages students to be independent and choose the correct method to find a solution.  Even more important, students that are then able to apply their mathematical understanding to practical situations (beyond the test) can often immediately see the benefits.

In the end my family used the tools that we had (fireplace, metal pot, water) along with other materials (coffee filter, ground coffee, cup) to create coffee that we all could enjoy.  If we were fully reliant on just the coffee maker and electrical power, we wouldn’t have had coffee to drink.  Understanding the details of the process and that there are multiple ways to find a solution is an important skill to have as adults and as students in the classroom.

The power came on and I believe we all found a newly acquired appreciation for the electrical grid in Michigan.  Our coffee story is unique and yet I feel as though it’s mathematically relevant as teachers will be back in the classroom to start the second half of the school year.  Enjoy the rest of 2013 and I look forward to a successful 2014.

Math: In Response to Your Question


I’ve been exploring the use of multiple solution problems in my math classes.  These types of problems often ask students to think critically and explain their mathematical processes thoroughly.  To be honest, these questions can be challenging for elementary students.  Most younger students expect or have been accustomed to finding one right answer throughout their academic career. Unfortunately, state and local standardized assessments often encourage this type of behavior through multiple choice questions.  This type of answer hunting can lead to limited explanations and more of a focus on only one mathematical strategy, therefore emphasizing test-taking strategies.  Encouraging students to hunt for only the answer often becomes a detriment to the learning process over time.  Moving beyond getting the one right answer should be encouraged and modeled.  Bruce Ferrington’s post on quality over quantity displays how the Japanese encourage multiple solutions and strategies to solve problems. This type of instruction seems to delve more into the problem solving properties of mathematics. Using this model, I decided to do something similar with my students.

I gave the following problem to the students:

How do you find the area of the octagon below?  Explain the steps and formulas that you used to solve the problem.

Octagon Problem

At first many students had questions.  The questions started out as procedural direction clarification and then started down the path of a) how much writing is required? b) how many points is this worth? c) how many steps are involved? d) Is there one right answer?  I eventually stopped the class and asked them to explain their method to find the area of the octagon, basically restating the question.  I also mentioned that they could use any of the formulas that we’ve discussed in class.  Still, more questions ensued.  Instead of answering their questions, I decided to propose a question back to them inorder to encourage independent mathematical thinking.  Here are a few of the Q and A’s that  took place:

SQ = Student Question         TA = Teacher Answer

SQ:  Where do I start?

TA:  What formulas have you learned that will help you in this problem?

SQ:  Do I need to solve for x?

TA:  Does the question ask for you to solve for x?

SQ:  Should I split up the octagon into different parts?

TA:  Do you think splitting up the octagon will help you?

SQ:  How do I know if the triangle is a right angle?

TA:  What have we learned about angle properties to help you answer that question?

Eventually, students began to think more about the mathematical process and less about finding an exact answer.  This evolution in problem solving was inspiring.  Students began to ask less questions and explain more of their thinking on paper.  At the end of the math session students were asked to present their answers.  It became apparent that there were multiple methods to solve the problem.  Even more important, students started to understand that their perseverance was contributing to their success.  The answer in itself was not the main goal, but the mathematical thinking was emphasized throughout the process.

Afterwards, students were asked to complete a math journal entry on how they felt about the activity.

Image Credit: Kreeti

Taking Math Outdoors

Math Outdoors

Recently I had an opportunity to attend an outdoor education trip with our elementary students. The trip took place over three days and was located in a very remote part of the state, away from high rises, city lights, cell phone signals, and televisions.  The trip focused on learning about birding, forest ecology, Native Americans, orienteering, and pioneering.  For many students this trip is a different learning experience.  It’s outside of the classroom and therefore a different learning environment for them. Acclimating to this environment took a bit of time for staff and students.

The adults were responsible to teach many of the concepts during hikes on campus.  Being outside is a great opportunity to introduce or highlight academic concepts that are generally taught through abstract means.  While talking about math outdoors, students expressed interest and asked questions that often led to additional mathematical questions.  Students that might not usually be fully engaged in a math lesson at school were shining on the hike. This experience led me to reflect on our current mathematical practices.  At times there’s a disconnect between what’s happening in the classroom and what’s occurring right outside of the doors to the school.  Teachers often attempt to bridge the gap, but self-directed student questions often come from real world experiences and curiosity.  Curiosity is often followed by questions.  Finding answers to those questions can lead students to find their passions (eg. #geniushour).  This motivation can be encouraged but not genuinely bought or sold.  Students decide how engaged they want to be and internal/intrinsic motivation often leads to learning experiences.

Below are some (of what I can remember) of the questions/topics that were discussed while on the trip:

Concepts

			

Math Debates in Elementary Classrooms

Learning through Conversations

Over the past few months I’ve dedicated a good amount of time to to having math conversations. These math conversations occur when the class is unsure of how to solve a problem or when disagreement ensues over what particular strategy should be used to tackle a problem.  The math conversations (or debates) allow students the freedom to openly discuss logical reasoning when solving particular problems.   These conversations can be sparked by the daily math objective or follow another student’s response to a question.  It’s not necessarily planned in my teacher planner as “math conversation” in yellow highlighter, but I do make time for these talks as I feel that they bring value and encourage student ownership.  The conversations also give insight to whether students grasp concepts and are able to articulate their responses accordingly.  Mathematical misconceptions can also be identified during this time.

During these conversations I have manipulatives, chart paper, whiteboards, iPads and computers nearby to assist in the discovery process.  I emphasize that there’s a certain protocol that’s used when we have these discussions.  Students are expected to be respectful and listen to the comments of their classmates.  To make sure the class is on task I decide to have a specific time limit dedicated to these math conversations.  Some days the conversation lasts 5 minutes, other days they may take upwards to 15-20 minutes.  When applicable, I might use an anchor chart to display the progress that we’ve made in answering the questions.  I should also mention that sometimes we don’t find an answer to the question.  Here are a few questions (from students) that have started math conversations this year:

  • Why is regrouping necessary? (2nd grade)
  • What can’t we divide by zero? (3rd grade)
  • Why are parentheses used in math? (3rd grade)
  • Why do we need a decimal point? (1st grade)
  • When do we need to round numbers? (2nd grade)
  • Why is a number to the negative exponent have 1 as the numerator? (5th grade)
  • Why do you have to balance an equation? (5th grade)
  • How does the partial products multiplication strategy work? (3rd grade)
  • Why do you inverse the second fraction when dividing fractions? (5th grade)
  • Why is area squared and volume cubed? (4th grade)

Above is just a sampling of a few of the math conversations that we’ve had.  Afterwards, students write in their journals about their experience finding the solution to the problem.

Of course this takes additional time in class, but I believe it’s time well spent.  The Common Core Standards  focus on depth of mathematical understanding, rather than breadth.  This allows opportunities to have these conversations that I feel are beneficial.  They also emphasize the standards of practice below.

  • CCSS.Math.Practice.MP1 – Making sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP3 – Construct viable arguments and critique the reasoning of others

Photo Credit:  Basketman


Do you have math conversations in your class?

Equivalent Fractions Tweak

Equivalent Fractions


A few days ago I started gathering resources to supplement a math unit on fractions.  The classroom was studying equivalent fractions and I thought there might be a variety of resources available on a few of the blogs that I regularly visit.  I generally follow the #mathchat hashtag  and find/share ideas that relate to mathematics.  While reading a few math blogs on fractions, I came across John Golden’s site that has some amazing ideas that can be used in math classroom.  His triangle pattern template sparked my interest.

Math Hombre

John provided a template that’s available on his site.  I printed out the template and began filling out each triangle with fractions.  I ended up with a sheet that looked like this.

Equivalent Fraction Puzzle

A template of this can be found here.

So what happened?

First a lot of brainstorming and error checking.  Then I decided to have students cut out the triangles and compile equivalent fractions.  This is what happened …

Students in fourth grade cut out each triangle and combined them to make equivalent fraction squares.  Students worked in collaborative pairs during the project.  I observed students using math vocabulary and having constructive conversations with each other to finish the assignment.

Before giving the assignment to a fifth grade class I decided to eliminate two triangles on the sheet above.  It was the job of the student to find what triangles were missing and create equivalent fractions to complete the squares.  The students were engaged in this activity from start to finish.  Some students even wrote the equivalent decimal next to each square.

photo 1

Overall this project took approximately 45 minutes to complete and it was worth every minute.  Students used the terms fraction, improper fraction, mixed number, numerator, denominator, multiplication, division, and pattern throughout the project.

Just as I did, feel free to tweak this project to best meet the needs of your students.

Teaching Algebra Through a Different Lens

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I recently taught a lesson on pan-balance equations.  In my curriculum pan-balances are taught as a precursor to more in-depth pre-algebra.   My students seemed to understand simple pan-balances and found that the balances (like an equation) needs to be balanced to work.  The majority of students had no problem with questions (like these) involving oranges, apples, paperclips, etc.

Day Two

During the next lesson I introduced the idea of variables with equations, like 2x + 4 = 18. Students seemed to understand, but less than the first lesson.  I brought everyone up to the classroom whiteboard and practiced many problems with the students.  The students who understood always seemed to raise their hands, while students who didn’t completely grasp the concepts tried to blend in with the carpet.  Students became less interested in what I was teaching when I started writing equations on the whiteboard.  Unfortunately, I felt like I was losing a battle here.  The more the students seemed to not understand, the more I felt the need for direct instruction.  Near the end of the lesson around half of the students seemed confident to proceed to the next algebra lesson – solving for x on both sides of the equation.

I had to change something.

Day Three

During the next day I decided to change up my instructional approach.  I remembered back to when I first learned algebra and the confusion that I used to experience.  My school memories of algebra started and ended by watching a chalkboard and overhead projector, as my teacher wrote and erased equations on the board.  This was the only way to learn algebra, or so I thought back then.  Using my experience  I decided to change the instructional medium.

I started my next algebra lesson with a quick review of pan-balances.  Students seemed to gain confidence as we had a conversation about the importance of using algebra in careers outside of the classroom.  We watched a quick BrainPop video on algebra and it’s uses.  Instead of using the whiteboard again, I decided to take out the iPads.  I already downloaded an app called Hands-On Equations a few weeks ago.  The students were quickly motivated and I modeled how to use the app under the document camera.

Hands-on-Equations

The class and I went through a few problems together until I thought they were ready to proceed.  I allowed the students 20 minutes to explore the app and lessons. The students were expected to complete at least three lessons and reflect on their experiences in their math journal.   What was interesting was that the students immediately took control of their own learning and utilized the app at their own pace.

Payoff

After approximately 20 minutes, I asked the students to write in their journal how they felt about their journey with pre-algebra.  The majority of responses were positive …

“I now understand why we use algebra”

“I never thought algebra could be so much fun”

“Having a picture of the balances helps me understand the concepts better.”

The Takeaway …

After hearing their responses and reflecting on the outcomes, I’m becoming more motivated to vary instruction to better meet the needs of my students.  Varying the instructional approach can give students multiple opportunities to grasp concepts that can be particularly challenging.  Your students may benefit from a bit of instructional change from time to time.


photo credit: ajaxofsalamis via photopin cc

Dice and Math Computation

Dice and Math


Since the beginning of the school year I’ve been searching for different ways to incorporate guided math in my classroom. Guided math has many benefits although organizing the groupings can bring a few challenges.  Guided math looks different depending on how the teacher implements the structure.  For example, one math group might be working with the teacher while two other groups are using math games or participating in problem based learning activities.  The groups will rotate according to a specific time schedule.  I’m finding that groups that are not with the teacher need specific instructions and expectations.

For the past few months I’ve been using dice games to emphasize number sense skills.  These dice games have peaked student interest and work well in increasing computation fluency.  I decided to collect multiple formative data pieces to validate whether the dice games were contributing to student success. By analyzing student data and observing over a period of time, I found that students were  becoming more fluent in adding, subtracting, and multiplying small/large numbers.

The games have worked for me, so I’m passing it along to others that might find it useful.  Needed materials and pdf files are below.

Materials

A variety of dice (6, 10, 20, 30, etc. dice)  Here are some examples:

photo 5

Click to Enlarge

Templates (in pdf form)

Roll to 150 (multiplication)

Roll to 125 (addition)

Roll to 100 (addition)

Roll to 45 (addition)

Roll from 50 (subtraction)

Roll from 95 (subtraction)

Roll from 35 (subtraction)

Student Data and Balance

Data and Balance

Data and Balance


Teachers in K-12 often use student data on a regular basis.  Student achievement data can be used to qualify students for reading, gifted, remedial, enrichment, acceleration, differentiation, and a variety of other services.  Recently, standardized testing data has been the forefront of educational trends and in the news.  Implementing a  balanced approach when looking at student data can keep stakeholders (educators and administrators) grounded in an understanding that the numbers behind the tests may give light to areas of strengths/needs.

Data isn’t evil

Assessing a student’s understanding of a specific concept isn’t necessarily a bad thing.  In fact, over the past few years I’ve grown to appreciate and utilize student achievement data more and more.  Whether the data is from a standardized test or not, the data can be helpful if used correctly. Moving data beyond just a number can benefit teachers and students.  Data can help teachers ask better questions and provide opportunities to reflect on how students learn best. Involving students in analyzing their own data can encourage student goal setting and ownership.

Having conversations with students about their data is powerful.

Have the conversation

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I’m definitely not an advocate for having additional standardized tests, although some seem more useful than others.  I find that assessments that give detailed feedback (e.g. areas that need strengthening, %ile compared to the norm, strength areas, next instructional steps, etc.) are more frequently used by teachers, compared to assessments the give little feedback.  Obviously, there isn’t a perfect test available for school purchase.  The assessments that a school uses should give detailed feedback that can be immediately used.

Do you hear a lot of negative talk in regard to standardized assessments?  Having a conversation about an assessment’s effectiveness in informing instruction may be needed. Instead of trash talking the assessments in general, educators and administrators should find assessments that work for them.  PLC teams should emphasize the importance of using formative assessments regularly.  I’ve found that teacher created formative assessments are some of the best ways to find areas that need strengthening and to identify differentiation opportunities.  The purpose of giving the assessments should be communicated to all stakeholders.  When teachers understand why the tests are given, (not just for VAM reasons), they may start to value the benefits of assessing students using a variety of tools (such as Common Core performance assessments).

Balance is needed

With teaching and in life, balance is needed.  Teaching is a profession that can be stressfull.  It has many teachers thinking right now, how many days till Spring Break??   Balancing assessments with instruction takes skill and patience.  Standardized tests are often at the forefront of school administrator’s minds.  One test shouldn’t be used to determine if success, or enough growth has been made to call that school year/class/school successful. Take a breath and look at assessments from a macro lens. A combination of formative, informal, formal, review checkpoints, activators, performance  (insert your assessment here), and even standardized assessments have their place in a school and can be beneficial to a certain extent.  The value of the data often depends on how it’s utilized.

Picture Credit: DigitalArt


“Your assumptions are your windows on the world. Scrub them off every once in a while, or the light won’t come in.” Isaac Asimov