Decomposing Fractions (2 Part Video Blog)

Cela ne veut pas difficile du tout!

Unless you can speak French, I'm sure you're wondering what I just said, right?

Well, there's a 4th grade fraction TEK (4.3A) that reads "represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b" and many of the teachers I've had conversations with have asked "what does that mean, exactly?"

Well, let's look at not only what that TEK means, but how to address it with our students!

Check out this Part 1 and Part 2 video on Decomposing Fractions.

By the way, the French statement above reads, "This isn't hard at all!" and ...it isn't! You got this!

New Year, New TEKS, New YOU!

Happy New Year!!!

Well, in the next 5 hours or so, we will be bringing in 2016 but for the past two years we have been tussling with these New TEKS!

Thought I'd share this firecracker, poppin' way to approach teaching 'determining the measure of an unknown angle...'. Okay, it might not be that big a deal, but 'tis the season for embracing new things.

Try embracing this idea...

Linking Unknown Angles to Strip Diagrams

Representing Addition of Fractions

I absolutely love that there are so many opportunities (with the NEW TEKS) for students to manipulate and draw! Not only does it add to the atmosphere of the classroom via engagement and discourse, but it also speaks more heavily to our kinesthetic and visual learners!

Adding and subtracting of fractions (with unlike denominators) in 5th grade begins with this premise as it sets the tone for students preparing to move more towards the abstract (algorithm) of adding/subtracting mixed numbers. I love that we get a chance to pull our colored pencils and graph-type paper to introduce this concept.

That's what makes this video blog quite special! I hope you find it fun to teach!!

Adding fractions with Unlike Denominators

Two-Digit by One-Digit multiplication with Strategies

I know what every Third Grade teacher out there is thinking:

"I would love to prepare my students for fourth grade...after all, no greater feeling than when our fourth grade teachers feel we did our job well!!"

Okay, maybe that's not exactly what you're thinking, but if you've ever received any accolades from the teachers in upper grades for how WELL you prepared their current students, then you understand this little feeling of joy i'm referring to. Of course, when we are teaching, we only think about our students 'passing some standardized test'; but friends...the picture is much bigger!

Move up a teach a higher grade and you will see what I mean. We can witness, first hand, the well-equipping of our students or the lack thereof, when you begin to teach students and they are quickly able to pick up the concept because they have retained the foundational skills in previous years. As a teacher who has taught everything from eighth grade down to fourth, I can say the feeling is amazing!

So third grade teachers, your fourth grade colleagues will be singing your praises when you appropriately disseminate the foundational strategies utilized in 2-digit by 1-digit multiplication. Should the vertical alignment go as planned...when those students pick up with this concept in fourth grade, they will feel quite successful! Thanks to you...

Multiplication Strategies (2-digit by 1-digit)

Dividing the Spoils!

In ancient times, "dividing the spoils" meant to split the loot (winnings) retrieved from a battle!

No worries my brave 5th grade teachers, we are prepared to win this battle against these NEW TEKS; only, there will be no dividing of the spoils! The only dividing taking place will be between unit fractions and whole numbers. So get your pencils and paper ready and let's go to war!

Representing Division of (Unit) Fractions and Whole Numbers

Practice with this fun link: Dividing Fractions Game

Help me DECOMPOSE this TEKS!

Did anyone want to throw in the towel when they realized our second and third graders were expected to "decompose numbers in more than one way"?

No? Just me!?

I don't buy it; you weren't too fond of this concept either. After all, those of us who taught older students certainly inferred that students were already struggling with place value in their younger years because when they show up in 4th and 5th grade, they're exhibiting misconceptions with place value. I remember sitting with some teachers in our first PLC before the school year began a few years back and deconstructing the following TEKS:

Use concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens and ones. (2.2A)

Compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate. (3.2A)

I'd like to submit, however, I believe this NEW standard will significantly assist our students with building a stronger foundation in mental math and their ability to fluently manipulate numbers as they develop in math. 

This video blog is dedicated to my 2nd and 3rd grade teachers!

[image from http://whoswhoandnew.blogspot.com/]

Partial Quotients Part III- Traditional Algorithm

Much like division being a complex concept for students to grasp, so teaching the concept can be quite daunting for teachers. For that reason, this video blog series centered around division has been "divided" (no pun intended) into three parts.

In efforts to avoid cliche and ill-advised "tricks" such as "DSMB: Does McDonalds Sell Cheeseburgers" to help students remember a myriad of steps, the video blogs I've created have been designed to assist with building the traditional algorithm from the ground-up.

If you haven't already seen Part I and II, it's imperative that you do, prior to watching this final video. After all, this blog is built on the premise that you as the teacher, have laid the foundation from parts 1 & 2, respectively.

With that being said, scroll back in my blog site to find those videos...otherwise, click play & enjoy!

Fractions on a Number Line

Happy Holidays! (In my native language, "Merry Christmas")

So what fraction of Christmas day do you spend with your family? What fraction of Christmas dinner are you able to eat? What fraction of the year is spent celebrating holidays? Okay, okay, okay...enough fraction questions already, right?

Well students aren't too fond of fractions either, but I think all of that is due to misunderstanding of the concept. Fractions don't have to be a tedious concept when we scaffold how students develop in this area. What fraction of a role do we, as educators, play in that developmental process? 1 out of 1!!

Although the idea of fractions begins as early as Kinder (and in some programs, Pre-K), in Texas, 3rd Grade is when students begin to see fractions in their numerical form and plotting fractional points on a number line. This can seem overwhelming for a Third grade teacher, but the fact that students are coming from 2nd grade with experience in fractions, and will move into 4th and 5th continually plotting fractions on various number lines (with decimals and graphs), this might help you breathe easier. Third grade teacher, you are only laying the foundation for fractions on number lines. But so we ensure we lay a great one...here's a quick gift from me to you! Merry Christmas!

Let's unwrap how to plot fractions on a number line!

Partial Quotients Part II- Area Models

You may have seen part I (Arrays) for developing representations for division of two-digit by one digit (which can also be used to do three-digit by one digit as well as four-digit by one digit).

In 4.4E, the expectation is also for students to build area models for up to four-digit by one digit problems!

Watch the video below for how to develop this representation with your fourth grade students.

Enjoy and let me know what you think!

Partial Quotients- Part I (Arrays)

Okay, I know nobody likes division! And I'm pretty sure if your students are like mine, they think it's the "hardest thing EVER!".

For this very reason (or reasons), I love the new standards in 3rd and 4th grade that allow students to return to the foundations of math and build models to represent the "long division" beast!

Check out this 4 minute video on 4.4E (fourth grade long division from a concrete and pictorial standpoint).

Tell me what you think!

Digital Science Fair!

Question: Can students create Science Fair projects using digital tools rather than science fair boards?

Hypothesis: Yes, I believe Science Fair projects can be created with digital tools because we can teach students to be more environmentally conscious, students are already acclimated to digital resources and there's an opportunity to share their product, globally!

I saw an amazing display of student work the other day when I was asked to judge the Digital Science Fair at a local school! Students in grades 3-5 had created their science fair project displays in Google Slides and sent them to their Science Specialist. After sorting them into folders in Google drive, we were able to log into google drive and view their displays. 

I was wow'ed even after I watched a Kinder class that had created their display in the Photo Story program and narrated the processes of their science fair class project under the direction and facilitation of their teacher. After scoring and selecting the top 3 winners from each grade level, we were able to walk down to the cafeteria where tables were set up and the 16 finalist were sitting on the left side with their laptops ready to present! 

Shortly thereafter, their student peers from the grade levels poured into the cafeteria and fanned out like crazed fans until they were sitting across from the finalist and viewing their presentations. With star-gazed eyes, 8, 9 and 10 year olds were deep in conversation, heavily engaged with their classmates about the projects that made the final cut! What a sight to behold!

Here's how (through the lens of the Scientific Method):  

Materials: Computers, digital cameras, Excel, iPod touch, iPad, voice recorders, iPhoto, iMovie, HDMI cable, iPhone or Smart phone, Rubric, Google doc (slides), Photostory program, science fair project references, and lab equipment for the experiments.

Sample Rubric

Research: Technology enables our students to assimilate knowledge more readily by employing a wide variety of media addressing all learning styles in our classrooms. We serve urban school children who are are primarily visual/kinesthetic learners who are often intimidated by written instructions. Using a digital camera, a computer, iPod touch, classroom iPads. I will produce school-wide video instructions showing students how they can document, produce and present their science fair project digitally. (source cited: http://www.weareteachers.com/lessons-resources/details/creating-digital-science-fair-projects)

Procedures: 

1. Students are given various research topics to choose from and/or are allowed to pick their own topic (with approval from teacher).

2. Students should be given a rubric and that rubric addressed in a thorough manner from the teacher so students know what the expectations are. 

3. Students do the Science Project either at home or in pairs/groups at school.

4. Students take pictures of their project using a digital camera or phone.

5. Students create their results and analyze their data (then create a graph in Excel or Word).

6. Students draw their conclusion.

7. Students create their project and digitally put it together using iPhoto, iMovie, Power Point (Google Slides) or Photo Story.

8. Students turn in their digital projects (via Google drive) to be scored/judged and finally, to present to their peers.

Data: 

Students hovering around the finalist to view their presentations.

This project had the attention of several students!

Conclusion: 
So, can YOUR school do Digital Science Fairs? Well, I've seen it done and although it may take some tweaking to differentiate it for the needs of YOUR campus, it IS, indeed, possible!

Number lines are strategies too!

I was planning with a 4th grade team the other day and after we discussed what the TEKS meant and what strategies we needewd to use to teach it, we dug into the idea of resources needed.

So the TEK was 4.3CD which states "determine if two given fractions are equivalent using a variety of methods" and "comparing two fractions with different numerators and different denominators and represent the comparison using symbols."

After we discussed using various fraction tiles, towers and circles for students to manipulate to find equivilencies for different fractions such as 1/2, 3/4, and 2/3 to name a few; a teacher brought up the fact that the students needed to also use number lines to identify equivalent fractions (based on the released questions and other resources utilized to unpack the standard).

She was absolutely correct! Students are bringing (or should be) in prior knowledge from Third grade with representing fractions on a number line via 3.3F. Benchmark fractions (0, 1/2 and 1) are a great place to start and utilize as a reference point. Conceptual understanding should include students knowledge of 0 as zero out of the total pieces and 1 whole as the entire number or set of pieces in the fraction (ie: 5/5).

Background knowledge (and pre-teaching or re-teaching) should also include building a number line that emphasizes counting the spaces between notches on a number line. Anyways, back to the thought-provoking idea a teacher came up with. She decided after teaching the initial concepts in her mini-lessons, she would create a station that allowed students to build number lines to compare fractions and/or find equivalent fractions.

So I reached into my stash of number lines and various fraction cards and we created some procedures to implement this activity as a station.

1. Have two students each pull a card.
2. Have both students predict (using benchmark reasoning) which fraction is larger (or smaller).
3. Have both students create/draw a number line (on pre-laminated sentence strips).
4. Use the number line to justify which fraction is largest by describing which benchmark fraction it's closest to.

Possible sentence stems:
"My fraction is __________, because its closer to _______."
"My fraction is closer to _______________, so it is ______________."

Enjoy the opportunity to engage in dynamic coversations with your colleagues (fellow teachers), as many great, simple station ideas can emerge from a clear understanding of the standards.

Don't be PARTIAL with Partial Products!

I simply adore following Donna Boucher on Math Coach's Corner. She's low-key my role model. I wanna be like her when I grow up!

Check her out: Math Coach's Corner


Recently she wrote a blog about how the new 3rd grade concept of two-by-one multiplication includes a critical strategy called partial products.

She nails it! But can you humor me as I add a representational component from the TEK (standard)?

Let me restate it:

"Use strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may include mental math, partial products, and the commutative, associative and distributive properties."

One of the critical components of students understanding algorithms, is the foundation of concrete and representational models. How powerful in the problem 14 x 6 is teaching them that 14 decomposes into 10 and 4 and they should multiply each component by(or compose equal groups of) 6.  

When students can draw, shade and label their model, how much more can they justify their product as well as peer-teach!?

So go read her blog...digest it and don't forget to add the models. It's a strategy that helps bridge the gap in connecting to the algorithm. And honestly, when the standard says "strategies may include...commutative, associative and distributive properties.", this representational model helps students internalize the commutative and distributive property without them ever knowing they are using them. They may not need to know those terms, for example:

(10 x 6) + (4 x 6) = (4 x 6) + (10 x 6) and students will often write them both ways on accident. But that opens the door for discussion about whether the product will be the same.

Exit Ticket:
Using the model, write a number sentence that would help you solve the product of 14 x 6.
 

Cell Phones: Get out of Jail Free!

So this "cell jail" theory dominated my timeline for some reason this week. It's the idea of having students turn in their cell phones into a "cage" of sorts (often utilizing some teaching related material) to keep them out of the hands and off the minds of students during instructional time. With some of the post comments, I read that teachers would use the "incentive" that students would get participation points for it or could use it as a time to charge their phone in one location. Ironically enough, the idea of it all arrested me and wouldn't allow me to rest without offering an educational alterative.

Imagine, if you will, these students in the real world in a few years. How would they analyze data, quickly retrieve a definition, share a message or thought, and collaborate with colleagues? Let's bring it a little closer to home. In less years than that, what application would be used to submit documents and retrieve their immediate feedback? In all honesty, it's not even in a few years...it's their current reality.

As students walk down the sidewalk, headed home or sitting in the stands at a football game; they are doing more than social networking and texting. They are collaborating with their families on dinner plans through groupme, comparison shopping for the tickets to their favorite concert, and collecting data on the next showing at the movie theatre. All by their phone.

So our challenge is how do we "get them out of jail"...for free?! How do we free our students from attending the isolation cell that is our classroom and release them to make the connection between their world and the learning they must attain? How do we, as educators rise to the occassion and bring learning to the student by helping them find value in core content. One of the process standards for math says we should be helping students see the value of math in everyday life! What better way to connect their math standards (for example) to everyday life than to present it through the means of utilizing their mobile devices? The Technology Application standards summarize that by late middle and high school, students should be colloborating and evaluating products that were created digitally by other students.

What better way than to demonstrate how the "smart phone" is in fact smarter than we think than to allow them the opportunity to explore those possibilities. Let's help them use it to the capacity for which it was designed! This may require some education on our parts, but isn't that what we are as teachers- life long learners?
Rather than be afraid of their open use, encourage it! Educational Conferences offer various seminars on how to incorporate technology into the curriculum. Try attending one the next time you're at a conference.

Here are some other "FREE" ways you can help release your students' cell phones from jail:

• Todays Meet (todaysmeet.com) - Enables discussion and empowers student voice! Great for facilitating discussions possibly while students are doing a silent reading activity or even while you're giving a lecture. 
• Padlet (padlet.com) - Allows students to collaborate and express their ideas about a common topic. Great for use in any subject.
• Kahoot! (kahoot.it) - Game based learning; great way to poll your class, give quizzes and play a game with immediate feedback. 
• Train the how to use their device responsibly and explain why. Allow THEM to set the norms and consequences for stepping outside of these boundaries as you assure them that your duty is to provide opportunities for usage. Trusting them, builds their trust in you!

These are just a few to get your started. I even heard of an experiment where students were measuring the height of different balls using a measuring stick. The problem was that the ball was bouncing too fast and they couldn't accurately read the height on the stick. The teacher had the students pull out their devices and use their video recorder to capture the process. Then she had them play back using the slow motion feature to capture a more accurate reading. What a powerful method of engagement, collaboration and integration! 

Kudos for that teacher; but you ARE that teacher! You are the one who uses technology everyday just like your students; not only that but you know what learning they're accountable for. So you are in the best position to train them to use their technology responsibly. Every meeting I go to, when I'm sitting in church, and when I'm driving, I'm challenged with the task of using my technology responsibly. Why not give your students a chance to practice, in class, a discipline they must already utilize in real life! 

Free their cell phone! 

Investigating Equivalencies

How many of you have ever taught equivalent fractions to students? If that's you, try to complete this phrase:

"What you do to the top, you must do to...."

This teaching phrase is commonly used as a "rule" to help students remember how to simplify a fraction (or what we used to call "reducing"). If we allow students to investigate and explore fractions in a concrete manner, they can create their own rules as early as third grade.

Now, in 3rd grade, equivalencies are already being determined on a number line (hopefully through exploration and utilization of fraction tiles)! *See 3.3G

In 4th grade, however is where we find ourselves reading TEK 4.3C "Determine if two given fractions are equivalent using a variety of methods." And almost immediately we think 'Let's make an anchor chart that tells the students to multiply or divide the top and bottom number by the same numeral to get a larger or smaller fraction'.

A few problems occur when we teach this way:

1. We teach a rule that kids feel the "need" to memorize. As a result, they feel defeatedd when they cannot memorize or recall that rule. 

2. We impose on students (yet another) rule in math, causing them to believe math is made up of a lot of rules that are impossible to remember. 

3. We become the ones imparting knowledge rather than giving students an opportunity to discover the relationships (not rules) that math creates!

Cathy Seely, author of Smarter than we Think, writes "If experience influences a person's intelligence, then the mathematical tasks and problems we present to students, and how we present them take on critical importance." I love this picture she paints of expanding a students capacity by allowing them to struggle. This concept addresses all of the aforementioned problems. 

So let's pose this problem to 4th grade students. 

Build/construct/compose an array of objects to reflect 3 out of 12 (3/12ths). Find another way to divide this array in order to find a fraction that is equivalent to three-twelfths. 

You may find this "struggle task" vague and complex for students. But allow them to struggle. The concrete model should assist them in finding equivalencies. Some students who struggle a little longer and harder may need a little assistance.

Once students have found that 1/4 (one-fourth) is a fraction that is equivalent to 3/12ths, have them explain why (justify their solution). It may help to have them record the two fractions side by side to help them prove their answer. Some students may extend their exploration to find that two out of eight also equals 3/12. With all of these fractions sitting side by side, they should be able to compose their own rule (relationship) between the three fractions. 

If students can visually see that one green chip out of 4 chips doubled is the same as two green chips out of 8 total chips due to a multiplicative pattern, they will determine that equivalencies are built upon multiplication. They will have created the information necessary to create an anchor chart (together as a class).

(credit: lawlerjoyinthejourney.blogspot.com)

This method addresses what Cathy Seely calls the "upside down" strategy and problems #1 & 2 from above. The students take ownership of their learning therefore releasing them from the pressure of having to memorize rules. Likewise, the concrete investigation seals their understanding and gives them the power to build on their discoveries. As uncomfortable as this may make the teacher by changing their role from teacher to facilitator, it's also powerful because students no longer see us as pouring knowledge into them, but rather bringing out what they know. This, in turn, significantly increases student efficacy.

Happy investigating!!

Fractions of Area?? In 3rd grade?

I'm sure some of these new math TEKS have taken many of you by storm! If you're not ready to throw in the towel, then you're a real trooper! As Math Educators, we are really on the "front line" out here because we are having to re-structure our thinking, all while teaching our students. It's almost the "blind leading the blind"...ALMOST! Hopefully, many of you are staying a few feet ahead of the students.

As an Instructional Specialist, I'm not suffering any less. I'm challenged to stay a few feet ahead of the teachers! So let's explore a TEK that shows up in the Geometry and Measurement Unit!

3.6E "Decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole and recognize that equal shares of identical wholes need not have the same shape."

In English please??? Right?

Layman's terms??: "Upon creating 2 flat shapes (using area models) that have the same product or area, separate/divide the shapes in different ways but so that both are separated equally. Do this so students understand that no matter how you separate a shape, its partner (same size, not same shape) shape will have the same amount of parts."

Let's explore this concept.

If I had 4 quarters and gave you two of them, I would have given you $0.50, right?
Well, if you had 10 dimes and you gave me five of those dimes, you would have given me 50 cents.
We both had the same value of money ($1.00); though we displayed our values in different ways (4 quarters, versus 10 dimes) we still both ended up with the same value ($0.50).

So how do we get our students to understand this concept?

Teacher: (Place students in pairs) Have students work with colored tiles (concrete) or colors and grids (representational) to display an array that creates 12. Encourage each partner in a pair to create an array different than their partner.

Students may make 1x12, 2x6, 3x4 and/or the commutative property of these arrays.

Teacher: Have students discuss whether they think dividing their perspective shapes in half will create the same amount of tiles. Simple YES or NO prediction. **Estimation opportunities are critical for students. It's a great way to get them to question their own thinking and justify their reasoning.

Students: Challenge students to divide their shapes in half (perhaps using a popsicle stick) and ask them to find out how many tiles are represented in this "half".

Teacher: Lead a discussion as to "why" each figure, though divided differently, might yield the same result in number of tiles? Have each student (in the pair) write a statement about their findings. Offer ELL students & struggling learners language support via sentence stems.

So when we look at the original version of the TEK, it may sound foreign to us and from that stand point a bit intimidating that we are essentially teaching our students how to find a fraction of a whole! And indeed we are! But here's the relief: we aren't teaching them an abstract concept without first laying the concrete foundation! One day the algorithm of multiplying a fraction by a whole number will make sense because they received a foundation in 3rd grade. Students are capable of understand myriads of things when given the opportunity to manipulate items. Are you giving your students that chance?

Here's a released STAAR question: 



Now, without this sample lesson I just shared, which answer do you think they would have chosen? I'm thinking A, due to spatial reasoning.
Students may be inclined (without the proper instruction and investigation) to assume that since the shaded portion in Figure M looks thicker than that in Figure N, that A is the answer.

But the true answer is C. With proper instruction (an investigative format coupled with a concrete standpoint), students can develop the cognitive reasoning that when both shapes have the same area and both have one-fourth selected (or separated) then the portion shaded in both figures represents an equal area. Third graders can "learn"; it is imperative that we, as teachers, remain lifelong learners so we can pull out the abilities of our students. Then and only then, will we witness the rewarding success of (not just our own practices, but) our students achieving what seems impossible!

Stop, drop and PIN!

Picture this: It's Saturday night and your lesson plans are due by Monday morning (if you're truly a teacher, you'll know this is really fictitious). So you open up your online planner and with intense excitement you minimize your screen and click on your bookmarked icon. You know the one...

After all, you just attended an amazing Math conference that had an hour long session dedicated strictly towards Pinning classroom ideas and networking with virtual teachers for some quick grabs! You head over to your Math classroom board to sift through your recent pins for a cute lesson that will engage your kids (keep them busy) and amaze your administrators (make them believe you're on top of your game) once the product is hanging on your bulletin board!
You cut and paste the "activity" in your planner and scroll your mouse over the SUBMIT button...but

WAIT!! STOP!! 

Step away from the button...

My friend, you are about to make a huge mistake. We are greatly privileged to be in one of the few professions where stealing is accepted rather than frowned upon. Yes, our livelyhood can easily be viewed as the practice by which plagerism (to a degree) isn't unlawful! Meaning yes, you are handed down creatively designed activities and sometimes at a rip off price ( thanks to heavenly sites such as Teacherpayteachers.com) but nevertheless they are pre-made and ready to implement! But you have forgotten the most critical component of planning...the goal!

Whether you're a freshman to teaching, still getting your sophomore/junior year credits up or cruising your super senior years, the following components are essential to your Pinterest success!

STOP!

1. Study your standards: Know them, what they mean and to what extent they must be taught. Utilize sites such as lead4ward to access, and download your grade level TEKS at a glance.

For example: K.2E states to generate a set of concrete models or draw a pictorial model that represents more than, less than or equal to a number (up to 20).

This lets you know that students (under no circumstance, even those advanced and gifted) should practice this skill beyond 20. The skill can be modified to be tougher (such as creating figuring out how much more or less or creating multiple numbers) but should not be pushed beyond 20.

DROP!

2. Facilitate an engaging mini-lesson! Remember YOU are the teacher, NOT Pinterest! Dr. Nicki Newton (author of "Guided Math in Action") says " you might teach the students a song"... (about more and less), you may "read a picture book, play a game or read a poem related to the topic". Get the Big Idea rolling and even create (as a class) a math anchor chart that students can reference later.

Utilize your curriculum and expertise as well as any additional curriculum resources. For example, you might consider displaying a dot card and have students place the same number of counters on their desk as is reflected in the dot card to emphasize " same" and discuss synonymous vocabulary such as "equal to". Next, you may ask them how to demonstrate more (greater than) and less (than) in relation to this amount and justify their answer. Discourse between you and the students as well as between students and their peers is critical at this stage.

PIN!

3. Now it's Pinterest time!! Pull out those manipulatives and search Pinterest for some activities that help drive the concept home! Don't be fooled though, utlitzing Pinterest activities take much thought in and of themselves. Should they be placed in a center? If so, when/how will I model how to use the center? What manipulative should I use? How will I differentiate the activity so users at different levels (novice, emergent, expert learners) can be challenged?

I'd be remiss if I didn't mention that a thoughtful pre-assessment (whether formal or informal) is imperative prior to planning. Knowing where your kids are and what they need drive this entire process! 

Take away? Pinterest is never the engine...its the caboose! Pinterest isnt the goal, its a tool that helps us reach goals! Pinterest should always serve as the icing on the cake. It should never replace the cake or play the role of the much needed ingredients used to compose the cake. Pinterest can be a powerful tool, so don't let anyone take that away from you or discourage you from its use. But be an educated user of the tool, otherwise it will waste your kids time and leave you frustrated!

Happy planning!! 

"I don't get it!"

Now I consider myself a person with a very high tolerance level but when I'm in the classroom, I find that few things are tolerable to me and one of them is the cop-out phrase "I don't get it!" You may think that's unacceptable for a teacher to hate the typical 'cry-for-help' phrase, but hear me out.

I always challenge my students to think positively rather than negatively. So I set up a standard at the beginning of the year. I suggest that we use the phrase, "Could you help me understand?" in place of "I don't get it!" Why? Well one phrase is often used to justify why they've given up. While the other suggests that I'm not willing to give up, if you can show me another way, I want to "get it".

Today, my (intervention) small group had a silent "I don't get it" moment. Let me set the stage for you:

So you've taught your whole group lesson on Perimeter and Area, fully equipped with hands on models, foldables and pair-share thinking questions. All of this lasted its full 30-45 minute time (since you're a 5th grade teacher and these concepts are review anyways).
Now you're in small groups with your 4-5 students and you're diving deeper into the concept. You come upon a problem that is a bit more rigorous and your goal is to hammer through it piece by piece via open discussion.

The problem goes like this:

Ms. Kim has a backyard with the dimensions 60ft by 40 ft. She has a flower garden in the corner of her backyard with the dimensions 15ft by 20ft. How much of Ms. Kim's backyard does NOT include her flower garden?

What do you do?

Let me share with you my journey through this rigorous discussion. To me this was the easier of the two questions I wanted to discuss. Based on the level of these students at my table, I was not shocked by their blank stares. My students were mentally numb and no matter of discussion was bringing forth a solution for this problem. So of course the obvious move was to switch into "INTERVENTION" mode and help students understand:

A. What the problem was asking us to do (find the grassy part of the backyard (minus the garden)).

B. Figure out a process by which to calculate this leftover section.

C. Chart out all of the steps (in correct order)...utilizing their Math Reference (STAAR) charts.

The discussion was quite toilsome. Below is the approach we took.

1. We looked at a piece of rectangular construction paper.

I asked "How does this piece of paper represent our backyard?"

2. Then we analyzed the problem by re-reading and pulling out components like "does NOT include her flower garden" that need interpreting. Using the construction paper, we drew a section to represent the garden and asked,

"What should we do to isolate the garden from the yard?"
STUDENT RESPONSE: Cut it off!!!

So we did. Then we discussed what the left over represents.
STUDENT RESPONSE: The part of the backyard that is NOT the garden.

3. We went back to the problem to make a connection.

"How do we demonstrate using math language, cutting the garden away from the backyard?"
STUDENT RESPONSE: Subtraction!!!

"Subtract what?", I followed up!
STUDENTS RESPONSE: Subtract the garden (area) from the backyard (area).

From here, we were able to decide what formula we needed.

Area (rectangular backyard) - Area (rectangular garden) = Area (leftover backyard)

 

I had students identify what formula they would use and justify why!

*This was very important with the smaller rectangle, because some of them thought we would use the AREA (square) formula simply because it LOOKED LIKE a square.

 

"What do we know about the sides of a square?", I asked. (That they are all the same)

"Are the sides of the garden the same?" (no)

"So what can we conclude?" (its a rectangle)

 

-------------------------------------------------------------------------------------------------------------------------

 

The next problem was even tougher for them to think through. But I could be a bit more understanding. Here's what it looked like:

 

A rectangular wall had a window in the middle of it. How will you determine the area of the window?

 

 

I wasn't even mad at one of my students who said, "Get a ruler and measure it!!"

That's the logical think to do, right? She got my vote but I'm sure everyone else at the table was mad when I shot down her idea with my response, "We would totally do that if they instructed us to use a ruler to measure it, but since they didn't, what other method should we use?"

 

No amount of THINK TIME was helping these students figure out a solution. So we took to a hands-on model to get their juices flowing.

 

So we used a series of linking cubes and a part-part-whole mat. I asked them to identify the "whole" from the linking cubes.

 

Then I related the linking cubes to the length of the wall. I had the students discuss how we could separate the yellow section (the part) from the purple section.

 

 

 

STUDENT RESPONSE: Cut off two purple ends!!

*and we did!

 

We translated our actions into a mathematical number sentence:

 

Whole - part = part

 

From here they were able to see the length of the wall as the "whole" and the window as the part that needed to be separated.

 

They discussed as a table, that they should:

 

A. Subtract 7 from 20, then subtract 7 again to find a leftover of 6. (20 - 7 - 7 = 6)

 

Then someone said, "Couldn't we just..."

 

B. Subtract 20 - 14?

 

So I posed their question to the group to vote! The we justified why these two approaches were similar.

 

Finally, the students were able to ascertain the length of the window. Now, in the original problem, the window was a square, so we were essentially done.

 

We labeled the lengths and widths of the square and calculated the Area of the window.

 

One young lady said, "why did they give us all that information if all they wanted us to do was find the area of the window?"

 

After we all had a good chuckle from her pertinent question, a student replied, "Because it was the information we needed to find the information we didn't have!"

 

WOW!!! What a learning moment! Now, could I expect them to do problems like this on their own? Not for this small group, but with more practice, they would eventually grasp the idea.

Here's the take-away. Don't expect all of your students to apply perimeter and area immediately! Students with slower processing skills need a more concrete association as they think through word problems. Be patient and open enough to build the bridge to their understandings.

My philosophy is, "Your light bulb may not go off when others' go off, but it WILL go off!"

 



 

 



When Teaching gets Frustrating...

Have you ever been standing at your dry erase board waiting for the class to give the answer to your math question? Waiting, in the intolerable silence; begging them by rehearsing the question in different ways almost literally giving away the answer. Not even your usual 'quick arm shooters' have their hands raised. You feel your blood boiling and you're moments away from giving up.

Later, in your small group you are asking students to decompose a number and the 5 students are sitting with linking cubes in front of them yet starring at you like a deer caught in headlights. Finally, you scream, in frustration, "COME ON GUYS, DE.COMP.OSE THIS NUMBER!!!!!!!!", as if that will somehow provoke them to spew out the right answer.

Yes, teaching (imparting knowledge into someone's brain) can be difficult and even frustrating. But when you're facilitating learning, the process is a bit more meticulous. Let's explore the process:

Let's say you ask your students to explain to you how they can mentally add $19 + $28.

1. First let's analyze the question I presented. I didn't start the lesson by telling them that we would learn how to add these two numbers. Rather I had them explain to me what they THINK they could do to add these two numbers without pencil and paper.

My objective (later) is to teach them the algorithm, but not without asking them to draw on their prior knowledge (hopefully of base ten and/or decomposing numbers, making ten, counting on, etc.). The idea is that I want them to explore the various strategies they may already be aware of.

2. After hearing their explanations (and hopefully you gave them opportunity to discuss with a partner or table of students so they can draw on the thinking of other students), you want them to share out. This allows you, THE FACILITATOR, to both correct their thinking and informally assess where they are.

If this step isn't taken seriously and intentionally planned, frustration can easily enter. You must KEEP CALM and allow various incorrect approaches!

3. Most importantly, you must be prepared for these incorrect approaches (answers). They must be premeditated. In other words, you should know all the responses that will be given and be ready for them. Yes, especially the incorrect ones!

So really, step 3 is pre-step 1! When you're lesson planning, you must think of how you will handle the responses-what intervention will be needed? You must think through how you will group your students based on their responses.

This step involves intervention planning (intervening when strategies aren't working).

A. If the students started at one and counted to nineteen then added 28 more...DON'T FUSS!!! This simply tells you that the students struggle with COUNTING ON and REASONING.

B. If the students counted on from 19...CELEBRATE THEM! That wasn't the best strategy but at

least you know they can SEQUENCE and COUNT ON! Their next step then is to help them apply UNITIZING and CONSERVATION of numbers.

C. If the students are able to decompose the 19 into 1 ten and 9 ones ...then decompose the 28 into 2 tens and 8 ones for a combination of 3 tens and 17 ones--well they are in a different and more advanced place! They may not know how to regroup the 17 ones but at least they understand DECOMPOSING!

*There are other answers that you could expect, such as:

MAKE TEN STRATEGY

Taking a "one" from the 28 and adding it to the 19 to make 20 (this is called the MAKE TEN strategy). This would restructure the problem to 20 + 27. Regrouping, then wouldn't be necessary.

The idea is to cut down on frustration on your part, by understanding the various places your students can and may be in their level of development. This can occur by reading articles on Numerical Development such as Subitizing by Douglas Clemens, 1999. You may also check out Teaching Student Centered Math by John Van de Walle.

Be prepared to take them to the NEXT level in their cognitive development.

"The best defense is a good offense!"

"Be proactive, rather than reactive!"

Where's the Value in Place Value?

I have a love-hate relationship with the New Math TEKS, as I'm sure many of you Math Educators can probably understand. Although, if I truly took your pulse, I'm willing to bet 'hate' would be the more dominant beat than love.

Well I love that the TEKS address the foundational and conceptual basis of Math (which in fact are NOT new at all, just new to us). I hate that there are little to no resources that really instruct teachers on how to disseminate the information to their students. But with the latter, I find it fun (call it the geek in me) to study different strategies and create in my mind the perfect classroom situation and centers for students to engage in to assist with their discovery of these understandings.

Big shout out to my 3rd grade teachers who are now helping students understand value of numbers to the hundred thousands place. Here's a few strategy and center ideas to help you out next school year! And if you happen to teach 4th grade, you may benefit from this as well.

The NEW TEK states:
3.2A (similar 4th grade TEK is 4.2A) -  Compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.

3.2B (similar 4th grade TEK is 4.2B)- Describe the mathematical relationships found in the base-10 place value system through the hundred thousands place.

*Note the only difference in the 3rd and 4th grade TEKS are verbs and the place value focus.

STRATEGY IDEA: So, during your 15-20 minute mini-lesson, have your students sit in cooperative groups and work with a set of Base Ten Blocks (distribute up to 1 thousand block, 3 hundred flats, 5 ten longs and 10 units per group/pair-this should help you keep the numbers you compose/decompose within a certain range).

1. Give your students the task of creating a ten long using only unit cubes. Have them verbally express how we could write this in a number sentence. [They may say 1 + 1 + 1... = 10] Assist them in creating a multiplication number sentence (because up to this point they've only covered multiplication in the contextual sense, the multiplication symbol hasn't been introduced formally).

 

 

Verbal sentence: "1 group of 10 units = a 10 long" 

Number sentence: 1 x 10 = 10

 

 

2. Repeat facilitation of this task by having them line up the ten longs to see how many it takes to create a hundred flat; how many hundred flats stacked create a thousand block. Again, have them create a verbal sentence and corresponding number sentence.

 

3. Students can have a place value chart drawn in their journals. They can draw an arrow from the ones place to the tens place and label the arrow "10 times". Then draw an arrow from the tens place to the hundreds place labeled "10 times" and so on; to show that the progression from right to left across the place value chart is based on multiples of 10. Allow them to extend the pattern to the hundred thousands place.

 

 

4. Finally, demonstrate how to build a number (such as 324) and trade one of the hundred flats for 10 ten longs. Have them count to see if the value remains the same:

 

Say: "One hundred, two hundred, two hundred ten, twenty, thirty..., three hundred, three hundred ten, three hundred twenty, three hundred twenty-one, ....three hundred twenty-four)."

 

*This will help them grasp the concept of 'regrouping' when subtraction!!!

 

**Then see if they can discover other relationships; can they determine a relationship between the units and the hundred flat?

 

Verbal sentence: "10 groups of 10 units = a 100 flat"

Number sentence: 10 x 10 = 100

 

 

CENTER IDEA:

Materials: Number generators (dice), index cards with different numbers written on them, Place value disks.

 

Place value disks

 

Create numbers on index cards (or let students generate numbers using dice) in standard form up to the hundred thousand place. Have students build that number using place value disks or place value blocks.

 

 

 

 

Students can then write the number in expanded notation.

(1 x 1,000) + (2 x 100) + (4 x 10) + 3 x 1)

 

What many don't know is that the standard implies that students must be able to represent the number in other ways. So have students trade out a place value (for its x 10 value equivalent) and rewrite the same numeral a different way.

 

 

Notice here how the thousand block has been traded for 10 hundred flats.

This makes a combination (total) of 12 (10 and 2) hundred flats.

So the new expanded form & notation, respectively, would be as follows:

 12 hundreds + 4 tens + 3 ones

(12 x 100) + (4 x 10) + (3 x 1)

 

OR

 

Notice here how one of the ten longs has been traded for 10 units.

This makes for a combination of 13 total units, which doesn't change the value of the number but does affect how its written:

1 thousand + 2 hundreds + 3 tens + 13 ones

(1 x 1,000) + (2 x 100) + (3 x 10) + (13 x 1)

 

 

Now ordinarily, I would advise one to use a center sheet so students can record their findings. Something like what you see below (but of course for larger numbers):

 





 





No disrespect for those types of sheets, I often use them! But I'm a bit of a "tree hugger" (I like to protect the number of copies I make) so I prefer dry erase boards (unless of course I'm collecting it for a grade). Then in that case, I would put to use the construction paper in my room and have students create a foldable and record their work on it, in chart form.

 

Can't wait for the Fall to roll around again so you guys can try this out and let me know how it works for your students! Happy Educating!