Happy Holidays!

Solve It! Friday- Task #15

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek15: There are 8 sugar cookies and 12 chocolate chip cookies.

Like this week's problem? 

Click the cover page below to view this holiday-themed pack in my TpT store!

Thursday Tool School: Understanding Fractions- Adding and Subtracting Fractions

Today's post will actually build on last week's post about decomposing fractions. Building a solid foundation for fractions in the upper elementary grades is essential for supporting students' understanding of operations with fractions in the middle grades. Last week, I discussed how important it is for students to be able to compose and decompose fractions. This week, I will extend this understanding to addition and subtraction stories.

Specifically, this activity address the following Common Core State Standard for Math: 
4.NF.B.3.D- Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 


The illustration below shows how to represent and make sense of the story problem displayed below. 

Requiring students to use multiple representations to represent the story will support the students' understanding of the connection between the verbal story, equation, and visual model. Depending on the interpretation of the verbal story, the equation and visual model may not be represented the way we might expect them to be. However, using multiple representations provides a strong foundation for the work to be done in middle and high school. 

When solving word problems, it is important for students to understand what story is being told. The Common Core State Standards for Math describe addition and subtraction of fractions as the joining and separating of parts. With this definition in mind, when we read the story problem in the picture above, we see that there was an initial amount of milk and then a part of the container was separated (removed), leaving a specific amount of milk in the carton. 

To write the problem as subtracting five-fourths from three and three-fourths misinterprets the problem. Writing an equation that matches the situation will help students understand how to make sense of the pictorial model and their solution. In the example above, notice that the pictorial model shows all parts removed except for the five-fourths that remained. This allows students to see that the part we need to find is the change from the initial amount to the leftover amount. 

It is important to note that this problem could be rewritten with the same numbers and the same context but the interpretation be different. What if Dewayne started with three and three-fourths cups of milk and drank five-fourths cups of them? It's a worthwhile exploration to do with students because many of them would have incorrectly assigned the equation for the situation above to the solution in the illustration above. 

As a general rule, when it comes to word problems, be sure to ask your students, "What story is the word problem telling?" Then encourage students to match their equations and visual models to their interpretation of the problem. Being able to successfully interpret the problem allows students to choose a model that correctly fits the situation and will lead to a successful response. 

Need more information about how operation situations vary? Check out my Operation Situations- Visual Math Word Problem Strategy Posters pack at my Teachers Pay Teachers Store. 

Sound Off! How do you use math tools to teach fractions in the classroom?

What I'm Reading Wednesday: Making Number Talks Matter- Chapter 8

Chapter 8 has to be one of my favorite chapters in the book because it involves fractions! The chapter is about making sense of fractions, decimals, and percents. While the majority of the chapter is about fractions, the authors do a great job of showing how number talks can be used to help students better understand decimal operations and the meaning of percents. 

Because teaching students to understand fractions is near and dear to my heart and continues to be one of the most difficult concepts for students, I decided to focus my visual today on comparing fractions. So often, our go-to strategies for comparing fractions are cross multiplication, or the butterfly method, and common denominators. The visual below shows how students can compare fractions using their reasoning skills and not with an algorithm. 

In addition, the authors go beyond the number talks above to include number talks for adding, subtracting, multiplying, and dividing fractions. This chapter is an important read for teachers because it helps us develop a deeper understanding of how to work with and reason about fractions, an understanding that can in turn be used to better support our students' understanding. 

The chapter concludes with several examples involving decimal operations and percents, an important read for upper elementary and middle school teachers. Being able to reason about decimals, percents, fractions, and the connection between them will serve our students well in more advanced mathematics courses. 

Like these strategies! Check out my Fraction Relationships and Comparison Strategies Collection at my Teachers Pay Teachers Store. 

Sound Off! How might using fraction number talks in the classroom support your students' ability to reason about fractional relationships?

Note: In order to honor the authors' work, I will only share my own personal experiences, thoughts, and reflections as related to the book's content. If something really strikes me, I will share a quote from time to time with the appropriate citations. 

Reference: Humphreys, C and Parker, R. (2016). Making Number Talks Matter. Portland, Maine: Stenhouse Publishers

Transformation Tuesday: Getting Started with Math Stations- Why Use Math Stations?

Today's post will be the final one of my "Getting Started with Math Stations" series. As the finale, I want to share all of the reasons why I love using stations in the classroom and answer a few burning questions that I receive regularly from teachers. The illustration below shows five reasons that I love using math stations. 

As I close out this series, I would like to address, or readdress, a few of the questions that I receive the most regarding stations:

1. How do I ensure that my students are working and not just playing around? 

It's important to be very specific about your expectations. Teaching students what math stations should look and sound like will help them better understand what is expected of them. Also, using a gradual release of control may help you, as the teacher, feel more confident about your students' ability to work independently. Read more here.

2. Using stations is an overwhelming task for me. How can I make the process flow more smoothly?

Organization is key! Taking a little extra time to set-up and get your system organized before beginning will save you tons of time in the end. Ideally, once your system is up and running, the only thing that needs to be done before each rotation is to re-fill the baskets and add the task directions. Once done, the task directions can just be printed and stuffed in the basket. Spend the time to make the first one and then it's done. Read more here.

3. I don't have time to make new resources every week, what ready-made or easy to implement resources can you recommend?

There are lots of resources that can be easily assembled once and then used over and over again. Using these types of resources allows the students to get started right away because they are familiar with the task and how to complete it. As a general rule, you may want to limit the addition of new resources in the rotation cycle to one per cycle. New resources take time for students to learn how to complete and you will need to review the directions thoroughly before beginning the rotation; however, using the resource during the learning cycle will help alleviate the need for this pre-rotation teaching time. Read more here. 

4. How do you keep up with all of the grading for the work that the students complete?
Because I view stations as an opportunity for review and practice, I do not grade station tasks. This does two things. It lessens the grading burden on me and allows the students to enjoy the station tasks without worrying about a grade. I purposefully create tasks that the students may not finish so that they have no excuse to stop working. Since some students work more slowly than others, knowing that the task has to be completed for grading purposes creates unnecessary anxiety. However, if you would like students to have more accountability, using a short formative assessment afterwards to assess the station rotation's included skills and content may provide more beneficial information about the students' understanding for both you and the students. Read more here. 

5. Where do you find the time?
Station rotations can be used in many ways throughout the learning process. In fact, they can replace something that you were already planning to do. For example, if I am teaching a lesson on multiplication and division of larger numbers, I can replace the independent work time with a station rotation. Or, I can teach students the multiplication and division strategies over a few days and then use a station rotation to provide independent practice. In this case, my stations may include basic algorithmic practice (using VersaTiles), a game with multiplication and division word problems, fact practice on the computer, and a teacher time activity involving estimation and assessing answers for reasonableness. 

In addition, I love to use stations to review. In fact, a few years ago, I created a massive station rotation to prepare for our state test with about 16 stations that reviewed all of our grade level content and skills. After I assigned each group a starting location, groups then worked at their own pace to complete the tasks to earn a completion sticker. Each station also included a multiple choice question that would better help the students prepare for the state test. The students had a week to work on the stations and collect the completion stickers. They loved it and I had the opportunity to float around and see how the students were doing. I called it the the Amazing Race: 5th Grade Math Edition! Read more here and here. 

If you have specific questions regarding how to use stations in the classroom, please post your comments in the section below. I would love to offer you more specific advice. 

New Product Alert! I will be launching my Tools for Organizing Math Stations pack in late December. You can be the first to know when it's available by going to The Routty Math Teacher Store on TeachersPayTeachers and clicking the green star under my store name. 

Sound Off! What burning questions do you have about station rotations?

See this series from the beginning! Click here!

Solve It! Friday- Task #14

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek14: This problem specifically asks for the total number of toys. For reference, I will provide the total for each toy. There are 59 dolls. There are 42 toy soldiers. There are 27 teddy bears. There are 14 sailboats. Altogether, there are 142 toys altogether.

Like this week's problem? 

Click the cover page below to view this holiday-themed pack in my TpT store!

Thursday Tool School: Understanding Fractions- Decomposing Fractions

Building a strong foundation for fractions in the early grades provides long-term support for the development of fraction operations in the upper grades. This understanding of operations of fractions begins around fourth grade. According to the Common Core State Standards for Math, students begin fraction operations by decomposing fractions into fractions with the same denominator in the fourth grade. 

Specifically, the related Common Core State Standard for Math states: 
4.NB.B.3.- Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 

4.NF.B.3.A.- Understand addition and subtraction of fractions as joining and separating parts referring to the same whole

4.NF.B.3.B- Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g. by using a visual fraction model. 

Understanding how to decompose fractions supports the following essential understandings: 

• understand the meaning and purpose of the numerator, denominator, and unit (or whole)
• addition of fractions is an extension of whole number addition and involves joining or combining disconnected parts
• addition sequences can represent different problem situations
• the interpretation of an problem situation can lead to different representations
• the denominators of the addends remain the same and are not added together
• addition and subtraction of fractions refer to the same unit, or whole

The activity I want to highlight today involves multiple ways to decompose fractions, using an 8-part spinner labeled with fractional quantities. Students spin the spinner and write a number sentence showing the decomposition of the fraction using fractions with the same denominator.

After spinning for the initial fraction and showing one decomposition (with an equation), students complete the following:

• show the decomposition another way
• justify the decomposition with a visual model
• write a story to match the equation

Consider having students make a poster like the one below to demonstrate their understanding. 

Make it a cooperative task! Assign each task a number and have students rotate their roles. 

1. Spin the spinner. Show how to decompose the fraction using an equation.
2. Show how to decompose the fraction another way.
3. Justify the decomposition with a visual model.
4. Write a story to match your problem. 

Note: My cooperative learning mats would be a great addition to this activity and would help students identify their roles. The mat can be rotated clockwise or counter-clockwise to complete each new fraction set. Check it out here or by clicking on the picture to the right. 

Sound Off! How do you use math tools to teach fractions in the classroom?

What I'm Reading Wednesday: Making Number Talks Matter- Chapter 7

Chapter 7 of Making Number Talks Matter is all about division strategies. What I love about this chapter is that many of the strategies can be applied to larger numbers. It's all about number sense! If our students understand the meaning of division, they can develop strategies that are useful to them. 

The authors also mention that this is the only number talk where students are allowed to use pencil and paper to help with the recording. However, in many cases, most of the work can still be done mentally.

The illustration below shows the four strategies highlighted in the chapter and is a quick overview of how students may articulate how they use each of the strategies, as titled in the text. I chose to use different problems than the ones in the book so that I could apply my own explanations to each strategy. 

Note: The authors offer that all of the strategies are intuitive for students except for the "Make a Tower" strategy which involves making a tower of multiplication facts until students find one that will help them divide with the least number of steps. I think of it as estimating quotients, using the closest product to the value of the two largest place values without going over. See the example below. 

In my area of the world, we tend to teach the "Chunk Out" strategy as a beginning division strategy. I call it partial quotients because students can keep taking out chunks until they reach the remainder. This strategy helps students understand the meaning of division and encourages them to find the most efficient chunks they can use. It's actually my favorite way to help students understand the division process. 

In the latter part of the chapter, the authors give a detailed explanation of how to develop the strategies above. This section includes a more in-depth description of the strategy, sample problems, and questions to probe when students use each strategy. In addition, throughout the chapter, the authors offer ways to use the same division strategies with decimals. 

Sound Off! What division strategies do you use?

Note: In order to honor the authors' work, I will only share my own personal experiences, thoughts, and reflections as related to the book's content. If something really strikes me, I will share a quote from time to time with the appropriate citations. 


Reference: Humphreys, C and Parker, R. (2016). Making Number Talks Matter. Portland, Maine: Stenhouse Publishers

Transformation Tuesday: Getting Started with Math Stations- Where Do Stations Fit in the Learning Cycle?

One of the questions I hear most often when I talk about using math stations is, "Where do they fit in your day-to-day math program?" One of the best qualities of math stations is their versatility. They can be used in a variety of ways to meet many different goals. For example, they can replace an old, dated math review. Or, they can be used to provide a variety of practice opportunities for a specific skill. In addition, they can give new life to an intervention or tutoring session. 

The table below identifies three instructional uses for math stations-- as a formative, summative, or responsive tool. Use the illustration to explore ways stations rotations can be used to support classroom assessment and provide essential information that can be used to support instructional decisions.

Another question that I receive is about grading. I am frequently asked whether or not I grade station tasks. The simple answer is no. I view station rotations as opportunities for review and do not use the tasks as opportunities to get additional grades for the gradebook. Believe it or not, this philosophy actually makes stations the most enjoyable instructional activity that we do for the students. The students know that the stations are about learning and nothing more. 

However, if I need some data to support instructional decisions, I use one of the strategies above. I either create a skill-based question to support the work done after each station task is complete or I create a short assessment to be completed by the class after the station rotation is finished. In both of these cases, the second case more so than the first, I may assign a grade for informative purposes and to allow the student, the students' family, and me to be on the same page with the students' strengths and areas for growth.  

Here's a challenge for you: As you read through this series, think about a time when a station rotation can be used to replace something that you are planning to do in an upcoming unit. In the meantime, check-out my Math Stations Board on Pinterest.

Sound Off! What burning questions do you have about station rotations?

Up Next: Why Use Math Stations?

Thursday Tool School: Understanding Fractions- Equivalence

"Thursday Tool School" on a Friday! End-of-the-semester graduate projects are really kicking my rear end right now. Happy to finally get this one posted! 

For the past month, I have been sharing ideas and activities for teaching fractions. To continue that support, I have decided to continue this series through the month of December to share more ways to help students develop a deeper understanding of fractions. 

Today's post specifically addresses the following Common Core State Standard for Math:

3.NF.3.A.3- Understand two fractions as equivalent (equal) if they are the same size, or the same point on the number line. 

Using math tools to show equivalence provides a natural opportunity for students to find equivalent fractions and develop the essential understanding that in order to be equivalent, the wholes must be the same size and the portion size must be the same as well. 

All too often, we encourage students to move to being able to generate equivalent fractions before they have had opportunities to recognize visual models of them. In today's activity, students explore creating equivalent fractions that have the same size whole and visually show the same covered area. 

This activity is called "Ways to Show ____" and allows students the opportunity to explore a set of fraction tools and create a poster of fraction pictures that are equivalent to a specific fraction. (Note: If you are concerned about your students' drawing skills, ask them to trace the fraction tools with a pencil to create the pictures.) 

The beauty of this activity is that students can use any available fraction tools to create their posters. This allows students the opportunity to see that the size of specific fractions differ based on the size of the whole. For example, half of the hexagon pattern block is smaller than half of the whole fraction bar. This helps students develop the essential understanding that the wholes must be the same size in order to compare them. 

This activity makes a great exploratory activity and can be used to help students make connections between the equivalent fractions and look for relationships between their numerators and denominators as well as compare them to the original fraction. Be careful though, in order to develop the most solid understanding of this skill, students must make the connections and discover the relationships on their own. 

This picture includes a set of fraction pattern blocks (brown right trapezoid and purple right triangle) that can be ordered from several math companies to extend the use of your pattern block sets. Contact your district's math manipulatives vendor to see if they are available. 

Free Resource Alert! Check out this freebie from my Teachers Pay Teachers store to help your students develop the understanding of the importance of using the same size whole to compare fractions. Click here or on the picture to download it from my store! 

Sound Off! How do you use math tools to teach fractions in the classroom?

Solve It! Friday- Task #13

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek13: Cheese = 1; Gift Box = 2; Mouse with Hat = 3; Mouse with Bow = 5

Like this week's problem? 

Freebie Alert! Click the cover page below to grab this freebie from my TpT store!

What I'm Reading Wednesday: Making Number Talks Matter- Chapter 6

Chapter 6 discusses mental math addition strategies. The authors do discourage the use of addition number talks when you first get started because many older students view addition as "too easy;" however, it is an excellent place to start to build confidence and to provide opportunities for students to build solid mental math skills. 

As in the previous chapters, the authors include ways to use mental math strategies to address operations with decimals and fractions. I found the addition strategies particularly useful and included them in my picture overview of the chapter.

The picture below is a quick overview of how students may articulate how they use each of the strategies, as titled in the text. I chose to use different problems than the ones in the book so that I could apply my own explanations to each strategy.

In the latter part of the chapter, the authors give a detailed explanation of how to develop the strategies above. This section includes a more in-depth description of the strategy, sample problems, and questions to probe when students use each strategy. 

Sound Off! What addition strategies do you use?

Note: In order to honor the authors' work, I will only share my own personal experiences, thoughts, and reflections as related to the book's content. If something really strikes me, I will share a quote from time to time with the appropriate citations. 


Reference: Humphreys, C and Parker, R. (2016). Making Number Talks Matter. Portland, Maine: Stenhouse Publishers

Transformation Tuesday: Getting Started with Math Stations- Who Participates and How to Manage Them

Today's post answers the question who participates in stations? The simple answer is any and everyone. However, participation may vary based on the purpose of the station rotation. For example, if the station rotation is to prepare for a test, everyone may be involved; however, if the purpose of the station rotation is for intervention or challenge, only a select group of students may be involved. I have used them in both ways and see the value in varying how and when a station rotation is used in the classroom. 

The next consideration is in regards to how students are grouped. Again, this decision is based on the purpose of the station rotation. For example, if the purpose of the station rotation is for intervention, grouping students by performance or ability level may be most appropriate. I use a weekly station rotation to review previously learned content and as a responsive tool to support students who have not mastered a recently taught skill. Because the re-teaching needs of my students vary, I group them by the skill I plan to work on with them. In other cases, I randomly place the students into groups and allow them to work with a variety of ability levels. As an additional strategy, I frequently use stations for after school tutoring groups. It allows me to structure this extra time in a manner that supports their learning needs in a fun and engaging way after a long day of learning. 

The last consideration is in regards to managing stations. Teachers often ask me how I get my students to complete their work when I am not standing over them managing them individually. Here's my response: It is the expectation-- completing the work is not optional. But, I know what you're thinking, my students are not independent workers; they will not be able to complete a station task without reminders of what to do and to keep working. My next response to you is that you have to teach your students to work independently and to follow station expectations. Then. Practice. Practice. Practice.  

The chart below shows a common set of station rotation expectations for my students. However, I usually develop them with the students before implementing a full station rotation.

To begin, I may only use a two-station rotation to give students the opportunity to practice the expectations and allow me a little more control over what the students are doing. Gradually, we move to more stations as the students demonstrate that they are able to do more work independently and I can trust that they will follow station expectations accordingly. Eventually, when we reach full station rotation mode, students are expected to self-manage their behavior and their groups as I am unavailable to answer questions or to redirect students when I am working with a small group (but I always have a watchful eye). 

Here are some other ideas I have tried or have seen others try over the years: 

* Ask 3, before Me- students must ask three other group members a question before interrupting the teacher to ask. 

* Focus on improving one station expectation during each set of station rotations and provide specific feedback on that goal. Reward students accordingly. 

* Reward groups who exhibit on-task behavior. 

* Use sticker charts to "catch them being good." 

* Include a formative assessment task at the end of the station to check for station completion and understanding (more about this in next week's blog post).

* Have students rate their own participation and behavior and ask for a written justification.

Many teachers believe that the best checking system to see whether or not students stay on task and complete the station work is to assign a grade. While I will admit that I have collected an assignment every once in a while because I needed a grade for something, station tasks in my classroom are not graded. This is mainly because I view station tasks as opportunities to practice, not to assign grades. I do, however, give formative assessments after stations that may result in a grade-- more on this topic next week.  

Here's a challenge for you: As you read through this series, think about a time when a station rotation can be used to replace something that you are planning to do in an upcoming unit. In the meantime, check-out my Math Stations Board on Pinterest.

Sound Off! How do you manage your students during station rotations? 

Up Next: Where Do Stations Fit in the Learning Cycle?

It's Time to Stock Up!

It's a great time to stock up on some of the fantastic finds you've got tagged on your wishlist!

Click here to go to The Routty Math Teacher Store!

While you're shopping, check out some of these favorite titles from my store. 
Click the cover to see it in my Teachers Pay Teachers Store.