Happy Turkey Day!

Solve It! Friday- Task #12

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:

#RMTSolveItWeek12: Since neither egg timer can run for the entire 15 minutes, both timers will need to be used. One solution would be to begin both timers at the same time. When the 7-minute timer is finished, there will still be 4 minutes remaining on the 11-minute timer. This is the beginning of the 15 minutes. When the 11-minute timer finishes, start it again to get 15 total minutes.  

Thursday Tool School: Understanding Fractions- Comparing Fractions

In the 1980s, A&W, an American restaurant chain, unveiled a new menu item-- a third of a pound burger meant to rival McDonald's popular Quarter Pounder. After creating an expensive marketing campaign and a fan-fare filled roll-out, the A&W's third pound burger failed. It was only after soliciting feedback from their focus groups that A&W executives began to understand why. The American public believed the third pound burger to be smaller than McDonald's quarter pound burger. Misunderstanding the four in one-fourth to be larger than the three in one-third caused Americans to believe that A&W was cheating them out of their money by charging a price for the burger that did not equal the value. 

This story speaks volumes about how fractions are perceived by students and adults alike. It's imperative that our students learn to understand fractions in the same way that they understand numbers in other forms. For this reason, today's activity will focus on comparing fractions by reasoning about their sizes. 

This activity addresses the following Common Core State Standard for Math: 
3.NF.A.3.D- Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 

Specifically, this activity addresses comparing fractions by reasoning about their size. Development of this skill begins with a conceptual model. When presented with a variety of visual fractions, students begin to understand that the more pieces the whole is partitioned into, the smaller the value of each piece. 

Here's an example: 

This is a great visual to pose the question "What do you notice?" and then chart the students' responses on chart paper. After exploring with fractions for enough time, students will begin to make connections and discover the relationships between fractional parts. 

It's important to actual have students build a fraction wall with fraction tiles or have them create their own with markers or colored pencils. This way, they have a reference tool they can refer back to so that they continue to reason and make conjectures about the size of various fractions. 

From the model, students develop the idea that the smaller the denominator, the larger the part and vice versa. Students can also make connections about the relationships between fractional parts, such as two one-eighth pieces equal one one-fourth piece. 

This is also a good time to explore comparing fractions with the same numerator. For example, four-fifths is greater than four-sixths because equal parts that are each one-fifth are larger than equal parts that are each one-sixth. 

Free Resource Alert! Click here to download a blank copy of this activity to use as a record of student learning. 

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

The purpose of this series is to help students build a strong foundation for comparing fractions through exploring the use of a variety of methods to make the comparisons. These explorations will provide students with the opportunity to think more deeply about the meaning of fractions and the relationship between them.

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

What I'm Reading Wednesday: Making Number Talks Matter- Chapter 5

Chapter 5 begins with a discussion about the importance of decreasing our focus on memorizing fact memorization and increasing students' experiences looking for "relationships between quantities" and understanding the meaning of multiplication (Humphreys and Parker, 2015, p. 62). Opportunities to explore the properties of numbers "will support their understanding of algebra later" (Humphreys and Parker, 2015, p. 62).  

The multiplication strategies in this chapter increase students' abilities to work with numbers mentally and decrease their reliance on the traditional algorithm-- a reliance the authors state results from limited experience "thinking with numbers" (Humphreys and Parker, 2015, p. 62). The multiplication strategies highlighted in the chapter are detailed in the illustration below. 

The authors offer the following big ideas in relation to the multiplication strategies used during number talks: 

* Use of the distributive property gives students the flexibility to make problems that seem hard much easier to solve.
* It is not necessary for students to break apart numbers into tens and ones. Students can also break factors into other addends.

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. At the end of the chapter, the authors illustrate how the strategies apply to fractions and decimals, as well as, a discussion on connecting algebra and arithmetic. 

A note about Algebra: Using these strategies during number talks provides students with experiences using the distributive, associative, and commutative properties. These are the very strategies that are essential to help students understand important algebra concepts. In addition, using multiplication problems during number talks gives us the opportunity to make connections between the actions students are using to solve the problems and the names of the properties. 

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- What Resources to Include

Last week, I shared the WHAT of station rotations with a few structures that could be implemented. (Missed the post? Read it here!) Today, I want to continue the WHAT of stations and share some of my favorite activities and tasks to integrate into station work. I mentioned in last week's post how I theme my stations. This allows me to just search for specific types of tasks, such as a game or a technology tool, to fill the basket. The beauty of using stations is that once the structure and organization strategy is in place, the only aspect that changes is the activities and tasks that students need to complete. 

Depending on the number of stations I am using, I typically include the following themed station activities: 
* Games
* Hands-on Activities
* Independent Tasks
* Computational Fluency
* Problem Solving
* Math 'n' Literature
* Technology

The chart below shows examples of station tasks that I may use to fill a particular basket. For example, for an independent task, I may choose to fill the basket with a VersaTiles activity that reviews a skill on which we have been working. You'll also notice the "Tech Tools" box. Whenever possible, I try to have a "Tech Tool" station where the students use a computer or some other technology tool to complete a task; however, if I do not have a station with this theme, I try to include a technology task (usually an online game) for the students to complete. Check out my favorite online games here.  

At times, the activity types may overlap, such as using a game during a computational fluency station; but that does not supersede the themed-station task. It may just mean that the students get two games during that set of rotations. 

Clipart by Phillip Martin 

I have many station-ready resources in my File Cabinet. Check it out here! In addition, you can find resources in my Teachers Pay Teachers store. Be sure to sign-up for my monthly newsletter, "Teaching Tidbits, where I include free station-ready resources. 

Guidelines for Station Rotation Resources

1. Use ready to go resources- I keep a variety of ready-made card and dice games, tiling task cards, file folder games, critical thinking tasks, and board games available to use. Note: Finding, assembling, and laminating station materials is a great parent volunteer or summer work task. The best thing-- once it's done, it's done! 

2. Use familiar activities and tasks- Using games and materials with which students are familiar allows them to start working immediately and cuts down on off-task behavior. I use a lot of games during lessons. Therefore, once it's been used with the class, it's likely to show up as a station task.  

3. Use routines that allow stations to run more smoothly- In order to help students gather resources more quickly, I keep common game materials in school boxes in a place where students can access them quickly. Read more about how I organize station materials here.

Getting Organized

One the biggest challenges of station rotations is staying organized. With multiple tasks and a short period of time, organization can make or break your success. Here are some ways that I stay organized: 

• Create a rotation chart so that students know where to go. This helps them stay on track and know in what order to rotate.
• I post a grouping chart so that students can remember their group members. This is especially helpful when I conduct stations over several days because some students tend to forget their groups. 
• I buy colored baskets and use them to separate the stations. I don't always label them because I frequently use the same baskets in other subjects; however, I always know what basket goes with each task. If you choose to use the same color baskets and you have a rotation theme, label each basket with the correct task type, i.e. games,  computational fluency, etc.
• I place most of my games in plastic bags so that they can be transported easily. If a task requires several different materials, I place them in plastic bags as well so they can be located easily. 
• I laminate as much as I can for durability.
• I keep miscellaneous paper, such as notebook paper, scratch paper (the backs of extra copies, unused worksheets, old school announcements/ advertisements), graph paper, and drawing paper in a central location.  
• I store items like dry erase materials in table tubs. Each table has its own tub of materials and depending on which table the task is assigned to, my students know to go and grab that particular tub. See the picture below. 
• Station tasks are always completed in the same location. When station time begins, students pick-up the correct tub from the shelf (if they're not out already) and take them to their station location. 

This picture shows my colored station baskets and dry erase tubs. 

Want more organization ideas, check-out my Math Stations Board on Pinterest.

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. 

Sound Off! What resources do you use for math stations? 

Up Next: Who Participates and How to Manage Them?

Solve It! Friday- Task #11

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:

#RMTSolveItWeek11: This question specifically asked about the weight of the carrots. However, I will provide the weight of all of the vegetables for your reference. Pumpkin = 18 pounds; Ears of Corn = 9 pounds; Onions = 12 pounds; Carrots = 4 pounds. 

Thursday Tool School: Understanding Fractions- Benchmark Fractions

Last week, I discussed how to use fraction tools to help students learn to connect a fractional part to the whole and then to the formal fraction notation. This week, I want to talk about using benchmark fractions to better help students make connections between the value of the fraction and the formal fraction notation. 

We all know that fraction concepts have plagued our students for many years. For some reason, students struggle to understand how to make sense of the value of a fraction. A few years back, in an effort to try to help my fourth graders really make connections between the value of a fraction and the formal fraction notation, I taught them how to compare the fraction to a benchmark. Each time I presented a fraction, I posed the question, "Is this fraction closer to zero, one-half, or one whole?" And, I often added, "How do you know?" 

It took some time, but I began to notice that my students' understanding of fractions developed into the deeper understanding I had envisioned. (An understanding that prevents the dreaded one-third plus one-third equals two-sixths because using benchmark fractions will allow students to see that one-half plus one-half equals one whole. Two-sixths is closer to zero, not one whole.)

It's important to note that students don't just develop this understanding without beginning with the conceptual models. Students need lots of opportunities to make the comparisons using fraction tools before being able to make a visual estimation from the formal notation. See the examples below of how to use fraction strips to compare to a benchmark. 

Today's resource supports the following Common Core State Standard for Math: 
5.NF.A.2- Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. 

Note: The one whole and two half strips are included for reference.

This conceptual model illustrates that the fraction one-eighth is closer to zero. 

This conceptual model illustrates that the fraction seven-twelfths is closer to one-half. 

This conceptual model illustrates that the fraction five-sixths is closer to one whole. 

Note: It is important to state here that confusion always arise with fractions that are close to one-fourth or three-fourths. I usually tell students to round them both up to one-half and one-whole for a quick reference; however, you may choose to make them benchmarks in and of themselves.   

After students have explored the conceptual model, they can record their learning on a graphic organizer. The example below show a graphic organizer where students record the formal fraction notation in the correct column and then "sum up" their learning by answering a question related to the what they notice about the relationship of the parts of the fraction (the numerator and the denominator) and the benchmark fraction. 

Blank Activity Sheet

Free Resource Alert! Click here to get a free copy of the blank recording sheet. 

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

What I'm Reading Wednesday: Making Number Talks Matter- Chapter 4

Chapter 4 of Making Number Talks Matter is about subtraction. The authors suggest beginning with subtraction because students in the upper grades may see addition problems as "too easy." As we all know too well, subtraction is a challenge for many of our students. Exposure to a variety of strategies may help students feel more flexible with subtraction problems. 

In addition, the authors stress the importance of students understanding that the meaning of subtraction includes both subtraction as the removal of something and as the distance between two numbers on the number line. Because students generally think of subtraction as "take away," the authors focused the number talk strategies in this chapter on subtraction as the distance between two numbers. 

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 a different problem than the one in the book so that I could apply my own explanations to each strategy.

While I have seen students use the "Add Instead" and "Decompose the Subtrahend" strategies, the other three strategies seem more difficult for students to use because of the thinking necessary to understand them. However, exposure to and the rehearsal of these skills will produce students with strong strategic thinking and computational skills. 

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. At the end of the chapter, the authors illustrate how the strategies apply to fractions, decimals, and integers. 

Sound Off! What subtraction 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- What Structure to Use

The next aspect of implementing math stations to consider is what structure to use. This decision is related to how you decide to set-up your stations. For me, the most difficult decision is how to structure my station rotations. However, there are a couple of questions that I need to ask myself when I am deciding what structure to use. I've included a short list below:

* How many students do I want to be in a group?

* How many tasks do I want to use?

* How much time is needed to complete each task?

* How much time do I have to devote to the station rotation, i.e one day, multiple days, etc?

Once I have determined the answer to these questions, I select a structure. The table below includes some examples of structures that I have used over the years.  

After I have decided on a structure, I determine how I want to theme the stations. In order to meet a variety of learning styles and to help me stay organized, I use the same types of stations each week. For example, for the 4-station rotation, I include a Teacher Station, a Math Facts and Computations Station, a Hands-on Activities Station, and an Individual Practice Station (the picture below shows an example). 

Note: While this picture is titled "Math Workshop," I did not use a workshop model;
 it was just what I titled it for the students. 

Organizing the station rotation this way allows me the opportunity to just fill-in the blanks on my station planning sheet. Each week, I know I need an activity to fit each station type.This saves planning time because I know I need a game (hands-on), independent practice (like a menu or a VersaTiles activity), and a computational fluency task (like an applet on the computer). How easy is that! 

Please join me next week for the continuation of this post to read about what tasks and activities to include and what methods to use to stay organized.   

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 structure your math stations? 

Up Next: What Resources to Include?

Solve It! Friday- Task #10

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! 

Extension: For students who need an additional challenge, add the following piece of information before asking the question, "Grandma Pat used 11 total bags." (Note: Adding an additional constraint restricts the solution to just one possible solution instead of two. You may choose to present this condition before or after posing the initial question.)

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:

#RMTSolveItWeek10: There are two possible solutions. Grandma Pat could have baked 10 cinnamon spice muffins and 27 blueberry muffins or 25 cinnamon spice muffins and 12 blueberry muffins. The answer to the extension is 10 cinnamon spice muffins and 27 blueberry muffins.

Thursday Tool School: Understanding Fractions- Parts and Wholes

Fractions. A single word that deflates the confidence of our most competent students and adults alike. In fact, I am sure that many readers are reliving their own fraction experiences as they read this post-- good or bad! I can't say that I myself don't have certain feelings about my own experiences with learning fractions, but over the years, I have developed a deeper understanding of fraction concepts through my experience as a teacher, a mathematics education graduate student, and now as a math coach-- which brings me to this month's series.

This month, I will share some of the understandings that I have developed over the years (and am still developing) with all of you. The really great news I want to share with you about fractions is that there are so many math tools you can use to reinforce these essential skills. With that in mind, each week in November, I will share a resource highlighting a fraction tool that can be used to address a specific Common Core Math Standard. 

Today's resource addresses the following Common Core State Standard for Math:
3.NF.A.1- Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a quantity a/b as the quantity formed by a parts as size 1/b.

I mentioned in an earlier post that I love to use pattern blocks to teach fractions. The yellow hexagon, red trapezoid, blue rhombus, and green triangle shapes fit together perfectly to model fractional relationships. The CCSSM standard above requires students to understand how to relate an individual piece to the whole using the formal fraction notation. 

The activity I created for this skill requires students to look at the fraction assigned to a specific piece and
a.) name the whole
b.) explain that the fraction's name is 1/b because it takes b parts to equal, make, or cover one whole.  

See the example below: 

I used pattern block for this initial activity, but many tools could and should be used to help students make the connection between the size of the part, named fraction, and the number of parts that make up the whole. For example, Cuisenaire rods can also be used to make this connection. It is important to note that it is essential for students to see the same size piece represent several different fractions based on the number of parts in the whole.  

See the example below: 

This activity would be great to use as a "record of learning" for an interactive notebook because it provides an example of a part, how and why it is named as 1/b, and references the relation to the whole. A blank copy of this resource is included below for you to add your own shapes and fractions. Students would then supply the picture of the whole and the explanation. 

Extension: As an additional challenge, try giving students the whole and the fraction and then allow them to determine the part. 

Blank Recording Sheet

Free Resource Alert! Click here for a free copy of the blank recording sheet. 

Note: While this activity does not emphasize using sets, it is important for students to complete the same process with sets of objects. Students tend to misunderstand fractions when presented as a set, so exposure to seeing fractions as parts of sets is important to fully understanding this skill. See the example below. 

Looking for more? Click here to check out a post I wrote in September about using pattern blocks to emphasize critical thinking with fractions. The post includes a freebie!

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

What I'm Reading Wednesday: Making Number Talks Matter- Chapter 3

Chapter 3 of Making Number Talks Matter by Cathy Humphreys and Ruth Parker is a list of guiding principles to support the purposeful decision making of teachers as they use number talks with students in the classroom. One of the guiding principles that the authors emphasize throughout this chapter is the importance of sense-making. The power of number talks comes from allowing students the space and time to create strategies that make sense to them. So, in the spirit of sense-making, I have chosen to present my Chapter 3 take-aways as a graphic with short summarizing big ideas. Enjoy!

Sound Off! How important are these big ideas to the success of number talks?

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- When to Use Them and How Often

Math Stations. One of the current hot button topics for educators and the sessions with the longest lines at any math conference. They are everywhere and gaining popularity. Math stations are not only fun for students but they can be adapted to address a whole host of learning styles-- more than we can typically address in a whole class setting. However, using math stations effectively in the classroom can seem like an overwhelming endeavor for beginners. 

To support those teachers who have been wanting to get started with math stations and to share some of my favorite tips and techniques with beginners and novices alike, each Transformation Tuesday post in November will feature the 5 Ws of math stations: Who, What, When, Where, and Why. Today's post will emphasize when using a math station format in the classroom may be most beneficial. 

One of the greatest benefits of using math stations is their versatility. They can be used in a variety of ways to meet a variety of needs. However, the first and most important thing to do is identify your goal. What do you want to accomplish? Answering this question will help you determine when a station rotation model may be more appropriate than a whole class direct teach. Stations are most beneficial when you want to review previously learned content, focus on new content in a small group setting, provide multiple activities for intervention and/ or challenge, and to reinforce/ practice a new skill in a variety of ways. I know what you're thinking-- Yes! Stations can be designed to suit multiple goals! Here's a little more about each one: 

• Preparing for a test? Compile a few activities that review test content and allow students to work on them in small groups to review instead of using a review sheet.
• Believe it or not, there are some teachers out there who teach the entire year's worth of content through stations. They meet with each group regularly and teach their content to the group at this time. The other stations would then be used to support this new learning or review previously learned material. 
• Because students are grouped, stations allow activities to be focused on a specific student need. For example, stations may be created to reinforce feedback/ data received on a recent assessment. Those students needing intervention can complete a specific set of station activities while those who do not need the intervention can complete challenge activities at each station. 
• Stations are great for independent practice! Imagine a set of stations to review each of your measurement skills, such as solving problems involving time, money, area, perimeter, and volume. Look at that-- reviewing five skills at once. Wow!

Once you've determined the purpose of your station rotation, the next question to answer is related to the frequency of your rotations. Stations can span from one class period to multiple class periods. For example, if you are reviewing for a test, a station rotation may be designed for a single class period or two; however, if the stations are being used to respond to assessment data, they may be used for one or more days weekly. 

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! When do you use math stations? How often do you use them?

Up Next: What Structure to Use?