MathReps

Ditch the C.U.B.E.S. Strategy in Math

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First of all, I want to acknowledge the tremendous impact of Robert Kaplinsky’s insights, which have resonated with many for years. His thought-provoking posts “Is Problem Solving Complex or Complicated?” and “Why Do We Have Word Problems?” are invaluable resources. Today, I want to not only express my appreciation for Kaplinsky’s work but also highlight an alternative strategy to the C.U.B.E.S. approach. We can broaden our problem-solving toolset by exploring new perspectives, and I believe this strategy will further enhance our problem-solving skills. Don’t miss out on the original posts: Is Problem Solving Complex or Complicated? and Why Do We Have Word Problems?.

This image shows what the acronym C.U.B.E.S. represents. Many teachers use this to teach an easy way for students can begin to tackle word problems. I, in fact, had an anchor chart displaying this very technique many years ago. As the saying goes, “Know better, do better.” I now know better and therefore do better. I know you might be asking yourself, “Okay, but I still don’t get why this isn’t ideal?”


Let’s consider the problem to the right. With our adult brains, we understand that it is a two-step problem. We understand we need to add the adults and children together and then subtract the number of males who were in attendance. Now imagine your 8-year-old self reading this and not yet having reading mastered let alone the comprehension skills to decode what is being asked. (Okay, maybe that was just me and my learning disability) You can see that I also went through the CUBES strategy. I circled the numbers, underlined the question, boxed keywords, and got rid of extra information. Arguably, the statement that everyone showed up might also be considered extra information to a young learner. An 8-year-old will start the CUBES process. They will look at the boxed words to find out what operation they need to use. And this is where we run into our first problem. It only says, “How many.” It doesn’t say, “how many more,” or “how many less.” Being around kids, we know that their default is to add. So what are they going to do? You guessed it, add 12, 25, and 15. Even if the question that is underlined is: How many females were there? chances are students are still going to add it all up because they are specifically looking for keywords like ‘in all’, ‘altogether’, ‘less’, ‘more’, etc.

Years ago, I used to teach my elementary students these strategies, as it was what I knew at the time. However, as I learned better strategies to help students, one of my favorite methods became the 3-Read Protocol. You can find more information about it in the blog post Mastering Mathematical Language Routine 6: Three Reads. This approach helps students concentrate on understanding the story, identifying units and quantities, and focusing on the question or task. By following the 3-Read Protocol, students can comprehend the context, which enables them to think critically instead of simply focusing on a procedure that may or may not work.

Finally, to help students effectively apply the information from the story, I highly recommend integrating MathReps into your classroom. MathReps are strategically practiced sets of skills that allow students to practice basic math concepts in similar skill clusters and receive immediate feedback. This approach promotes automaticity and fluency, freeing students to focus on what is needed rather than both what to do and how to do it. You can delve deeper into MathReps by reading a series of blog posts. I suggest starting with The Power of MathReps.

It Doesn’t Always Start Off Smoothly

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When I first started attending tech and innovative teaching conferences, I felt completely overwhelmed. It’s common to feel this way when surrounded by so much new information and ideas. Many of us are eager to try everything at once, but then struggle to figure out where to begin and how it will all come together. I have to admit, I often wondered if I could even pull off some of the amazing things I saw. It took me far too long to realize that presenters, including myself now, tend to showcase only the very best. Starting something new is rarely a smooth process; in fact, it’s often quite messy and can even be painful at times. But embracing the challenges is all part of the journey. And that’s what this post is about.

Messy MathReps

When presenting on MathReps, I am very open and honest about the time it took my 5th-grade class to complete the first MathRep – 45 minutes, in case you were wondering. As a teacher, I am acutely aware of the learning curve involved in implementing MathReps. Some students may breeze through independently and accurately within three days, while others may require weeks of support. This variability is the inevitable, albeit messy, aspect of introducing MathReps or any new routine/learning opportunity.

Why do I bring this up? I understand the frustration of the messiness of beginning something new and wanting to revert back to a previous routine or way. The idea of starting something new after the school year has started can be daunting. There are pacing guides, expectations, benchmark assessments, and a million other tasks that weigh on us. We often hear that we should give it time, but all too often it doesn’t feel as if we have the time to give. But some things are worth spending time on.

Why Is MathReps Worth It?

MathReps is undeniably worth the investment due to the proven effectiveness of repetition in skill acquisition. As with any new skill, whether it is teaching, riding a bike, or knitting, initial proficiency is typically lacking. Through consistent practice and the repetition of key actions, one can develop heightened confidence, risk-taking, and greater speed. Similarly, MathReps specifically aims to provide students with ample opportunities for repetitive practice, thereby enabling mathematical skills to become second nature. Once proficiency is achieved, students can then apply their knowledge in varied contexts, such as solving word problems. Without this foundational knowledge, tackling word problems becomes an overwhelming task. For instance, even if a student knows that they need to multiply a two-digit number by a three-digit number, without understanding the process of multiplication, they are left unable to proceed effectively. It is akin to attempting to walk before mastering the skill of crawling. Through consistent practice and the connections facilitated by MathReps, students are empowered to enhance their confidence, adaptability in thinking, and fluency in their mathematical abilities.

More 2nd-grade MathReps on Wipebook

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After sharing my post about the amazing NBT standards on Wipebook, the response has been incredible! So many people have reached out, eager to discover what other fantastic MathReps are available on this innovative platform. And guess what? I’ve got another gem for you – skip counting! This incredible resource delves into three NBT.A standards, making it an essential tool for mastering mathematical concepts. Let’s dive in and explore this MathRep on Wipebook and the standards found on it!

  • 2.NBT.A.1: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases
  • 2.NBT.A.2: Count within 1000; skip-count by 5s, 10s, and 100s.
  • 2.NBT.A.3: Read and write numbers to 1000 using base-ten numerals, number names, and expanded form
  • 2.OA.C.3:Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends

Bonuses of Using MathReps on Wipebooks

  • Concept Connection: Using MathReps allows students to make connections between concepts, helping them master the skills necessary to access higher thinking skills.
  • Reusable Surface: MathReps on Wipebooks provides a reusable surface, allowing students to practice multiple repetitions in one go.
  • Student Engagement: Teachers who have implemented MathReps on Wipebooks have found great success, with students cheering when asked to complete them.

Watch the video below that explains how to implement MathReps in your classroom.

2nd Grade NBT MathRep

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The captivating world of MathReps is even more exciting with this second-grade MathReps on Wipebook! What sets this apart is its eco-friendly design that saves paper, its reusable surface allowing for multiple reps in one go, and its seamless integration with a scanning app. Not to mention, it covers three of the four NBT standards in second grade. Be sure to check out the accompanying video for a closer look at this game-changing MathRep on Wipebook!

Mastering Mathematical Language Routine 7: Compare and Connect

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In the previous Mathematical Language Routine (MLR) discussions, we explored a variety of essential skills. MLR 1 focused on enhancing our understanding by revisiting and reinforcing key concepts, making our knowledge “Stronger and Clearer Each Time.” We then moved on to MLR 2, where we delved into the crucial skill of “Collecting and Displaying” data effectively. Building on this foundation, MLR 3 emphasized the importance of “Critiquing, Correcting, and Clarifying” our models and methodologies for optimal results. In MLR 4, we explored the “Information Gap” and how to use this strategy to be thoughtful of the information needed to solve problems. Continuing this journey, MLR 5 introduced the skill of “Co-Crafting Questions and Problems” collaboratively to foster innovative approaches and insights. Finally, in MLR 6, we explored the technique of “Three Reads,” emphasizing the significance of multiple reads in order to enhance student understanding. Let’s now embark on our next MLR discussion, MLR 7 Compare and Connect.

MLR 7: “Compare and Connect,” has the purpose of fostering students’ meta-awareness in their exploration of different mathematical approaches, representations, concepts, examples, and language. Through this MLR, students are encouraged to reflect on and verbally respond to these comparisons. This involves analyzing why certain mathematical actions or statements are done in a particular way, identifying and explaining connections between various mathematical representations or methods, and pondering how one idea relates to others in terms of both concepts and language. To support this learning process, teachers should model their thinking aloud when addressing these questions. This routine allows students to engage in rich mathematical conversations. We will explore two ways in which to accomplish this.

Which one doesn't belong

Getting students to engage in discussions about math, make connections, and consider different perspectives can be quite challenging. I often encounter students who simply say, “It was in my brain” or “My brain told me the answer.” However, by modeling and encouraging metacognitive awareness, students can begin to make connections on their own. One effective routine that focuses on linguistic skills is called ‘Which One Doesn’t Belong‘. This activity can be done in groups, in pairs, or as a whole class. Students are presented with four images, equations, numbers, graphs, or geometric shapes, and they are asked to identify a commonality among three of them and explain their reasoning. The interesting twist is that any combination of three out of the four options can be correct. For example, in the orange example, one could argue that the three triangles go together and the hexagon is the odd one out. Alternatively, one could justify grouping all the white-filled shapes while excluding the shaded shape. This activity is both enjoyable for students and provides the opportunity to hear and consider different viewpoints.

Another interesting activity that aligns well with this MLR is the Math EduProtocol Sous Chef from The EduProtocols Field Guide Math Edition (Chapter 9, page 56). In this activity, students are grouped together to solve a problem using different approaches and then present their work to the class. For instance, if students were given the task of solving 4 x 6 in third grade, one student might use equal groups, another could opt for repeated addition, a third student may create an array, while the last student represents the equation with the area model. Through this activity, students can establish connections with previously learned concepts and broaden their understanding. There are numerous ways to implement Sous Chef, but the central focus remains on fostering connections among ideas and encouraging students to share their thought processes orally.

In conclusion, incorporating this MLR into your math class will greatly benefit your students. It will help them enhance their meta-awareness, make connections between different concepts, and foster a deeper understanding of the subject. While we have explored two approaches to this MLR, there are numerous other equally powerful techniques available. In our next discussion, we will delve into MLR 8: Discussion Supports, which focuses on stimulating rich and meaningful conversations in the classroom.

Mastering Mathematical Language Routine 3: Critique, Correct, and Clarify

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Welcome to the fourth installment of our series, where we delve into the fascinating realm of Mathematical Language Routines (MLRs). In our previous discussions on MLR 1: Stronger and Clearer Each Time and MLR 2: Collect and Display, we explored the crucial role they play in cultivating critical thinking skills and fostering a deep understanding of mathematical concepts. Let’s continue our journey toward mathematical success by exploring the next MLR in line.

MLR 3: Critique, Correct, and Clarify is a routine designed to enhance mathematical writing and discussions. The primary purpose of MLR 3 is to foster a culture of critique and improvement in mathematical conversations. By engaging in this routine, students are encouraged to actively evaluate, correct, and articulate mathematical concepts with clarity. Through collaborative groups or partner talks, students can refine their thinking as they work together. To introduce this routine, teachers can model it by providing a predetermined piece of writing for critique, ensuring that it includes common errors and vague language to encourage more precise language. This approach empowers students to identify and rectify mistakes while enhancing their ability to clarify their ideas effectively.

Beginning this routine can be tricky, especially since it involves critiquing and correcting another person’s work. However, there are strategies that can help create a safe space where students feel comfortable critiquing and correcting each other’s mathematical reasoning.

At the middle school and high school levels, it is a bit easier as students change classes, and using an example from another class can happen – with names removed.

For elementary-level students, making up a problem/solution that they can use to critique is advisable. It’s important to ensure that the problem/solution contains common errors related to the content being studied.

Once this routine is established, it will become easier for students to seek out peer feedback. Teachers play a crucial role in creating a safe environment where students feel encouraged to seek out one another for critiquing. By implementing these strategies, teachers can foster a supportive and collaborative atmosphere for students to improve their mathematical reasoning skills together.

One Math EduProtocol that works well with this MLR is Nacho Problem. This EduProtocol was developed by Ligia Ayala-Rodriguez with the intention of addressing common errors exhibited by students. The main concept behind Nacho Problem is to task students with identifying and explaining the errors they encounter. Let’s take a look at an example to better understand how it works:

Second-Grade Nacho Problem Example

Ms. Daines needs to drive to San Jose which is 109 miles away. Along the way she stopped in Salinas which is 48 miles away. When she began driving from Salinas, how far away was Ms. Daines from San Jose? The work was provided but no explanation was given. This allowed for students to critique and analyze the provided work, find the error, and clarify their reasoning.

In this example, which was taken from an introductory lesson using Nacho Problem, the wording is kept basic and straightforward. However, as students progress with this EduProtocol, their written expression and complexity will naturally grow.

The beauty of Nacho Problem lies in its simplicity and effectiveness. By encouraging students to find errors and explain their reasoning, it promotes a deeper understanding of mathematical concepts. So, if you’re looking for an educational approach that fosters critical thinking and problem-solving skills, Nacho Problem is definitely worth considering, but not your only option. Sometimes, always, never is another good approach to this MLR.

Incorporating MLR 3 into your math class can greatly enhance your students’ understanding and written communication skills, which are vital for their success. This instructional approach can be implemented as early as kindergarten, allowing students to develop the valuable ability to analyze others’ work critically. This fosters a deeper comprehension of mathematical concepts and empowers them to ask more precise and insightful questions. In our next discussion, we will explore MLR 4: Information Gap, where students are encouraged to engage in critical thinking by identifying the necessary information to solve word problems.

Mastering Mathematical Language Routine 1: Stronger and Clearer Each Time

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Mathematical Language Routines (MLRs) play a crucial role in enhancing students’ comprehension and communication skills in mathematics. Developed to meet the diverse language needs of learners, these frameworks have become an invaluable tool in promoting a deeper understanding of mathematical concepts. In this series, we will explore each MLR in detail, starting with MLR 1: “Stronger and Clearer Each Time.”

Mathematical Language Routine 1: Stronger and Clearer Each Time

MLR 1: “Stronger and Clearer Each Time” focuses on refining students’ ideas and communication through various activities. By incorporating writing, listening, explaining, and integrating new language, students are encouraged to continually improve their understanding of mathematical concepts. This routine, often conducted in pairs, provides students with the opportunity to collaborate and build upon each other’s ideas, fostering a culture of shared learning and growth.

Throughout this series, we will delve into the different structures and strategies that can be employed within MLR 1, unveiling how this routine nurtures students’ confidence and fluency in mathematics. Join us as we explore the remarkable impact of MLR 1 and its profound influence on students’ language development and mathematical achievements.

The purpose of this routine is to foster the refinement of students’ verbal and written output through structured conversation and revision. By engaging in this process, students can enhance both their thinking and their expression of it.

In this routine, students initially work individually or in groups, gradually progressing towards partner work. This approach allows students to acclimate to the task and build their confidence. For those who may be less familiar with writing, explaining, and refining their thoughts, supportive strategies can be implemented to ensure their success.

Once the structures are in place, it is crucial for students to recognize the ultimate goal, which is either a deep understanding of the concept or the ability to articulate it like an expert. The listener’s role becomes significant as they ask clarifying questions, enabling a comprehensive understanding of the speaker’s thoughts. Simultaneously, the speaker benefits from this exchange, refining their thinking more clearly.

To encourage thorough responses, it is valuable to have students switch partners multiple times during the routine. By engaging in back-and-forth conversation, with equal emphasis on speaking and listening, students not only refine their thoughts but also strengthen their language and reasoning skills. The iterative nature of this process reinforces the importance of pressing for details and encourages the continual refinement of ideas.

Convince Me That, by Daniel Kaufmann, is a highly effective protocol that teachers can implement in their math lessons to foster deeper understanding and engagement among students. To successfully introduce and implement this routine, educators can follow these step-by-step guidelines:

  1. Introduce the Problem: Begin by presenting a math problem along with its solution to the students. For instance, students can be asked to explain why 3 x 4 equals 12.
  2. Form Partners or Small Groups: Divide the students into pairs or small groups to facilitate collaborative learning. This structure encourages peer interaction and promotes the sharing of ideas.
  3. Restrict Algorithmic Thinking: Emphasize that students should focus on concrete or pictorial methods rather than relying on algorithms. This restriction encourages students to think deeply about the problem and explore alternative approaches.
  4. Initiate Individual Thinking: Give students time to think individually about the problem and develop their own explanations for the solution. This step helps to build independence and promotes critical thinking skills.
  5. Structured Pairing: After individual thinking, partners or group members should share their explanations with each other. This process enables students to refine their understanding through constructive discussions and peer feedback.
  6. Revise Written Responses: Encourage students to revise and improve their written explanations based on the feedback received during the structured pairing phase. This step promotes self-reflection and reinforces learning.

To facilitate the refinement process and prompt students effectively, here are some examples suitable for Math Learning Routine (MLR) 1:

  • “Convince your partner why the sum of any two even numbers is always even.”
  • “Explain to your group why dividing by zero is undefined and cannot result in a finite number.”
  • “Justify why the product of any number and zero is always zero.”

These prompts stimulate students to think critically, apply their knowledge, and refine their explanations. By implementing the Convince Me That routine with these strategies and prompts, educators can foster deeper conversations, encourage active learning, and enable students to demonstrate a more profound understanding of mathematical concepts.

For a more detailed explanation and implementation guidelines, you can refer to Chapter 19 of The EduProtocols Field Guide Math Edition. This invaluable resource offers comprehensive insights and practical tips for effectively utilizing the Convince Me That routine in math classrooms.

In conclusion, the implementation of MLR 1 has proven to be highly beneficial for students. It provides them with a structured platform to refine their thinking, improve their communication skills, and deepen their understanding of the subject matter. By engaging in the collaborative and iterative process of MLR 1, students are empowered to develop clearer and more coherent responses.

We invite you to stay engaged with our series and continue exploring the world of Mathematical Language Routines. The second routine in our series, MLR2: “Collect and Display”, has a specific purpose. It aims to capture students’ oral words and phrases and transform them into a stable, collective reference. The main goal is to preserve the language that students use and use it as a reference point for developing their mathematical language.

Enhancing Math Discourse: Introducing the ‘Mastering Mathematical Language Routines’ Series

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Unleash the Power of Mathematical Language: Introducing the Game-Changing 8 Routines

Calling all educators! Prepare to embark on a transformative educational journey unlike any other. I am thrilled to bring you a powerful series on Mathematical Language Routines – the game-changers that will revolutionize your math classroom. Crafted by the brilliant minds at Stanford University and wholeheartedly adopted by the esteemed CA Department of Education in the newly adopted 2023 Math Frameworks, these routines are set to redefine the way we teach and learn mathematics.

But what exactly are these Mathematical Language Routines? There are eight carefully designed techniques that provide a powerful framework for enhancing language acquisition in the context of mathematical learning. These routines have proven to be instrumental in bridging the gap between mathematics and language, ensuring that students develop a deep understanding of both. Throughout this captivating series, we will delve into each routine, unpacking their unique benefits and offering practical guidance on how to implement them effectively in your classroom.

In this series, we will delve into 8 Mathematical Language Routines (MLRs) that can effortlessly enhance your math classroom experience, no matter the age group you teach. Let’s start with

MLR 1: Stronger and Clearer Each Time – In this routine, students write and share their responses to math problems verbally. They eagerly listen to valuable feedback, which further enriches and refines their responses.

MLR 2: Collect and Display – As students explain their thoughts and processes, the teacher captures the language they use. This serves as a helpful tool for further clarification.

MLR 3: Critique, Correct, and Clarify – Think of this as an in-depth analysis of errors to enhance learning.

MLR 4: Information Gap – Students form two groups, each with partial information, and must collaborate to solve a problem by obtaining the missing pieces of information.

MLR 5: Co-Craft Questions and Problems – Here, students generate questions and problems based on real-life scenarios, akin to the engaging nature of 3-Act Math Tasks.

MLR 6: Three Reads – This routine involves reading a problem three times, each time with a specific purpose. It is particularly effective for dissecting story or word problems.

MLR 7: Compare and Connect – Students compare, discuss, and connect their understandings with those of their peers.

MLR 8: Discussion Supports – Supportive sentence frames, thoughtfully organized into categories, can enhance students’ participation in discussions.

Get ready to embark on an exciting journey that will revolutionize your teaching practice. Together, we will explore the immense potential of these MLRs, empowering you to unleash the true capabilities of your students in mathematics and beyond. Over the past two years, I have gathered a wealth of knowledge in this area, and I can’t wait to share it with you. Join me as we dive into this exhilarating series, equipping you with the tools and skills to excel, ensuring your students’ success. Come back for more captivating insights that will leave you inspired and eager for the next installment.

Number Paths: The How Pt. 3

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Discover the enchantment of number paths, where the elegance of simplicity meets the practicality of organization. Often, these paths showcase numbers encased within colored boxes, thoughtfully grouped in sets of five. Visualize orange encompassing numbers 1 to 5, while red houses 6 to 10. This delightful arrangement resonates harmoniously with the familiar 5-frame and 10-frame charts that our young learners explore. To enhance convenience and ease, a palette of subdued grays has also been created, ensuring ease for copying purposes. Allow yourself to be captivated by this delightful approach to number path aesthetics.

To effectively utilize these materials with students, it is advisable to laminate them for the purpose of writing. There are various activities that can be done using individual number paths. For instance, students can demonstrate numbers, and perform addition and subtraction. When students are asked to locate numbers on the number path, it is crucial for them to not only encircle the specified number (e.g. 4), but also the entire set of numbers that surround it. They should be encouraged to encircle numbers 1, 2, 3, and 4 as a group. For more detailed insights, The Recovering Traditionalist has an excellent blog post on this topic. This approach helps students develop an understanding of how numbers are organized and represented as a series of objects, and it supports their journey toward comparing numbers.

To ignite the excitement of young learners in primary classrooms, it can be incredibly inspiring for teachers to create a captivating number path either on the classroom floor or in an outdoor play area. A fantastic strategy to kickstart their journey is to encourage students to use colorful unifix cubes to represent each number along the path. Once students gain confidence in their number sense, a wonderful way to keep the momentum going is to challenge them further. Here’s a fun idea: have students cover up the numbers on the number path, then roll a dice or spin a spinner (depending on the numbers involved) and uncover the corresponding numbers. If you want to wow your students with an amazing interactive experience, check out Toy Theater, which offers an engaging number path simulation that you can seamlessly use in whole group activities. These concrete examples will surely empower primary teachers to embark on their number path journey with enthusiasm and immediate action.

No matter where you embark on your journey with number paths – whether it’s through simulations, utilizing large paths on the floor, or implementing individual paths – your primary students will undoubtedly develop a profound comprehension of numbers and enhance their overall number sense. Moreover, by incorporating number paths early on, students will be equipped with the necessary foundation to thrive in 2nd grade and seamlessly transition to working with number lines. Embrace the power of number paths, and empower your students on their mathematical voyage!

Number Paths: The Why Pt. 2

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As we explored in the previous post on number paths, which can be found at this link, these remarkable tools have proven invaluable for TK-1 students. However, it is important to note that number paths begin with the number 1 and do not include zero. Moreover, they exclusively display whole numbers. Drawing upon insights from The Math Collaborative, these intriguing pathways enable students to gain a deeper understanding of number relationships and their spatial positioning in relation to one another. It’s a remarkable journey into the world of numbers!

Children need to explore how numbers relate to one another in order to build a robust and flexible number sense.

The Math Collaborative

Number paths versus number lines: Why is one preferred over the other? This is a common question that often arises when teaching mathematics to young students. The reason behind using number paths instead of number lines lies in the fact that number lines are primarily measurement-based, which is a concept introduced formally in 2nd grade. However, number lines go beyond mere counting of tick marks. They help students understand that there are numbers between these marks and that the distance between them remains constant. In essence, number lines play a vital role in strengthening students’ grasp of number sense and are incredibly useful for developing a strong mathematical foundation and that foundation begins with number paths.

Join me next time when we talk about strategies and share resources.