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5 Practices Lesson Planning

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"Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first."

By Kristin Gray

I've come to think that approaching a lesson plan is like approaching a 500-piece puzzle. It's hard to know where to start. It takes time. It involves endless trial and error. But when it's finished - when all the pieces have been put in place - there's a sense of pride and accomplishment.

I'm a math teacher, and when I shared this reasoning with other teachers for the first time, I included a lot of beautiful words around mathematical connections, but in the middle somewhere, I used the phrase "making connections explicit." The room seemed to recoil. I paused to reflect on why teachers were withdrawing. I asked them, "Was it the word 'explicit?'" There were nods throughout the room.

Trust me, I get it. The word "explicit" has developed a cringe-worthy connotation among peers in the math community. In the past, if I were asked to paint a picture of a math teacher engaging in "explicit" teaching, I would have painted either a teacher standing up at the front of the room, telling students how to solve a problem, or a teacher crouched down next to a struggling student, "correcting" the way the student approached the problem because they were taking the "long way there."

These scenarios make us cringe, because we know they don't lead students to true understanding. I think there are parallels in teaching other subject areas as well. Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first.

In a math classroom, an "explicit" teacher tells students how to approach math, typically by way of a procedure. There's no discovery. But, when we purposefully plan using a problem-based curriculum in our classroom - setting up investigations for our students, designing and implementing routines and fostering intentional talk - we can change what it means to be "explicit."

When we plan and teach lessons - in any subject area - we make hundreds of decisions around a specific learning goal. We make decisions about the questions we'll ask to help students see connections. We make decisions about which particular pieces of student work to share at the end. In each of these choices, we are being explicit about the idea we want to bring to light.

We are explicitly planning, not explicitly teaching.

I've had many moments of frustration when planning and teaching a problem-based curriculum. Ideas can go unnoticed. Connections can go unmade. I sometimes used to spend the end of class standing at the board, frantically trying to explain something, because I was worried students were about to leave without getting "the point" of the lesson. Somehow, the kids didn't read my mind! Moments like this came about because my planning wasn't explicit enough.

I've found Mary Kay Stein and Peg Smith's "5 Practices Framework" a useful tool for explicit planning. The framework forces me to think continuously about the goal, choose an activity that supports that goal, plan questions to move students toward the goal and sequence student work in a way that creates a productive, purposeful discussion toward an explicit idea. I believe this could work in any subject area, as a way of supporting our students in their development of true understanding.
Mary Kay Stein and Peg Smith's "5 Practices Framework:"

1. Anticipate what students will do - what strategies they will use - in solving a challenge.

2. Monitor student work as they approach the challenge in class.

3. Select students whose strategies are worth discussing in class.

4. Sequence those students' presentations to maximize their potential to increase students' learning.

5. Connect the strategies and ideas in a way that helps students understand the concepts learned.

Mary Kay and Peg also describe a Practice 0 of Choosing a Mathematical Goal and Appropriate Task in their book - like selecting the right puzzle to put together on a particular day.

This framework can apply across various contexts and subject areas, but I think it's embodied in the open education resource (OER) curriculum by Illustrative Mathematics. Each lesson and unit tells a mathematical story in which students arrive at a specific mathematical landing point. While students may not all arrive in the same way or at the same time, the problems and discussions are structured to ensure students don't leave the day's work without understanding the goal.

Explicit planning requires us to understand the final picture - the goal - before the puzzles pieces can come together. It requires us to carefully connect each piece, because one missing piece will leave the final picture unfinished. While planning lessons for our unique groups of students, we may not at first see exactly how each puzzle piece will connect, and sometimes we can feel stuck. But our colleagues and students can help until all the connections are clear - and we're proud of the final picture we see.
Next Steps

Check out the video to see teachers learning about these 5 Practices in a recent IM professional development.*

Find a sample lesson plan written using the 5 Practices here: Grade 6, Unit 3, Lesson 4, Activity 1: Road Trip in Illustrative Mathematics 6-8 Math Curriculum.

If you are reading the 5 Practices book , I would love to hear your personal connections and reflections as you read.

If you are using the 5 Practices to plan for activities in your classroom, I would love to hear about your experiences.

to learn more about professional development of the 5 Practices for your classroom or school.

*Professional Development based on NCTM's 5 Practices for Orchestrating Productive Mathematics Discussions by Smith and Stein.

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The 5 Practices Framework: Explicit Planning vs Explicit Teaching - IM CERTIFIED® BLOG

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"Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first." By Kristin Gray I've come to think that approaching a lesson plan is like approaching a 500-piece puzzle.

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Classify solutions to pairs of linear equations.MDC

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for ‘Career and College Readiness’ at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routin…

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Task-based curriculum emphasizing the SMPs

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professional development for common core state standards

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Classify solutions to pairs of linear equations.MDC

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Task-based curriculum emphasizing the SMPs

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professional development for common core state standards

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surrounded and covered.6G.Problem of the Month

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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08.Three Potential Goal Statements for a Multiplication Lesson

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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Most Misunderstood Standards in Grade 6 Math

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Most Misunderstood Standards in Grade 6 Math

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Over the past five years I have had the privilege of visiting many mathematics classrooms, over 1,000 and counting. People who know me well know I am either talking about mathematics instruction, reading about mathematics instruction, or planning for mathematics instruction. I define myself as an instructional nerd! One of the many perks of my classroom visits is the opportunity to provide feedback to teachers, instructional coaches, and administrators. I use the Instructional Practice Guide for Coaching as my lens for lesson observation and feedback. You can learn more about the coaching tool here.

During my visits, I definitely have noticed some trends in instruction. There are some standards, regardless of the teacher, the school, or the district that seem to be commonly misunderstood. Somehow the intent of these standards is lost during instruction. I do not believe this is ever done intentionally, but could be a byproduct of many factors. Maybe it is because old standards are still lingering in the current curriculum; maybe lessons and/or materials have been recycled or repurposed from before the Common Core State Standards; or maybe there is a lack of understanding of what the standard actually means. Regardless of the reason, I would like us to learn and grow our brains together! I hope through this series of posts, a broader network of math educators will be better equipped to plan for and implement standards-based instruction.

I would like to look more in-depth at two of these standards, starting with my favorite Grade 6 math standard, 6.NS.A.1. It is such a beautiful standard if understood and instructed for understanding! Let's look at this task from Illustrative Mathematics.

Each of the solution paths shown above relies on visual models over the algorithm for division of a fraction by a fraction. No need for "Keep, Change, Flip" as students are making sense of fraction division through context using models! There are problems where models do not make sense. I mean, who wants to draw a model for 5 2/5 ÷ 3/7? So, if your learning goal is focused on using visual models, be strategic with the number choice used in your problems!

The next standard of interest is 6.G.A.1. The intent of this standard is to have students composing and decomposing figures in order to find the area of composite figures. These Illustrative Mathematics and EngageNY tasks are examples of the type of tasks that students should be engaging in to meet the intent of this standard. Both problems expect students to decompose the given figure into rectangles and/or triangles in order to find the area of the shape. There is no need to require students memorize the area of a trapezoid! They should decompose it into shapes for which they know how to find the area!

The suggestions given here are just a start to figuring out this very challenging profession of teaching students mathematics. I hope my experiences in many middle school classrooms will help grow your brain around these standards and now that you know better, you will do better in ensuring each student learns math at a high level. Next month, Most Misunderstood Middle School Mathematics Standards in Grade 7. Stay tuned and #InstructUP!

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Most Misunderstood Middle School Mathematics Standards In Grade 6 - Achieve the Core Aligned Materials

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Over the past five years I have had the privilege of visiting many mathematics classrooms, over 1,000 and counting. People who know me well know I am either talking about mathematics instruction, reading about mathematics instruction, or planning for mathematics instruction. I define myself as an instructional nerd!

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Factors Associated with the Maintenance and Decline of High

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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Illustrative Mathematics Free Curriculum (OER)

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Illustrative Mathematics Free Curriculum (OER)

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Math Curriculum Overview - Open Up Resources

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Open Up Resources’ core Math and ELA curricula are all:

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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Learning Trajectories- Dr. Jere Confrey, et.al:  CCSSM trajectories

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Learning Trajectories- Dr. Jere Confrey, et.al:  CCSSM trajectories

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Welcome to TurnOnCCMath.net! The GISMO research team at North Carolina State University has developed 18 learning trajectories that unpack the K-8 Common Core State Standards for Mathematics from the standpoint of student learning, and elaborate the underlying scientific research in mathematics education. The hexagon map illustrates these learning trajectories, standard by standard.

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Illustrative Mathematics Free Curriculum (OER)

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Illustrative Mathematics Free Curriculum (OER)

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Math Curriculum Overview - Open Up Resources

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Open Up Resources’ core Math and ELA curricula are all:

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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MA Curriculum Unit & Lesson Plan templates

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Most Misunderstand Standards in Grade 7 Math

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Most Misunderstand Standards in Grade 7 Math

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In my last post we explored the Most Misunderstood Middle School Mathematics Standards in Grade 6. I have loved the conversations I've engaged in with math educators about the post and welcome more people to let me know your thoughts! I can be found on Twitter here: @FLMathNinja.

In this post, we are going to take a dive into the most misunderstood middle school standards in Grade 7. Again, this is not about evaluating or judging teachers, but rather looking at the practice of mathematics instruction in Grade 7 holistically and learning more together about the standards. With that said, these are not the only misunderstood standards, but are those that I have seen the most over the years while supporting teachers.

I would like to look more in-depth at two Grade 7 standards, starting with 7.RP.A.2a. This standard gives two examples of methods for determining if a proportional relationship exists by analyzing tables or graphs. Note that using a strategy such as cross products is not part of this standard! I am going to use an example from a lesson available through the OER EngageNY that could be used to meet the first part of this standard.

In this example, we would want students to reason with sets of ratios to determine if they are equivalent. They could compare different sets of ratios such as 10:4 and 5:2 in the table. Students could also use what they learned from Grade 6 to determine the unit rate from each scenario and use that reasoning to justify if ratios are equivalent or not. In this example, students could reason 10 ounces cost $4, or $0.40 per ounce; 5 ounces costs $2, or $0.40 per ounce. Students could continue using unit rate reasoning for the remaining ratios (12.5 ounces costs $5; 8 ounces cost $3.20) to determine if they also have a unit rate of $0.40 per ounce.

What I also really like about the Frozen Yogurt example is that the table is provided for students to analyze. Sometimes I have seen students struggling to complete the table from a given context or expression. Before long, the bell has rung and students did not spend much time, if any at all, deciding if the two quantities were in a proportional relationship. Keeping a spotlight on the intent of the standard will help you make instructional decisions to maximize the time students spend working on grade-level standards.

The next standard I would like to discuss is 7.EE.B.4a. The standard begins with "Solve word problems leading to equations of the form px + q = r and p(x + q) = r." For this standard, I have seen a wide range of approaches, from focusing on one-step equations to not including problems of the form p(x + q) = r. When given an equation such as 5(2+x) many students automatically perform the procedure for distributing without making sense of the structure. This is a chance for teachers to highlight MP 7 (Look for and make use of structure) to set students up for high school work in Algebra . For example, let's analyze this opening exercise from EngageNY. An equation that can be used to represent this situation is where h = hours per day that soccer was practiced. One way to solve this problem for h is to distribute the 5 and then continue by solving the equation. However, if students are given the opportunity to make sense of the situation and structure of the equation, they could reason the number of practice hours in one day can be found by dividing 20 by 5. After this step, it would be easy to figure out the value of h for h + 2= 4 . This type of reasoning is often missing from the Grade 7 classroom. If you want to see an example of a great lesson that highlights this type of thinking, the newly released Illustrative Mathematics curriculum by Open Up Resources has included a lesson on Different Options for Solving One Equation (accessible with a free account). Again, we should have students make sense of the structure of the equation before choosing a method to solve.

As I said in my last post, the suggestions given here are just a start to figuring out this very challenging profession of teaching students mathematics. I hope my experiences in many middle school classrooms will help grow your brain around these standards and now that you know better, you will do better in ensuring each student learns math at a high level. Next month, Most Misunderstood Middle School Mathematics Standards in Grade 8. Stay tuned and #InstructUP!

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Most Misunderstood Middle School Mathematics Standards in Grade 7 - Achieve the Core Aligned Materials

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In my last post we explored the Most Misunderstood Middle School Mathematics Standards in Grade 6. I have loved the conversations I've engaged in with math educators about the post and welcome more people to let me know your thoughts! I can be found on Twitter here: @FLMathNinja.

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Applications of Pythagorean Theorem- Lesson 8.10

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Applications of Pythagorean Theorem- Lesson 8.10

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Most Misunderstood Standards in Grade 8 Math

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Most Misunderstood Standards in Grade 8 Math

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In my last posts we explored the Most Misunderstood Middle School Mathematics Standards in Grade 6 and I have loved the conversations I've engaged in with math educators about the posts and welcome more people to let me know your thoughts! I can be found on Twitter here: .

In this post, we are going to dive into the most misunderstood middle school standards in Grade 8. Again, this is not about evaluating or judging teachers, but looking at the practice of mathematics instruction in Grade 8 holistically, and learning more together about the standards. With that said, these are not the only misunderstood standards, but are those that I have seen the most over the years while supporting teachers.

I would like to go more in-depth with three of these Grade 8 Standards, starting with 8.EE.B.6, Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

When thinking about the aspect of Rigor to be emphasized during instruction, teachers should consider this very much a conceptual standard. In my experience of classroom observations and instructional coaching, sometimes instruction for this standard leads with the procedure of calculating slope using the formula, which strips students of the opportunity to make sense of the meaning of slope and the strong connection it has to their work with Ratio and Proportional Reasoning in grade 7 and transformations in 8.G.A.

In this from Open Up Resources, students are asked to explain why their two assigned triangles are similar. One goal of this lesson is for students to realize no matter how they construct a right triangle with a line segment on line AE as the hypotenuse, they will obtain a constant ratio (when dividing the vertical distance by the horizontal distance). After this key discovery by the students, the teacher can set up the conditions to engage students in Math Practice 8: Look for and express regularity in repeated reasoning. Through repeated calculations of determining the slope using similar triangles, the students will be able to make sense of the standard formula for slope and use it as an alternative method for determining slope.

The next standard I would like to discuss is 8.F.B.4, Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

For whatever reason, instruction seems to be limited to constructing the function only and does not focus on interpreting the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

In this lesson by EngageNY, students have to not only construct the function to model the situation, but they must also interpret the rate of change and the initial value within the context. The openness of the problem (Exercises 1-6) allows students to use their prior understanding from the Expression and Equations Domain to engage in this task using multiple representations (table, graph, function, verbal description).

According to the Focus by Grade Level, Geometry does not show up as Major Work in Middle School until Grade 8. With that in mind, I would like to address the last standard from the Geometry Domain, 8.G.B.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. In my experience in classrooms, instruction typically is limited to two dimensions and does not extend the Pythagorean Theorem to three dimensions.

Here is a nice task in the Grade 8 Curriculum from Open Up Resources that gets at what students are expected to be able to do when applying the Pythagorean Theorem in three dimensions. Students need to use the Pythagorean Theorem twice in this context: once for finding the length of the hypotenuse of the triangle created by the base, which in turn becomes a leg of the second right triangle used to find the length of the blue diagonal. Using a two-dimensional drawing of a three-dimensional figure can be difficult for some students to visualize, so be sure to have physical models of the rectangular prisms handy as students grapple with this task!

As I said in my last post, the suggestions given here are just a start to figuring out this very challenging profession of teaching mathematics. I hope my experiences in many middle school classrooms will help grow your brain around these standards and, now that you know better, you will do better in ensuring each student learns math at a high level. Continue to #InstructUP!

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Most Misunderstood Middle School Mathematics Standards in Grade 8 - Achieve the Core Aligned Materials

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In my last posts we explored the Most Misunderstood Middle School Mathematics Standards in Grade 6 and Grade 7. I have loved the conversations I've engaged in with math educators about the posts and welcome more people to let me know your thoughts! I can be found on Twitter here: @FLMathNinja.

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PA Core Vs. Common Core Standards: Grade 6

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PA Core Vs. Common Core Standards: Grade 6

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PA Core Vs. Common Core Standards: Grade 7

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PA Core Vs. Common Core Standards: Grade 7

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(*not all tasks are created equal!)

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My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

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Linear Functions

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PA Core Vs. Common Core Standards: Grade 8

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PA Core Vs. Common Core Standards: Grade 8

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for ‘Career and College Readiness’ at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routin…

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for ‘Career and College Readiness’ at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routin…

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Mathematics Georgia Standards of Excellence (GSE) 6-8

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Mathematics Georgia Standards of Excellence (GSE) 6-8

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These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

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Task-based curriculum emphasizing the SMPs

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professional development for common core state standards

# data[].attributes.body
(*not all tasks are created equal!)

# data[].attributes.attachment_link.title
My Fav Places to go to Find Rich Math Tasks created by Margie Pearse @pearse_margie & Lane Walker @LaneWalker2

# data[].attributes.headline
Illustrative Mathematics Free Curriculum (OER)

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Illustrative Mathematics Free Curriculum (OER)

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Math Curriculum Overview - Open Up Resources

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Open Up Resources’ core Math and ELA curricula are all:

# data[].attributes.headline
Illustrative Mathematics Free Curriculum (OER)

# data[].attributes.subject
Illustrative Mathematics Free Curriculum (OER)

# data[].attributes.attachment_link.title
Math Curriculum Overview - Open Up Resources

# data[].attributes.attachment_link.description
Open Up Resources’ core Math and ELA curricula are all:

# data[].attributes.headline
Mathematics Georgia Standards of Excellence (GSE) 6-8

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Mathematics Georgia Standards of Excellence (GSE) 6-8

# data[].attributes.attachment_link.description
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

# data[].attributes.attachment_link.description
Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for ‘Career and College Readiness’ at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routin…

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Mathematics Georgia Standards of Excellence (GSE) 6-8

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Mathematics Georgia Standards of Excellence (GSE) 6-8

# data[].attributes.attachment_link.description
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

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Math Talk: Teacher Questions & Sentence Starters

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Math Talk: Teacher Questions & Sentence Starters

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03.3 Act Lesson Template WCPS ELE Version (1)

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Week(s) of Inspirational Math

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Choose your own maths adventure with our interactive tools that allow you to build a custom playlist of inspirational activities!

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8.F.4

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Daily Math Routines that address the SMPs 

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Daily Math Routines that address the SMPs 