Fifth Grade Standards Survey: Using Data to Teach Math Well

This is a snapshot of what some of the survey data looks like. 

I created a fifth grade math standards survey with one question that encompassed the main idea of each fifth grade standard. Students took the survey this week, and this morning I spent time analyzing the data. As usual, the data demonstrates a range from students who have little knowledge of the fifth grade standards to students who are very familiar with the standards' content.  The data also revealed that there are areas of the fifth grade standards that most of the class is very familiar with and areas that only a few have experience with.

I will use this data, in part, as I plan the program for the year ahead. I hope to employ Boaler's "floor to ceiling" approach by creating a number of learning experiences that begin at the review level and extend far beyond fifth grade if students want to take it there.

I have shared the data with the entire teaching team including paraeducators who will be providing a lot of one-to-one support, grade-level teachers who will lead "Response to Intervention" sessions, the math coach who will work with us, the math curriculum director, and the building principal. We all spend time at Professional Learning Community (PLC) meetings looking at data, both informal, formal, and observational, to discuss ways that we can teach each child well.

I will also share this data with family members at the fall conference as one way to demonstrate to family members the standards students are learning and where their child currently falls with regard to those standards. The data will provide a good discussion point about how we can work together to support each child's continued math development throughout fifth grade. I will also use this assessment at the end of the year as well for some or all to assess student growth. I may not use it with students who have already demonstrated precision with most or all of the questions as I will use other assessments for those students.

This process of giving the survey copied below is one way to know who my class is and what they need with regard to developing a strong math foundation. Feel free to take the survey below if you'd like and don't hesitate to provide me with feedback. I will continue to develop this approach.

Analyzing Students' Work Habits, Attitude, and Social Skills Survey

I took all of the work habits and social skill standards from our report card and a few more standards that I include in my teaching and added them to a Google form student survey. Students took the survey. Just taking the survey gave students a chance to think about the language and standards with regard to expectations and attitudes and behaviors for learning success.

Now I'll take the results that represent about 2/3 of the student response (there's still more students to take the survey and I'll update the response accordingly), and give students a chance to analyze the data.

With their math teams, I'll have students study the data chart below. Similar to the way our teaching team analyses data, I'll ask students to first share what they notice. Then we'll share questions that arise about the data. After that we'll discuss how this data set might affect individual's and the class's learning and teaching this year.

As I look at the data, my next steps include the following:

• Recognize that most students have great attitudes and work skills.
• Look for the individual students who answered "rarely" or "never" to see if there are any patterns there. If it's the same student, I want to sit down and talk to that student. If it's a few students, I may have a small group devoted to this.
• Look at the "other" comments and those that made those comments. I want to take their words seriously and see how that might affect my teaching and learning.
• Look closely at the areas where there was more scatter, and talk to students about how we might help the whole class to achieve in those areas.
• Think about categories that are not included in this survey--categories that are integral to learning well today. What would you add?

There were also a few errors in the survey that I need to correct. Students will take this survey including the new questions added again in January before we mark the progress reports. I'll be interested to see how the explicit teaching and learning we'll do together now will affect the results then. 

1. Learning Math Begins with a Good Attitude

This With Math I Can website has many helpful links
for teaching and learning math. 

Today officially begins the math year. After spending the first few school days on team projects, today we'll begin our math focus.

Today's theme is that our attitudes affect math learning engagement and success.

We'll start with the the With Math I Can video listed on this page.

We'll then talk a bit about the video and a few organizational matters.

After that students will review the day's online learning menu and then follow the menu which includes completing an attitudinal/behavioral self assessment and math warm-up games.

Teamwork Matters: Making Math T-Shirts

Start the year by focusing on the fact that students learn better when they work together. Emphasize that your math class is a math team, and as a team the way you work together matters. 

Ask students if they belong to any teams. Then ask them to tell you what's important when it comes to successful teamwork. After that, you may want to show one of my all time favorite teamwork videos, and ask students to look for the attributes of successful teamwork as they watch the video:

After the film, acknowledge that like the Panyee soccer team, most teams have T-shirts. Say that the class will have some fun with numbers as they make paper T-shirts and a class bulletin board that celebrates your dynamic math team.

To make the T-shirts, do the following:

1. Set up a computer and printer in the classroom. Have students take a good picture of their face and print it. I use PhotoBooth for this. Students go to the picture stand, take their photo, and print it one after another.
2. Hand out the number activity guide which includes a T-shirt template.
3. Allow students to work with calculators to complete and check their work. 
4. As students work, observe their efforts and listen to their conversation. This observation will serve as a good initial assessment of student effort and need--an assessment that will inform future lessons.
5. Have students complete the task in pencil, check the work with a friend, and then check with you. Coach students with regard to correct completion of the work. When complete, have students write over the numbers and words with a fine line pen, write their number on their t-shirt and add the fact data to the front of the T-shirt or just below, tape their picture on top and hand it in for a class bulletin board that will display all the students pictures with a title such as "Teamwork Matters: Marvelous Mathematicians."
6. When students are done, they can work together on the enrichment activities included on the number activity guide. 

This project is a great start-of-year activity, one that provides assessment, fun with numbers, vocabulary/fact review, and an emphasis on teamwork. I recommend. 

Know Your Students: Assess Fact Skill and Knowledge

That Quiz is free and easy to use for simple student assessments and practice. 

While recent math research and writing points to a need to lessen our obsession with fact knowledge, an obsession that serves to make many students so nervous that they end up disliking math, it's still important to develop fact facility and fluidity. When students know their math facts well, they're more able to make connections, compute, and solve math problems.

How can math teachers do this in a sensitive and productive way.

First, it's important to assess students' fact knowledge. Who knows their facts and who still struggles with basic fact knowledge at fifth grade.

To assess this skill in an easy way that provides good data, I like to use That Quiz. That Quiz is a simple, free to use math practice and assessment site.

I simply put all students names into the program and create/assign assessments and practice sessions (outlined below). Generally I create a number of assessments that move from simple and easy to access to more complex and challenging. I have students start with the simpler assessments and move up as they can. Later I assess the assessment results by the number of problems completed, the time it took, and the complexity achieved. That gives me good data about who knows their facts with ease and who is still struggling.

After that I'll review fact results with individual students and parents, set goals with students, and work with the children to gain fact skill and depth of fact understanding through a varied array of investigations, practice, games, and projects as part of the fifth grade math program and during Response to Intervention and Math Tech times.

Fact Assessment Sets for Fifth Grade
50 addition facts
50 subtraction facts
50 addition/subtraction facts
50 multiplication facts
50 division facts
50 multiplication/division facts
100 multiplication/division/addition/subtraction facts
100 facts with simple variables
100 facts with more complex variables

Making Teaching/Learning Goals Explicit

Too often goal setting doesn't result in good work. This is often true because the goal setting does not identify the success criteria and tracking processes upfront.

To set a goal and move towards that goal with careful thought and attention is an amazing process. To honestly identify the goal's movement from start to finish provides information that helps us to reach the goal with strength, and reaching a goal with strength means that we're doing the work we set out to do.

As I think about my teaching/learning goals, goals that are mandated by the Massachusetts' Educator Evaluation system, I want to clearly identify the success criteria and tracking process.

Student Learning Goal
Students will progress in the following areas:

• Identified fifth grade math standards
• Identified learning-to-learn mindsets and behavior with a focus on collaboration, character, perseverance, organization, empathy, confidence, self-awareness, and self-advocacy

Success Criteria/Tracking
Formal and informal formative and summative assessments will demonstrate growth. Assessments will include the following:

• Math Progress: Content, Concept, Skill Assessments, Project Work, and Oral/Written Reflections.
• Learning-to-Learn Mindsets and Behavior Assessments: Observation, conversation, reflection, project work.
• With acknowledgement of new ESSA requirements, tracking will identify growth for all students including the categories of economically disadvantaged, English Language Learners, Special Needs, and Historically Underserved Cultural/Racial Groups.

Teaching/Learning Path

The path to meet these goals will include the following elements:

• Explicit introduction, discussion, and goal setting related to each goal.
• Transparent, inclusive student-teacher project work, practice, reflection, revision, and assessments as part of the overall learning/teaching process
• Open, honest communication with the learning team related to teach goal and the activity related to that goal
• Regular parent/student meetings and communication related to individual coaching for each goal. 
• Extra support where needed to support children's needs related to the goals above. 

I look forward to explicitly sharing the goals, success criteria, and tracking processes above with students as we learn. I will not explicitly discuss the ESSA goal categories with the whole class to avoid any kind of judgement, prejudice, or name calling. However, I will pay attention to those categories as I teach and look for ways that I can build greater cultural proficiency to make good growth for all students including those specifically identified by ESSA. 

A tight focus as noted makes the teaching/learning year more interesting and focused on what I can do for and with children. We know that good progress mostly depends on the relationships we make with students and the time and attention we invest in each child. Our good work matters in this regard. 

Math Homework Study and Routines

“I know you've heard it a thousand times before. But it's true - hard work pays off. If you want to be good, you have to practice, practice, practice.” -  Ray Bradbury

Many studies report that there is little gain from homework, yet I'm not ready to give up on math homework altogether as I believe homework still has value with regard to building good study habits, practice, and independent or family time to think deeply about math ideas.

I do want to heed the homework research and results though by treating homework routines with greater care, differentiation, and simplicity. I don't want homework to turn into added struggle for students or families. Therefore I will establish a positive homework routine beginning on the first day of school.

The routine I'll foster includes the following actions and goals:

• Students will receive a back-to-back weekly assignment sheet (see below) each Wednesday.
• A math practice challenge problem or activity will be assigned for three school day nights: Wednesday, Thursday, Monday, and Tuesday. The time to complete the practice problem will be a 5-10 minutes.
• There will be a 5-10 minute online math tech activity suggested for each day.
• Enrichment activities will be shared as well and those options will be optional.
• Parents or guardians will be asked to sign the sheet each week on Tuesday night to be returned on Wednesday. 
• The assignment sheet will be due each Wednesday and I will review and respond to the students' efforts as well as their weekly class efforts on the weekly sheet and return sheets by the following Monday. 

There are many reasons why I want to establish and embed this routine into the math program.

First, I want a way to think carefully about, and respond to, each child's efforts each week. This simple sheet will give me a chance to do that. Children thrive when they get regular, positive and helpful response.

Next, I want family members to regularly know about the math concept, knowledge, and skill we are focusing on. Although I do explain that in the newsletter, parents are more likely to look closely and engage in a conversation with their child when they have to sign off on a child's paper.

And, I'll use this review, in part, to inform the program as I'll gather important data about student effort, mathematical thinking, and need. I'll be able to use that information for program design and coaching meetings as I work to teach all children well. 

Establishing weekly study and practice routines helps students to learn math well and see the connection between good routines, practice, and learning well. 

What routines do you establish at the start of each school year to promote student success and regular, positive teacher/parent response? How do you coach students in ways that support these routines with success?

Teaching Fifth Grade Math: What Do Students Know?

To assess learning and teaching, it's essential to evaluate what students know at the start of the year. Using the standards and our school progress report as a guide, I created two early year assessments. The first assessment assesses students' attitudes, efforts, and skill. The first time students take that assessment they'll have to think about the attributes related to their learning skills and mindset. The second assessment asks students to answer a host of questions that represent the fifth grade standards (see an example below).

I will review the data with students, colleagues, and parents at the start of the year, and use the data and parent, student, and collegial response to inform the math teaching/learning program for the year ahead.

I welcome you to try out the assessments as I've created the assessment linked above and the one below as trial copies. If you're a fifth grade teacher and have ideas about change, I welcome your thoughts.

Mathematics in Massachusetts: Preparing for Next Generation MCAS

Membership in The Association of Teachers of Mathematics in Massachusetts is one avenue to keeping up with
expectations and changes related to Next Generation MCAS and other state-wide math initiatives.

I like to stay on top of new initiatives. I don't have to backtrack and re-step to meet new policies and protocol. Hence I've signed on to a number of good resources that inform me about upcoming changes and expectations. Massachusetts' Department of Elementary and Secondary Education does a good job with communication. The Commissioner, Mitchell Chester, writes a weekly newsletter that keeps all educators in the state abreast of what's happening with lots of links, details, and opportunities for learning and leading.

Similarly there are many agencies in the state that help to inform and prepare educators as well. Recently I joined one of those associations, Association of Teachers of Mathematics in Massachusetts (ATMIM). ATMIM just sent out an update so that educators can prepare for our upcoming assessment changes for 2016-2017 school year. I respond to the update notes below:

• ". . .all schools will administer the computer-based versions of the ELA and math tests in grades 4 and 8. Fo grades 3, 5, 6, and 7, schools may elect wither the computer-based or the paper-based tests next spring." The note goes on to say that the state encourages schools to use computer-based testing at as many grades as possible. Fortunately, our schools are well equipped to give the computer-based tests. Khan Academy is free and provides student with good practice for these online tests. TenMarks is another program that provides students with good practice for these kinds of tests. Schools may want to look closely at their computer-use schedules so that students have adequate use of computers for this practice and learning. Also, I recommend that schools invite families in early in the year to learn how to use Khan Academy so that they can support their children in this study. This computer time should be in addition to the core program as I don't believe computer practice/learning programs should replace the creative, collaborative, and critical thinking skills-oriented math programs that make up the core program. 
• The state offers support with regard to preparation for these tests. It's my guess that it's best to reach out for this support, if possible, during the lazy days of summer when they're likely to be less busy. "Department staff in our Office of Digital Learning (odl@doe.mass.edu) and in Student Assessment Services (assessment@doe.mass.edu) are available to answer questions from districts and provide additional assistance."

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Early School Year Focus on Landmark Numbers

Last year I began the math year with a focus on landmark numbers. The introduction proved to be helpful with regard to creating a community of math learners who had common language, patterns, tools, resources, and routines.

I will weave the landmark number lessons into the early year math curriculum again this year. In the days ahead I'll revisit this unit to revise, add, and enrich. In the meantime if you have ideas for change and enrichment, let me know.

Eleven: Design Math Learning for Student Engagement and Success

Good learning design impacts student engagement, empowerment, and education in math and all
other disciplines.

How does a teacher design learning well?

It's important to develop a learning design process. Most educators prefer the backend design method, a process brought to life in education by Wiggins and McTighe in their book, Understanding by Design. John Hattie further exemplifies this approach in his wonderful book about how to teach well, Visible Learning for Teachers, Maximizing Impact on Learning. 

Over many years, I've crafted a design process that I use to design units of study. It is a successful approach that takes into account research from the books and expertise mentioned above as well as many experiences and professional learning events I've engaged in.

As I continue to write this math book/blog, I'll utilize the design information, approach, and resources I outline in the presentation below. As you think about your math units of study, I recommend that you also use some or all of the information below to organize and implement good design to reach and teach children well.

Eight: Assess Attitudes, Effort, and Skill

Ruth Charney in her wonderful book, Teaching Children to Care, Classroom Management for Ethical and Academic Growth, points to the need for educators to be explicit about every classroom expectation.

The system I work in explicitly outlines the social skills, work habits, and math effort elements they expect students to reach for throughout the school year. It's important that students and their families understand these expectations explicitly in order to reach the goal of exhibiting and employing these attributes to support successful learning. One way to explicitly relay and quantify these attributes is to present students with a survey to complete at the start, middle, and end of the year.

The initial survey results can be used to inform a teacher and students about his/her student team. The same survey can be used to relay and discuss a student's overall attitude and behavior related to learning with family members and the student during early year conferences.

I plan to use the survey below, a survey that includes the areas our district has decided to target with regard to social skills, work habits, and math effort. I have also added a couple more questions related to knowledge, growth mindset, and math attitude--areas that affect student knowledge and application of the attributes listed.

I will review the survey questions with students before they take the survey. As I review the questions, students and I will discuss each area so that everyone understands the questions. We will also discuss the survey process and how that helps students to learn well and achieve their goals. I'll explain that the survey results will be shared with family members at the first conference too, and I'll urge students to be honest so that as the year moves forward they'll be able to see their growth in this regard.

I've made two copies of the survey, one for students and one for you to try out should you like to see how the survey works and what questions are included. If you have any ideas or thoughts related to this survey, please let me know as I'll likely make a few edits prior to the school year.

Seven: Know the Math Content

You will be able to teach math well if you know the content well.

Too often elementary school teachers don't have a solid foundation of math concept, knowledge, and skill, and realistically, for many educators, it's been some time since they studied math with depth and it's also likely that when you learned math you may have learned it mainly by memorization rather than deep, conceptual study, the kind of study we utilize with students today.

There are many paths to learning the content well in preparation for the school year.

One of my favorite teaching/learning vehicles is Khan Academy. I recommend that you get to know this platform well and do all the exercises associated with the grade level that you will teach in the year ahead. Khan Academy utilizes the latest research with regard to teaching and learning math and develops student understanding in a blended way using mathematical language, models, real-world problems, and many exercises. You may complete the whole grade level section during the summer or plan to complete each unit section prior to teaching that unit. Khan Academy is a terrific resource for teachers and students everywhere when it comes to deep understanding of math concept, knowledge, and skill. Even better, Khan Academy is free and accessible anywhere that you have a computer and WIFI.

Another way to learn the standards and content is to take a course at a local university. When the new standards were announced a few years ago, I signed up for a course right away and it was an awesome, collaborative way to gain expertise with regard to the standards.

Further there are multiple standards-based platforms online that help you to gain a firm understanding of all the standards and how to teach those standards, and system resources offer support too.

To know the content well prepares you well for the teaching and learning ahead. Summer is a great time to shore up content knowledge and skill. Once you gain expertise in the grade-level standards, its important to grow your knowledge of the foundation skills, concept, and knowledge leading up to your grade level and the mathematical content that your grade-level knowledge leads to--the multiple grade-level standards beyond the grade you teach.

Three: Know Your Math Progress Report

Creating 2D, 3D, and animated math models is an integral
part of teaching and learning math today. 

Taking a close look at the report card or progress report helps an educator to define the priorities that are important with regard to his/her teaching and the organization he or she works for. I analyzed our fifth grade report card below with regard to social development, work habits, math effort, and math skills, concept, and knowledge expectations. The analysis points to a large number of lessons that should occur early in the year and throughout the year so students are well aware of what's expected and how they can reach those expectations.

Jo Boaler, author of Mathematical Mindsets, is not a big fan of homework, however she recommends that reflective homework assignments can be very helpful and positive. To begin the year, I may include lessons related to social development, work habits, and math effort during each lesson. I also may ask students to reflect on these elements as part of their nightly homework to build capacity as well as practice and knowledge of reflective practice--a skill that's a valuable element of any learner's toolbox.

As a new or existing math teacher, take some time this summer to review and analyze your school's report card or progress report. Think about the standards students will be scored on. In systems like the one I work in scores are simply progress notes such as "progressing towards expectations, meets expectations, or exceeds expectations" and in other systems students are still scored with old-time traditional grades. I am a fan of standards-based reports as they foster a progressive, learner's mindset. The kind of mindset that demonstrates to learners that we're all on the continuum and our good efforts help us to move closer to our goals (standards).

Below I analyze each element of the report card that I have to grade in the year ahead. The analysis demonstrates to me a good order and process for teaching each concept. Not all required standards are included on our reports, but the standards with greatest priority are included.

Social Development

Perseveres when faced with challenges:
What does this look like? How does one persevere? When have you persevered in the past? To introduce this topic, I may present an open task and have students work on it with a focus on their needed perseverance. We'll reflect later on what that felt like, and how they motivated themselves and each other to keep going.

Uses positive strategies to resolve conflict:
We'll review the "Take the first step" process of conflict resolution which is to talk to the person you're having the conflict with. The second step is to get help. We'll brainstorm the types of conflicts that happen at school and the many ways one can resolve those conflicts.

Accepts responsibility for own actions:
What do you do when you make a mistake or do something wrong? How do you deal with that?

Follows school and classroom rules:
First we'll create a class constitution with our rules and protocols. Then we'll talk about ways one can work to follow those rules.

Plays cooperatively with peers:
What does it mean to play cooperatively? What does this look like?

Work Habits

Demonstrates initiative:
What is initiative? How does one demonstrate that in school?

Seeks help as needed:
Don't stay stuck! When you need help, seek it out. How can one do this?

Manages time well:
What does it mean to manage time? What strategies do you use to do this.

Works independently and productively:
What id independent work? When might a student be asked to work independently? What does "working productively" mean? What does that look like?

Transitions in a timely and appropriate manner:
What does transition mean? When do we transition? What is appropriate behavior for transition?

Follows directions, routines, and procedures:
First we need to establish routines and procedures and make those routines and procedures explicit. Similarly all directions need to be clear and explicit. So to follow directions, routines, and procedures, first students need to understand what they are. Next students need to brainstorm together the strategies they use to follow directions, routines, and procedures well.

Mathematics Effort

Participation:
What does optimal participation look like in the math classroom? How can we encourage one another to participate? What hinders student participation?

Asks for Clarification:
What does the word "clarification" mean? What are the ways that you might ask for clarification during class or after class? What happens if you ask, and you're still not clear?

Attends to Work:
What's expected with regard to "attending to work" and what does that look like in a classroom? Will this look different during different kinds of learning experiences?

Perseveres when Challenged:
What does perseverance look like in the math classroom? What behaviors are exhibited when you are persevering, and what activities are not exhibited in this regard?

Attends to precision while applying mathematical concepts:
Why is precision important in mathematics? What strategies can one use to be precise?

Math Concept, Knowledge, and Skill

Communicate mathematical thinking orally and in writing using math vocabulary:
Math vocabulary study is integral to math understanding. Every unit should begin with a review of the vocabulary. Vocabulary should be clearly visible in the classroom and accessible for at-home study too. Students should regularly express their mathematically thinking with written or spoken words. Creation of math scripts and videos support this work which ties in nicely with the Standards of Mathematical Practice.

Represent and interpret data using graphing skills:
Initially this is a great standard to meet when making early year infographics to help a class get to know one another. The line plot is the featured graph for fifth grade at this time. Later in the year this standard can be reviewed as you solve fraction problems and chart fraction/decimal data related to real-world problem solving.

Automatically recalls multiplication and division facts through 12:
What do we know about the numbers one through twelve? What factors and multiples can we identify related to these numbers? Which of these numbers are square numbers, composite numbers, prime numbers, or perfect numbers? Where are these numbers present in our culture, and what other names do we have for these numbers? How can we gain automaticity with facts? Why is this important? How do we do with this now, and how can we develop this skill?

Coordinate Grid Knowledge and Use: Able to graph points on the coordinate plane to solve problems.
Understand how to graph points, and then how to use graphing to determine numerical relationships. Student enjoy learning this and practicing it. This study can be embedded throughout the year as students analyze numerical relationships and solve problems.

Classifies two-dimensional figures based on their properties with particular attention to parallel, intersecting, and perpendicular lines:
Students enjoy this study which can be taught well using drawing, tangrams, geoblocks, and online games/venues.

Understand the concepts of volume and applies formulas to solve problems:
Since volume problems essentially include simple numbers, this is a great unit to introduce as you review basic facts. It's also good to review area and perimeter at this time before you get to area models of multiplication and division with larger numbers. Further this unit comes nicely after an initial review of two-dimension geometry since understanding those shapes and how to talk about them informs the volume, area, and perimeter unit.

Use positive and negative numbers to describe quantities:
This concept fits nicely into initial number talk at the start of the year when we look carefully at the numbers 1-12 and where those numbers fit on the number line. Again this can be reviewed during money-related word problems.

Writes and interprets numerical expressions using parentheses:
This is a great skill to teach as you teach students to use calculators effectively. This is also a great teaching goal to master when you are working with "easy numbers" and warming up math year skills. Though it is scheduled on our reports for Term Two, most teach this early in the year as it can be useful and practiced throughout the year as number concepts, knowledge, and skill become more sophisticated.

Generates, extends, and compares patterns and relationships:
Again this is scheduled for term two, but should be a focus of every math lesson with the following questions: What patterns do you see? How do the patterns compare? How might you extend the pattern? What relationships do you see? How would you define the relationship? Can you represent the pattern and/or relationship using a numerical or algebraic expression? Can you graph the relationship or pattern?

Understands the base ten place value system from millions to thousandths?
What are the parts of the place value system? How do these parts work together? Why was the base ten place value system invented? What value does it have in our culture? How do the values of numbers change as you move up and down the place value system? How do the values of numbers change when you add, subtract, multiply, or divide by base ten numbers?

Perform addition, subtraction, multiplication operations with multi-digit whole numbers?
Why do we use the word "operation" when we talk about addition, subtraction, and multiplication? Why do we add, subtract, and multiply? How can we perform those operations effectively and efficiently? Is there one way to do this? Is there a best way or does this depend on when and why we are performing the operation? What 2D, 3D, and animated models can we make to demonstrate these operations?

Perform division operation with multi-digit whole numbers?
What does it mean to divide? When would we use this operation? How can you divide one number by another? What words do we use to describe the numbers associated with division? What models can we make to depict division?

Performs operations with decimals to the hundredths:
When do we need or want to add, subtract, multiply, or divide decimal numbers? Why is this important? How can we perform these operations? Working with money and problem solving is an important art of this study.

Convert standard measurement units within a given measurement system:
This study fits nicely with new science standards that included working with matter. Many experiments can be done that find students converting measurements while exploring matter. This also alines well with base ten system study as students look at models of the base ten system and match those models to metric models.

Interpret a fraction as division of the numerator by the denominator:
1/2 = 1 divided by 2 which equals .5, both 1/2 and .5 are equivalent to half of the whole. Proper fractions like decimals demonstrate part of a whole. We can easily turn a fraction into a decimal by dividing the numerator by the denominator. Sometimes it's easier to work with fractions and sometimes it's easier to work with decimals. A good way to practice this skill is to begin by making a fraction, decimal, percent, and ratio equivalency chart. Later it's good to look at how this skill relates to meaningful fraction numbers with real-time problem solving and project work.

Add and subtract fractions with unlike denominators and solve word problems with addition and subtraction of fractions:
Use a signature story to describe and discuss this skill: "Amy and Paul were making cookies. They each had a table of ingredients. The cookies called for 3/4 cup of sugar. In the end, Paul had 1/8 cup of sugar let and Amy had 2/6 cups of sugar left. Together do they have enough sugar left to make one more batch of cookies?" How would we figure this out? What can we do to figure out if they have enough sugar left? Later students can make up their own stories and demonstrate this skill conceptually. Lots of fraction model wok supports this effort as well.

Applies and extends previous understandings of multiplication to multiply a fraction or whole number by a fraction and solves related real-world problems.
This work also benefits from signature stories. Begin with multiplying a whole number by a fraction: "Emily wanted to give every child 1/2 a piece of paper for the art project. There were twelve children. How many whole pieces of paper did Emily need? What if Emily wanted to give each child 1/3 piece of paper, then how many sheets of paper would she need?" Draw a model of this story. Solve the problem with a model and mathematically. Make up your own problem, solve with model and numbers. Present your problem to the class.

Next, work with multiplying a fraction by a fraction using the area model. Again use a signature story: "I bought a big rectangular cake for my dad's birthday. After the party I had 1/2 of the cake left. I wanted to give my brother 1/4 of the half cake. What portion of the original cake did I give to my brother?" Show how this problem looks with a model, then solve the problem mathematically. Have students make up their own problems to solve with the area model (similar to the cake), picture model, and using the fraction multiplication algorithm.

Applies and extends previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions:
Introduce with a story: "I was going to the fireworks with my family. I had three pieces of licorice, but there were eight people in the car. I decided to split the licorice into 1/4 pieces. If I did that, would I have enough licorice pieces for all people in the car?" How can we write this problem as an expression? What would this problem look like as a model? Do I have enough pieces if I divide 3 by 1/4? Or "I had 1/4 of a pizza left. I want to split it into 3 equal pieces. What size pieces will I end up with?" What does this problem look like as an expression and model? Students can make up their own fraction division problems to solve with friends and present to the class.

Understand what a mixed number is and solve related world problems:
Discuss mixed numbers. Brainstorm when we use mixed numbers in real-time. Look at the many ways we can write and/or model a mixed number. Add, subtract, multiply, and divide mixed numbers. Solve and create real-world mixed number problems.