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**math**. Show all posts## Thinking Outside the Box with Math

December 18, 2019

It’s funny to think about changing instruction just for engagement. But that is what I did with better support my differentiation efforts. Oh, I should mention my principle LOVED the engagement on a recent walkthrough.

Last time I mentioned spending more time looking at and using more “science” than “art” in my elementary resource room. Mostly, because I have no programming. That let me down what could have been a rabbit hole to find some sort of small group instruction but not sit and get. I mean even in my eight student math group, I have the same range you would find in a classroom and all at least two years behind.

Would it surprise you to know, that most special education resource rooms only do some version of sit and get? Differentiated but limit independent skills practice. Many times all these guys need is a reteach and time to practice—think guided release from Fisher and Frey. But what if you have kiddos who need more direction instruction—what do you do then? Bore one or move to fast for them to get the skill.

What this “idea” MUST have: guided direct instruction, varied independent practice, engagement, and easy to put together (both time and money).

Visible Learning research stresses:

Creating math centers has helped meet students' individual needs and continued to challenge everyone without the fear of failure and create an environment where risks are celebrated. I have found that thinking outside of the box is what has motivated students to do their best and reach for challenges and be more accepting with grappling with the material they don’t understand. But it didn’t come at the cost of having success criteria that pushes them to focus on their progress in math.

I’m not sure it means they changed their minds about math and they know like it but I do know they work harder during our math time. They ask more questions. They take more risks. But

Last time I mentioned spending more time looking at and using more “science” than “art” in my elementary resource room. Mostly, because I have no programming. That let me down what could have been a rabbit hole to find some sort of small group instruction but not sit and get. I mean even in my eight student math group, I have the same range you would find in a classroom and all at least two years behind.

Would it surprise you to know, that most special education resource rooms only do some version of sit and get? Differentiated but limit independent skills practice. Many times all these guys need is a reteach and time to practice—think guided release from Fisher and Frey. But what if you have kiddos who need more direction instruction—what do you do then? Bore one or move to fast for them to get the skill.

What this “idea” MUST have: guided direct instruction, varied independent practice, engagement, and easy to put together (both time and money).

Visible Learning research stresses:

- Focusing on progress: shifting from focusing on what teachers are doing to what students are learning
- Errors are welcome: creating a classroom where errors facilitate learning and growth
- Explicit success criteria: students know the learning intentions of each lesson and the criteria for success
- The right level of challenge: teachers set challenging goals, and offer students opportunities for deliberate practice to meet those challenges

Creating math centers has helped meet students' individual needs and continued to challenge everyone without the fear of failure and create an environment where risks are celebrated. I have found that thinking outside of the box is what has motivated students to do their best and reach for challenges and be more accepting with grappling with the material they don’t understand. But it didn’t come at the cost of having success criteria that pushes them to focus on their progress in math.

I’m not sure it means they changed their minds about math and they know like it but I do know they work harder during our math time. They ask more questions. They take more risks. But

Math centers have become a fun way for my students to gain independence in the classroom while reinforcing the concepts taught back in their general education classrooms.

Math centers allow them to practice a math topic in a variety of ways--each one focuses on the same skill allowing student s to gain independence while working towards mastery.

They have four centers:

- Direct Instruction
- Independent Skill Practice
- Technology
- Games

Students visit all four centers twice over the course of a week. Direct instruction is teacher-directed and I provide instruction on the current math skill using guided release.

Independent skill practice is either current skill or past skills depending on where they happen to be on their way to skill mastery. But this station like technology and games is totally independent practice. Unlike Direct Instruction, this means its differentiation depending on where the student is on their learning math skills.

I'm very fortunate to have iPads, which means they have a math app folder from which they choose how they want to spend that rotation time. I change the apps with each skill change, so there is allows something different there.

The Games station doesn't always change when skills change. It depends, with our current skill, money, I slowly changed out the games as I taught the new ones.

I hope I have given you an idea of how you can change up your math group.

Chat soon,

Labels:differentiation,math | 0
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## Math Preschool Style

April 30, 2017

Preschooler, experiencing the world through play as they explore and learn with great enthusiasm. Giving preschoolers a solid foundation in early math literacy is critical to their future academic success, not to mention how important it is to their day-to-day functioning.

The California Mathematics Council maintains a For Families section at its website (www.cmc-math.org/family/main.html). Here you will find articles on mathematics education issues of interest to parents, hands-on activities to do at home and information on how to host your own Family Math event at your preschool or education center.

The Math Forum (www.mathforum.org) is a web portal to everything “mathematics.” Here you can ask Dr. Math questions and get answers! You will also find weekly and monthly math challenges, Internet math hunts, and math resources organized by grade level.

Head Start–Early Childhood Learning and Knowledge Center (www.eclkc.ohs.acf.hhs.gov/hslc) is linked to the federal Head Start Program. Here you will find information about government programs for early learning, including resources that are available to families.

Thinkfinity (www.thinkfinity.org) is a project of the Verizon Foundation. This website has more than 55,000 resources—including many that focus on math—that have been screened by educators to ensure that content is accurate, up-to-date, unbiased, and appropriate for students. The resources on this website are grouped by grade level and subject area.

PBS Parents, the early education website of the Public Broadcasting Service (www.pbs.org/parents/education/math/activities), offers numerous resources, including the stages of mathematics learning listed for babies through second grade children. It is also a rich source of math activities to do at home

Math at Play (www.mathatplay.org) offers multimedia resources for anyone who works with children from birth to age five. Here you can explore early mathematical development and the important ways that caregivers nurture children’s understanding of math concepts through social-emotional relationships, language, everyday play experiences, materials, and teaching.

Let’s Read Math (www.letsreadmath.com/math-and-childrens-literature/ preschool/) wants to make parents and families aware of the growing body of children’s literature with themes related to mathematics. Here you will find a long annotated list of live links to preschool children’s books with math themes, listed by title, author, and mathematics topic.

#### How preschoolers learn the many aspects of math

Most preschoolers, even without guidance from adults, are naturally interested in math as it exists in the world around them. They learn math best by engaging in dynamic, hands-on games and projects. Preschoolers love to ask questions and play games that involve the many aspects of math. The table below lists the key aspects of preschool math, along with simple games and activities you can use to help your child learn them.#### Math Games and Activities

- Count food items at snack time (e.g., 5 crackers, 20 raisins, 10 baby carrots)
- Use a calendar to count down the days to a birthday or special holiday. Help your child see the connection between a numeral like "5," the word "five," and five days on the calendar.
- Practice simple addition and subtraction using small toys and blocks.
- Play simple board games where your child moves a game piece from one position to the next.
- Have your child name the shapes of cookie cutters or blocks.
- Arrange cookie cutters in patterns on a cookie sheet or placemat. A simple pattern might be: star-circle-star-circle.
- Let your child help you measure ingredients for a simple recipe - preferably a favorite!
- Measure your child's height every month or so, showing how you use a yardstick or tape measure. Mark his or her height on a "growth chart" or a mark on a door frame. Do the same with any siblings. Help your child compare his or her own height to previous months and also to their siblings' heights.
- Talk through games and daily activities that involve math concepts.
- Have your child name numbers and shapes.
- Help them understand and express comparisons like more than/less than, bigger/smaller, and near/far.
- Play games where you direct your child to jump forward and back, to run far from you or stay nearby.
- Use songs with corresponding movements to teach concepts like in and out, up and down, and round and round.

#### Website Ideas

The Early Math Learning website (www. earlymathlearning.com) includes free downloads of PDF files of this Early Learning Math at Home booklet as well as individual chapters. Additional articles and resources for families will be added regularly.The California Mathematics Council maintains a For Families section at its website (www.cmc-math.org/family/main.html). Here you will find articles on mathematics education issues of interest to parents, hands-on activities to do at home and information on how to host your own Family Math event at your preschool or education center.

The Math Forum (www.mathforum.org) is a web portal to everything “mathematics.” Here you can ask Dr. Math questions and get answers! You will also find weekly and monthly math challenges, Internet math hunts, and math resources organized by grade level.

Head Start–Early Childhood Learning and Knowledge Center (www.eclkc.ohs.acf.hhs.gov/hslc) is linked to the federal Head Start Program. Here you will find information about government programs for early learning, including resources that are available to families.

Thinkfinity (www.thinkfinity.org) is a project of the Verizon Foundation. This website has more than 55,000 resources—including many that focus on math—that have been screened by educators to ensure that content is accurate, up-to-date, unbiased, and appropriate for students. The resources on this website are grouped by grade level and subject area.

PBS Parents, the early education website of the Public Broadcasting Service (www.pbs.org/parents/education/math/activities), offers numerous resources, including the stages of mathematics learning listed for babies through second grade children. It is also a rich source of math activities to do at home

Math at Play (www.mathatplay.org) offers multimedia resources for anyone who works with children from birth to age five. Here you can explore early mathematical development and the important ways that caregivers nurture children’s understanding of math concepts through social-emotional relationships, language, everyday play experiences, materials, and teaching.

Let’s Read Math (www.letsreadmath.com/math-and-childrens-literature/ preschool/) wants to make parents and families aware of the growing body of children’s literature with themes related to mathematics. Here you will find a long annotated list of live links to preschool children’s books with math themes, listed by title, author, and mathematics topic.

Labels:math,parents,preschool,technology | 0
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## Best Practices: Number Sense

October 02, 2016

Number sense begins very early and must be a focus of primary math. This is the solid foundation in math that all kids need.

A sense of numbers is critical for primary students to develop math problem solving skills.

The National Council of Teachers of Mathematics increasingly calls for districts to give more attention to building this skill, and studies have found that number sense accounts for 66% of the variance in first grade math achievement. The council have also addressed five critical areas that are characteristic of students who have good number sense:

Having a sense of how numbers work is a very broad topic that covers all numerical thinking. At its core, it is making sense of math concepts and mathematical reasoning.

Operationally, it is counting skills, having number knowledge, using estimation, and the ability to use problem solving strategies.

Knowing the why of how numbers work is of utmost importance, and children should not be shown the how until they understand the "why." Techniques such as using ten frames and using concrete models to show place value concepts are daily necessities for young children.

Inquiry-based approaches (such as math dice games) to teaching children mathematics should be utilized as primary teaching methods in the early grades.

This is not to say that explicit teaching of sense of numbers skills is not essential, especially for those students from low socio-economic status. We absolutely need to do this.

It is saying that teachers should provide multiple opportunities for students to experience numbers and make connections before putting the pencil to paper.

Carefully consider your objectives and the type of learners in your room when choosing a math game to include. NCTM also suggests you consider:

Seven ways teachers can directly impact a developing sense of number.

1. Link school math to real-world experiences

Present students with situations that relate to both inside and outside classroom experiences. Students need to recognize that numbers are useful for solving problems.

2. Model different computing methods

Focus on what methods make sense for different situations. There is no one right way to compute. We need our students to be flexible thinkers.

3. Mental Math

Real life requires mental computation. Students need to be able to move numbers around in their heads and discuss their strategies.

4. Discuss Strategies

Students must be able to explain their reasoning. This not only will give you insight into how they think, but also will help the children to cement their own ideas and reevaluate them.

5. Estimate

This should be embedded in problem solving. This is not referring to textbook rounding. Real life estimation is about making sense of a problem and using anchor numbers to base reasoning on.

6. Question Students About Reasoning Strategies

All the time, not just when they make a mistake. Constantly probing sends several important messages: your ideas are valued, math is about reasoning, and there are always alternative ways to look at a problem.

7. Measuring Activities

When teaching children mathematics, measuring activities should be front and center. Make students verify estimates through doing.

A sense of numbers is critical for primary students to develop math problem solving skills.

The National Council of Teachers of Mathematics increasingly calls for districts to give more attention to building this skill, and studies have found that number sense accounts for 66% of the variance in first grade math achievement. The council have also addressed five critical areas that are characteristic of students who have good number sense:

- Number Meaning
- Relationships Between Numbers
- Number Magnitude
- Operations Involving Numbers
- Referents for Numbers/Quantities (referents are words or phrases that denote what something stands for)

__WHAT IS NUMBER SENSE IN CHILDREN?__Having a sense of how numbers work is a very broad topic that covers all numerical thinking. At its core, it is making sense of math concepts and mathematical reasoning.

Operationally, it is counting skills, having number knowledge, using estimation, and the ability to use problem solving strategies.

Knowing the why of how numbers work is of utmost importance, and children should not be shown the how until they understand the "why." Techniques such as using ten frames and using concrete models to show place value concepts are daily necessities for young children.

Inquiry-based approaches (such as math dice games) to teaching children mathematics should be utilized as primary teaching methods in the early grades.

This is not to say that explicit teaching of sense of numbers skills is not essential, especially for those students from low socio-economic status. We absolutely need to do this.

It is saying that teachers should provide multiple opportunities for students to experience numbers and make connections before putting the pencil to paper.

Carefully consider your objectives and the type of learners in your room when choosing a math game to include. NCTM also suggests you consider:

- the type of mathematical practices involved in each game (there should be more than one)
- how feedback will be given
- does the game encourage competition, collaboration and communication?
- the types of strategies students will have to use to solve a puzzle or to win

Seven ways teachers can directly impact a developing sense of number.

1. Link school math to real-world experiences

Present students with situations that relate to both inside and outside classroom experiences. Students need to recognize that numbers are useful for solving problems.

2. Model different computing methods

Focus on what methods make sense for different situations. There is no one right way to compute. We need our students to be flexible thinkers.

3. Mental Math

Real life requires mental computation. Students need to be able to move numbers around in their heads and discuss their strategies.

4. Discuss Strategies

Students must be able to explain their reasoning. This not only will give you insight into how they think, but also will help the children to cement their own ideas and reevaluate them.

5. Estimate

This should be embedded in problem solving. This is not referring to textbook rounding. Real life estimation is about making sense of a problem and using anchor numbers to base reasoning on.

6. Question Students About Reasoning Strategies

All the time, not just when they make a mistake. Constantly probing sends several important messages: your ideas are valued, math is about reasoning, and there are always alternative ways to look at a problem.

7. Measuring Activities

When teaching children mathematics, measuring activities should be front and center. Make students verify estimates through doing.

Labels:math,teaching | 0
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## Websites to Support Math Planning

July 27, 2016

Planing for specific and targeted math instruction is a challenge and some days a pain. I work to make sure my instruction resources are free. I also work with these ideas in mind--even when I think I know which direction I need to go in next.

Mathematics interventions at the Tier 2 level of a multi-tier prevention system must incorporate six instructional principles:

- Instructional explicitness
- Instructional design that eases the learning challenge
- A strong conceptual basis for procedures that are taught
- An emphasis on drill and practice
- Cumulative review as part of drill and practice
- Motivators to help students regulate their attention and behavior and to work hard

This is a collection of websites I use to plan math instruction to differentiate and help student’s access core instruction.

- The Illustrative Mathematics Project connects mathematical tasks to each of the standards. Bill McCallum, a lead writer of the Common Core State Standards, helped create the site to show the range and types of mathematical work the standards are designed to foster in students.
- The Arizona Academic Content Standards contain explanations and examples for each of the standards created by teachers with the help of Bill McCallum a lead writer of the Common Core State Standards.
- Achieve the Core is the website for the organization Student Achievement Partners (SAP) founded by David Coleman and Jason Zimba, two of the lead writers of the Common Core State Standards. The website shares free, open-source resources to support Common Core implementation at the classroom, district, and state level. The steal these tools link includes information on the key instructional shifts for math and guidance for focusing math instruction.

- The Model Content Frameworks from Partnership for Assessment of Readiness for College and Careers (PARCC) were developed through a state-led process of content experts in PARCC member states and members of the Common Core State Standards writing team. The Model Content Frameworks are designed help curriculum developers and teachers as they work to implement the standards in their states and districts.
- The What Works Clearinghouse (WWC) has released a new Practice Guide: Teaching Math to Young Children. From naming shapes to counting, many children show an interest in math before they enter a classroom. Teachers can build on this curiosity with five recommendations from the WWC in this practice guide. The guide is geared toward teachers, administrators, and other educators who want to build a strong foundation for later math learning.

The Common Core State Standards were built on mathematical progressions. This website provides links to narrative documents describing the progression of a mathematical topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

- What Works Clearinghouse released a practice guide, Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. In addition, Doing What Works has developed professional development resources associated with the practice guide for Response to Intervention in Elementary-Middle Math.
- The Colorado English Language Proficiency Standards provide educators with an invaluable resource for working with not only English Language Learners in mathematics but developing mathematical language in all students. The Can Do descriptors are particularly helpful entry point to the standards.
- Open source Mathematics materials for English Language Learners, released by Understanding Language, were developed using research-based principles for designing mathematics instructional materials and tasks from two publicly accessible curriculum projects, Inside Mathematics and the Mathematics Assessment Project. Each lesson supports students in learning to communicate about a mathematical problem they have solved, to read and understand word problems, or to incorporate mathematical vocabulary in a problem solving activity.

Labels:lesson plan,math,special education,technology | 0
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## Preschool Math Summer Ideas

July 15, 2016

Preschooler, experiencing the world through play as they explore and learn with great enthusiasm. Giving preschoolers a solid foundation in early math literacy is critical to their future academic success, not to mention how important it is to their day-to-day functioning.

####

Most preschoolers, even without guidance from adults, are naturally interested in math as it exists in the world around them. They learn math best by engaging in dynamic, hands-on games and projects. Preschoolers love to ask questions and play games that involve the many aspects of math. The table below lists the key aspects of preschool math, along with simple games and activities you can use to help your child learn them.

The Early Math Learning website (www. earlymathlearning.com) includes free downloads of PDF files of this Early Learning Math at Home booklet as well as individual chapters. Additonal articles and resources for families will be added regularly.

The California Mathematics Council maintains a For Families section at its website (www.cmc-math.org/family/main.html). Here you will find articles on mathematics education issues of interest to parents, hands-on activities to do at home, and information on how to host your own Family Math event at your preschool or education center.

The Math Forum (www.mathforum.org) is a web portal to everything “mathematics.” Here you can ask Dr. Math questions and get answers! You will also find weekly and monthly math challenges, Internet math hunts, and math resources organized by grade level.

Head Start–Early Childhood Learning and Knowledge Center (www.eclkc.ohs.acf.hhs.gov/hslc) is linked to the federal Head Start Program. Here you will find information about government programs for early learning, including resources that are available to families.

Thinkfinity (www.thinkfinity.org) is a project of the Verizon Foundation. This website has more than 55,000 resources—including many that focus on math—that have been screened by educators to ensure that content is accurate, up-to-date, unbiased, and appropriate for students. The resources on this website are grouped by grade level and subject area.

PBS Parents, the early education website of the Public Broadcasting Service (www.pbs.org/parents/education/math/activities), offers numerous resources, including the stages of mathematics learning listed for babies through second grade children. It is also a rich source of math activities to do at home

Math at Play (www.mathatplay.org) offers multimedia resources for anyone who works with children from birth to age five. Here you can explore early mathematical development and the important ways that caregivers nurture children’s understanding of math concepts through social-emotional relationships, language, everyday play experiences, materials, and teaching.

Let’s Read Math (www.letsreadmath.com/math-and-childrens-literature/ preschool/) wants to make parents and families aware of the growing body of children’s literature with themes related to mathematics. Here you will find a long annotated list of live links to preschool children’s books with math themes, listed by title, author, and mathematics topic.

####

How preschoolers learn the many aspects of math

Most preschoolers, even without guidance from adults, are naturally interested in math as it exists in the world around them. They learn math best by engaging in dynamic, hands-on games and projects. Preschoolers love to ask questions and play games that involve the many aspects of math. The table below lists the key aspects of preschool math, along with simple games and activities you can use to help your child learn them.#### Math Games and Activities

- Count food items at snack time (e.g., 5 crackers, 20 raisins, 10 baby carrots)
- Use a calendar to count down the days to a birthday or special holiday. Help your child see the connection between a numeral like "5," the word "five," and five days on the calendar.
- Practice simple addition and subtraction using small toys and blocks.
- Play simple board games where your child moves a game piece from one position to the next.
- Have your child name the shapes of cookie cutters or blocks.
- Arrange cookie cutters in patterns on a cookie sheet or placemat. A simple pattern might be: star-circle-star-circle.
- Let your child help you measure ingredients for a simple recipe - preferably a favorite!
- Measure your child's height every month or so, showing how you use a yardstick or tape measure. Mark his or her height on a "growth chart" or a mark on a door frame. Do the same with any siblings. Help your child compare his or her own height to previous months and also to their siblings' heights.
- Talk through games and daily activities that involve math concepts.
- Have your child name numbers and shapes.
- Help them understand and express comparisons like more than/less than, bigger/smaller, and near/far.
- Play games where you direct your child to jump forward and back, to run far from you or stay nearby.
- Use songs with corresponding movements to teach concepts like in and out, up and down, and round and round.

#### Website Ideas

The Early Math Learning website (www. earlymathlearning.com) includes free downloads of PDF files of this Early Learning Math at Home booklet as well as individual chapters. Additonal articles and resources for families will be added regularly.

The California Mathematics Council maintains a For Families section at its website (www.cmc-math.org/family/main.html). Here you will find articles on mathematics education issues of interest to parents, hands-on activities to do at home, and information on how to host your own Family Math event at your preschool or education center.

The Math Forum (www.mathforum.org) is a web portal to everything “mathematics.” Here you can ask Dr. Math questions and get answers! You will also find weekly and monthly math challenges, Internet math hunts, and math resources organized by grade level.

Head Start–Early Childhood Learning and Knowledge Center (www.eclkc.ohs.acf.hhs.gov/hslc) is linked to the federal Head Start Program. Here you will find information about government programs for early learning, including resources that are available to families.

Thinkfinity (www.thinkfinity.org) is a project of the Verizon Foundation. This website has more than 55,000 resources—including many that focus on math—that have been screened by educators to ensure that content is accurate, up-to-date, unbiased, and appropriate for students. The resources on this website are grouped by grade level and subject area.

PBS Parents, the early education website of the Public Broadcasting Service (www.pbs.org/parents/education/math/activities), offers numerous resources, including the stages of mathematics learning listed for babies through second grade children. It is also a rich source of math activities to do at home

Math at Play (www.mathatplay.org) offers multimedia resources for anyone who works with children from birth to age five. Here you can explore early mathematical development and the important ways that caregivers nurture children’s understanding of math concepts through social-emotional relationships, language, everyday play experiences, materials, and teaching.

Let’s Read Math (www.letsreadmath.com/math-and-childrens-literature/ preschool/) wants to make parents and families aware of the growing body of children’s literature with themes related to mathematics. Here you will find a long annotated list of live links to preschool children’s books with math themes, listed by title, author, and mathematics topic.

Labels:math,parents,technology | 0
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## May Pinterest Pick 3

May 03, 2016

I think Colorado has decided its time for Spring. Or maybe its just this week since by Thursday it's going to be 80. I'm not sure though.

As its the end of the year, I'm thinking ahead to the fall and things I want to change. The big thing is--creating pathways to personalized learning. This is a big on my teacher rubric. This is not a small idea because I also need to integrate technology into this grand plan. Oh, I almost forgot IEP goals still drive instruction.

I'm not a fan of handing student's an iPad just to have them play a game or something else that's just plug and play. Student's have to do something with them--technology has to be a jumping off point to something even better. SAMR provides that. A big piece of the teacher rubric in students using technology in a meaningful way. I see students of a limited time. Students have to use them but I want them to do more than just replace a task for a task. Tat's harder than you may think. These guys have tons of apps but not clue what any of them do. Yup-fun times. So, as the year winds down they are going to become familiar with different apps and what they can do with them. Of course, they love this idea but they have not seen a rubric attached to their work.

The big push is coming in the form of personalize learning. I'm not totally sold because I'm not sure how this meshes with IEP goals and the like. However, with the reading I've done it doesn't seem to be a totally bad idea. This is something I will play with this month before leaving on break. I really like that this idea is ground in differentiated instruction. Any more its the hallmark of great things regardless of who is watching.

Just in case you didn't know, the TpT site-wide Teacher Appreciation Sale is this Tuesday and Wednesday! Everything in my store will be 20% and you can get an additional 10% off by using the promo code CELEBRATE at checkout. This a great time to load up on bundles as they are already discounted, so with the sale you save...well, a bundle! You might also want to check out no-prep Interactive math picture book or my Errorless Sentence Stems.

Have a great week. Happy shopping.

As its the end of the year, I'm thinking ahead to the fall and things I want to change. The big thing is--creating pathways to personalized learning. This is a big on my teacher rubric. This is not a small idea because I also need to integrate technology into this grand plan. Oh, I almost forgot IEP goals still drive instruction.

One thing that is big with my teacher rubric is student goal setting. The point being the instruction is student driven. I'm not sure if the IEP goals and student driven learning go hand in hand but I'm game to take it out to play. I like this idea because it's a SMART goal minus the SMART goal language. Students can focus on an IEP goal and set a short term outcome. The hard part is right know I don't have tons of extra time but next year the team is looking at moving to a three week instruction with the fourth week being progress monitoring. This idea is used with out SLPs this year but I'm thinking it may be worth trying next year.

I'm not a fan of handing student's an iPad just to have them play a game or something else that's just plug and play. Student's have to do something with them--technology has to be a jumping off point to something even better. SAMR provides that. A big piece of the teacher rubric in students using technology in a meaningful way. I see students of a limited time. Students have to use them but I want them to do more than just replace a task for a task. Tat's harder than you may think. These guys have tons of apps but not clue what any of them do. Yup-fun times. So, as the year winds down they are going to become familiar with different apps and what they can do with them. Of course, they love this idea but they have not seen a rubric attached to their work.

The big push is coming in the form of personalize learning. I'm not totally sold because I'm not sure how this meshes with IEP goals and the like. However, with the reading I've done it doesn't seem to be a totally bad idea. This is something I will play with this month before leaving on break. I really like that this idea is ground in differentiated instruction. Any more its the hallmark of great things regardless of who is watching.

Just in case you didn't know, the TpT site-wide Teacher Appreciation Sale is this Tuesday and Wednesday! Everything in my store will be 20% and you can get an additional 10% off by using the promo code CELEBRATE at checkout. This a great time to load up on bundles as they are already discounted, so with the sale you save...well, a bundle! You might also want to check out no-prep Interactive math picture book or my Errorless Sentence Stems.

Have a great week. Happy shopping.

Labels:Linking Party,math,Pinterest,writing | 3
comments

## RTI Activities for Your Math Class & Giveaway

November 29, 2015

I’m always asked what are simple things that teachers can do in their rooms to support RTI in math. These four are easy to do and don’t require tons of up-front work and meet the learning needs of all the learners in your room.

1) Math Journaling

Implementing a math journal allows your students to "think about their thinking" (metacognition) and record it in a way that makes sense to them. This journaling process gives you a window into each student's mind to determine where he or she needs help or enrichment.

Encourage students to draw, write and calculate in a math journal to solve problems, work through processes, and explain their actions. Assign math journals once a day, once a week or even once a month to create an invaluable, ongoing formative assessment.

In respect to RTI, you can differentiate journal assignments for Tier 1 students by providing open-ended questions, like "How would you quickly count all of the toes in this classroom?" Differentiate further for Tier 2 and Tier 3 students by asking more concrete questions, based on the concepts they are currently working on.

Math journals are a great way for students to show critical thinking and their problem solving skills.

Looking for good examples of a math journal?

Check out: Pinterest user Susan Cardin's "Math Journal" board.

2) Manipulatives

Consider a kindergarten classroom. It's likely stocked with colorful bins full of plastic toys, connecting cubes, blocks and three-dimensional shapes. Now, somewhere along the way to middle school those toys got left behind, but the cubes, blocks, and three-dimensional shapes still serve as valuable manipulative materials.

Manipulatives help students of all ages learn and understand math concepts, from counting to multiplication and division. Break out these manipulatives -- foregoing toys in an effort to respect the maturity of eighth graders -- to introduce more complex math concepts in a way students can see and touch (and talk about).

These manipulatives do not necessarily have to be concrete either! Recent educational technology developments even allow students to use virtual manipulatives on a touchscreen or laptop.

Your students will benefit from "seeing" math concepts in a new way. As they progress, some Tier 1 students will likely leave the tactile manipulatives behind as they "get it." Tier 2 and 3 students can continue to refer back to the objects (virtual and/or physical) for to help form better understandings and reinforce prior knowledge.

Check out: Megan Campbell's "Math Lessons,Manipulatives, & Ideas" board showcases a nice variety of manipulative ideas for math students of all ages and ranges.

3) Introduce and Review Math Vocabulary

As you know, math is its own language. Beginning in the early grades, your students learned terms like "sum", "difference", or "addend". These words (hopefully) became part of their everyday vocabulary. However, these mathematics terms often require revisiting and scaffolding, regardless of the student's current learning level and goals.

Post a running list of math vocabulary in the classroom and review it often. Going back to strategy one, ask students to journal about specific terms and real world application. It will be interesting to see how each student uniquely describes the term "factor" or "exponent." Allow students to draw, diagram or provide examples of terms rather than memorizing a textbook definition.

Learning the vocabulary will help all students become more familiar with math concepts. In respect to your RTI model, you can stratify the complexity of the terms and the method of reviews between the tiers. For example, Tier 1 students might be best suited to learn more complex terms, as necessary, while Tier 2 and 3 students can continue to revisit learned terms via differentiated modalities as they develop needed comprehension. Plus, most state assessments use math vocabulary changing it or watering it down will cause confusion later on.

Check out: "Math Vocabulary Builders" Pinterest Board from Carol Camp for great math vocab activities and ideas!

4) Think Aloud

When teaching, or re-teaching, math concepts, using a "Think Aloud" activity is a great method for students to understand, hear, and see what's going on in your head as you solve the problem or work through a mathematical process.

Walk students through several examples by thinking aloud each step of the way. Encourage struggling students to model the "think aloud" process by asking them to explain each step as they go. This can be done in a whole-class, small group, or partner setting.

While Tier 1 students often "get it" without further explanation, thinking aloud helps break complex processes down into manageable steps for Tier 2 and 3 students. Also, by hearing and seeing explanations from their peers, students often have "light bulb" moments that may not have clicked during your teacher-led instruction. I use Think Alouds several times a week-I even work to get my students to lead them!

I hope you find something to take back to your class. Be sure to fill out the Rafflecopter to get a Broncos Magnet and a 25 dollar gift certificate to Teachers pay Teachers--just in time for Cyber Monday. Don't forget everything on my site will also be on sale!!

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## What is Composing and Decomposing? (And why is it important to Computational Fluency)?

July 19, 2015

One task that I find students struggle with is seeing the trees in the forest--breaking apart numbers to small ones. This skill is the beginning of place value in the kindergarten and first grade but becomes a powerful addition and multiplication strategy later on. Common Core Standard 2.NTB.B.5 moves students to using the break apart strategy on an Open Number Line to add and subtract.

When talking about computational fluency, many of the current articles use the terms composing and decomposing numbers. These are terms that may not be familiar to most parents. They are really not anything new. These terms refer to the idea that numbers can be put together or broken apart to make other numbers.

For instance, the number ten can be broken up (decomposed) in many ways.

10 = 5 + 5

10 = 4 + 6

10 = 3 + 3 + 2 + 2

This may seem like a simple idea, but to a child just learning about numbers it is not simple at all. We want to foster this understanding because it is a critical understanding in terms of becoming computational fluency.

When solving the problem 28 + 45 a student decomposed 28 into 20 + 5 + 3.

Can you see how that would make this problem easier to solve?

28 + 45 = 20 + 5 + 3 + 45 = (45 + 5) + 20 + 3 = 73

This skill can even be helpful when learning basic facts. For instance, when doing 7 + 8, a student might decompose the 7 or the 8 to make ten and extras.

7 + 8 = 5 + (2 + 8) = 5 + 10 = 15

Students have developed a firm understanding of place value of two-digit numbers and to subtract multiples of ten, and are ready to add and subtract within 100 (including the case of adding or subtracting a two-digit number and a one-digit number, and two two-digit numbers). First students are given problems where regrouping is not necessary, and later, problems where regrouping is necessary. Further, students understand that in addition and subtraction, digits in the ones place are added and subtracted; digits in the tens place are added and subtracted; and sometimes regrouping is necessary. In addition, sometimes we must regroup ten ones to form an additional ten, and in subtraction, sometimes we must break a ten into ten ones. The eventual goal of this standard is fluency. This will not happen all at once; students will build gradually towards having procedures and strategies by which they can fluently add an subtract, including standard algorithms and skip-counting up or down.

In a given addition and subtraction problems, ask students to identify which digits are in the one and tens positions. In addition, they should be able to identify the digits in the ones and tens positions.

Provide students with a variety of manipulatives and technologies (such as base-ten blocks or drawings) which can aid in their practice of addition and subtraction through 100.

Use properties of addition to make addition more fluid. For example, 64 + 8 can be thought of as (62 + 2) + 8, reordering addends allows (2 + 62) + 8, regrouping addends yields 2 + (62 + 8) which gives 2 + 70, which is equal to 72. This is what is meant by using a strategy with the properties of the operations. Another strategy would be using skip-counting for addition (e.g., 58 + 15 can be found by skip-counting up from 58 by ten, and then by five. An additional strategy is to break down place value: 58 + 15 can be thought of as 50 + 10 + 8 + 5, which is 60 + 13, or 73.

Students should have at least one algorithm in place that is robust and works in all cases, but they also should be encouraged to use alternative strategies if they can do so quickly and accurately.

When talking about computational fluency, many of the current articles use the terms composing and decomposing numbers. These are terms that may not be familiar to most parents. They are really not anything new. These terms refer to the idea that numbers can be put together or broken apart to make other numbers.

For instance, the number ten can be broken up (decomposed) in many ways.

10 = 5 + 5

10 = 4 + 6

10 = 3 + 3 + 2 + 2

This may seem like a simple idea, but to a child just learning about numbers it is not simple at all. We want to foster this understanding because it is a critical understanding in terms of becoming computational fluency.

When solving the problem 28 + 45 a student decomposed 28 into 20 + 5 + 3.

Can you see how that would make this problem easier to solve?

28 + 45 = 20 + 5 + 3 + 45 = (45 + 5) + 20 + 3 = 73

This skill can even be helpful when learning basic facts. For instance, when doing 7 + 8, a student might decompose the 7 or the 8 to make ten and extras.

7 + 8 = 5 + (2 + 8) = 5 + 10 = 15

Students have developed a firm understanding of place value of two-digit numbers and to subtract multiples of ten, and are ready to add and subtract within 100 (including the case of adding or subtracting a two-digit number and a one-digit number, and two two-digit numbers). First students are given problems where regrouping is not necessary, and later, problems where regrouping is necessary. Further, students understand that in addition and subtraction, digits in the ones place are added and subtracted; digits in the tens place are added and subtracted; and sometimes regrouping is necessary. In addition, sometimes we must regroup ten ones to form an additional ten, and in subtraction, sometimes we must break a ten into ten ones. The eventual goal of this standard is fluency. This will not happen all at once; students will build gradually towards having procedures and strategies by which they can fluently add an subtract, including standard algorithms and skip-counting up or down.

In a given addition and subtraction problems, ask students to identify which digits are in the one and tens positions. In addition, they should be able to identify the digits in the ones and tens positions.

Provide students with a variety of manipulatives and technologies (such as base-ten blocks or drawings) which can aid in their practice of addition and subtraction through 100.

Use properties of addition to make addition more fluid. For example, 64 + 8 can be thought of as (62 + 2) + 8, reordering addends allows (2 + 62) + 8, regrouping addends yields 2 + (62 + 8) which gives 2 + 70, which is equal to 72. This is what is meant by using a strategy with the properties of the operations. Another strategy would be using skip-counting for addition (e.g., 58 + 15 can be found by skip-counting up from 58 by ten, and then by five. An additional strategy is to break down place value: 58 + 15 can be thought of as 50 + 10 + 8 + 5, which is 60 + 13, or 73.

Students should have at least one algorithm in place that is robust and works in all cases, but they also should be encouraged to use alternative strategies if they can do so quickly and accurately.

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## The Importance of Memorizing the Times Tables (plus freebie)

July 10, 2015

One of the hardest math concepts is learning multiplication is an essential part of a student’s elementary education. It’s the foundation for more complex math student encounter as they move through school.

multiplication and memorizing the times tables are building blocks for other math topics taught in school - higher learning such as division, long multiplication, fractions and algebra. Students who have not memorized the times tables will find these levels of math much more difficult than they need to be. Students who have not mastered their tables will very often fall behind in math and begin to lose confidence. Multiplication is used in our daily lives. You might need it when doubling a recipe, determining a discount at a store or figuring out our expected arrival time when traveling.

Have fun together in this process. One of my favorite ways is to use games! It's always a good review and opportunity for the whole family to exercise their brains.

2. Explain why it is important.

3. Demonstrate what fast recall is.

4. Be interested in math yourself.

5. Find out what facts they already know.

6. Involve your child in the goal setting process.

7. Focus primarily on the facts they need to learn.

8. Use a chart to monitor progress.

9. Provide encouragement along the way.

10. Spend quality time together practicing.

11. Acknowledge their success.

12. And most importantly: Have fun!

#### Why memorize the times tables?

Just like everything in life to do more complex task like walking before running learningmultiplication and memorizing the times tables are building blocks for other math topics taught in school - higher learning such as division, long multiplication, fractions and algebra. Students who have not memorized the times tables will find these levels of math much more difficult than they need to be. Students who have not mastered their tables will very often fall behind in math and begin to lose confidence. Multiplication is used in our daily lives. You might need it when doubling a recipe, determining a discount at a store or figuring out our expected arrival time when traveling.

#### Calculators?

Calculators are great tools for figuring out complex calculations. However, using a calculator takes much longer for simple facts and can result in keying errors. Students who rely on calculators are also weak in estimating skills and are unaware of wrong answers that occur from keying mistakes. In Colorado and on most high stakes tests calculators are not allowed in many tests and admission exams—they have to use paper and don’t have the time to work them the using strategies.#### Understanding or memorization or both?

It's not one or the other, it's both. A student must understand and memorize the facts. Early on, a student needs to understand what multiplication is - the grouping of sets, repeated addition, and a faster way of adding. Show them this with an assortment of manipulatives, by skip counting and by using arrays. As they master the basics, expand upon this concept by creating interesting word problems. Eventually comes a time to highlight the importance of rapid recall. Students need to know that they should recall the answer instantaneously. This is why quizzing and practicing need to happen at the same time.Have fun together in this process. One of my favorite ways is to use games! It's always a good review and opportunity for the whole family to exercise their brains.

#### How Parents Can Help Their Child Memorize the Times Tables

1. Make sure there is understanding.2. Explain why it is important.

3. Demonstrate what fast recall is.

4. Be interested in math yourself.

5. Find out what facts they already know.

6. Involve your child in the goal setting process.

7. Focus primarily on the facts they need to learn.

8. Use a chart to monitor progress.

9. Provide encouragement along the way.

10. Spend quality time together practicing.

11. Acknowledge their success.

12. And most importantly: Have fun!

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## Preschoolers and Number Sense: Summertime Ideas

June 14, 2015

One of the most important math skills students need to learn is number sense. It is the bases for more completed math skills we learn through elementary. I have included some games you can play this summer to build number sense while having fun!

Preschool number activities often involve counting, but merely reciting the number words isn't enough. Kids also need to develop "number sense," an intuitive feeling for the actual quantity associated with a given number.That's where these activities can help. Inspired by research, the following games encourage kids to think about several key concepts, including:

CCSS.MATH.CONTENT.K.CC.A.1: Count to 100 by ones and by tens.

CCSS.MATH.CONTENT.K.CC.A.2: Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

CCSS.MATH.CONTENT.K.CC.A.3: Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). K.CC.B:

Count to tell the number of objects.

CCSS.MATH.CONTENT.K.CC.B.4: Understand the relationship between numbers and quantities; connect counting to cardinality.

CCSS.MATH.CONTENT.K.CC.B.4.A: When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

CCSS.MATH.CONTENT.K.CC.B.4.B: Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

CCSS.MATH.CONTENT.K.CC.B.4.C: Understand that each successive number name refers to a quantity that is one larger.

CCSS.MATH.CONTENT.K.CC.B.5: Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

Compare numbers.

CCSS.MATH.CONTENT.K.CC.C.6: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

CCSS.MATH.CONTENT.K.CC.C.7: Compare two numbers between 1 and 10 presented as written numerals.

Most activities use a set of cards and counting tokens. Here’s what you need to get started. Preparing for preschool number activities:

Cards will be used in two ways, (1) as displays of dots for kids to count, and (2) as templates for kids to cover with tokens. Make your cards from heavy-stock writing paper, marking each with an Arabic numeral (1-10) and the corresponding number of dots.

Make your dots conspicuous, and space them far enough apart that your child can easily place one and only one token on top of each dot. The larger your tokens, the larger your cards will need to be.

In addition, you might make multiple cards for the same number--each card bearing dots arranged in different configurations. For example, one “three” card might show three dots arranged in a triangular configuration. Another might show the dots arranged in a line. Still another might show the dots that appear to have been placed randomly. But whatever your configuration, leave enough space between dots for your child to place a token over each dot.

Tokens

Kids can use a variety of objects for tokens, but keep in mind two points.

1. Children under the age of three years are at special risk of choking, so choose big tokens. According to the U.S. Consumer Product Safety Commission, a ball-shaped object is unsafe if it is smaller than a 1.75” diameter golf ball. Other objects are unsafe if they can fit inside a tube with a diameter of 1.25” inches.

2. Kids can get distracted if your tokens are too interesting, so it's best to avoid the fancy plastic frogs or spiders

Start small. It’s important to adjust the game to your child’s attention span and developmental level. For beginners, this means counting tasks that focus on very small numbers (up to 3 or 4).

Keep it fun. If it’s not playful and fun, it’s time to stop. Be patient. It takes young children about a year to learn how the counting system works.

Place a card, face up, before your child. Then ask your child to place the correct number of tokens on the card—one token over each dot.

After the child has finished the task, replace the card and tokens and start again with a new card. Once your child has got the hang of this, you can modify the game by helping your child count each token as he puts it in place.

Choose two cards, each displaying a different number of dots, taking care that the cards differ by a ratio of at least 2:1. For instance, try 1 vs. 2, 2 vs. 4, and 2 vs. 5. You can also try larger numbers, like 6 vs. 12.

Next, set one card in front of your child and the other in front of you. Have your child cover all the dots with tokens (pretending they are cookies) and ask her

After she answers you, you can count to check the answer. But I’d skip this step if you are working with larger numbers (like 6 vs. 12) that are beyond your child’s current grasp. You don’t want to make this game feel like a tedious exercise.

As your child becomes better at this game, you can try somewhat smaller ratios (like 5 vs. 9).

And for another variant, ask your child to compare the total amount of cookies shared between you with the cookies represented on another, third card. In recent experiments, adults who practiced making these sorts of “guesstimates” experienced a boost in their basic arithmetic skills.

Instead of playing with the tokens, have your child place the cards side-by-side in correct numeric sequence. For beginners, try this with very small numbers (1, 2, 3) and with numbers that vary by a large degree (e.g., 1, 3, 6, 12).

Choose three toy creatures as party attendees and have your child set the table—providing one and only plate, cup, and spoon to each toy. Then give your child a set of “cookies” (tokens or real edibles) and ask her to share these among the party guests so they each receive the same amount. Make it simple by giving your child 6 or 9 tokens so that none will be left over.

As always, go at your child’s pace and quit if it isn’t fun. If your child makes a mistake and gives one creature too many tokens, you can play the part of another creature and complain. You can also play the part of tea party host and deliberately make a mistake. Ask for your child’s help? Did someone get too many tokens? Or not enough? Have your child fix it. Once your child gets the hang of things, try providing him with one token too many and discuss what to do about this "leftover." One solution is to divide the remainder into three equal bits. But your child may come up with other, non-mathematical solutions, like eating the extra bit himself.

Play the basic game as described above, but instead of having your child place the tokens directly over the dots, have your child place the tokens alongside the card. Ask your child to arrange his tokens in the same pattern that is illustrated on the card. And count!

For this game, use cards bearing dots only--no numerals. To play, place two cards--each bearing the same number of dots, but arranged in different patterns--side by side. Ask your child to recreate each pattern using his tokens. When she’s done, help her count the number of tokens in each pattern. The patterns look different, but they use the same number of dots/tokens.

Even before kids master counting, they can learn about the concepts of addition and subtraction. Have a puppet “bake cookies” (a set of tokens) and ask your child to count the cookies (helping if necessary). Then then have the puppet bake one more cookie and add it to the set. Are there more cookies or fewer cookies now? Ask your child to predict how many cookies are left. Then count again to check the answer. Try the same thing with subtraction by having the puppet eat a cookie.

Don’t expect answers that are precise and correct. But you may find that your child is good at getting the gist. When researchers asked 3-, 4- and 5-year olds to perform similar tasks, they found that 90% of the predictions went in the right direction.

As your child begins to master the first few number words, you can also try these research-tested preschool number activities for teaching kids about the number line. Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.

Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions:

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added.

Increase the number of dots on the cards.

Memory Match (5-7 years) 2 players

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions

Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. They then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (example: If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions:

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Preschool number activities often involve counting, but merely reciting the number words isn't enough. Kids also need to develop "number sense," an intuitive feeling for the actual quantity associated with a given number.That's where these activities can help. Inspired by research, the following games encourage kids to think about several key concepts, including:

- Relative magnitudes
- The one-to-one principle of counting and cardinality (two sets are equal if the items in each set can be matched, one-to-one, with no items left over)
- The one-to-one principle of counting (each item to be counted is counted once and only once)
- The stable order principle (number words must be recited in the same order)
- The principle of increasing magnitudes (the later number words refer to greater cardinality)
- The cardinal principle

#### Common Core Standards These Games Target:

Know number names and the count sequence.CCSS.MATH.CONTENT.K.CC.A.1: Count to 100 by ones and by tens.

CCSS.MATH.CONTENT.K.CC.A.2: Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

CCSS.MATH.CONTENT.K.CC.A.3: Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). K.CC.B:

Count to tell the number of objects.

CCSS.MATH.CONTENT.K.CC.B.4: Understand the relationship between numbers and quantities; connect counting to cardinality.

CCSS.MATH.CONTENT.K.CC.B.4.A: When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

CCSS.MATH.CONTENT.K.CC.B.4.B: Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

CCSS.MATH.CONTENT.K.CC.B.4.C: Understand that each successive number name refers to a quantity that is one larger.

CCSS.MATH.CONTENT.K.CC.B.5: Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

Compare numbers.

CCSS.MATH.CONTENT.K.CC.C.6: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

CCSS.MATH.CONTENT.K.CC.C.7: Compare two numbers between 1 and 10 presented as written numerals.

Most activities use a set of cards and counting tokens. Here’s what you need to get started. Preparing for preschool number activities:

__Cards__Cards will be used in two ways, (1) as displays of dots for kids to count, and (2) as templates for kids to cover with tokens. Make your cards from heavy-stock writing paper, marking each with an Arabic numeral (1-10) and the corresponding number of dots.

Make your dots conspicuous, and space them far enough apart that your child can easily place one and only one token on top of each dot. The larger your tokens, the larger your cards will need to be.

In addition, you might make multiple cards for the same number--each card bearing dots arranged in different configurations. For example, one “three” card might show three dots arranged in a triangular configuration. Another might show the dots arranged in a line. Still another might show the dots that appear to have been placed randomly. But whatever your configuration, leave enough space between dots for your child to place a token over each dot.

Tokens

Kids can use a variety of objects for tokens, but keep in mind two points.

1. Children under the age of three years are at special risk of choking, so choose big tokens. According to the U.S. Consumer Product Safety Commission, a ball-shaped object is unsafe if it is smaller than a 1.75” diameter golf ball. Other objects are unsafe if they can fit inside a tube with a diameter of 1.25” inches.

2. Kids can get distracted if your tokens are too interesting, so it's best to avoid the fancy plastic frogs or spiders

#### Games to play

One you have your cards and tokens, you can play any of the preschool number activities below. As you play, keep in mind the points raised in my evidence-based guide to preschool math lessons:Start small. It’s important to adjust the game to your child’s attention span and developmental level. For beginners, this means counting tasks that focus on very small numbers (up to 3 or 4).

Keep it fun. If it’s not playful and fun, it’s time to stop. Be patient. It takes young children about a year to learn how the counting system works.

__The basic game: One-to-one matching__Place a card, face up, before your child. Then ask your child to place the correct number of tokens on the card—one token over each dot.

After the child has finished the task, replace the card and tokens and start again with a new card. Once your child has got the hang of this, you can modify the game by helping your child count each token as he puts it in place.

__The Tea Party: Relative magnitudes__Choose two cards, each displaying a different number of dots, taking care that the cards differ by a ratio of at least 2:1. For instance, try 1 vs. 2, 2 vs. 4, and 2 vs. 5. You can also try larger numbers, like 6 vs. 12.

Next, set one card in front of your child and the other in front of you. Have your child cover all the dots with tokens (pretending they are cookies) and ask her

__“Which of us has more cookies?”__After she answers you, you can count to check the answer. But I’d skip this step if you are working with larger numbers (like 6 vs. 12) that are beyond your child’s current grasp. You don’t want to make this game feel like a tedious exercise.

As your child becomes better at this game, you can try somewhat smaller ratios (like 5 vs. 9).

And for another variant, ask your child to compare the total amount of cookies shared between you with the cookies represented on another, third card. In recent experiments, adults who practiced making these sorts of “guesstimates” experienced a boost in their basic arithmetic skills.

__Bigger and bigger: Increasing magnitudes__Instead of playing with the tokens, have your child place the cards side-by-side in correct numeric sequence. For beginners, try this with very small numbers (1, 2, 3) and with numbers that vary by a large degree (e.g., 1, 3, 6, 12).

__Sharing at the tea party: The one-to-one principle__Choose three toy creatures as party attendees and have your child set the table—providing one and only plate, cup, and spoon to each toy. Then give your child a set of “cookies” (tokens or real edibles) and ask her to share these among the party guests so they each receive the same amount. Make it simple by giving your child 6 or 9 tokens so that none will be left over.

As always, go at your child’s pace and quit if it isn’t fun. If your child makes a mistake and gives one creature too many tokens, you can play the part of another creature and complain. You can also play the part of tea party host and deliberately make a mistake. Ask for your child’s help? Did someone get too many tokens? Or not enough? Have your child fix it. Once your child gets the hang of things, try providing him with one token too many and discuss what to do about this "leftover." One solution is to divide the remainder into three equal bits. But your child may come up with other, non-mathematical solutions, like eating the extra bit himself.

__Matching patterns: Counting__Play the basic game as described above, but instead of having your child place the tokens directly over the dots, have your child place the tokens alongside the card. Ask your child to arrange his tokens in the same pattern that is illustrated on the card. And count!

__Matching patterns: Conservation of number__For this game, use cards bearing dots only--no numerals. To play, place two cards--each bearing the same number of dots, but arranged in different patterns--side by side. Ask your child to recreate each pattern using his tokens. When she’s done, help her count the number of tokens in each pattern. The patterns look different, but they use the same number of dots/tokens.

__The cookie maker: Making predictions about changes to a set__Even before kids master counting, they can learn about the concepts of addition and subtraction. Have a puppet “bake cookies” (a set of tokens) and ask your child to count the cookies (helping if necessary). Then then have the puppet bake one more cookie and add it to the set. Are there more cookies or fewer cookies now? Ask your child to predict how many cookies are left. Then count again to check the answer. Try the same thing with subtraction by having the puppet eat a cookie.

Don’t expect answers that are precise and correct. But you may find that your child is good at getting the gist. When researchers asked 3-, 4- and 5-year olds to perform similar tasks, they found that 90% of the predictions went in the right direction.

__The Big Race: Increasing magnitudes and the number line__As your child begins to master the first few number words, you can also try these research-tested preschool number activities for teaching kids about the number line. Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.

__Deal and Copy (4-5 years) 3-4 players__Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions:

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added.

Increase the number of dots on the cards.

Memory Match (5-7 years) 2 players

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions

- Increase the number of pairs of cards used.
- Use a greater number of dots on the cards.
- Pair a dot card with a numeral card.

__What's the Difference? (7-8 years) 2-4 players__Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. They then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (example: If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions:

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Have a great time playing this games this summer that target important math skills. Happy playing!

Labels:math,parents,preschool,summer | 1 comments

## What is Mental Math?

January 12, 2015

For me mental math plays a huge part of building number sense and a students ability to work math in their heads. Some days most of my math block is spent doing mental math and other days it may only be 3 minutes of an activity. I have listed some for my students favorite. They work great for interventions and RTI.

Mental math is the main form of calculation used by most people and the simplest way of doing many calculations. Research has shown that in daily life at least 75% of all calculations are done mentally by adults. However, unfortunately due to the emphasis on written computation in many classrooms, many children believe that the correct way to calculate a simple subtraction fact such as 200-3 is to do it in the written form.

Through regular experiences with mental math children come to realize that many calculations are in fact easier to perform mentally. In addition, when using mental math children almost always use a method which they understand (unlike with written computation) and are encouraged to think actively about relationships involving the particular numbers they are dealing with.

In order to be effective Mental Math sessions should:

- occur on a daily basis (5-10 minutes per day)
- encourage ‘having a go’ on the part of all students
- emphasize how answers were arrived at rather than only whether they are correct
- Promote oral discussion
- allow students to see that there are many ways to arrive at a correct answer rather than one correct way
- build up a dense web of connections between numbers and number facts
- emphasize active understanding and use of place value

Following are some possible activities for K-5 classrooms:

#### Fill the Hundreds Chart:

On day one display a Hundreds Pocket Chart with only 5-6 pockets filled with the correct numerals. Leave all other pockets blank. Select 3 numerals and 3 students. Ask each student to place his/her numeral in its correct pocket and to explain the strategy they used to help them complete this task. Repeat the above with 3 numbers and 3 students per day until all pockets are filled. Take note of students who use a count by one strategy and those who demonstrate an awareness of the base ten patterns underlying the chart. Select numbers based on your knowledge of individual student’s number sense (e.g. you may select a number immediately before or after a number that is already on the board for one child and a number that is 10 or 11 more than a placed number for another child who you feel has a good understanding of the base ten pattern).#### Possible questions to involve other students:

Yesterday we had __ numbers on our number chart and today we added 3 more. How many numbers do we now have on our number chart? How do you know?If there are __ numbers on our number chart how many more numbers do we need to add to fill our chart? Ask several students to explain the strategy used to solve this problem.

We now have ____ numbers on our number chart. If we continue to add 3 numbers every day how many more days/weeks will it take to fill our number chart? Explain your thinking.

#### Today’s Number is…

Select a number for the day (e.g. 8) and write it on the board or chart paper. Ask students to suggest calculations for which the number is the answer. Write students' suggestions in 4 columns (addition examples, subtraction, multiplication and division). After 8 or 10 responses, focus in on particular columns or types of responses that you would like more of. For example,"Give me some more addition examples", "Give me some ways which use three numbers", "Give me an example using parentheses" etc.#### What's My Number

Select a number between 1 and 100 and write it down without revealing it to your students. Have students take turns to ask questions to which you can only answer ‘yes’ or ‘no’. Record each question and answer on chart paper. For example:Is it greater than 30? No

Is it an even number? Yes

Is it a multiple of 3? No

Does it have a 4 in the ones place?...

After 3 or 4 questions ask, “What is the smallest number it could still be? What is the largest? Discuss why it is better to ask a question such as "Is it an odd number?" than "Is it 34?" early in the game. To ensure that all students are involved have them use individual laminated 100 charts with dry erase markers to mark off numbers after each question is asked. Keep going until the number has been named correctly. During the game you may also want to keep track of how many questions are asked before the number is named. Next time you play challenge students to guess the number with fewer questions.

'Friendly' number activities

Give a number less than 10. Students must respond with an addition fact that will make the number up to 10. For example, if today's target number is 10 and you say 6 the student must respond with "6 + 4 = 10". Vary the target number e.g. 20, 50, 100, 200, 1000 etc. to suit students' ability level.

Labels:math,parents,RTI | 0
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## What is Number Sense?

September 28, 2014

A person's ability to use and understand numbers:

• knowing their relative values,

• how to use them to make judgments,

• how to use them in flexible ways when adding, subtracting, multiplying or dividing

• how to develop useful strategies when counting, measuring or estimating.

a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The National Council of Teachers identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

To begin with, early number activities are best done with movable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single color stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognized without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognized. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.

Card D could be called a random arrangement, though in reality it has been quite deliberately organized to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.

Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added. Increase the number of dots on the cards.

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions

Increase the number of pairs of cards used. Use a greater number of dots on the cards. Pair a dot card with a numeral card.

Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Number Sense plays into how well order students grasp onto the more difficult concepts such as rounding, place value, and learning the basic math facts. Look for more information to come. Have a great week!

• knowing their relative values,

• how to use them to make judgments,

• how to use them in flexible ways when adding, subtracting, multiplying or dividing

• how to develop useful strategies when counting, measuring or estimating.

### What is number sense?

The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well-organized conceptual framework of number information that enablesa person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The National Council of Teachers identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

- mental calculation
- computational estimation
- judging the relative magnitude of numbers
- recognizing part-whole relationships and place value concepts and;
- problem solving

#### How does number sense begin?

An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding. Piaget called this ability to instantaneously recognize the number of objects in a small group 'subitizing'. As mental powers develop, usually by about the age of four, groups of four can be recognized without counting. It is thought that the maximum number for subitizing, even for most adults, is five. This skill appears to be based on the mind's ability to form stable mental images of patterns and associate them with a number. Therefore, it may be possible to recognize more than five objects if they are arranged in a particular way or practice and memorization takes place. A simple example of this is six dots arranged in two rows of three, as on dice or playing cards. Because this image is familiar, six can be instantly recognized when presented this way.Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

#### What teaching strategies promote early number sense?

Learning to count with understanding is a crucial number skill, but other skills, such as perceiving subgroups, need to develop alongside counting to provide a firm foundation for number sense. By simply presenting objects (such as stamps on a flashcard) in various arrangements, different mental strategies can be prompted. For example, showing six stamps in a cluster of four and a pair prompts the combination of 'four and two makes six'. If the four is not subitised, it may be seen as 'two and two and two makes six'. This arrangement is obviously a little more complex than two groups of three. So different arrangements will prompt different strategies, and these strategies will vary from person to person.
If mental strategies such as these are to be encouraged (and just counting discouraged) then an element of speed is necessary. Seeing the objects for only a few seconds challenges the mind to find strategies other than counting. It is also important to have children reflect on and share their strategies. This is helpful in three ways:

- verbalizing a strategy brings the strategy to a conscious level and allows the person to learn about their own thinking;
- it provides other children with the opportunity to pick up new strategies;
- the teacher can assess the type of thinking being used and adjust the type of arrangement, level of difficulty or speed of presentation accordingly.

To begin with, early number activities are best done with movable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single color stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognized without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognized. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.

Card D could be called a random arrangement, though in reality it has been quite deliberately organized to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.

Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

#### What games can assist development of early number sense?

Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.__Deal and Copy (4-5 years) 3-4 players__Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added. Increase the number of dots on the cards.

__Memory Match (5-7 years) 2 players__Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions

Increase the number of pairs of cards used. Use a greater number of dots on the cards. Pair a dot card with a numeral card.

__What's the Difference? (7-8 years) 2-4 players__Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Number Sense plays into how well order students grasp onto the more difficult concepts such as rounding, place value, and learning the basic math facts. Look for more information to come. Have a great week!

Labels:common core,lesson plan,math | 0
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## Math Big Ideas

September 19, 2014

Concrete - Representational - Abstract: Sequence of Instruction

The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

What is it?

What are the critical elements of this strategy?

When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).

How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?

The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.

Understanding manipulatives (concrete objects)

To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives.

General types of math manipulatives:

Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.).

Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1

Suggestions for using Discrete & Continuous materials with students who have learning problems:

Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch. As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used.

Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997):

Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean).

Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each).

Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks).

Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1

Suggestions for using proportional and non-proportional manipulatives with students who have learning problems:

Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values.

Examples of manipulatives (concrete objects)

Suggested manipulatives are listed according to math concept/skill area. Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones.

Colored chips

Beans

Unifix cubes

Golf tees

Skittles or other candy pieces

Packaging popcorn

Popsicle sticks/tongue depressors

Description of use: Students can use these concrete materials to count, to add, and to subtract. Students can count by pointing to objects and counting aloud. Students can add by counting objects, putting them in one group and then counting the total. Students can subtract by removing objects from a group and then counting how many are left.

Base 10 cubes/blocks

Beans and bean sticks

Popsicle sticks & rubber bands for bundling

Unifix cubes (individual cubes can be combined to represent "tens")

Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds."

Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container)

Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers

.

Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.).

Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers. When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution.

Fraction pieces (circles, half-circles, quarter-circles, etc.)

Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc.

Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block).

Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces.

Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape.

Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc.

Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2).

Suggestions for using manipulatives:

At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level.

Drawing is not a "crutch" for students that they will use forever. It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level.

Examples of drawing solutions by math concept level

The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples.

A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving.

Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

Students who are not successful solving problems at the abstract level may:

Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding).

Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding).

Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences).

- Have difficulty with basic facts/memory problems

Suggestions:

The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

What is it?

- Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, beans and bean sticks, pattern blocks).
- Students are provided many opportunities to practice and demonstrate mastery using concrete materials
- The math concept/skill is next modeled at the representational (semi-concrete) level which involves drawing pictures that represent the concrete objects previously used (e.g. tallies, dots, circles, stamps that imprint pictures for counting)
- Students are provided many opportunities to practice and demonstrate mastery by drawing solutions
- The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols)
- Students are provided many opportunities to practice and demonstrate mastery at the abstract level before moving to a new math concept/skill.

What are the critical elements of this strategy?

- Use appropriate concrete objects to teach particular math concept/skill (see Concrete Level of Understanding/Understanding Manipulatives-Examples of manipulatives by math concept area). Teach concrete understanding first.
- Use appropriate drawing techniques or appropriate picture representations of concrete objects (see Representational Level of Understanding/Examples of drawing solutions by math concept area). Teach representational understanding second.
- Use appropriate strategies for assisting students to move to the abstract level of understanding for a particular math concept/skill (see Abstract Level of Understanding/Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers).

When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).

- Provide students many practice opportunities using concrete objects.
- When students demonstrate mastery of skill by using concrete objects, describe & model how to perform the skill by drawing or with pictures that represent concrete objects (representational level of understanding).
- Provide many practice opportunities where students draw their solutions or use pictures to problem-solve.
- When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding).
- Provide many opportunities for students to practice performing the skill using only numbers and symbols.

How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?

- Helps passive learner to make meaningful connections
- Teaches conceptual understanding by connecting concrete understanding to abstract math process
- By linking learning experiences from concrete-to-representational-to-abstract levels of understanding, the teacher provides a graduated framework for students to make meaningful connections.
- Blends conceptual and procedural understanding in structured way

#### Concrete

What is it?The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.

Understanding manipulatives (concrete objects)

To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives.

General types of math manipulatives:

Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.).

Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1

Suggestions for using Discrete & Continuous materials with students who have learning problems:

Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch. As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used.

Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997):

Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean).

Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each).

Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks).

Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1

Suggestions for using proportional and non-proportional manipulatives with students who have learning problems:

Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values.

Examples of manipulatives (concrete objects)

Suggested manipulatives are listed according to math concept/skill area. Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones.

__Counting/Basic Addition & Subtraction Pictures__Colored chips

Beans

Unifix cubes

Golf tees

Skittles or other candy pieces

Packaging popcorn

Popsicle sticks/tongue depressors

Description of use: Students can use these concrete materials to count, to add, and to subtract. Students can count by pointing to objects and counting aloud. Students can add by counting objects, putting them in one group and then counting the total. Students can subtract by removing objects from a group and then counting how many are left.

__Place Value Pictures__Base 10 cubes/blocks

Beans and bean sticks

Popsicle sticks & rubber bands for bundling

Unifix cubes (individual cubes can be combined to represent "tens")

Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds."

__Multiplication/Division Pictures__Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container)

Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers

.

__Positive & Negative Integers Pictures__Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.).

Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers. When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution.

__Fractions Pictures__Fraction pieces (circles, half-circles, quarter-circles, etc.)

Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc.

Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block).

Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces.

__Geometry Pictures__Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape.

__Beginning Algebra Pictures__Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc.

Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2).

Suggestions for using manipulatives:

- Talk with your students about how manipulatives help to learn math.
- Set ground rules for using manipulatives.
- Develop a system for storing manipulatives.
- Allow time for your students to explore manipulatives before beginning instruction.
- Encourage students to learn names of the manipulatives they use.
- Provide students time to describe the manipulatives they use orally or in writing. Model this as appropriate.
- Introduce manipulatives to parents

#### Representational

What is it?At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level.

Drawing is not a "crutch" for students that they will use forever. It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level.

Examples of drawing solutions by math concept level

The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples.

#### Abstract

What is it?A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving.

Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

Students who are not successful solving problems at the abstract level may:

- Not understand the concept behind the skill

Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding).

Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding).

Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences).

- Have difficulty with basic facts/memory problems

Suggestions:

- Regularly provide student with a variety of practice activities focusing on basic facts. Facilitate independent practice by encouraging students to draw solutions when needed (see the student practice strategies Instructional Games, Self-correcting Materials, Structured Cooperative Learning Groups, and Structured Peer Tutoring).
- Conduct regular one-minute timings and chart student performance. Set goals with student and frequently review chart with student to emphasize progress. Focus on particular fact families that are most problematic first, then slowly incorporate a variety of facts as the student demonstrates competence (see evaluation strategy Continuous Monitoring & Charting of Student Performance).
- Teach student regular patterns that occur throughout addition, subtraction, multiplication, & division facts (e.g. "doubles" in multiplication, 9's rule - add 10 & subtract one, etc.)
- Provide student a calculator or table when they are solving multiple-step problems.

__Repeat procedural mistakes____Suggestions:__

- Provide fewer #'s of problems per page.
- Provide fewer numbers of problems when assigning paper & pencil practice/homework.
- Provide ample space for student writing, cueing, & drawing.
- Provide problems that are already written on learning sheets rather than requiring students to copy problems from board or textbook.
- Provide structure: turn lined paper sideways to create straight columns; allow student to use dry-erase boards/lap chalkboards that allow mistakes to be wiped away cleanly; color cue symbols; for multi-step problems, draw color-cued lines that signal students where to write and what operation to use; provide boxes that represent where numerals should be placed; provide visual directional cues in a sample problem; provide a sample problem, completed step by step at top of learning sheet.
- Provide strategy cue cards that student can use to recall the correct procedure for solving problem.
- Provide a variety of practice activities that require modes of expression other than only writing

#### The Big Idea

Student learning & mastery greatly depends on the number of opportunities a student has to respond!! The more opportunities for successful practice that you provide (i.e. practice that doesn't negatively impact student learning characteristics), the more likely it is that your student will develop mastery of that skill.
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