**math**. Show all posts

**math**. Show all posts

## A Path to Ensuring Mastery in Addition and Subtraction for Math Success

Do you happen to know how many standards students have to master to be able to solve basic multiplication and division equations?

I went looking and it’s **more than 15**!

I’m talking about place value, counting, and solving addition and subtraction facts fluently.

These skills are the foundations and need to be taught to mastery!

Do you need help getting your students to master place value or counting skills or addition and subtraction fluency?

If so, you're in luck! In this blog post, we will discuss how to teach these skills and ways to teach these concepts.

The first way is drill and kill practice. This is a traditional approach that many teachers use. The second way is with place-value games. Games are a great way to engage students and help them learn in a fun way. Finally, the third way is to give students time to demonstrate mastery in a variety of different ways.

#### Why Worry about it??

First–if you don't, who will!! Passing the buck doesn’t help anyone and when they get to 3rd grade your students will drown and the teacher who has them will give up.

Thank you for being part of my soap box.

Mastering multiplication and division requires a strong foundation in several basic mathematical skills, notably place value, counting, and addition & subtraction. Each of these skills plays a crucial role in understanding and performing multiplication and division effectively.

#### Place Value

Place value is fundamental in mathematics as it helps in understanding the significance of digits in a number based on their position.Break Down Numbers: Multiplication and division often involve breaking down larger numbers into smaller, more manageable parts. For example, understanding that 234 is 200 + 30 + 4 allows for easier mental multiplication and division using distributive properties.

Align Numbers Properly: When multiplying or dividing multi-digit numbers, place value ensures that digits are aligned correctly, which is crucial for obtaining accurate results. Misalignment can lead to significant errors.

#### Counting

Think for a second, can you students count by 1s past 50 without starting at 1. Or can they skip count by 5s starting at 65. Or counting by 100s starting at 200?

Counting is a foundational skill that underpins many mathematical concepts, including multiplication and division.

Counting is one of those skills that starts in preschool and gets more complex as students move through the grades. But it is also a standard that we think students have mastered or understand and walk away from before there is data to show they can count.

In Kindergarten, students are to count to 100 in both 1s and 10s. First grade, students are extending the counting sequence 120. Not to mention plus 10s or minus 10s. In Second grade students need to count by 100s and skip count by 5s, 10s, and 100s.

Students don't get counting or skip counting with calendar math. They need more. They need to count everything. Not just by 1s starting at zero or one but starting at 14 or 46 or 98.

Do you have students that don't know what number comes after 100 or 110?

You need this!

In this product, you will find student worksheets to get students working on counting by 1s, 5s, 10s, and writing numbers passed 100. And like all my activities--progress monitoring to support interventions and the RTI process.

Understanding Multiples: Multiplication can be viewed as repeated addition. For instance, 4 x 3 can be thought of as 4 counted three times (4 + 4 + 4). Similarly, division involves understanding how many times a number can be subtracted from another number, essentially counting in reverse.

Skip Counting: Skip counting (counting by 2s, 3s, 4s, etc.) is a direct application of counting that helps in learning multiplication tables and understanding the concept of grouping in division.

Patterns Recognition: Counting aids in recognizing numerical patterns, which is essential for mastering multiplication tables and identifying factors and multiples.

#### Addition & Subtraction

Addition and subtraction are the building blocks of multiplication and division:

In second grade there is a tiny standard that where multiplication starts. It's 2.OA.C--students start to learn about arrays and start using skip counting to solve multiplication facts of 2s, 5s, and 10s. This set of activities will help you build students capacity in using games and number talks.

Foundation of Multiplication: Multiplication is essentially repeated addition. For example, 5 x 4 can be seen as adding 5 four times (5 + 5 + 5 + 5). A solid grasp of addition makes this concept more intuitive.

Division as Repeated Subtraction: Division can be conceptualized as repeated subtraction. For example, 20 divided by 4 can be understood by subtracting 4 from 20 repeatedly until reaching zero, counting the number of subtractions made.

Handling Remainders: Division often results in remainders. Proficiency in subtraction is necessary to understand and calculate what is left over after dividing.

#### Interconnectedness of Skills

The interconnectedness of place value, counting, and addition & subtraction with multiplication and division highlights the importance of these basic skills. Mastering them provides a strong mathematical foundation, enabling students to tackle more complex problems with confidence. Understanding place value ensures accurate computation, counting fosters an intuitive grasp of numerical relationships, and addition & subtraction form the operational basis for both multiplication and division.#### BUNDLE

I have you covered with a growing bundle. All these products are bundled together in my store, so you can start the year off strong and build those necessary skills to ensure your students master all the skills they need to understand multiplication.

## Thinking Outside the Box with Math

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

- Direct Instruction
- Independent Skill Practice
- Technology
- Games

## Math Preschool Style

#### 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.

## Best Practices: Number Sense

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.

## Websites to Support Math Planning

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

- 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.

- 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.

## Preschool Math Summer Ideas

####

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.

## May Pinterest Pick 3

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.

## RTI Activities for Your Math Class & Giveaway

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!!

a Rafflecopter giveaway

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

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.

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

#### 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!

## Preschoolers and Number Sense: Summertime Ideas

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.

## What is Mental Math?

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.

## What is Number Sense?

• 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.- 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!

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