2019 - Down River Resources | Your Elementary Math Guide
Students that have "number sense" can solve a problem in a variety of ways. Flexibility with numbers is what helps these students show their sensational number sense.

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.

Most Number Sense routines take an average of about 10 minutes and are incorporated at the beginning of the math block.

There's four common Number Sense routines that are used daily in elementary math classrooms across the world. Okay, friend, here's what you've been waiting for...

Number Sense Routines: Estimation Task, Number Talks, Which One Doesn't Belong, and Other Number Sense Routines. (Notebook with these words written in an organizational chart.)

The Best Number Sense Routines for the Elementary Math Classroom

1. Which One Doesn't Belong? (WODB)

Which One Doesn’t Belong? is a routine that revolves around an thoughtfully designed four image set. Each of the quadrants can be a correct answer to the question “Which one doesn’t belong?”

Because all their answers are right answers, students naturally shift their focus to justifications and arguments based on properties. A teacher can facilitate rich discussions and teach mathematical argumentation using Which One Doesn’t Belong? You can use this routine to listen closely and respectfully to students’ ideas.

Resources for Which One Doesn't Belong?

Guiding Questions: What do you notice? What makes all the items alike? What makes them different? Which one doesn’t belong? Can you share your reasoning to justify your answer?

Image Website: http://wodb.ca/index.html

Twitter Hashtag: #wodb

2. Number Talks

Number talks are brief discussions (5–15 minutes) that focus on student solutions for a single, carefully chosen mental math computation problem or a number string, a series of related problems. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board.

Resources for Number Talks

Guiding Questions: How did you solve that? Explain why you did that? How did that help you solve the problem?

Book Recommendations: This is my FAVORITE book for Number Talks (by Sherry Parrish) and is appropriate for all elementary grades.

I really appreciate the myriad of suggest problems to use to help students practice specific strategies. They are all laid out in this book. It's like a huge resource bank for Number Talks. There's also video footage of variety of Number Talks, which helpful for newbies to this routine.

There is another series of book for upper grades that I recommend: Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 3-10. I have found valuable information in this book, but has less examples than the first text I recommended. It is typically less expensive that Sherry's book.

Twitter Hashtag: #numbertalks

3. Estimation Tasks

Estimation is a skill that occurs naturally but one that needs to be nurtured and massaged as the child's brain develops. Think: use or lose it. Estimating a quantity weaves in mental computations that require problem solving and critical thinking.There are a variety of estimation tasks that students can engage in including physical models and virtual models.

The most common estimation task for grades K-2 is an estimation jar. Fill a jar with similar objects (examples: cereal, marbles, mini erasers, etc.). Have students take one scoop out of the jar. Then, ask students to estimate how many scoops it will take to empty the jar. Record students' thinking and responses.

Resource for Estimation Tasks

Website Recommendation

Estimation 180 is best for upper elementary math. Estimation 180 provides students with opportunities to strengthen their number sense and mathematical thinking through the use of engaging visuals and rich discourse. The visuals at Estimation 180 allow students to engage in mathematical conversations where students are encouraged to support their mathematical claims with evidence and reasoning.

4. Other Number Sense Routines

This is a broad category for every other Number Sense routine.

The most popular books for these routines are Number Sense Routines (see below for links).

Resources for Other Number Sense Routines

Book Recommendations: These are two excellent books that help teachers understand the learning trajectories for the number sense routines.

Find the primary book here: Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3.

Find the upper elementary book here: Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3-5,

I hope this list of Number Sense routines and suggested resources gets you thinking about implementing OR diversifying your Number Sense routine in your classroom.

What number sense routines do you currently use in your classroom? 
What routine would you like to add to your repertoire?

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Pinterest Image: Down River Resources (Notebook picture)

We want to motivate students, encourage active learning in the classroom while we develop critical-thinking, communication, and decision-making skills. Group work can be an effective method to do all of these things. When looking more specifically at the 21st Century Skills, (12 abilities that we want to instill in our students to prepare for careers in the Information Age) group work enables us to build critical thinking, creativity, collaboration, communication, flexibility, leadership, initiative, productivity, and social skills. Wow! Nine of the 12 21st Century Skills can be addressed by simply using group work in the classroom! What does productive group work look like in the classroom? What does productive group work sound like in the classroom? We will be diving into student group work! 

The Best Skills for Cooperative Learning Groups 

It is imperative that in the first weeks of school that you intentionally plan and spend time working on how to work well in groups with your students. We cannot assume that our students come into our classroom with the appropriate social skills needed for productive cooperative groups. 

You want your students to be able to:

✅ listen to each other
✅ respect each other
✅ build on each others’ ideas

Things you might see when students work well in groups:

πŸ”Ž leaning in and working in the middle of the table
πŸ”Ž sticking together
πŸ”Ž following team roles

Things you might hear when students work well in groups:

πŸ‘‚ equal air time
πŸ‘‚ silence when speaker is talking
πŸ‘‚ asking each other a lot of questions

Model and practice

When first starting out using groups, you must have students model the correct behaviors (listed above). Have students practice working in groups. When finished, have the classroom gathered together and label specific behaviors that were aligned to the classroom expectations. 

If you requested that all of the group members lean in, praise a group specifically for leaning in and working in the middle of the table. 

By specifically labeling the correct behaviors, you are reinforcing the specific expectations for productive group work.

How can you provide support with building on each others' ideas? 

  • Begin to ask more open-ended questions that may have more than one solution.
  • Give students question stems and encourage them to ask questions within their groups.
  • Practice questioning between partners and groups.

Students love using these sentence stems in the classroom. 

Using Accountable Talk in the Math Classroom

Supporting language and vocabulary development is crucial in the math classroom. Educators need to explicitly teach and pay attention to the quality of talk in the classroom. English Language Learners will be supported, as well as ALL learners. This resource will help you keep mathematicians engaged in math conversations as they become proficient in "speaking math."

I hope this post inspires you to incorporate group work in your classroom. If you are interested in using my math sentence starters for questioning in group work, you can find them in my TpT shop!

What would you add to this list? What do you find the most difficult part to practice? 

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Kindergarten math and first grade math homework can be confusing and frustration to parents and teachers at times. My favorite messages come from these adults regarding questions on homework pages. Come 4:00 p.m., my cellphone starts blowing up with pictures of homework pages! These parents and teachers know that my love of math runs deep, but my love for helping others understand, especially the students who are depending on them to answers runs deeper! Most recently, I received one of these after-school text messages with a question from first grade geometry. The question read: "I am not a rectangle. I am not a triangle. What shape am I?" Four answer choices were given including a circle, square, rectangle, and triangle. How can you answer this question? More importantly, how you can you explain this answer to your child at home or in your classroom? Read more to learn about why a square is considered a special rectangle.

Math 101: Why is a Square a Special Rectangle?

If you haven't read between the lines yet, the answer to the question is a square.

I am not a rectangle. I am not a triangle. What shape am I?

The child on the other end of the text message instantly crossed out a rectangle and triangle.

I was especially excited to see this because it is evident that the child's parent and/or teacher had taught them the importance of eliminating answer choices!

The circle and square were remaining. The father instantly thought:

 It must be the circle.

Why is that? Why did the parent deduce the circle from those two choices?

The human brain is constantly looking for patterns. The child just eliminated two prominent shapes that have characteristics similar to the square. The rectangle and the triangle have vertices and edges. The brain sends the message that the square fits into a similar pattern; therefore, it must be the circle. 

Right and wrong.

There's more to this question that meets the eye, as with many word problems.

Though "circle" is in fact the correct answer, the justification needs a little work.

This question is testing the knowledge of a first grader, in this instance, if they know that a square is considered a special type of rectangle.

The father instantly proclaimed:

When I was in school, a square was a square and a rectangle was a rectangle!
Though I can't discount this statement {we do know a lot more mathematical explanation in education these days}, this is a common frustration point with many teachers and parents alike.

To learn more about this mysterious "special rectangle," we need to revisit the attributes or properties of shapes, particularly quadrilaterals.

Attributes or Properties of a Quadrilateral 

It's really easy to get lost within all of the academic vocabulary of kindergarten, first grade, and second grade geometry.

Here's the basics:


A quadrilateral has four sides, is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Both rectangles and squares are quadrilaterals. Both shapes are two-dimensional four-sided closed figures with straight sides.

Rectangles, squares, trapezoids, rhombuses, and parallelograms are all part of the quadrilateral family.

What makes rectangles and squares unique within this family?


A rectangle is a quadrilateral. It's sides intersect at 90 degree angles. A rectangle has opposite sides which are congruent, or have the exact same length. . The diagonals, are mutually bisecting, or cut each other in half.


A square is a quadrilateral. It's sides also intersect at 90 degree angle. Similar to a rectangle, its opposite sides are congruent, but ALL of its sides are congruent, or have the exact same length. The diagonals, are mutually bisecting, or cut each other in half. The square also has perpendicular bisecting diagonals.

A rectangle is also classified as a square when both pairs of opposite sides are the same length; thus, a square is a special rectangle. 

Hierarchy of Quadrilaterals

In this illustration above, the rectangle has the properties identified within the red quadrilateral.

The square contains ALL of the properties of the rectangle AND the properties listed solely within the square figure. 

This further depicts that a square is a special type of rectangle.

If you are still following this, you now have passed kindergarten math in Texas! Yeehaw! This just so happens to be first grade math in pretty much the rest of the country.

Content Standards Addressed: TEKS K.6A
The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties. The student is expected to: identify two-dimensional shapes, including circles, triangles, rectangles, and squares as special rectangles

I hope this post inspires you to dust off your college textbook and learn more math lingo, or gives me the privilege to be a bookmarked site for future mathematical assistance.

What is the most confusing homework problem you've seen?

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Comparing numbers can be difficult. Young mathematicians, especially in kindergarten, just become proficient in learning the quantity of numbers before diving into comparing those numbers. For decades, teachers have relied on the 'ole alligator analogy for teaching students to compare numbers and draw comparison symbols. You will find numerous blog posts and printables with this cute alligator, even suggesting to use him with decimals and percentages. Have you or your students become dependent on drawing teeth on comparison symbols? Students think, "Is the bigger value eating the smaller one?" or "Is it the value it already chopped on?" This becomes confusing, especially for younger students. While this can be a cute anchor chart, students are not internalizing the meaning of the comparison symbols. This misnomer is confusing children and frustrating their future teachers. I know what you are thinking, if the alligator analogy isn't the best for students to learn, what can I teach my students so they can retain the correct meaning for these symbols? Read more to find a strategy for teaching comparing numbers and correctly explaining comparison symbols. 

Why You Need to Ditch the Alligator When Comparing Numbers Blog Post by Down River Resources

There's A Better Way To Compare Numbers

Thinking about how a child learns vocabulary, it's easy to roll back time and reflect on how a young child learns their name. How does a child learn their name? Parents have identified the child by their name. The parents and others around the child use their name in context. With repeated exposure to their name, the young child memorizes it. Soon they are able to repeat their name and identify themselves as such.

Children are innately logical and literal. If we say "alligator," children think of the lizard-like swamp animal, not a mathematical symbol. 

If adults call a comparison symbol an "alligator," we are mislabeling a content-specific word for students.

I truly believe this is what happened in my early education. I learned tricks in school, not actual mathematics. Eventually, there comes a point when tricks no longer suffice for students and can lead to failure in advanced mathematics, which is the WHY behind everything I do with Down River Resources.

There is a big movement, especially in upper elementary, middle, and high schools to break students away from these misrepresentations.

Trust me, I know it is not ANYONE'S intention to teach incorrectly, often times, we are victims of our own education. Educators teach the way they were taught unless they have since learned differently.

How Do I Teach Comparisons?

Use a word problem to introduce two sets of numbers. If you are comparing numbers 9 and 7, it may sound like this:

Kaila and Marco were born on the same day. Kaila has 9 birthday candles on her cake. Marco has 7 candles on her birthday cake. Use a comparsion symbol (>, <, or =) to compare 9 and 7.

Ideally, with exposure to symbols, such as in the word problem above, students memorize the meaning. Just as students hear comparative language, such as "equal to," "greater than," and "less than," they need to see the symbols that represent these phrases too.

After modeling this language and its corresponding symbols, the students should practice saying and writing them too!

If you are still stuck on the alligator so the students have a scaffold for learning the comparison symbol, I have a strategy to help. 

Sometimes before students can internalize the meaning of the symbol, it helps to actually analyze the shape of the comparison symbol.

Why You Need to Ditch the Alligator When Comparing Numbers and Strategy Support
Think about an equal symbol. 

The line segments are parallel; the bars are the same distance apart on both sides. (See the "equal to" image.)

Now, let's take a closer look at the symbols that represent inequalities. 

The segments, or bars, are tilted when using the inequalities. There is a smaller side and a larger sides.

The GREATER number is next to the wider end, while the LESSER number is next to the narrower end. 

Please note the language used in that statement above. Use GREATER instead of 'bigger,' and LESS rather than 'smaller.'

Why You Need to Ditch the Alligator When Comparing Numbers and Strategy Support As students move into advanced mathematics, students will need to apply this skill to integers. Calling -6 'bigger' than -16 creates confusion for students.

As you can see below, EVERY grade level focuses on comparisons to some degree. 

Let's commit to focus on teaching mathematics, not just the tricks!

Math Content Standards Addressed: 

Why You Need to Ditch the Alligator When Comparing Numbers and Strategy Support - TEKS K.2G Compare sets of objects up to at least 20 in each set using comparative language.
- TEKS K.2H Use comparative language to describe two numbers up to 20 presented as written numerals.

First Grade
- TEKS 1.2D Generate a number that is greater than or less than a given whole number up to 120.
- TEKS 1.2E Use place value to compare whole numbers up to 120 using comparative language.
- TEKS 1.2F Order whole numbers up to 120 using place value and open number lines.
- TEKS 1.2G Represent the comparison of two numbers to 100 using the symbols >, <, or =.

Second Grade
- TEKS 2.2C Generate a number that is greater than or less than a given whole number up to 1,200.
- TEKS 2.2D Use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and symbols (>, <, or =).

Third Grade
- TEKS 3.2D Compare and order whole numbers up to 100,000 and represent comparisons using the symbols >, <, or =.

Fourth Grade
TEKS 4.2 Compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =.
- TEKS 4.3D Compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <.

Fifth Grade
TEKS 5.2B Compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.

I hope this post inspires you to use math language when teaching students how to compare numbers, if you want to download some free mats to use when comparing numbers, you can find them here.

Were you taught the alligator analogy?

Get FREE Comparing Numbers Mats Delivered Straight to Your Inbox! Join Down River Resources as a Valued Partner!

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Why You Need to Ditch the Alligator When Comparing Numbers Strategy

One of the best things about teaching history during Women's History Month is the accessibility of quality children's literature or picture books! Students are much more likely to learn about the important events, people, and places which make our world unique through an accessible text. Throughout our history women have made valuable contributions. No matter what their role, women's experiences remain an important and sometimes overlooked aspect of our history. I've included some historical texts and a few inspirations book for young girls. These picture books value the diverse experience of women and provide inspiration for the young girl sitting in our classroom or being raised within our home.

Finding the Best Picture Books for Women's History Month

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Girls come in all different colors and sizes. They delight and amaze us. They're full of surprises. Girls can do anything they want to do. And if YOU are a girl . . . You can do these things too! 
- Girls Can Do Anything

Shaking Things Up: 14 Young Women Who Changed the World

Shaking Things Up introduces fourteen revolutionary young women—each paired with a noteworthy female artist—to the next generation of activists, trail-blazers, and rabble-rousers. This book has beautiful illustrations and is sure to inspire your budding artists too!

In this book of poems, you will find Mary Anning, who was just thirteen when she unearthed a prehistoric fossil. You’ll meet Ruby Bridges, the brave six year old who helped end segregation in the South. And Maya Lin, who at twenty-one won a competition to create a war memorial, and then had to appear before Congress to defend her right to create.

And those are just a few of the young women included in this book. Readers will also hear about Molly Williams, Annette Kellerman, Nellie Bly, Pura BelprΓ¨, Frida Kahlo, Jacqueline and Eileen Nearne, Frances Moore LappΓ©, Mae Jemison, Angela Zhang, and Malala Yousafzai—all whose stories will enthrall and inspire. This poetry collection was written, illustrated, edited, and designed by women and includes an author’s note, a timeline, and additional resources.

With artwork by notable artists including Selina Alko, Sophie Blackall, Lisa Brown, Hadley Hooper, Emily Winfield Martin, Oge Mora, Julie Morstad, Sara Palacios, LeUyen Pham, Erin Robinson, Isabel Roxas, Shadra Strickland, and Melissa Sweet.

This book is best suited for upper elementary students.

She Persisted: 13 American Women Who Changed the World

Chelsea Clinton introduces eager students who are ready to take on the world to thirteen inspirational women who never took no for an answer, and who always, inevitably and without fail, persisted.

Throughout American history, there have always been women who have spoken out for what's right, even when they have to fight to be heard. In She Persisted, Chelsea Clinton celebrates thirteen American women who helped shape our country through their tenacity, sometimes through speaking out, sometimes by staying seated, sometimes by captivating an audience. They all certainly persisted.

This book features: Harriet Tubman, Helen Keller, Clara Lemlich, Nellie Bly, Virginia Apgar, Maria Tallchief, Claudette Colvin, Ruby Bridges, Margaret Chase Smith, Sally Ride, Florence Griffith Joyner, Oprah Winfrey, Sonia Sotomayor—and one special cameo.

Dear Girl, 

Dear Girl, is a remarkable love letter written for the special girl in your life; a gentle reminder that she’s powerful, strong, and holds a valuable place in the world.

Through this charming text and stunning illustrations, any girl reading this book will feel that she's great just the way she is—whether she enjoys jumping in a muddy puddle, has a face full of freckles, or dances on table tops.

Dear Girl, encourages girls to always be themselves and to love who they are—inside and out.

I love text letter written inside of  book jacket. This surely gives you the sentiment that is beautifully captured within the pages of this text:

Dear Girl,
This book is for you.
Wonderful, smart, beautiful you.
If you ever need a reminder, just turn to any page in this book and know that you are special and you are loved.
—Amy and Paris

If you are a mother, aunt, grandmother, or someone special to a lucky girl, this book is the perfect gift!

Girls Can Do Anything

This enchanting book is all about the things girls can do. Whether she dreams of being a vet that heals people's pets, a firefighter that braves the flames, an astronaut floating in deep, dark space, or a fearless jungle explorer, there's nothing that a girl can't do. Girls Can Do Anything!

One size definitely does not fit all in this book: charming depictions of girls being scruffy or fancy, neat or messy, and everything in between are explored and celebrated, because each girl is unique and unlike all others. 

Empower young girls everywhere and let them know that being told "you're such a girl" is the greatest compliment of all!

One of my favorite things about this book is that girls of every walk of life are represented. There's a girl in a wheelchair and girls of color in this text. If representation matters to you when selecting a text this book surely fits the bill. 

Girl, You're Amazing!

An upbeat, rhyming tribute to girls offers readers encouragement to build confidence and self-esteem while whimsical paintings celebrate the many things that girls of every age can do.

Girl, You're Amazing! has you chanting that beautiful sentence to your favorite girl all day long!

This book also celebrates diversity. If you are looking for a book that celebrates the uniqueness of all girls, this book is for you! The illustrator aims for inclusion in her hip gouaches, which feature girls of all races and appearances. Asymmetrical faces, fashionably mismatched patterns, and a quirky palette of colors adorn each unique page.

Another great thing about this text is that it has a companion, Boy, You're Amazing! With all this girl talk, we need to remember that EACH child is unique and special no matter their gender!

I hope this post inspires you to find the perfect picture book to celebrate girls and women this March and beyond! I look forward to adding to this list as I find more books that celebrate girls.

What is your favorite female picture book?

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STAAR testing season is right around the corner! The best STAAR test prep strategy is high-quality instruction throughout the school year, but many teachers like to add a test prep session to their schedule for good measure! {Whatever helps, right?!} Whether you test prep throughout the school year or in the final weeks before the STAAR test, or other standards-based assessment, you can gamify your test prep materials to improve student motivation and engagement! Do your students want to learn? Are your students making the effort to learn? Are they enjoying the process and doing well? Are your students motivated to learn or are they dragging their heels? Gamifying test prep might be just what your students need, especially if you are reading this right around Spring Break?! Continuing reading to find a simple way to increase motivation and engagement during test prep!

How To Increase Motivation and Engagement During Test Prep

Test Prep Games are Good for the Brain

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The social component of working on a game in a group leads to LOADS of benefits on brain function including:

- Activate neurotransmission
- Increase brain plasticity
- Rewires
- Mitigates brain inflammation
- Mitigates deleterious effects of oxidative stress

These are just the benefits of the social components of gamifying educational content.

Adding learning games in your classroom will boost intrinsic and extrinsic motivation, engagement, and learning outcomes for learner!

With so many benefits, what are you waiting for friend?

My favorite way of gamifying the classroom is by simply adding a popular game to test prep sessions.  

Want to do it too?

Here's How To Test Prep with Ease:

Collect a game or gameboard. Goodwill and Savers are the best cost-savings options. I often find games on clearance too!

Use word problems or sample test questions as the educational basis of the game. You can use questions from worksheets, released test questions, or specific game cards to practice heavily-tested standards. (In Texas, we call these the readiness standards.)

Rigorous game cards

The class can be divided into two teams. The team can work together to solve the problem.

If the team gets the problem correct, the team has a turn to play the game. Students within each team can take turns being the “player” for their team by following the game’s instructions.

If the team does not get the problem correct, they forfeit their turn. {You can also create a “Steal” option. The other team can generate a solution for an extra turn. You can give teams a couple extra chances to use through the game, three strikes, etc.}

Connect Four Shots

Are you ready to try this strategy, but need test-like questions to minimize your prep time?

I'll keep you afloat with my math test prep bundles that are just waiting for you to print. These will surely add some challenge into your math test prep sessions. They can be used during testing season or throughout the year for a spiral review. The bundles come with printable game boards OR you can add your own physical game too. 

I hope this post inspires you to gamify your test prep sessions, if you need Connect Four Shots,
you can find it here.

The rigorous test prep bundles can be found in my TpT shop:

What are some other ways you have motivated and engaged your students during test prep sessions?

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