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HK Workshop #4 - Visual Mathematics: Geometry and Measurement.pptx

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Students identify, describe, analyze, compare, and compose two- and three- dimensional shapes. Counting, addition, and subtraction are revisited in the geometric contexts.

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Copy of Copy of BTC Curricular Math Task Sheet K-12 

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I'm Sarah Carter, a high school math teacher who passionately believes math equals love. Through this blog, I'm on a mission to equip other teachers to create inviting and engaging classrooms where learning mathematics is fun.

I currently teach AP Calculus AB, Pre-Calculus, and Statistics at Coweta High School in Coweta, OK. Over my eleven-year teaching career, I have taught both math and science at the secondary level. If you want to know more about my career in education, check out my About Me page.

THANKSGIVING MATH ACTIVITIES

MOST POPULAR ACTIVITIES

MOST RECENT BLOG POSTS

If you're looking for a traditional blog reading experience, I recommend checking out my BLOG page. You can find it by following the previous link or by clicking on the Blog Tab at the top of the page. This page will list all of my blog posts (over 1300 of them!) in chronological order with the most recent blog post first.

This blog started as a way for me to make sense of my student teaching experiences in the middle school and high school math classroom. It's evolved over the past decade to become a way to share free resources for engaging and fun math activities with teachers around the world.

Looking for resources for a specific math teaching topic? I recommend navigating the blog using the tabs at the top of the page or by using the search box. You can easily find and read all of my posts about polynomials, puzzles, or posters this way.

FREE PRINTABLE MATH RESOURCES

PRINTABLE MATH PUZZLES

I am a passionate proponent of incorporating puzzles into the mathematics classroom. Over the past five years, I have compiled a collection of hundreds of free printable puzzles that have been used in math classrooms around the world with students ranging from elementary school to high school as well as adults! You can find out more about how to access these free puzzles on my puzzles page!

FREE PRINTABLE MATH POSTERS AND CLASSROOM DECORATIONS

My high school math classroom is my happy place, and I love decorating it to display my love of math to my students. I have created many math posters over the years that I am sharing here on my blog. I recommend checking out my entire math poster collection - ten years' worth of math posters!

FUN AND CREATIVE MATH PRACTICE ACTIVITIES

I love creating fun, interactive, and hands-on activities. Be sure to check out some of these activities while you are here!

GAMES

Looking to infuse some fun into your math classroom? Check out these free printable games! You and your students are sure to have a ton of fun!

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I'm Sarah Carter, a high school math teacher who passionately believes math equals love. Through this blog, I'm on a mission to equip other teachers to create

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Chapter 5 - How We Ask Questions in a BTC

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Chapter 5 - How We Ask Questions in a BTC

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Robert Kaplinsky

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Open Middle Math Depth Of Knowledge Matrices - Robert Kaplinsky

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If you love Open Middle problems, then you’ll love these matrices that compare problems across the grade levels at different Depth of Knowledge levels. Use the button below to download high quality, printable PDFs of them all or preview what you’ll get by scrolling below. Click on a link below to jump to the…

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Solving equations using algebra tiles: with pictures!

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There are so many cool ways to use algebra tiles in math. Last week I wrote a post about using them to factor . In this post I wanted to...

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Algebra Tiles: The Complete Guide for Students, Teachers & Parents

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Teaching & Learning Cycle - Unpacking a Unit

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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R. Kaplinsky

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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Bee-Bot is an exciting new robot designed for use by young children. This colorful, easy-to-operate, and friendly little robot is a perfect tool for teaching sequencing, estimation, problem-solving,…

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Every student deserves to have the opportunity to problem-solve and engage in genuine mathematical thinking. Rich tasks are designed to make these rich learning experiences possible. We've written these tasks to launch quickly, engage students, and promote the habits of mind mathematicians need: perseverance & pattern-seeking, courage & curiosity, organization & communication.

This free video PD series will help you get the most out of the tasks below.

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Every student deserves to have the opportunity to problem-solve and engage in genuine mathematical thinking. Rich tasks are designed to make these rich learning experiences possible. We've written these tasks to launch quickly, engage students, and promote the habits of mind mathematicians need: perseverance & pattern-seeking, courage & curiosity, organization & communication.

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By Paul Lockhart

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How do you co-create rubrics with students?

Co-creating rubrics is when a teacher collaborates with students to create a rubric assessment tool together. The theory is that students gain a deeper understanding of what is expected from them if they help create the rubric.

When you co-create a rubric with your class, it's kind of like you and your class are working together to create a building block replica of a famous painting.

Everyone in the class contributes some building blocks - including the teacher.

You're working together towards an end goal - let's say you're trying to re-create a building block version of the Mona Lisa.

You work as a team to talk about key elements that you have to include in your replica.

Through the discussion process, you realize that some blocks fit better than others.

You go back and forth discussing with your students to create something that is more than the sum of its' parts.

Over time, you might tweak your structure to make it even better and more accurate..

In the end, you class might even end up with a Lego reconstruction of the Mona Lisa like this. (Made by Nathan Sawaya)

Now imagine the same process of collaboration, but with ideas. And the goal is to create an assessment tool for an assignment.

Everyone in the class contributes some ideas - including the teacher.

You're working together towards an end goal - creating an assessment rubric for a major assignment.

You work as a team to talk about key elements that you have to include in your rubric.

Through the discussion process, you realize that some ideas fit better than others.

You go back and forth discussing with your students to create something that is more than the sum of its' parts.

Over time, you might tweak your rubric to make it even better and more accurate.

But it's one thing to build with building blocks because they're tangible and more concrete. It's a lot harder when you have to build ideas together in a group.

So, here's a free lesson plan on how to teach students to effectively co-create rubrics! Use it to learn the process, and then co-create rubrics with your own subject material and lessons!

How to Co-Create Rubrics with Students - Table of Contents

Co-Creating Rubrics with Students... Ummm, why?

Shouldn't the teacher just tell the students what to do and make the rubric themselves? After all, they are the teacher, and students are, well, students.

Sometimes, we get stuck in a fixed mindset view of teachers.

This kind of philosophy often revolves around the following ways to look at teachers:

But, what if we tried to have a growth mindset view of teachers.

A growth mindset revolves around the idea that through effort and strategies, we can become better.

If we think about teaching with a Growth Mindset, then we might accept the following:

And, by the way, we can shift our mindset from a fixed mindset to a growth mindset just by phrasing things differently. (Check out this post for the magic phrase teachers should use at school to put students into a growth mindset.)

We're entering an era where teachers are looking at ways to incorporate student voice into their lessons.

But, there are authentic ways and not so authentic ways to incorporate student voice into learning.

Is there a difference? Yes.

What's a rubric (in a school / academic setting)?

First the basics: Rubrics are an assessment tool that help students and teachers look at a piece of work and give it a grade.

They've been in style for more than 20 years. Rubrics come with all sorts of different styles, descriptors, and level of detail, but they all basically use a set of criteria to judge a piece of work.

The term "rubric" refers to a scoring guide used to evaluate the quality of students' constructed responses (written compositions, oral presentations, or science projects).
SOURCE: W. James Popham What's Wrong-and What's Right-with Rubrics

Traditionally, rubrics are made by teachers (or publishers of educational resources), but there's a focus now on co-creating rubrics with your students.

What does it mean to co-create rubrics with students?

Normally, rubrics are made by the teacher.

Co-creating rubrics is when you and your class agree on what the rubric for an assignment will look like.

In other words, you collaborate and create the rubric together. Sounds simple right?

Spending class time talking about rubric criteria sounds like it can be relatively easy. The teacher...

leads a whole-class conversation

asks students what they think the success criteria looks like (i.e. What do you need to do in this assignment (to get a Level 3)

tells students if their suggestions are right or wrong,

puts the correct student answers on the board

hands out the rubric that was already photocopied

But this is not collaboration. (It's co-ordination. It might look like you're incorporating student voice into the rubric creation process, but the rubric was already created by the teacher and the teacher is "teaching" the rubric to their students.)

Co-creating rubrics well with authentic student voice is harder than it looks. (And, more time-consuming.)

That's because genuinely co-creating something requires collaboration. And, collaboration is not the same as teamwork.

Benefits to co-creating rubrics with your class:

Before we talk about how co-creating something requires collaboration (and not just teamwork), let's briefly take a moment to look at the many benefits to co-creating rubrics with your class:

It's authentic learning because students create something that will actually be used to assess their work.

You often get more buy-in from students because they were a part of the process. They are invested in the process.

Students get to see you model how to collaborate. (And collaboration is an important 21st-century skill that students need to be competitive in the global economy.)

Students gain a deeper understanding of what is required in the assessment. Because they had to decide what was the key aspects of the task and to convince other people that this was indeed the most important core features.

One of the most important benefits to co-creating rubrics is that students get to engage in good citizenship. They contribute to the class community and they help create an assessment tool that will help the class to learn.

We often focus on student learning of academic curriculum. (And, we should. We're teachers. Our product is "selling" curriculum content to our students.)

But, our students need more than academic content (which is ever-changing and often easily google-able.)

We need to prepare our students with 21st-century learning and that includes teaching them how to be good citizens. (We're talking how to contribute to the classroom community, and not just service-projects and think-local, act global volunteerism.)

Three Styles of Teamwork:

Very briefly, when we are working groups, there are three styles of teamwork:

1. Collaboration is when the group works together to create new knowledge.

The group has a common goal, but the solution to the problem isn't predetermined.

Watch this clip from the Big Bang theory - the group has a common goal of getting to the movie theatre, but the solution to the problem is created during the brainstorming session:

2. Coordination happens when one person is telling others what to do.

The group achieves a common goal because everyone does their part. No new knowledge is created - this is more about the execution of a plan.

Usually, there is one person or a group of people coordinating the team. (For example, military or orchestra.)

3. Cooperation happens when everyone helps each other out and shares resources but there's no common goal.

If anything new is created, it's because of individual effort as opposed to the group working together collaboratively.

For example, think about teachers planning in a subject group or division meeting before the start of the school year.

So, you were co-operating with your colleagues and helped them out, but there's isn't usually a common goal to run the same classroom program.

(Note - there are times when there are teaching teams that work together to collaborate and co-create a common program. But, what happens when deep down inside, you disagree with your colleague's teaching style or philosophy?

Chances are those "team planning" sessions are really co-operation sessions where people are working side-by-side, but creating their own program.

FYI: We talk about the difference between collaboration, cooperation, and coordination a lot more in our unit about Collaboration:

Collaboration means something new is created.

(In this case, a new understanding of the success criteria for the rubric.)
Collaboration means :

a new understanding is developed

based on input from everyone.

If the teacher already has the rubric planned out before the class starts to "co-create" a rubric, then this is most likely an example of coordination. (A new understanding of success criteria wasn't created in the process.)

This is what often happens when a teacher asks students about what should be on the rubric. (Yes, I've done this myself):

The teacher asks a series of leading questions to guide students towards their rubric expectations.

If a student gives a different answer than what was expected, the teacher redirects the conversation towards the desired answer.

One could argue that the students are playing a game called "guess what the teacher wants" and they're learning that there is a right answer and a wrong answer.

That's one of the reasons why it's so hard for students to be creative and innovative - because as teachers, we often train them to give us the most efficient, precise, direct answer at school.

By constantly creating a world where there is only one right answer (or best answer), we reinforce a culture where students don't want to take risks. This comes back to Educational and Business experts talking about the need to embed 21st-century global competencies into our day-to-day teaching practice to make sure students can answer un-Googleable questions.

We don't always have time to co-create every rubric with students

Even though I'm sharing this lesson plan about how to co-create rubric with students, there are still times when I am "co-creating rubrics" with my students but really, I'm not. It's just a class discussion about what could be on the rubric.

Often times, I find myself coordinating (instead of collaborating) when I don't have enough time.

If I teach multiple classes, and each class creates their own different rubric, that's a lot more work for me to be able to assess. It's easier if I use a similar rubric with all of my classes.

In this case, what I do is:

1. Still go through the co-creating rubric process with all of my classes (using the process below)

2. Explain to my classes that I'm going to have to make some executive decisions to combine all three class rubrics into one.

3. I combine all three rubrics on my own trying to synthesize and embed language from all three classes into a master rubric.

4. Then I do a coordination lesson where I walk students through the final rubric (because it'll be similar yet slightly different from the rubric that the students came up with in each class.)

Empathize with your students. Co-creating rubrics is hard work.

What the teamwork process looks like really depends on two things:

1. Who is in the driver's seat controlling the meeting, and

2. whether this is a task suited for collaboration.

If you're performing Tchaikovsky's 1812 Overture, you probably want a conductor co-ordinating all of the musicians. Probably not the time for collaboration and creating new knowledge.

If you're co-creating rubrics with students where you are trying to create new knowledge (like how to assess an assignment), that's an opportunity for collaboration because there are different points of view about what a rubric should be assessing.

But, remember: If you are doing a lesson on co-creating rubrics with your students, you're the one in the driver's seat of the lesson. You're leading the class and you have the power.

So, it's important to be empathetic to your students because collaboration can be tough.

Change is hard for everyone. (Teachers can have fixed mindsets, too!)

As teachers, it's hard to hear that we need to do something new - like co-creating rubrics with our class.

It's also hard to hear that we're not doing something right and that just because we have a class discussion about rubrics, it doesn't mean we captured student voice into the rubric authentically.

I get it. It's hard to hear constructive feedback. And we feel like we have to be the expert - after all, we're the teacher.

In fact, some of us have been teaching the same material for decades. Some of us have been "co-creating rubrics" with students for years and somehow after the "co-creating lesson" we always end up with the exact same rubric that we've been using for years.

(You know, the one that was already photocopied before the collaboration lesson began?) If this was collaboration, what was the new knowledge and understanding that was created in that group?

In order to think critically about the value of co-creating rubrics with students - and by thinking critically, I mean making an informed decision that is based on analysis and criteria, instead of first impressions and emotion - we need to try to keep an open mind to all possibilities.

So, in an effort to try to put teachers at ease (and less defensive), let's think about some of our experiences in staff meetings, professional development workshops and school or school board / school district initiatives.

Principals, vice principals, and central teachers running PD workshops can think about this from their perspective of running the meeting.

The rest of us can think about this next scenario from the perspective of the teachers sitting in the meeting.

By the way, we dive deeper into critical thinking strategies (like how to be open-minded) in our 5 weeks of Critical Thinking lesson package.

Before trying to co-create a rubric with your students, think about a time when someone tried to co-create something with you

Have you ever gone to a staff meeting where you were asked to help create a school mission statement, philosophy or school improvement plan?

Have you ever been asked for your input because the leaders "want to hear your ideas," but you know from previous experience that it doesn't really matter what you say because the people running the meeting have already thought of a solution that they're simply trying to lead you to in a way that you think that you came up with the idea on your own?

I have. In fact, I can distinctly remember one specific high school Vice Principal who would constantly try to paraphrase the teacher's ideas in a completely different way to go in line with what they wanted us to say.

To be fair, it's not their fault.

If you're co-creating a school vision or school improvement plan with your admin, it's probably really not an example of collaboration where new knowledge and possibilities are explored. It's most likely an exercise in co-ordination where the principal or workshop facilitator is leading you towards their end game.

How do I know this?

Because higher up the food chain, some superintendents and directors and other people have created a Board Improvement Plan with their vision about how their school board or school district is going to prepare students for the 21st century.

And to achieve this goal, the school system needs all of the individual School Improvement Plans to line up with the Board Improvement Plan.

District school boards help set direction and provide support for the school improvement planning process. They should establish vision and mission statements for the board and board improvement plans, and communicate them to schools so that schools can use them as a context for their improvement planning."
SOURCE: Education Improvement Commission, Ontario Ministry of Education School Improvement Planning: A Handbook for Principals, Teachers, and School Councils (Page 12)

And generally speaking, schools have very little time at staff meetings for big picture stuff. We all know that teachers are swamped with everything that we have to do in addition to all the things that get piled on to us. So, there's very little time to collaborate in an authentic way where the end goal is truly open.

So, you get maybe 30 minutes dedicated in a staff meeting to talk about your school vision.

Question for the principals and the people running this meeting:

Do you have an idea about what your school vision should be?

If the group decides to go in a completely different direction (but still aligns with the Board improvement plan), are you open minded and comfortable enough to go with a change in direction?

Question for the teachers sitting in the meeting:

How often do we sit disengaged in that meeting?

Yes, there are always a few informal leaders in the crowd. The one who try to rally the crowd and spearhead initiatives. So, if something new is created in this meeting - is it because the group worked together (like in the Big Bang theory collaboration clip) to come up with something new, or was something new created because of the individual effort put in by a few key people in the crowd?

Is this collaboration?

To be fair, the Education system is a huge machine involving provincial / state politicians, consultants, international expertise, school boards and districts, and then individual schools with students, teachers, and principals.

Like any large bureaucracy, ensuring a common vision involves coordination between all of the parts.

"Schools and boards do not operate in isolation. Improvement planning is most successful when [1] school and classroom-based plans and initiatives are closely aligned with board-level plans and [2] school and board plans are lined with province-wide policies, initiatives and achievement targets."
SOURCE: EQAO Guide to School and Board Improvement Planning (Pg 2)

While there's certainly an opportunity for collaboration at various levels to create how the big picture vision will be implemented, I wonder how many meetings are run in a truly collaborative way - or because of time constraints and leadership style, things are implemented through coordination or cooperation, and not collaboration.

Don't Worry. We're not alone. Business leaders also tell their group what to think

Simon Sinek (of TED talk and Start with Why fame) is an organizational leader and motivational speaker.

Teachers, Curriculum Leaders and Principals aren't the only ones who lead by telling people what they think first (and guiding the group towards coming up on their own with that answer.)

"I see it in board rooms every day of the week. Even people who consider themselves to be good leaders who may actually be decent leaders, will walk into a room and say here's the problem and here's what I think but I'm interested in your opinion, so let's go around the room. It's too late"
SOURCE: Simon Sinek - Five Rules to Follow as you Find Your Spark

Sinek tells an audience of youth change leaders that they'll be told all their lives to learn to listen, but he would emphasize the importance of learning to be the last to speak.

SPEAKING LAST is an important strategy for some of our students to use, so we included it in our free lesson plan on co-creating rubrics with students.

How to Co-Create Rubrics with Students - FREE LESSON

ABOUT this LESSON:

The goal of this lesson is to introduce students and teachers to a collaborative process for co-creating rubrics.

Although you can co-create rubrics in any subject, we chose to use Character (perseverance, resilience) as the topic to co-create a rubric for.

Once students get used to the process, it gets easier and quicker.

Character is important because it's a universal transferable skill (21st Century Skill / Competency) that can apply to any subject, job, or relationship.

We suggest you do this lesson before you co-create a rubric for your subject material. Students get to learn about CHARACTER as well as how to co-create rubrics (COLLABORATION)

In this TASTE of CHALLENGE: Co-Creating Rubrics package, students work in groups:

to learn about Character

to co-create a rubric for assessing Character

NOTE: Answer Key not provided as student answers will vary based on class discussions.

Strategies used in the Co-Creating Rubric lesson plan to encourage authentic COLLABORATION

1. Independent Thinking Strategy: ASK "WHY or HOW"

When we are answering questions, it's very easy to simply skim and scan words from a handout and copy them down without really thinking about it. One strategy to get a deeper understanding is to teach students to ask "why or how" ...

Look at your idea / answer. Ask yourself Why is this important? How is this important?

The key part here is that you have to try to answer your own question.

NOTE: You need to ask a question that you don't know the answer to. Otherwise, it's not helping you to dig deeper into the answer.

Take the answer is to your why / how question and ask yourself why / how that answer is important. The more you play the why or how game, the deeper your answer gets.

2. Group Thinking Strategy: IDEA VOLLEYBALL

A way to encourage group collaboration is to play "idea volleyball"

3. Group Thinking Strategy: PEEP

1. P. Give an answer. (What's your Point?)

2. E. Support your answer by literally saying the phrase, "because it says right here..." and pointing at your Evidence

3. E. Explain how your evidence supports your point

Although we are collaborating to figure out the key "look fors", we also need to use critical thinking to determine which of the proposed group look fors really get to the main idea. Supporting ideas becomes important:

4. Group Thinking Strategy: SPEAK OUT

If you disagree with the ideas in the rubric that is being co-created, you need to speak out and say something.

we're asking people to try to be assertive and to be open to other ideas as long as they don't cross the lines they set for themselves.

This may be a good time to point out the goal is not to be aggressive and to win at all costs and not listen to the other side

Nor is it a time to be passive-aggressive and agree with things snarkily or passive (and to not speak up when we disagree, but simply go along)

More information about assertiveness and the 4 communication styles is included in our COMMUNICATION Chapter.

As this is a co-created rubric, the key is to create something new that everyone can live with. In terms of communication styles:

5. Group Thinking Strategy: SPEAK LAST

Nelson Mandela's father was a tribal leader. He would bring Nelson Mandela along.

They would always sit in a circle The leader would always be the last to speak.

Simon Sinek talks about the importance of speaking last based on Nelson Mandela's philosophy of leadership:

Listen to others, ask questions, but keep your opinions to yourself - don't give non-verbal feedback in a way that says you agree or disagree.

As teachers, of course, be positive and receptive to the speaker in an age appropriate way to encourage students to share their ideas.

But the goal in co-creating rubrics is to share the power and to hear other people share their thinking in a way that their ideas can actually shape the end product..

You need to learn to be the last to speak.

Two different versions of a student self assessment rubric template to help students assess how well they demonstrate Character.

Two different versions of a class tracking sheet template to help the teacher assess how well students demonstrate Character.

29 pages outlining detailed lessons for 2 days with the suggested time required for each lesson and the range of slides that go with each day / lesson.

Handout with notes about Character and space for students to show what Character is.

Handout with "Look-Fors" graphic organizer for students to identify key success criteria (Look-Fors) as well as to brainstorm and record the key criteria as determined by their small group and whole class.

What do we get in this FREE LESSON?

A link to the Google Slideshow so you can show it to your students right away. (Get started in seconds!).

A link to a version of the Google Slideshow that you can make a copy of the presentation. (This way you can edit / tweak the content to fix your exact classroom needs.

A powerpoint file that you can download (PPT) and modify. (Edit the presentation to fit your needs, and use the presentation when the internet is down!)

We provide handouts:

We provide 3 different versions of the lesson slideshow (147 slides):

NOTE: This TASTE of CHALLENGE is part of a larger lesson

The slides from this Taste of Challenge are taken from our Chapter 7. How to Teach 21st Century Skills to Students lesson package, specifically the following slides:

For more information about this larger lesson on co-creating rubrics with all of the 6 Cs as well as another activity comparing and contrasting the 6 Cs to find similarities and differences, please read this post:

LESSON PLAN OVERVIEW:

Introduction (slides 1-14) - 5 minutes

Introduce concept of 6Cs / Transferable skills (slides 1-4)

Similar concepts grouped in slightly different ways / groups (slides 5-14)

Part 1. Notes - Show Don't Tell

Introduction Handout 4a Notes (Show don't tell) - 10 minutes

How to "show, don't tell" (slides 15-21) - 5 minutes

Example of a strong answer (slides 22-27) - 5 minutes

Character - 35 minutes

Overview (slides 28-36) - 15 minutes

Work Period (slide 37) - 20 minutes

We base this schedule on a 50 minute period and try to provide around 45-50 min of content per lesson. Depending on your teaching style and classroom dynamics, you may find things go longer or shorter than the suggested times. Some discussion points are included in some slide notes.

Introduction to Look Fors (slides 38 - 44) - 10 minutes

Student ("Penguin") example (slide 38-40)

What is a Look For (slides 41 - 43)

Handouts (slide 44) - Handout 4b - Look Fors

Co-creating assessment criteria for Character (slides 45 - 54) - 45 minutes

Overview (slide 45)

Step 1. Independent Thinking (slides 46 - 47) - 10 min

Step 2. Group Work (slides 48 - 49) - 15 min

Step 3. Key Look Fors (slides 50 - 51) - 5 min

Step 4. Class Look Fors (slides 52 - 54) - 15 min

Self-Assessment prep (slides 55 - 56) - 5 min

DAY/LESSON 1 - What are the 6Cs / Notes - Show don't tell (slides 1-37) - 50 MIN

DAY/LESSON 2 - Look Fors / Co-Creating criteria (slides 38-56) - 60 MIN

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How do you co-create rubrics with students? Agree? Disagree? Leave a comment below

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How to Co-Create Rubrics with Students (FREE Lesson Plan, handouts, slideshow) - Educircles.org

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Co-creating rubrics is when a teacher collaborates with students to create a rubric assessment tool together. The theory is that students gain a deeper understanding of what is expected from them if they help create the rubric.

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The development of spatial and geometric thinking: The Importance of Instruction | NRICH

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This is part of our Secondary Curriculum collection of favourite rich tasks arranged by topic.

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Due to school firewalls and security restrictions, students may not be able to access resources directly from this site. To give a student access, please first save a copy of the resource to a teacher-owned Google account, then connect the student to the teacher-owned copy. See the video tutorial on "How to share Google files with students" for more information.

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Arrow Cards: 4 digits

Arrow cards support students in reading multi-digit numbers and connecting a number's standard form to its expanded form. The Jamboard "Arrow Cards within 999" also includes a variety of quantitative representations such as ten frame cards, 10x10 arrays, and bundles & sticks.

Last Revised: 01/11/22

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Take out the numbers to make word problems make sense Learn More Try one tomorrow Primary (K-2) Older Students Explore ready-made problems Problem Banks More Problems This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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KG- 8th

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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What is Data Collection? Data collection is the process of gathering, analyzing, and determining intentional lesson planning next steps of accurate data from various sources to find answers to learning trends and student abilities. There are various steps to collecting data in order to intentional

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Intro, Ch 1 and Ch 2

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The Progression of Fractions by Graham Fletcher

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The Progression of Fractions by Graham Fletcher

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Watch this 7 minute video for great visual representations to teach and learn fractions.

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I'm excited to share the 4th installment of the Making Sense Series which explores meaning, equivalence, and comparison of fractions. Fractions are the gatekeeper of algebraic thinking and pr...

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Effective Teaching Practices for Mathematics

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Effective Teaching Practices for Mathematics

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NCTM Principles to Actions

www.mathcoachscorner.com

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Podcast: Exploring The Impact of Principles to Actions

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Podcast: Exploring The Impact of Principles to Actions

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Exploring the Impact of the Landmark Publication: Principles to Actions

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National Council of Teachers of Mathematics - Adding It All Up

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National Council of Teachers of Mathematics

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Exploring the Impact of the Landmark Publication: Principles to Actions

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By Jo Boaler

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CCSS K-8 Instructional Focus & Fluency Charts by grade

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CCSS K-8 Instructional Focus & Fluency Charts by grade

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Copy of BTC Non-Curricular Tasks Sharing (K-12)

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… efforts to intensify attention to the traditional mathematics curriculum do not necessarily lead to increased competency with quantitative data and numbers. While perhaps surprising to many in the public, this conclusion follows from a simple recognition that is, unlike mathematics, numeracy does not so much lead upwards in an ascending pursuit of abstraction as it moves outward toward an ever richer engagement with life’s diverse contexts and situations.

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The progression of addition and subtraction

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The progression of addition and subtraction

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The Progression of Addition and Subtraction HD

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Progression Of Early Number and Counting

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Progression Of Early Number and Counting

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Progression of Early Number and Counting

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By Steve Leinwand

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Presented by Steve Leinwand, Principal Research Analyst at the American Institutes for Research
Sponsored by Conceptua Math
Watch the Recording If you attended the live session, you'll be emailed a CE certificate within 24 hours of the edWebinar. If you view the recording and would like a CE certificate, join the Building Understanding in Mathematics community and go to the edWebinar Archives folder to take the CE quiz.
This edWebinar is part one of our three-part series, Making Your Math Class a Model to Behold, which will address the steps you can take as a teacher to make your mathematics classroom as successful as possible for you and your students. It will focus on the research-based strategies around the Standards for Mathematical Practice. The series will specifically focus on the upper-elementary mathematics classroom, but will apply to all K-8 grade levels.

The 3rd Standard for Mathematical Practice asks students to construct viable arguments and critique the reasoning of others. Many regard this as the single most stimulating of the 8 SMP's, but how can you ace it every day? This session focuses on the expectations for enacting SMP3. Mathematics expert Steve Leinwand provides simple yet practical guidelines and methods for every teacher, from the novice to the seasoned pro. He also will provide guidance for the math coach or principal on how to see that a teacher is making strides with this standard.

Educators are encouraged to share this recording with their principals and/or math coaches, as the content will help everyone to be on the same page.

This edWebinar is part of the Making Your Math Class a Model to Behold Series:
Students Making Viable Arguments in Math: SMP 3 in Action
Low-Entry, High-Exit Math Tasks that Keep Every Student Engaged
Formative Assessment with Mathematical Models
About the Presenter
Steve Leinwand is a dynamic presenter and thought leader in the mathematics education community. He has over 35 years of leadership experience in mathematics education and is a principal research analyst at the American Institutes for Research (AIR). Steve currently serves as a mathematics expert on a wide range of projects that focus on high quality mathematics instruction, turning around underperforming schools, evaluating programs, developing assessments and providing technical assistance. He has spoken and written about effectively implementing the Common Core State Standards in Mathematics, differentiated learning, and "What Every School Leader Needs to Know about Making Math Work for All Students."
Join the Building Understanding in Mathematics community to network with educators, participate in online discussions, receive invitations to upcoming edWebinars, view past edWebinars, take a quiz to receive a CE certificate for a past edWebinar, and access free resources. Watch the Recording
Conceptua Math is an upper-elementary on-line curriculum that uniquely blends personal and social student learning using visual models and adaptive teaching techniques within the classroom.

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Students Making Viable Arguments in Math: SMP 3 in Action

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Mathematics expert Steve Leinwand provides simple yet practical guidelines and methods for every teacher, from the novice to the seasoned pro.

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by Robert Kaplinsky

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What Isn't Mathematical Modeling? - Robert Kaplinsky

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Many teachers reading the Standards for Mathematical Practice are confused by what "Model with Mathematics" means. Here are some common misconceptions.

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by Graham Fletcher

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By Graham Fletcher, posted April 11, 2016 -
When the Common Core State Standards (CCSS) were released in 2010, a heavy emphasis was placed on the eight Standards for Mathematical Practice (SMP). In fact, more emphasis was probably placed on the SMP than the content standards themselves. Through personal experience, I have learned that for all the emphasis placed on the SMP, the modeling with mathematics standard has proved most elusive in its understanding.

What Is Not Modeling with Mathematics

As elementary school teachers, we have misinterpreted the term model to mean simply the use of manipulatives, a misunderstanding that is causing our students to miss the mark when it comes to modeling with mathematics. The questions then arise, What is modeling with mathematics? and How can we make a cognizant effort to be sure it is taking place in our elementary school classrooms?

In short, the use of manipulatives does not ensure that modeling with mathematics is taking place. If the mathematics is not contextualized, modeling with mathematics cannot exist. Modeling with mathematics does not mean, "I do, we do, you do."

Many times, elementary school problem solving engages students in a front-loaded process; that is, all the information required to solve a problem is freely handed over to students. Traditional word problems clearly specify the "givens," the "goals," and the "legal" solution steps for students; the interpretation processes for the child have been minimized or eliminated (English 2004, p. 207). When this happens, students tend to locate the quantities in the problem, identify a keyword, apply the operation, and call it day. Modeling activities avoid this pitfall because they ask that both the teacher and student to approach the mathematics differently.

What Is Modeling with Mathematics

Mathematical modeling is used to interpret real-world situations in mathematical formats (English, Fox, and Watters 2005). In elementary school, this could be as simple as writing an equation to represent a contextualized problem. But in looking at the CCSS high school modeling strand, I suggest that modeling requires more than just a simple equation. Here are three actions that begin to ensure our students are engaged in modeling:

1. Identify the problem, or pose a question.

2. Make an estimate.

3. Identify the variables needed to solve, and answer the problem or question posed.

I rarely observe these actions taking place in an elementary classroom. We usually give students the questions to solve, refrain from giving them opportunities to estimate, and almost always give the information they need to solve the problem or question. Mathematical modeling emphasizes the fact that "thinking mathematically" is about interpreting situations mathematically at least as much as it is about computing (Lesh and Lehrer 2003, p. 111). So how can we get students to begin thinking mathematically and engage in the components required when modeling with mathematics?

Three-Act Tasks

In 2010, Dan Meyer introduced the world to three-act tasks, and the result of his work has empowered teachers to become more effective storytellers on the path to mathematical modeling.

Act 1-Posing the Question

Watch the video.

Just as students would if they were to watch this video, begin to mathematize what you're seeing. Begin to notice and wonder things.

You have now just identified the problem or posed a question. For the purpose of this blog post, we'll stick with the question, How many Whoppers are in the jar?

Your homework

Estimate how many Whoppers are in the jar, and post your guess in the comment section. Be sure to justify your reasoning. In addition to making an estimate, to determine how many Whoppers are inside the jar, be sure to identify the information ( aka variables) that you will need. This information will be provided in our next post (check back on April 25, 2016).

The lack of mathematical modeling in elementary schools is a cause for concern because it limits our students' ability to reason algebraically and to see the everyday usefulness of mathematics. By simply posing a video in place of a traditional word problem, we are engaging in the process of mathematical modeling.

Your Turn

Join us next time when we look to wrap up this modeling problem and share the benefits of including three-act tasks into your teaching repertoire. Until then, we want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

References

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

English, Lyn. 2004. "Mathematical Modeling in the Primary School." In Mathematics Education for the Third Millennium: TOWARD 2010, Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 1, pp. 207-14. Sydney, Australia: MERGA.

English, Lyn, Jillian Fox, and James Watters. 2005. "Problem Posing and Solving with Mathematical Modeling." Teaching Children Mathematics 12 (October): 156-63.

Lesh, Richard, and Richard Lehrer. 2003. "Models and Modeling Perspectives on the Development of Students and Teachers." Mathematical Thinking and Learning 5 (2 and 3): 109-29.

Meyer, Dan. 2015. "Missing the Promise of Mathematical Modeling." Mathematics Teacher 108 (April): 578-83.

Graham Fletcher has worked in education for over ten years as a classroom teacher, math coach, and district math specialist. He graduated from the University of Georgia, where he earned his specialist degree in Math Education. Fletcher's passion for conceptual understanding through problem-based lessons has led him to present internationally and throughout the United States. He continues to be an advocate for best practice and a change agent for elementary school mathematics.

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Modeling with Mathematics through Three-Act Tasks

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By Graham Fletcher, posted April 11, 2016 - When the Common Core State

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Previous Math Matters!

Next Mathematical Practices Activity

The Common Core Mathematical Practices were designed to guide teachers in helping children to develop a deep understanding of mathematical concepts. At first glance, they might seem like lofty goals for preschoolers. But a look "under the hood” can provide insights into why these eight practices can be so valuable and how they naturally fit into the preschool classroom. You’ll be surprised by how many traditional preschool skills they support as well!

Why are the Mathematical Practices important for preschool?

The mathematical practices:

represent a way of thinking about mathematics that all children can benefit from.

are based on years of research by the National Council of Teachers of Mathematics and the National Research Council.

provide teachers with a way to engage children in thinking about mathematics instead of just doing mathematics.

provide children with tools that will be useful all of their lives.

Common Core Mathematical Practices

MP1. Make sense of problems and persevere in solving them.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.

MP1. Make sense of problems and persevere in solving them

Wow—developing problem solving skills and perseverance! What terrific goals for preschoolers! For young children to make sense of problems and persevere in solving them in the math world means that they have to think about a variety of ways to solve the problem (such as counting on their fingers, using teddy bears, asking friends to help, or measuring) and then try out the solution that seems to be most promising. If the first solution doesn’t work, they can go back to the drawing board (that’s where we see their perseverance!). Making sense of problems also means making sense of the mathematics of the problem. What quantities are involved? Are we putting together sets? Taking them apart? Are we balancing more than one block? Do the two triangles fit in the square? If not, why not?

MP2. Reason abstractly and quantitatively

We often think, mistakenly, that children do not have the ability to think abstractly. It is true that initially children reason with quantities in a very concrete way: “If I have four trains and I have to share fairly with Simon, I’ll only have two left!” Later, children learn that these sorts of quantity questions can relate to numbers themselves in a more abstract way, for example, that one plus any whole number equals the next number. Likewise, children can use number words and the terms more or less to describe differences between unequal sets of train cars.

MP3. Construct viable arguments and critique the reasoning of others

Anyone who has ever argued with a four-year-old over whether bedtime is reasonable in the light of a midsummer evening, or exactly how many pieces of Halloween candy are enough, understands that children can develop negotiating skills that involve sophisticated arguments full of astute critiques of an adult’s reasoning. Preschool children can develop these very same skills in mathematics (and use them in a more healthy and constructive manner!). Why isn’t that a square? Well, because all the sides aren’t the same. Why isn’t a split of six tunnel blocks and four tunnel blocks fair? Well, because you have more. These are the rudiments of later thoughtful and proficient reasoning skills.

MP4. Model with mathematics

Preschoolers begin by counting with concrete objects like teddy bear counters, play dishes, and cars, and by joining and taking away from sets of objects. Later they develop more abstract mathematical thinking, for example, knowing that being “five years old” is older than being “four years old.” However, for both types of thinking, modeling with mathematics is a perfect fit in preschool. Ask children to “Show me how you know that.” Even early in preschool, children can become very adept at using objects and drawings to illustrate their thinking and problem-solving activities.

MP5. Use appropriate tools strategically

Preschoolers love tools! Teachers can introduce children to appropriate tools for solving a variety of mathematical problems. Classroom, small group, or one-on-one conversations about what sorts of tools are good for measuring weight, height, length, and temperature can also introduce vocabulary useful in multiple settings (heavier, lighter, smaller, longer, infinitesimal). Initially, young children use non-standard tools to measure and do comparisons. For instance, they will measure the length of a block road by counting how many same-sized rectangular blocks were used to make the road, or they will hold two pumpkins to estimate which is heavier.

MP6. Attend to precision

You may have noticed that as children go through their last year of preschool, rules become increasingly important to them. Just watch them play a game out on the playground. Negotiating rules and requiring adherence to those rules can take almost as much time as playing the game! In these activities, preciseness is common, and obviously within their realm of ability. While some mathematical activities require estimation, a lot of mathematical problem-solving requires precision: in most cases, numbers are not interchangeable. Ask a five-year-old if he is four years old and you’ll get an earful or laughter at your silly question. So you can have conversations about when estimation is useful and when precision is really required. When engaging in mathematical problem-solving, encourage your children to communicate precisely.

MP7. Look for and make use of structure

Children notice structure early in life. They use structure to assign animals to categories: “Has a tail, four legs, fur, and barks… must be a dog.” And to pick out clothes that they like: “I want the striped one, not the polka-dot one.” They can sort shapes and create a table setting so that there is a plate, fork, and cup for each person at the table. Before long, they begin to investigate the patterns in number; for instance, anyone who is one year younger than them can be described as their previous age in the number sequence (“I’m one year older than you! I used to be four, too”). Our job is to provide opportunities in which we can support children as they notice increasingly sophisticated structures in mathematics (“You have three crackers and Samuel has two, if I switched your plates, will there still be five all together?”).

MP8. Look for and express regularity in repeated reasoning

Can young children really do this? Yes! One-plus-one-is-two-plus-one-is-three! Anytime we add one, it is just the next number in the counting sequence! If we match identical square pattern blocks side to side, every square fits up directly against four other square pattern blocks. If we split each square into two triangles, we will always have two triangles for every square. Count the squares and then count the triangles, and you will always have twice as many squares as triangles. Young children are very perceptive and have a lot of fun looking for patterns in their environments.

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You may have noticed that as children go through their last year of preschool, rules become increasingly important to them. Just watch them play a game out on the playground. Negotiating rules and requiring adherence to those rules can take almost as much time as playing the game!

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What teachers and students do in a balanced math program.

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What the SMP are and are not

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Variety of problems - modeling & problem solving

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Variety of problems - modeling & problem solving

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Padlet: Assessing Mathematics that Make Sense

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Padlet: Assessing Mathematics that Make Sense

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Edutopia article: Building a Thinking Classroom

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Edutopia article: Building a Thinking Classroom

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June used it the next day. It did not go well. A forest of arms immediately shot up, and June moved frantically around the room answering questions. Many students gave up quickly, so June also spent much effort trying to motivate them to keep going. In general, there was some work attempted when June was close by and encouraging the students, but as soon as she left the trying stopped. This continued for the whole period.

The following day I was back with a new problem. The results were as abysmal as they had been on the first day. The same was true the third day. Over the course of three 40-minute classes, we had seen little improvement in the students' efforts to solve the problems, and no improvements in their abilities to do so. So June decided it was time to give up.

I wanted to understand why the results had been so poor, so I stayed to observe June and her students in their normal routines. After three full days of observation, I began to discern a pattern. That the students were lacking in effort was immediately obvious, but what took time for me to realize was that the students were not thinking. More alarming was the realization that June's teaching was predicated on an assumption that the students either could not or would not think.

Once I realized this, I proceeded to visit 40 other mathematics classes in a number of schools. In each class, I saw the same thing-an assumption, implicit in the teaching, that the students either could not or would not think. Under such conditions it was unreasonable to expect that students were going to be able to spontaneously engage in problem solving.

This motivated me to find a way to build, within these same classrooms, a culture of thinking. I wanted to build what I now call a thinking classroom-one that's not only conducive to thinking but also occasions thinking, a space inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion.

Over 14 years, and with the help of over 400 K-12 teachers, I've been engaged in a massive design-based research project to identify the variables that determine the degree to which a classroom is a thinking or non-thinking one, and to identify the pedagogies that maximize the effect of each of these variables in building thinking classrooms. From this research emerged a collection of 14 variables and corresponding optimal pedagogies that offer a prescriptive framework for teachers to build a thinking classroom.

1. The type of tasks used: Lessons should begin with good problem solving tasks. In the beginning of the school year, these tasks need to be highly engaging, non-curricular tasks. Later these are gradually replaced with curricular problem solving tasks that then permeate the entirety of the lesson.

2. How tasks are given to students: As much as possible, tasks should be given verbally. If there are data, diagrams, or long expressions in the task, these can be written or projected on a wall, but instructions should still be given verbally.

3. How groups are formed: At the beginning of every class, a visibly random method should be used to create groups of three students who will work together for the duration of the class.

4. Student work space: Groups should stand and work on vertical non-permanent surfaces such as whiteboards, blackboards, or windows. This makes the work visible to the teacher and other groups.

5. Room organization: The classroom should be de-fronted, with desks placed in a random configuration around the room-away from the walls-and the teacher addressing the class from a variety of locations within the room.

6. How questions are answered: Students ask only three types of questions: proximity questions, asked when the teacher is close; "stop thinking" questions-like "Is this right?" or "Will this be on the test?"; and "keep thinking" questions-ones that students ask in order to be able to get back to work. The teacher should answer only the third type of question.

7. How hints and extensions are used: The teacher should maintain student engagement through a judicious and timely use of hints and extensions to maintain a balance between the challenge of the task and the abilities of the students working on it.

8. Student autonomy: Students should interact with other groups frequently, for the purposes of both extending their work and getting help. As much as possible, the teacher should encourage this interaction by directing students toward other groups when they're stuck or need an extension.

9. When and how a teacher levels their classroom: When every group has passed a minimum threshold, the teacher should pull the students together to debrief what they have been doing. This should begin at a level that every student in the room can participate in.

10. Student notes: Students should write thoughtful notes to their future selves. They should have autonomy as to what goes in the notes and how they're formatted. The notes should be based on the work already on the boards done by their own group, another group, or a combination.

11. Practice questions: Students should be assigned four to six questions to check their understanding. They should have freedom to work on these questions in self-selected groups or on their own, and on the vertical non-permanent surfaces or at their desks. The questions should not be marked or checked for completeness-they're for the students' self-evaluation.

12. Formative assessment: Formative assessment should be focused primarily on informing students about where they are and where they're going in their learning. This will require a number of different activities, from observation to check-your-understanding questions to unmarked quizzes where the teacher helps students decode their demonstrated understandings.

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Over more than a decade, the author has developed a 14-point plan for encouraging students to engage deeply with math content.

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If we want our students to think, we need to give them something to think about-something that will not only require thinking but also encourage thinking. In mathematics, this comes in the form of a task, and having the right task is important. The research revealed that we have to give thinking tasks .

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K-8 Non-Curricular Tasks to start off the year!

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If you buy from a link in this post, I may earn a commission. This does not affect the price you pay. As an Amazon Associate I earn from qualifying purchases.
Posted September 24, 2018 By Presh Talwalkar. Read , or .

I adapted this problem from a similar one for elementary students, as tweeted by China Focus.

How tall is the table?
Homework In China - How Tall Is The Table?
The problem is meant for children, but surely it will stump some adults. Can you figure it out? Watch the video for a solution.

(Pretty much all posts are transcribed quickly after I make the videos for them-please let me know if there are any typos/errors and I will correct them, thanks).

One way to solve the problem is algebra. We can make equations based on the height of objects. From the picture on the left we have:

cat + table - turtle = 170

From the equation on the right we have:

turtle + table - cat = 130

To solve for the table, we add these two equations, which then gives:

2(table) = 300

Then we can readily solve the height of the table is 150 cm.

I'm not sure if the students learned algebra yet, so I wanted to derive another method as well.

There is a neat way to solve the problem visually, by stacking the diagrams.

The height from the top of the cat at the bottom to the top of the cat at the top is 300. This is exactly twice the height of a table, as can be seen by removing irrelevant elements in the diagram, and shifting down by the height of the cat:
Source
Thus the height of the table is half of 300, or 150.

China Focus tweet
https://twitter.com/China__Focus/status/1039338689187332096

Published by

PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

(As you might expect, the links for my books go to their listings on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.)

By way of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. It's been quite a journey! I thank everyone that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.

I studied Economics and Mathematics at Stanford University.

People often ask how I make the videos. Like many YouTubers I use popular software to prepare my videos. You can search for animation software tutorials on YouTube to learn how to make videos. Be prepared--animation is time consuming and software can be expensive!

Feel free to send me an email [email protected] . I get so many emails that I may not reply, but I save all suggestions for puzzles/video topics.

MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2023.

(US and worldwide links)
https://mindyourdecisions.com/blog/my-books

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)

Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.

Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.2/5 stars on 33 reviews)

Math Puzzles Volume 3 is the third in the series. (rated 4.2/5 stars on 29 reviews)

KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription you will get access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

MERCHANDISE

Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.

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Homework In China - How Tall Is The Table?

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If you buy from a link in this post, I may earn a commission. This does not affect the price you pay. As an Amazon Associate I earn from qualifying purchases. Posted September 24, 2018 By Presh Talwalkar. Read , or . I adapted this problem from a similar one for elementary students, as tweeted by China Focus.

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Blog - Caty Romero

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Last week, I had a great time teaching & learning division of fractions with one of my teachers. I have not stopped thinking about how models really provide a visual representation of what dividing fractions really mean. Our students use concrete models to gain understanding of what fractions are, how to put them together, take them apart, and take parts "of" it. It is true that when we come to dividing fractions, it gets tricky. This is why models play such an important part.

Here is a picture of a concrete model that shows how to divide ½ by ¼ . Using a concrete manipulative, such as these fraction tiles, allow students to identify the ½ in a whole, and the ¼ in a whole. Then they will think "How many fourths FIT in a half?" or "How many groups of 1 fourth FIT in a half?"

Using fractions tiles is essential when teaching & learning fractions. The great thing about these manipulatives, is that the whole is already identified, and students can use the little fraction tiles to figure out how to divide fractions. These concrete tool gives students different visual representations so they can picture what a half is, or what a fourth is, etc.

THERE IS NO NEED TO INVERT FRACTIONS!

There is no need to teach students strategies such as "flip" the fraction to solve division of fractions. When we provide opportunities for students to visualize what fractions are, they build conceptual understanding, therefore, they can make sense of what dividing fractions is all about.

In this next problem, students solve 4/5 ÷ 1/10 , they understand that this equation means "how many one tenths fit into four fifths?"

Students used a great app called "fraction manipulatives" created by Braining camp, and they used a pictorial representation of what 4/5 looks like, then they made sense of how many 1/10 fit in it. They came to the conclusion that there are eight 1/10 in there. Isn't this great? No tricks, no rules, no algorithms! Students can see what the answer is, by trusting the model!

To build conceptual understanding, students should move from CONCRETE manipulatives, to PICTORIAL representations, to finally move into more ABSTRACT strategies.

Here is one more example: 3/4 ÷ 3/12 or how many 3/12 fit into 3/4?

Then we moved into another problem that brought some productive struggle. Just like we planned in our lesson.

Here is what happened:

The fraction tiles work when the fractions are alike, but when working with fifths and eights, it can get messy, my favorite part of a math lesson, when it gets messy!

"There is a remainder" they said. Students need to understand what to do with this part that is left.

It is not easy to manipulate eights and fifths in the same whole, but a pictorial model can do that.

After they use a pictorial representation, they can start making groups of 1/8 in their model. Students will be able to visualize and understand that 1/8 of the rectangle is equivalent to 5/40. How many groups of 5/40 are in 3/5?

They will count 4 groups and some leftover. The left over is 4/5. Because 5 of those will make 1 more group, since they do not have 5, that becomes its own fraction.

Concrete manipulatives and pictorial representations (models) are made to build conceptual understanding. That is why we need to use problems that make sense, such as ½ divided by ¼, or 4/5 divided by 1/10. To provide our students with opportunities to visualize and build understanding of what dividing fractions really mean.

In grade 6, when we get to fractions like 3/5 divided by 1/8, students can start making connections, and with the conceptual understanding they built in grade 5, start using other strategies that are more efficient for that grade level. They will be able to answer questions like: Why? How do you know? How does that work? How does that look like?
Models are used to build conceptual understanding. To understand WHY a strategy works and HOW it works.
This reflection made me remember a student in grade 5 last year, who could multiply and divide fractions by multiplying numerator by numerator, and denominator by denominator, without making any sense, just getting the answer quickly. She could divide by inverting the fraction.

I asked her if she wanted to play with fractions tiles, and she said yes. I gave her a problem with fourths and twelves. She started making a whole with fourths, then she made a whole with twelves. Then she solved 3/4 by 4/12. She moved the pieces, put the twelves inside the fourths, and she gasped! She got very quiet. I asked "what is happening?" She looked at me, and said: "Miss Caty, is this why 3/12 is equivalent to 1/4? Because 3 of these fit in 1 fourth? Is this how dividing fractions was invented?"

I said: "Yes. Someone had so much fun playing and understanding fractions with models, that they created a shortcut. The short cut is to invert fractions."

Let's explore a lesson using an upside-down approach to teach math. It is basically the opposite from the traditional I-do-you-do approach, where the teacher lectures the class with a specific way to solve a problem depending on the content, and students repeat the same steps the teacher just showed, but using different numbers. In this upside-down approach, or Student-Centered approach, the students do all the thinking, all the problem solving, all the modeling, and all the reasoning to solve a problem.

Some teachers have been finding elements of this upside down approach really effective, for example naming the strategies after the student that first came up with it, is really effective.

It might not sound like a big deal, but students get so excited about trying "their peer's strategies" so much more than using the decomposing strategy (even though it is the same strategy.)

However, there are some other elements of this same approach, that could be more challenging, for example how to make sure that the lesson was effective if the work in students' notebook is unfinished or extremely messy, or if students chose to solve problems with a strategy that is not the most efficient.

Let's explore a 2nd grade lesson on solving problems of addition and subtraction with re-grouping with my friend and colleague Richard Ziegler.

Number Talk

Mr. Ziegler's first move is to start with a quick number talk, to warm up his students' brains. Sometimes, he uses an equation such as 26 + 36 and students solve it in their head. Only mentally. Then he asks students to share with each other how they solved.

After that, Mr. Ziegler asks a few students to explain how they solved. They explain step by step, and he uses numbers and pictures to represent the strategy that the students are describing. It is an art!

Sometimes Mr. Ziegler uses a math talk, it could be an image, and the questions could be: What do you notice? What are you wondering? How do you know? Convince your partner. The process is the same, he asks students to think about it on their own, mentally, then share with a partner, and finally, to share with the class.

Check out the books "How Many?" and "Which One Doesn't Belong" by Christopher Danielson. They are great resources for math talks.

I will write another blog on number talks soon. Stay tuned!

Mini-lesson

The easiest way to use numberless problems is to use the slow-release function in a power point presentation. This will allow you to release information about the problem little by little.

If you do not want to use a computer, you could create some strips of paper and use them little by little, just as the slow-release function works on the power point slides.

The mini-lesson starts here. What are dragonflies? Have you seen them? What are toads? There are pictures on the slides to make the context very clear to students. There are no numbers yet. Students are not trying to guess answers quickly. They are wondering about this real situation.

Release one more part of the problem.

Mr. Ziegler then asks: What can the numbers be? What numbers could not be? Students here explore estimation. It is about building number sense.

Students start talking about many possibilities. They make sense about the fact that the snake ate more than the toad. How many more? "We do not know yet" they say.

Students are engaged, they want to know more, how many? How many? Please tell us Mr. Ziegler!
"How do you make students engage in math class? Give them a problem to care about!" Steve Leinwand at MSIS 2017.
The next slide reveals that the snake ate 26 more dragonflies than the toad ate. How many did the toad eat? The toad ate 39.

Mr. Ziegler pauses. "Hmmm. What can the question be?"

Multiple hands go up!

This is an essential teaching move, as students learn that there are many answers that can be found with this information, they begin to understand that there is not only one answer to a problem, it depends on what we are figuring out. There are many questions out there, and also many answers.

Students come up with all sort of questions:

*How many dragonflies did the snake eat?

*How many dragonflies did they eat altogether?

*Why do they eat the dragonflies? Why don't they eat grass?

Some questions don't make sense. Get ready for that. Ha!

Mr. Ziegler then reveals the question!

"Ahh, I knew it" says one student.

This was the mini-lesson. The goal is to spend 8-10 minutes talking about the context of the problem. Having some student to student discourse about what the problem is about, and what are we going to figure out. The teacher is a facilitator.

At this point the teacher has not asked students to use any specific strategy. The questions are about the context, and having a clear idea about what is it that we want to find out.

Students can solve this problem in many different ways. They can use manipulatives, or draw models, or use numbers flexibly. Students already have ideas about how to solve problems, let's hear them out.

This problem has 3 different combinations of numbers. Students can choose to start with the first set of numbers, and move on to another set. Or they can solve just the first set. It is not about quantity of addition facts, but quality of problem solving strategies. Students are encouraged to use different strategies to solve the different sets. This problem allows for differentiation.

Active Engagement﻿

"Is everyone clear about what we need to do now?"

"Off you go," says Mr. Ziegler.

Students now go to their tables and work in pairs. They can use concrete manipulatives, drawings, or numbers.

The role of the teacher now is to walk around and identify who needs support, or who has some clarifying questions. It is here, where the teacher can pull a small group, and perhaps review a strategy that was shared the day before and that this group might need to review again.

The teacher can also observe if there are students that need an extra challenge, maybe there are students that solved all 3 sets of numbers.They can come up with another question using the same context. There are opportunities of differentiation when using numberless problems, and also by extending the questions.

The teacher could be talking to different groups or, individual students and checking for understanding. Also this is the opportunity to individually ask students to try new strategies. For instance, if a student has been using the same strategy for the past few lessons, the teacher would suggest he/she tries a new strategy as he/she seems to have mastered the previous one.

Mr. Ziegler chooses to sit with a group of students and is asking them questions about the strategy they are going to use. I hear him ask a student if he could also use a number line to solve this problem. His voice is very melow and friendly, it sounds like an invitation, not a challenge. The student says "yes, of course I can!" Mr. Ziegler is such a talented teacher.

This student solved using the decomposing strategy, and then took a risk and solved with a different strategy.

I sat with a student that used the number line to solve all three sets of numbers. I asked her in a few different ways if she would be willing to try a new strategy, but she said no. She really liked using the number line, she said. Later in the day, during our debrief session, I made sure to let Mr. Ziegler know about this, so he could use his amazing and inviting voice next time so he can encourage this girl to try a new strategy.

This student chose to use the same strategy for all three sets of numbers.

Share-Out

"It is time to come back to the carpet" says Mr. Ziegler. "Now we are all going to share some of the strategies we use in 2-RZ."

This is the most essential part of the lesson because it is here that students will be exposed to a variety of strategies, and the teacher might introduce a new strategy.

During the active engagement part of the lesson, the teacher is walking around and talking to different groups or students to check for understanding, and also choosing the strategies that will be shared today in order to hit the learning target.

Students come up to the front and show their work under the document camera or any other device that helps students have a good visual. It is recommended to have 3 or 4 students share their strategies, this gives the rest of the group a variety of different ways they can try the next day.

If a student comes up with a strategy for the first time, the teacher might ask the class if they should call this strategy "Laura's strategy" or the name of the student that came up with that particular strategy. Believe me, naming the strategies with students' names makes the strategy suddenly very, very polular!

If most students used a similar strategy or not a very efficient strategy, it is here where the teacher can share an efficient strategy that he/she would have used. This strategy then, is one more option, but not the only option. Making an emphasis on how efficient this strategy might be, and/or giving the new strategy a special name, will make your students really want to try it.

Mr. Ziegler had 2 students share their strategy. The first student shared how she solved using pictorial base ten blocks. She explained how she added the tens, and made 5 tens. Then she explained how she added the ones and got 15 ones, so she made another ten and added it to the 5 tens she previously had. She explained how she ended up with 6 tens and 5 ones, which is 65 dragonflies.

Student #1 used a pictorial base ten blocks strategy.

The second student came to the front and showed how he had solved the problem by using the decomposing strategy. He explained step by step, how very similar to her classmate, he had added the tens first, then the ones, and added a new ten. He also found that 65 dragonflies was the solution.

Student #2 solved by using the decomposing strategy.

Mr. Ziegler then came and did his magic! He showed to the class how these two strategies are connected! You can see on the picture, how he is showing both strategies at the same time, and allowing students to make the conenction, because both strategies are based on place value. They just had different visual representations, that is all. Magic.

Mr. Ziegler then guided students to see the connection between both place value strategies.

If you run out of time, or you don't have enough time in your lessons, you can start you lesson with the "share out" part from the day before. If you have enough time during your lesson (some teachers have 60 minutes for each lesson) you could try to start with a slide for "Some of thework from yesterday...." and you could have a quick teaching point about a very efficient strategy before starting your lesson for today. Then do all the other parts of the lesson. This is ideal as you get 3 teaching points throughout your lesson.
This could be the overall structure of a Student-Centered Lesson:
Part 1- Some of the work from yesterday...... (teaching point) 4-5 min

Part 2- Mini lesson (what is the problem about/context) 8-10 min

Part 3-Active engagement (teaching points to individual students or small groups) 15-20 min

Part 4-Share out (learning happens here! Multiple teaching points) 10-15 min

Something to consider....

Before you try this approach to teaching math, you need to be comfortable with students trying the different strategies at different times.

One of the biggest concerns is that sometimes you want your students to try a specific strategy right away, that day, so you can assess their work and see if they get it. But with this approach, this might take a few days. So if you are patient, your students will start making sense of what are the most efficient strategies to use, and will be able to explain it themselves, and apply it.

This approach allows them to own their learning. It ensures a balance between content standards and mathematical practices, which makes a balanced math approach to learning.

Connection to Mathematical Practices

Which math practices are used on a lesson like this?

They use problem solving skills, by choosing the most efficient strategy on their own. Making sense of what the problem is about to choose how to go about it. (The opposite of following a set of rules.)

They use modeling to solve problems, by showing how their strategy works with a visual representation. This representation helps them illustrate the problem and find the answer.

They use reasoning & communicating skills, by convincing their classmates about how their strategy works.

Give it a try and let me know how it went!
If you want to dive deep into this approach, read the book "Teaching Student Centered Mathematics" by John A. Van de Walle

As teachers, we continue to learn new approaches to explore mathematics and are faced with new opportunities to provide the best learning experiences for our students. We are constantly participating in different professional learning to keep our teaching practices in line with the most up-to-date research on how students best learn. We collaborate constantly with fellow teachers, administrators, and instructional coaches to learn from each other, and to make sure our classrooms are true learning environments.

作为老师，我们一直在学习对数学进行探索的新方法，同时还面临着向我们的学生提供最佳学习体验的机会。我们不断参加各种职业培训，以在给学生提供最好的学习体验这件事上让我们的教学实践跟上最新的研究成果。我们不断与其他老师、行政管理人员以及指导教练合作，相互学习，来保证我们的课堂是真正适合学习的环境。

I am also a parent of twins in elementary. As parents, we want to be informed and involved in our child's education, so we can help at home. We engage with our kids asking questions about what they learned at school at every dinner conversation. We want to be a positive influence on our kid's emotional and academic life.

正是因为这些原因，作为上海美国学校浦西校区的小学部数学指导教练，作为正在读四年级的双胞胎的家长，我开始参与到我们的家长社区中来。在过去两年的时间里，我们举办了多场"数学交流会"，介绍了我们在上海美国学校教授和学习数学的方法。自第一场活动吸引了90位家长以来，这个活动的反响非常好。

These first meetings were about sharing with our parent community that recalling facts, procedures, and formulas is merely arithmetic, and while it seems to "work" for elementary and middle school, it does not prepare students to solve real mathematical problems, which is the main goal of mathematics: to solve "messy" or "unfamiliar" complex problems. Mathematics is about making connections and solving problems by making sense of numbers, by using numbers flexibly, by using concrete and visual representations to build understanding. When students truly understand a concept and they can visualize it, use a model or representation, and can explain it, then they are ready to make a connection to a new math concept where they can apply their knowledge accordingly to solve the new "messy" problem.

在这些活动中，我们让学生家长认识到，记住数学事实、程序和公式仅仅是停留在算术层面，尽管这似乎对于小学和中学来说"奏效"，但这并不能让学生解决真正的数学问题，解决"棘手的"或"不熟悉的"复杂问题才是数学教学的主要目的。数学就是要对数字进行理解，以灵活的方式使用数字，使用有形和直观的方法帮助理解，从而建立联系，解决问题。当学生真正理解一个概念，能够对其进行可视化呈现，使用模型或表征，并对其进行解释的时候，他们就能够与全新的数学概念建立联系，运用他们学到的知识解决新出现的"棘手"问题。

Parents continued to want to learn more about this conceptual approach to learning mathematics, so we invited them to participate in four sessions called "Math in March."

家长们希望进一步了解数学学习的概念化方法，于是我们邀请他们参加了四场名为"三月数学"的活动。

We started by showing the progression of addition and subtraction strategies our students use to build understanding in solving problems. Parents were engaged using a variety of CONCRETE manipulatives, to make sense of place value when adding and subtracting without the need to use any rules or tricks such as "carry" or "borrow". They were able to use place value strategies to understand how to make tens, or how to "unbundle" a ten when subtracting. They continued to use PICTORIAL representations such as tape diagrams or number lines to model the problems. They understood the difference between modeling and fake modeling (which is just drawing an answer found by an algorithm). Using visual representations is how we really MODEL a problem, then we can choose other strategies to help us solve.

首先我们向家长们展示学生们用来解决问题的加减策略。家长们使用一系列"有形"教具在加减运算过程中对位值进行理解，无需使用"进位"、"借位"等任何规则或技巧。他们能够使用位值策略理解如何加到十，或者是在进行减法运算时如何对十进行"拆分"。随后，他们使用尺图、数轴等"图形"表示对问题进行建模。他们理解了真建模和假建模（只是用算法求出答案）之间的差别。我们使用直观的方法对问题进行建模，随后我们可以选择其他策略帮助我们解决问题。

We moved onto ABSTRACT strategies and then expanded into multiplication and division, and in later sessions into fractions. (Fractions were a hit with the parent community. They were so excited to use fraction tiles, unifix cubes, number lines, area models, and tape diagrams to understand how to add, subtract, multiply and divide fractions conceptually!)

接着我们介绍了"抽象"的方法以及乘除法和分数运算。（分数深受家长喜爱。他们非常兴奋地使用分数瓦片、方块积木、数轴、面积模型和尺图理解如何从概念上对分数进行加减乘除！）

The constant sounds of "ahhhhh" and "ohhhhh" were just fabulous when parents were shown a visual representation of what ½ x ¼ looks like! Parents were able to see past the formula and tricks and able to see in a new, clearer way what one half of one fourth really looks like, and why the product is one eighth. The room was loud, Chinese and English language, iPhone flashes, unifix cubes everywhere! Our parents were doing mathematics!

当看到 ½ × ¼ 的直观表示时，家长们发出了一阵阵的惊叹声。家长们能够透过公式和技巧以更加清晰的全新方式看到四分之一的二分之一到底长什么样以及乘积为什么是八分之一。家长们议论纷纷，有说中文的，也有说英文的。大家纷纷拿出手机拍照，到处都是方块积木。看哪，家长们在学数学！

Annie Barnard, mother of 5th grade and kindergarten students, said of the experience, "Most of us were impressed with simple questions we had been taught in the past and knew how to do, but never got to understand why. You taught us to visualize the math. Seeing the answer make sense in front of our eyes really gave us a good vibe for the first time, and got us wanting to learn math again. We're now helping each other understand how to solve math problems without worrying about not being able to remember the right way - because there isn't one way to solve math, there are many ways."

作为五年级学生和幼儿园学生的妈妈，Annie Barnard表示："让我们大多数人印象深刻的是那些我们过去学过并且知道如何解答但从不知道其中缘由的简单问题。我们在用直观的方法学习数学。亲眼看到答案呈现在眼前真可谓让我们眼前一亮，也让大家升起了再次学习数学渴望。我们现在能够相互帮助，理解如何解答数学问题，而不用担心记不住正确方法，因为解答数学问题并不只有一种方法，而是有很多方法。"

What can parents do at home to support their child to feel successful in mathematics?

家长们在家里能够做些什么，从而让他们的孩子在数学上获得成就感呢？

*If you find a mistake in their homework, instead of saying, "That is the wrong answer, do it again." Try saying, "guide me through this problem, how did you get that answer?" Most likely, when explaining their strategy, they will catch their own mistake.

*如果你在他们的家庭作业中发现了错误，不要说"这个答案不对，重做一遍"，而是要说"给我讲讲这个问题，你是如何得到这个答案的"。大多数情况下，他们在解释做题策略时，就会发现自己的错误。

*When doing homework, ask: "Can you think of a different way to solve that?"

*在做家庭作业时，你可以说："你能不能想出解这道题的另一种方法？"

*Play with numbers, for example: find all the different ways to make 24, or all the different ways to solve 16 x 9.

*玩数字游戏，例如：找到得到24的所有方法或解答16 × 9的所有方法。

*Solve math puzzles.

*解答数学难题。

*Play math games to practice fluency, fun games, not flashcards!

*通过玩数学游戏做到熟练掌握，要玩有趣的游戏，而不是数字卡片！

Across both campuses, at SAS we use a balanced approach to teaching and learning mathematics. Teachers and students not only focus on concepts and procedures, they focus on the use of visual representations to model real-world situations, they problem solve, they communicate and reason in their thinking, and make connections to their future learning.

在上海美国学校的两个校区，我们使用一种两者兼顾的方法开展数学教学。老师和学生不仅专注于概念和程序，还专注于使用直观方法来对现实生活情形进行建模，解决问题，沟通并推理他们的想法，与未来学习建立联系。

If you are introducing fractions in your class, allow students to use unifix cubes to make sense of what fractions really are and look like.

You can start with fractions that have the same denominators so students understand that the whole is made of a number of parts. A fraction is part of a whole. You can have 1 whole divided in 4 parts, and you can have 1 whole divided in 8 parts. The most important thing is for students to understand that the wholes have to be the same size. For instance 1/2 small pizza and 1/2 large pizza do not make 1 whole pizza because the wholes are differen sizes.

After students have had a lot of opportunities to explore adding a lot of fractions with the same denominators, you could start posing problems with "alike" denominators. For instance start with a problem like 1/4 + 1/8. Give students unifix cubes and let them figure out how they could represent these 2 fractions using the same whole as a reference.

After they represent these 2 fractions with the same whole, they can add the 2 fractions. When students work in pairs, they can help each other to make sense of how to represent eights and fourth using the same whole. It might take some time, but they will get there.

Step 1- Ask students: using unifix cubes, how can you make 1 whole that can be divided in "eights" and also in "fourths"?

Ask students to SHOW how their whole can be divided in fourths and eights. Some students might make a whole with 4 unifix cubes only, and when they try and divide that in eights, they will realize it doesn't work. They will try again. Do not solve for them.

Have students explain their thinking, and share with the class how did they get to that conclusion.

After they have their fractions represented, they can identify 1/8 and 1/4.

Step 2- Ask students to show 1/8 of the first whole, and 1/4 of the second whole. Label each fraction. Students will get a visual representations of what both fractions look like related to one another. It has to make sense to them.

Step 3- Ask students to put their fractions together and find an answer. If students answer 3/4, ask them to go back to what they identified as 1/4, ask: "Does that make sense?" They can discuss and reflect on it, and come to understand that each of those 3 unifix cubes look just like the 1/8 from step 2.

This whole time, students do not need to use any paper or pencil, and defnitely no need to learn a set of rules to find the common denominator. That will come later. Students will make sense of it on their own. Just give it time.

Concrete & Pictorial Representations

Here is one more example to use in your class, to allow students to make sense of adding fractions with alike denominators without using memorized steps. These strategies allow students to understand why if you are adding thirds and sixths, the answer can't be ninths! Because they have never seen ninths in the representations they are creating.

Step 1 - Pose the problem on the board. 4/6 + 1/3. Ask them to use unifix cubes to represent sixths and thirds using the same size whole. Let them struggle a bit. They can work in pairs, and talk to their partners. They will start with thirds and eventually will make sense of using 6 unifix cubes to also represent thirds. Have students explain to each other how they came up to that conclusion. Explaining their thinking is for their own learning, some students finish making sense of what they learned, when they hear themselves explain, when they organize their thoughts, and make sense of it in words. Learning is fascinating!

Step 2 - Ask students to label their fractions.

Step 3- What is the total fraction now? Have students explain why their answers is 1 whole. Have them show how these 2 fractions make 1 whole. This is a perfect moment to ask students if they could come up with other models to represent this same problem. Perhaps they can use fraction tiles, or fraction circles. They can also draw the fractions if you do not have other manipulatives in your class.

There are many concrete manipulatives to allow students to build conceptual understanding of fractions.

Finally, it is time to use pictorial representations. The bar model or tape diagram, is a recomended model to represent fractions. It is easier for students to use a rectangular model than a circular model to be more accurate with the equal number of parts.

Most students will start using 2 tape diagrams, and that is ok, as long as the 2 wholes are the same size.

Eventually they should start partitioning the same whole in both sixths and thirds. They can use 2 different color markers to see their work. This is a good strategy for when they start sbtracting fractions.

Use a variety of good problems to help students see where we use fractions in the real world. Here are some ideas:

If Billy has 35 cookies and shares 1/5 of them, what fraction does he have left?

If I read ¾ of my book, and I have 80 pages left, how many pages is my book?

I am running a 3 km race. I am almost there. I have 1/6 left. How much do I have to run?

Is the sum of 4/5 and 7/8 under or over one? Explain how you know.

Enjoy playing fractions with your students!

I have been teaching for about 29 years. Most of those years in elementary. Math has always been my most favorite class to teach, but as most teachers did in the past, I used the I-do-you-do approach. I did my best at explaining how to solve problems using manipulatives and pictures, and quickly moved into abstract strategies using tricks, and helping my students memorize steps. I remember how subtracting with re-grouping was always a "hot topic" in my classes.

I thought I was doing justice to my job by taking extra time to review the steps to re-group when solving subtraction. I was patient and used small group instruction to review the steps. I offered extra sessions during recess to review steps for subtracting using the algorithm efficiently. I thought this was best practice. This was back in the early 2000's.

It wasn't until several years later, when I started using a student-centered-approach to teaching and learning math. That was one of the best years in my long journey as a teacher. That was the year when my students dicovered on their own, what re-grouping really means, and why we use it. That was the first year, that re-grouping made sense!

It is ideal to teach addition and subtraction strategies at the same time. Math problems can be solved by adding or subtracting. Use concrete manipulatives for students to see what re-grouping looks like.

When students can use concrete manipulatives, like these base ten blocks, they can make sense of what "take-away" looks like. For instance, when taking away 128 from 274, students will start by taking away the hundreds, to then take away the tens.

Then they have 154 left. When they want to take away 8 from the ones, they will realize they do no thave enough tens, so they organically will "exchange" a ten for 10 ones, and take the 8 ones from there. At this point, there are no rules or tricks. Students know that they have enough to take away 8, they can SEE that they do not have enough ones but they have enough tens.

It is important to allow students to work in pairs, so they discuss this, so if one of them is not thinking of a strategy, the other student will probably come up with it, and all of a sudden, no one knows who came up with it, it was their work combined!

After that students just see what they have left. They have 1 hundred, 4 tens, and 6 ones.

It is important to allow students to use manipulatives for a few days, so they really learn to trust the process of using these base ten blocks to find the answer. A lot of students do not trust this process and quickly use an algorithm to find the solution, an dthen just represent the answer with the base ten blocks. That is fake modeling. That is not using manipulatives to build understanding.

After students get use to using manipulatives, and understanding how to make sense of place value, they can explore other strategies like "subtracting by parts" where they decompose the number, and subtract part by part.

When students develop strong number sense, and use numbers flexibly, they could come up with other strategies like "subtracting by place." It is ideal to allow this strategy to organically emerge from one of the students, instead of teaching it. When they use their own words to explain a new strategy, it suddenly makes sense.

I have come to realize that when your main focus on the lesson the problem, with engaging context, students really care about the context, and feel free to explore different strategies. They are not focused on using one strategy efficiently, they care about the math that is involved to figure out that good problem.

Today, I do not have to teach the algorithm strategy, nor do I teach any tricks and steps to solve a subtraction problem, because when students build understanding of using place value strategies with base ten blocks, and using decomposing strategies, they make sense of how the algorithm works. They figure out that algorithm tricks like "borrow from the "neighbor", is nothing but realizing that if you don't have enough ones, well, then you take it from the tens!

Do you have students that solve ALL problems using only the traditional algorithm? Do they stop and think about the strategy that could make the most sense for that particular problem?

Here are some strategies that support students in building understanding of what adding really means, before using "carry-on" as a set of rules to be followed to solve any problem.

It is essential for students to move from CONCRETE manipulative, to PICTORIAL strategies, to end in more ABSTRACT strategies. This progression will bring understanding and fluency, as students will have a variety of mental strategies to solve problems.

It is important to allow students to explore concrete manipulative to understand how to bundle ones into tens, and tens into hundreds when it is necessary.

When solving a problem with manipulative, students tend to start putting together the hundreds, then the tens, and the ones.

When students get more than 10 ones, they immediately re-group them and put them with the other place value that it belongs.

Then they count how many of each and find an answer. There are no rules or procedures, the concrete manipulative allows them to make sense of what they have in all.

After using these concrete experiences to make sense of addition with regrouping and making sense of place value, students can move into pictorial strategies. One strategy could be using the place value chart. Students can use real place value discs, or draw the place value discs.

Then they make a connection with the previous concrete strategy, and will know what to do with more than 10 ones.

Use the term "bundle" for students to understand that 1 ten is a bundle of ten ones.

Students can make a connection between using the manipulative and using this place value chart. It is not a new strategy, it is a connection. Same strategy different visual representation.

Now it is time to move into more abstract strategies but still based on place value, not based on a memorized set of rules. For example this expanded form strategy is another visual representation of the exact same thing they did using the place value chart.

Again, students will likely start by adding from left to right (unlike the algorithm), and will make sense of what to do with more than 10 tens, and more than 10 ones. They had the previous experience of "exchanging" the manipulative, and "bundling" the place value discs. They do not know what re-grouping even means, but they understand what the value of these numbers are in all.

Finally, students can make a connection with another visual representation. This abstract strategy is usually called partial sums.

Students can see the same values than before, they will start by adding 300 and 110, and can mentally solve using the visual representation they have built previously, and come up with 410, then mentally again add 13 by breaking it down into 10 and 3.

When students become fluent with these strategies, they will eventually make a connection with the traditional algorithm, and they will finally understand that "carry" is nothing more than "bundling", but bundling makes sense.

Even if the math program you use in your school encourages students to solve 15+ addition problems using the standard algorithm, you could make a decision and create an environment of exploring the beauty of mathematics, and allow your students to solve 3 or 4 addition problems using CONCRETE-PICTORIAL-ABSTRACT strategies to really understand how math works, and how strategies based on place value are all connected!
"As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. But teachers and curriculum writers are often unable to access important research and this has meant that unproductive and counter-productive classroom practices continue." Jo Boaler, Fluency Without Fear.

By inviting children to solve problems in their own ways, we are initiating them into the community of mathematicians who engage in structuring and modeling their "lived worlds" mathematically. Fosnot and Jacob, 2007
Teaching mathematics through problem solving requires you to think about the types of tasks you pose to students, how you facilitate discourse in your classroom, and how you support students use of a variety of representations as tools for problem solving, reasoning, and communication.

This is a different approach from "do-as-I-show-you" approach where the teacher shows all the mathematics, demonstrates strategies to solve a problem, and then students just have to practice that exact same skill/strategy, perhaps using a similar problem.

Teaching mathematics through problem solving means that students solve problems to learn new mathematics through real contexts, problems, situations, and strategies and models that allow them to build concept and make connections on their own.

The main difference between the traditional approach "I-do-you-do" and teaching through problem solving, is that the problem is presented at the beginning of the lesson, and the skills, strategies and ideas emerge when students are working on the problem. The teacher listens to students' responses and examine their work, determining the moment to extend students' thinking and providing targeted feedback.

Here are the 4 essential moves in a math lesson using a student-centered approach or problem-solving approach:

Number Talk (5-7 min) (Connection)

The mini-lesson starts with a Number Talk. The main purpose of a Number Talk is:

*to build number sense, and

*to provide opportunities for students to explain their thinking and respond to the mathematical thinking of others.

Please refer to the document Int§roducing Number Talks. Or watch this video with Sherry Parrish to gain understanding about how Number Talks can build fluency with your students.

Here are some videos of Number Talks so you can observe some of the main teaching moves.

The role of the teacher during a number talk is crucial. He/she needs to listen carefully to the way student is explaining his/her reasoning, then use a visual representation of what the student said. Other students also share their strategies, and the teacher represents those strategies as well. Students then can visualize a variety of strategies to solve a problem. They learn how to use numbers flexibly, there is not just one way to solve a problem. When students have a variety if strategies in their math tool box, they can solve any problem, they can make connections with mathematical concepts.

There could be multiple answers for these kind of math talks. Students make sense of their answer and convince the class using their reasoning and math vocabulary.

There are a variety of resources that can be used for Math Talks. Note: the main difference between Number Talks and Math Talks, is that one allows students to use numbers flexibly leading them to fluency, develop number sense, and opportunities to communicate and reason with mathematics; the other allows for communicating and reasoning, building arguments to critique the reasoning of others, the use of logical thinking, and the ability to recognize different attributes to shapes and other figures and make sense of the mathematics involved.

2. Using problems to teach (5-7 min)

Problems that can serve as effective tasks or activities for students to solve have common features. Use the following points as a guide to assess if the problem/task has the potential to be a genuine problem:

*Problem should be appropriate to their current understanding, and yet still find it challenging and interesting.

*The main focus of the problem should allow students to do the mathematics they need to learn, the emphasis should be on making sense of the problem, and developing the understanding of the mathematics. Any context should not overshadow the mathematics to be learned.

*Problems must require justification, students explain why their solution makes sense. It is not enough when the teacher tells them their answer is correct.

*Ideally, a problem/task should have multiple entries. For example "find 3 factors whose product is 108", instead of just "multiplying 3 numbers. "

The most important part of the mini-lesson is to avoid teaching tricks or shortcuts, or plain algorithms. Our goal is always to help guide students to understand why the math works (conceptual understanding). And most importantly how different mathematical concepts/ideas are connected! "Math is a connected subject" Jo Boaler's video

"Students can learn mathematics through exploring and solving contextual and mathematical problems vs. students can learn to apply mathematics only after they have mastered the basic skills." By Steve Leinwand author of Principles to Action.

3. Active Involvement (20 min)

This is the opportunity for students to work with partners or independently on the problem, making connections of what they know, and trying to use the strategy that makes sense to them. Always making sure to represent the problem with a visual representation. It can be any model that helps student understand what the problem is about.

The job of the teacher during this time, is to walk around asking questions to students to guide them in the right direction, but without telling too much. Allowing students to come up with their own solutions and justifications.

Teacher can clarify any questions around the problem, not the solution.

Teacher emphasizes reasoning to make sense of the problem/task.

Teacher encourages student-student dialogue to help build a sense of self.

Some lessons will include a rich task, or a project based learning, or a number problem (find 3 numbers whose product is 108). There are a variety of learning target tasks to choose from, for each grade level on the Assessment Live Binders website created by Erma Anderson and Project AERO.

Again, keep in mind that some lessons will follow a different structure depending on the learning target for that day. Regardless of instructional design, the teacher should not be doing the thinking, reasoning, and connection building; it must be the students who are engaged in these activities

4. Share (10-12 min) (Link)

The most crucial part of the lesson is here. This is where the teaching/learning happens, not only learning from teacher, but learning from peers reaching their unique "zone of proximal development" (Vygotsky, 1978).

We bring back our students to share how they solved their problem. Sometimes they share with a partner first, to make sure they are using the right vocabulary, and to make sure they make sense of their answer. Then a few of them can share with the rest of the class. But sharing with a partner first is helpful so everyone has the opportunity to share.

"Talk to each other and the teacher about ideas - Why did I choose this method? Does it work in other cases? How is the method similar or different to methods other people used?" Jo Boaler's article "How Students Should Be Taught Mathematics."

Students make sense of their solution. The teacher listens and makes connections between different strategies that students are sharing. Teacher paraphrases the strategy student described, perhaps linking it with an efficient strategy.
"It is a misperception that student-centered classrooms don't include any lecturing. At times it's essential the teacher share his or her expertise with the larger group. Students could drive the discussion and the teacher guides and facilitates the learning." Trevor MacKenzie
If the target for today's lesson was to introduce the use a number line, for example, this is where the teacher will share that strategy as another possible way to solve today's problem!

This could also be a good time for any formative assessment, using See Saw, using exit slips, or any kind of evidence of what they learned today.
References.
"Teaching Student-Centered Mathematics" Table 2.1 page 26, Van de Walle, Karp, Lovin, Bay-Williams

"Number Talks", Sherry Parrish

"How Students Should be Taught Mathematics: Reflections from Research and Practice" Jo Boaler

"Erma Anderson, Project AERO Assessments live binders?

"Principles to Action", Steve Leinwand

" Turning Teaching Upside Down ", by Cathy Seeley

"Four Inquiry Qualities At The Heart of Student-Centered Teaching"

By Trevor MacKenzie

"The Zone of Proximal Development" Vygotsky, 1978

*** Here is a link to my favorite places to plan Math at SAS padlet, you will find a variety of resources, videos, articles, etc. By Caty Romero

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Last week, I had a great time teaching & learning division of fractions with one of my teachers. I have not stopped thinking about how models really provide a visual representation of what dividing fractions really mean. Our students use concrete models to gain understanding of what fractions are, how to put them together, take them apart, and take parts "of" it.

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Padlet - Caty Romero

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F-10 Curriculum | V9 Australian Curriculum

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Padlet - Turning Teaching Math Upside Down

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Padlet - Turning Teaching Math Upside Down

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A variety of resources to teach & learn mathematics

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Video: Flexible Thinking and conceptual understanding

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Video: Flexible Thinking and conceptual understanding

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Flexible Thinking & Conceptual Understanding