3 Act: The Arcade

This might be one of my new favorite tasks.  It might be the experience of seeing the attendant turn the adding machine around once my son and I had mentally calculated the total or it might just be because I got to play (and win) Coin Dozer (my absolute favorite game)!  Please tell me what you think by commenting on how you will or have used it in your classroom!

When we go to the arcade, my son doesn’t like to wait to turn in his tickets so we always end up with several separate sheets of ticket totals by the end of our visit.

The task below could be used for CCSS 4.NBT.B.4 and MLS 4.NBT.A.5.

I love this problem because act 3 allows the student to engage in MP3 “construct viable arguments and critique the reasoning of others” as well as MP6 “attend to precision.”

Act 1:

What do you notice?  What do you wonder?

How many tickets does he have? Write a too high estimate, write a too low estimate.

Act 2:

Act 3:

The attendant came up with this total.  Is he correct or incorrect?

Would you be happy or unhappy with this total?  Why?

(Side note, when I asked him if I could take a picture of it for a math lesson he said “I rounded up for you.”  I think there might be some great talk around this too:)

For example:

What does he mean he rounded up?  What was he rounding to?  The nearest 10, the nearest 100 or something else?

Update!

Here is the actual total with calculator strokes.  I think this is important for students to see once they finish their calculations and discuss.

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3 Act: Sealed With __ Hearts

My son was working on his Valentines tonight and said, “how am I supposed to get these hearts to stick?  There are supposed to be 3 on each one.”  I said, “you are probably only supposed to put one on each” and he said “no there are 48 hearts and 16 cards so there are exactly enough for 3 on each.”

Well, that got my brain going when I looked at that perfect little array on the sticker sheet and I jumped up to film:)

Best Fit CCSS Standard(s) 4.OA.A.2, 5.NBT.B.6

Act 1:

What do you notice?  What do you wonder?

How many heart stickers will there be on each Valentine?

Estimate.  Give a too high and a too low answer.

Act 2:

What information do you need to solve the problem?

Choose the image you prefer for this part of the task.

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Act 3:

Were you correct?

 

 

Algebra in First Grade?!

One of my favorite sites for promoting mathematical discourse and allowing students to see patterns and connections in mathematics is Fawn Nguyen’s ww.visualpatterns.org.

I first learned of this website from a Twitter chat that I regularly participate in and couldn’t wait to try it out with my students.  I img_0788.jpgdecided to try it out in our small groups and simply ask students the following questions:

 

 

  • What do you notice?
  • What do you wonder?
  • What do you think will come next in the pattern?  Can you draw it?
  • Explain your thinking.

It didn’t take long for the room to be full of chatter around the task.  The website quickly became a favorite and students would ask me throughout the day if we would be using it in small group.

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In my opinion, the greatest challenge in teaching today is student motivation…and if you can find that magical tool that combines deep conversation, connections to mathematical concepts AND your students are excited about participating…you run with it for as far and long as you can!

 

 

One thing that I have noticed about students over the years is when they understand, when they TRULY understand a mathematical concept, they are eager to share their learning with others if given the chance to do so in a creative way.

The interesting part is that many times my students who are reluctant to share at first aren’t students who struggle with a math concept, they are students who understand the concept, but don’t want to be singled out by a teacher who always calls on them. They have come from classrooms that value answers and they have already lost their excitement for learning.

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However, if we stop making mathematics about answers and start making it about problems that can have creative solutions, those students are just as excited to share as others because they get to show what they really care about…who they are and what they have to offer the group; Not a one word answer to someone else’s problem.

 

One of the many great things about Fawn’s website is that the tasks are low floor, high ceiling.  Every student in the classroom can enter the task and begin talking about the mathematics they see.  Look at this example.  Each student in my classroom could count the number of objects, tell me what shapes they were, notice that there were more each time, etc.  and the students who needed a challenge could easily set to work figuring out what the 7th image would look like while the rest of the students continued their conversations.  It really is beautiful to see.  Everyone working on the same task and every conversation a little different.

During this particular task I heard:

 

“No the fourth shape would have four on the bottom row because each time they added a row to the bottom with one more”

 “They just added one more to each row.”

“There is 1 triangle in the first one and 3 in the second one…”

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And students couldn’t wait to get up and explain their thinking to their peers.  They even established ways to label their thinking to make it more clear to each other.

The amazing thing about these tasks is that they are giving students the opportunity to see patterns and relationships between numbers in a visual way so that they can create connections between concepts.

Math classrooms should be creative spaces for students to explore, connect, argue and explain.  Places where students can be engaged in problems that are exciting and stimulating.  When I find a resource that can support that goal, it’s a win!

 

 

 

Storytelling in Math Class?

I created a new session for a conference in St. Louis this summer after having great success in our classroom with stop motion animation as a means for exploring word problems.   I was so excited about the engagement and conversations I was hearing that I really wanted to share the experiences, so I got to work.  Pretty soon it occurred to me that my presentation and the excitement that I had wasn’t about Stop Motion or any other tech tool.  It was about the powerful mathematics that was happening in our classroom because we were telling stories to make sense of mathematical ideas. This is what I tried to capture in the session. I was a little concerned that attendees may be upset that we didn’t actually address Stop Motion until the end of the session, but hoped for the best.

What I wasn’t prepared for was the absolutely fantastic conversations I was hearing with teachers as they began to tell stories; The way the room started buzzing and the smiles I saw on their faces when they used materials to “be silly” with each other.

While I was walking the room and listening to conversations, I heard one teacher say to another, “That’s not a story.  You just put some animals down and said what they were!” I quickly grabbed the opportunity to call this conversation out because it is exactly what happens with students.  Our first graders are obsessed with rules and procedures.  They want to make sure that everyone is being treated fairly and “following the rules.”  If one of the students had done that with their objects, the other would most definitely call them out on it.  We discussed how this opened up an opportunity for not only powerful conversations about math, but that we just learned that some students weren’t clear on what constitutes a story.  This student (and others) might need to revisit the elements of a good story.

I also saw many teachers using their animals to hunt each other or kill the other animals. It’s always interesting to me to see the variation in beliefs about this type of behavior with teachers.  Some teachers feel very strongly about keeping it “kid appropriate” and others just go all in.  Kids are not going to allow their imagination to be stifled.  I intentionally include knights and dragons and predators and prey in the materials I put out for students.  In my opinion, we have got to allow students to be creative and not stifle their creativity.  Yes, there are definitely instances when we might have to step in, but one of the great things about these activities is that students are able to see how math connects to their world.  I want to leave that as open-ended as possible for them.   The minute I tell a student that they can’t tell a story “that way,” I feel like I have put them in a box and told them their imagination is not valid or worse yet, “not good enough” or “teacher-approved.”

Teachers ultimately have to decide what is and is not appropriate in their classroom.   However, I would challenge everyone to step back and take an unbiased look at the rules and procedures in your classroom.  Are you staying consistent in beliefs and practice?

For the session I decided to focus on eight different ways to use storytelling to develop context for word problems.   In my opinion, this allows students opportunities to build experiences with the language that are used in situations in daily life, but also in math word problems.

Story Telling

Allow students time to explore and tell stories.

During this section of the session, teachers were given many different types of materials and allowed to tell stories.  There need be no math involved, they were just supposed to “play with the objects” and make up stories.  This is much like when we introduce a new manipulative and allow students to explore it before we ask them to use in the context of learning or when we use a team building lesson to introduce a new Kagan structure.

PBL Connections

Connect to a PBL Unit or use scenarios familiar to students.

We discussed several ways to connect story telling and word problems to PBL concepts such as following up a video or lesson with a connected math problem or using an image to discuss possible stories that could be told using math language.

Flip the Task

Provide students the answer and have them come up with the question.

This is one of my favorite things to do and so easy to differentiate.  The answer is 12, what is the question?  This could be used in Pre-K all the way through high school and have such a range of mathematical concepts.

Focus on Strategies

Assign a strategy that students must use in their story.

For younger students in particular, this is a great way to firm up their understanding about which strategies to use and why.  For example, teachers were asked to tell a story that illustrated the making ten strategy.

Some teachers used animals to demonstrate multiple addends and grouping them to make a ten first, some used animals that added to ten (just two addends), and some used tenths using pattern blocks to explore fractions.  There were so many representations of this strategy and so many powerful stories!

Provide a Visual

Let students decide how to make a story from a visual.

What’s the saying?  A picture is worth a thousand words?  If that is indeed true, then I bet we can pull out at least a thousand words from a picture.  We played with using a picture as a springboard for storytelling and allowing students to take their own pictures (easy to connect to PBL unit).

Provide the Equation

Students decide how to model an equation.

Starting with the abstract representation of a story is a great way to get students involved in backwards design.  What context would apply to this?  The possibilities are almost limitless.

Use Resources

Change independent practice worksheets into storytelling opportunities.

The fact is at the end of the day, all student will be exposed to problems at some point either in their resources texts or on standardized tests that are written as a word problem that may or may not be written in a context familiar to them.  However the idea is that we can use story problems from those texts to have students explore a problem or two more deeply through storytelling and modeling instead of assigning several problems that aren’t invested in.

Let them Eat Cake! (or candy)

Refreshing the items in your Stop Motion station can add a fresh perspective to story telling.  Sometimes it only takes a trip to the dollar store to re-create the excitement around story telling in math class!

Here is the presentation:

 

3 Act: Not Enough Eggs

One of the most common math problems I run into on a regular basis is how to modify a recipe according to the number of ingredients in my pantry.  I often decide to bake on a whim and don’t always have enough ingredients to complete the entire recipe.  Today I was starting a batch of my kids favorite blueberry muffins, but when I went to put in the eggs, I realized I only had 3 eggs and the recipe called for 4.  I could have went to the store, but it was 5 degrees outside and I wasn’t up for the trip.  So instead, I decided to modify the recipe to fit the parameters of 3 eggs.

Missouri Learning Standard(s) 5.NF.B.6, 5.NF.A.1.

CCSS 5.NF.B.3.

Act 1

What do you notice?  What do you wonder?

How much of each ingredient will I need to make the recipe using 3 eggs?

You can provide the recipe ingredients in Act 2 to have students do some estimation about what fraction of the ingredients will be needed?

Act 2

What information do you need to help you solve the problem?

Full Recipe

Blueberry Act 2

If scaffolding is needed you can use these pictures to help visualize fractions.

Act 2 Blueberry Visual

Act 2 Blueberry Visual 2

Act 3

Some of my pics didn’t turn out, so for now I just have amounts next to the pics of the ingredient.  I will fix this when I get a chance to take pics of the measurements:)

 

Flipping Instruction in K-5 Math Classrooms

Flipped instruction is becoming more and more prevalent now that technology is no longer a limiting factor for most schools, but many still have lots of questions and concerns about the practice.

It’s a mistake to look at flipped instruction and face to face instruction as separate entities.  We should instead discuss effective mathematics instruction holistically.  Effective instruction  whether face-to-face or flipped should provide opportunities for  students to make sense of mathematics.  This includes worthwhile tasks that form connections between the world and the mathematics concepts being explored.

It is also important to view mathematics understanding as a continuum not just between concepts but the acquisition of number knowledge.  Students acquire deep understanding by experiencing new concepts through activities that offer concrete, pictoral, and abstract representations.  This website offers a good overview of the CPA teaching approach.  Students need to experience mathematics through inquiry and investigation and build conceptual understanding through manipulation of mathematical ideas. Using the CPA approach to teaching mathematics insures that all students have a deep understanding of concepts.  Employers aren’t look for employees who can follow mathematical procedures, software programs and calculators are available for that.  They need employees who can recognize and analyze patterns, brainstorm and investigate viable solutions to complex problems, think and reason critically and collaborate with others.

I recently watched the movie Hidden Figures about a group of ladies who performed mathematical computations for NASA.  Once the computer technology surpassed the need for their computational skills, the ladies needed to adapt to a new environment by learning computer coding and applied mathematics.  Procedural mathematics is not an employable skill.  Companies are looking for people who can think and reason and our math classrooms need to base our instruction on that need.

It is equally imperative that math blocks consist of opportunities for students to engage in the math practice standards and collaborate with other students on rich mathematical tasks.  Gone are the days of students working in isolation; Our world is one of collaboration and problem solving and we need to provide those experiences for students.

It is only once we understand effective teaching practices that we can begin to translate that knowledge into a virtual environment.  Too often we put technology before pedagogy and try to fit the learning into a tech-shaped box.  Instead we should start with powerful instruction and application and find virtual tools that enhance student learning.

Flipped Instruction

So what exactly is flipped instruction?  Flipped instruction started as videos being used for students to watch at home as homework so they would be able to complete exercises and tasks at school where they could receive teacher support.  Many teachers now flip their instruction within the school day by having students watch the lesson as bellwork or in a station while the teacher is meeting with small groups to open up instructional time for teachers to support students at their point of difficulty.flippedgraphic(web960px)

The topic of flipping instruction in elementary school is somewhat controversial; It’s similar to suggesting math stations in a secondary classroom.  There is a misconception that math stations are only beneficial for elementary students and the same is true for flipped instruction; Many teachers think it can only be pulled off in high school.  Although this is a common misconception, I would like to note that any quality flipped lesson (elementary or secondary) will incorporate opportunities for students to interact with mathematics either with concrete materials or with models.  This might be by having students pause the video to model a problem with base 10 blocks or a virtual manipulative or by completing a Desmos, Geogebra or Illuminations activity or simulation.

It is worthwhile to reflect on interactions elementary students choose to have with technology in their spare time.  My son and his friends spend hours researching topics and watching them on Youtube, from watching gameplay on how to use Redstone in Minecraft to tracing the lineage of the Jedi Order in Star Wars.  Why is it a stretch to think this would be effective in the classroom as well?

Why Don’t More Teachers Flip Instruction?

It can be really intimidating to record yourself on camera.   I have flipped numerous lessons at multiple grade levels, and it is definitely a different feeling to know that your instruction is being recorded and archived.  You are more aware of your every word, more intentional about your language, and more tight with instructional strategies.  In my opinion, that is a great thing!

We’ve all been there; we’ve planned the perfect lesson and decided on exactly what to say and then we start talking and nothing comes out right.  When you record a lesson for students, you have the opportunity to capture your best teaching.  It’s like the difference between live television and pre-recorded TV shows.  You have the opportunity to edit and make them more refined because you are editing for the best take.

Additional Benefits of Flipped Instruction

Many schools require all students in each grade-level to teach content areas at the same time with the same objective.  We could debate all day about this practice, but the key concept to take away is that we want to make sure we provide all students with a rigorous curriculum while still allowing for personalized instruction.

Flipped lessons are a great way to allow students to have access to the same grade-level content, but also get the interventions they need to continue to grow as a learner.  This can be done by allowing students who have mastered the concept to complete their work and move onto other concepts or to be given opportunities to apply their understanding.  Students struggling with the concept can receive additional intervention with the teacher because she has time to meet with them individually or in small groups.

A common misconception is that “struggling learner” means that a student has a low IQ or doesn’t have a particular subject ability.  An equally damaging misconception is that “gifted” students don’t need intervention and are gifted in all areas.  The truth is that all students have different strengths and experiences and a student who struggles with number sense may very likely soar through geometry.

Again, the point of flipping instruction is to open up additional instructional time for the teacher to support both struggling students and those who are ready to deepen their understanding by allowing more time for small group and individual intervention during math class.

Review and Reference

Many students who have “mastered” concepts often need review at times.  As adults, we sometimes fail to realize that things we take for granted such as looking up a conversion chart on our telephones when buying something sold in mL is a learned behavior in how to navigate mathematics.  The same goes for when we need to install a new dryer and we have to look up a youtube video on how to convert the outlet.  Students need to know how to find information when they need it and one way to do that is to provide them with past lessons to review conceptual and procedural knowledge.  I’m not embarrassed to say that I’ve had to look up the procedure for the long division algorithm as an adult math teacher because I have accepted the fact that my brain doesn’t have to hold that information because I rarely use it.  I use mental strategies and tools such as the calculator on my phone in my daily life to approximate and divide, not a written algorithm.

Other Ideas

There are lots of ways that teachers use flipped instruction in their classroom besides lecture and mini-lessons including:

  • Game directions
  • Introduction to virtual Tools
  • Tutorials on how to use new software
  • Classroom expectations
  • Classroom and school tours for new students
  • Parent videos explaining a math concept

All of these have the same goal, to streamline procedural information or review so that the teacher has more time to work with students on individual skills.  Primary teachers who implement a workshop approach in their classrooms often struggle the most with interruptions to small group instruction due to the fact that many of their students cannot read tasks.  Flipping those tasks allows students to remain on task and engaged while the teacher is working with intervention groups.

Tips for flipping instruction:

  1. Revisit your flipped lessons regularly.  Don’t keep the same lesson if you have found better ways to teach it.  Refine your work and you will stay on the cutting edge of research.
  2. Use relevant and precise vocabulary
  3. Keep videos short (No longer than 7 minutes)
  4. Include an interactive component
  5. Provide multiple representations and models
  6. Provide images and graphics only when relevant to the mathematics
  7. Talk in a conversational tone
  8. Follow up with an opportunity to practice the skill learned and receive feedback
  9. Survey your students – what is working, what needs to change?

Image from: https://facultyinnovate.utexas.edu/flipped-classroom

 

 

 

3 Act: It’s Not Just a Mint

Anytime I see multiple sizes of products, I immediately think of what task I could make.  I’ve had these tic tacs on my desk for about three months and finally decided to sit down and film today.  This task aligns to CCSS 5.NBT.B.6 if using long division as your objective.  It aligns to 5.NF.B.3 if using dividing whole numbers leading to answers in the form of a fraction or mixed number.  I’ve included different images for Act 3 for each.

Act 1

What do you notice?  What do you wonder?

Possible questions:

How many small boxes fit in a large box?

Give a too high and too low estimate.

Act 2

What information do you need to solve the problem?

It's Not Just a Mint Act 2

Act 3

It's Not Just a Mint Act 3 RemainderIt's Not Just a Mint Act 3 Mixed Number

As always, I would love feedback on how this could be better!