Numberless Word Problems

I started using Numberless Word Problems in my math classes late last year, and immediately, I was hooked.  I loved how it made my students look at work problems in a different way - they really had to think about what the problem was asking ... and develop a plan to solve it - without being distracted by the actual numbers in the problem, or just taking the numbers in the problem and adding or multiplying them together.  The conversations that were happening between my students were fascinating - they were far more willing to take risks with their partners or small groups because the numbers were eliminated at first - students weren't worried about their math skills, or getting the "answer" wrong.  Engagement was definitely high and students were eager to share what strategies they used to "solve" the numberless problems.

Numberless word problems are an excellent way to get your students thinking about WHAT the word problem is actually asking them to do and developing strategies for solving the problem.  Using these kinds of word problems provides the essential scaffolding that helps students develop a stronger understanding of how the numbers work together and how to make a plan to solve the problem.

 I would use these problems as a mini-ish lesson (takes a little longer than the 10 minute lesson) when beginning a particular concept (operations, measurement, fractions, etc.), when teaching a problem-solving strategy (using smaller numbers, simpler problem, identifying the operation, etc.), or when practicing algorithms or strategies for multiplication and division. These are great for differentiation when practicing these strategies because students are working with "their own" numbers for the first problems; encourage students to use numbers that are reasonable, but that they are comfortable working with.

   

I would do these activities as a whole class - projecting the problems so everyone could see.  I would display the first slide - a chosen problem with all numbers removed like the example on the right.  You can make up a problem or use any word problem you have that fits your learning goal for numberless word problems (grab a textbook or worksheet) - just remove the numbers - and make sure you separate the problems (don't have all 3 on one page - students shouldn't see the third problem until the very end). 

After I displayed the slide I would them let my students talk (or use whiteboards or their journals) with their elbow buddy or small group, asking: 

• "What do you know about the problem?" (here I would look for answers like numbers are increasing per day)
• "What do you need to know?" (what are the points? is there a pattern? how many points on Thursday?)
• "What is a possible solution?" (here I ask the students to try to come up with an equation related to the learning goal (in this case - adding and subtracting large numbers) using their own set of reasonable numbers - they may come up with something like 100 + 200 + 300 + _____ + 500 = 1500, or 250 + 300 = 500, or something completely different.)  At this point I ask students to share some of their equations with the class and we discuss if the answer makes sense and if it is reasonable. This is a great time to check for misconceptions, too - make sure students understand the importance of the word increasing ... and that we are looking for a specific answer for a specific day.

     

I would then show the second problem - same problem, same question, with some information added to it.  This time students discuss:

• "What changed in the problem?"  "Does this change your equation?"  How?" 
• By asking them again, "What do you need to know?" they narrow down what information is still missing and realize that the actual question they need to solve did not change, but HOW they go about solving the problem may change - they may need to change their equation (here I would look for students realizing this was a multi-step problem - that they would probably have to add points together and subtract from the total to find Thursday - reminding students of the learning goal while they work through these questions often helps them with choosing the correct steps).
• Students work with their partners to change their original equation (either by substituting in the new information for their number and re-evaluating, or by using a new equation altogether.  Again, students are asked to share how their thinking evolved or what made them change their thinking.

     

The third problem I show has all the information they need to solve the problem.  But instead of having students work with their partner or small group, I have them work independently for the first few minutes ... then I usually allow them to partner up - making sure they "prove" their solution to each other.  That independent part is important to help them become risk-takers and attempt difficult things and use their problem-solving strategies and make a plan to start ... and all those great things we want them to do when working independently. I choose one or two pairs or groups to share their work at the end and model how they solved the problem.   

I've been making some Numberless Word Problem resources to use in my classroom.  Each resource contains 10 different scaffolded problems, plus an extra "challenge" page that has 3 - 4 more word problems (not numberless) that review other math concepts related to the word problem.  There are also problem-solving templates, learning goal and success criteria to display, and an answer key.  Each resource also contains a self-reflection form, as well as peer and teacher feedback forms for formative assessment opportunities, and a rubric for more formal assessments.  Having students actively use the feedback forms to make improvements on their next problems by explaining how they used the feedback on their self-reflection forms makes these word problems an excellent portfolio piece or piece of assessment evidence.





You can take a peek at these resources by clicking HERE:

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5 Activities for Teaching Problem-Solving

This "5 Activities for Teaching" post (click the link for more 5 Activity Ideas) is all about Problem-Solving in Math (and there just may be 6 activities in this one).  As our testing relies heavily on our students' ability to problem-solve and analyze and solve word problems, we have a heavy focus on problem-solving all year long.  This post aims to give you some new ideas to get your students digging deep into word problems, on their way to becoming problem-solving masters.  

1.  Go Numberless - 

I recently started using Numberless Word Problems in my classroom and it has changed the way we look at word problems.  By taking the numbers out of the question, students work with the actual problem itself - deepening their understanding of what the problem actually means and how to go about solving it.  It is fantastic for differentiation because students start to substitute their own numbers into the problem (working with numbers they are comfortable with using) as a way of scaffolding the process.  You can read more about Numberless Word Problems and see exactly how I use them in my classroom in my blog post HERE.

2.  Use CUBES to help students analyze the problem before they begin solving - 

Before I began using CUBES with problem-solving, I would watch in frustration as my students plucked numbers from the question and began solving without really thinking about WHAT they needed to solve.  Sometimes they would miss some of the steps in a multi-step question, and other times their answers weren't actually answering what the question asked.  By introducing CUBES, my students now actually slow down and examine the question fully first ... leading to much greater success in problem-solving activities.  I introduce CUBES very early in the school year.  I hang an anchor chart in the classroom, and we complete an entry in our Interactive Math Journals (click HERE to see this resource in greater detail).

I also created sets of concept-related Word Problem Activities using CUBES.  These resources contain a checklist for CUBES right on the page that students need to complete before they begin solving the problem.  I LOVE using these pages for quick formative assessments in the classroom - and they make a great portfolio piece to keep parents informed of what we are doing in the classroom, and how their child is progressing.  There are 2 different versions of each word problem so you can easily provide differentiation for your students or provide some extra practice for students who need some reinforcement with the concepts.

3.  Establish a set of success criteria for the steps to solving the problem - 

Once my students know how to analyze the word problem, it's time to start talking about how to SOLVE the problem.  Because we have a HUGE focus on communicating HOW the students solve the problem, just showing their work isn't enough - they have to show their thinking, too.  Using my Building Better Responses in Math has been the key to this in my classroom.  This resource breaks down the problem-solving process step-by-step and creates a set of easy-to-follow success criteria for students, ensuring that they have not only solved the problem, but also communicated their thinking during the process.  We start out with our first success criteria, and every time we master a criteria (usually every two to three weeks), we add another criteria to our board (there are 9 criteria in all).  I also post EVIDENCE (a student-created exemplar) alongside the criteria each time we add a new one.  This gives the students a model to reference.  This resource also comes with printables for students to practice each criteria in isolation, as well as pages for scaffolding the steps each time they add a new one.  All pages contain a checklist for students so they can be sure they are completing each step.  

4.  Work Backwards - 

I LOVE this open-ended activity for getting students to think about word problems.  I start with an answer, and get each student to write a word problem for the answer on a sticky note.  They quickly check in with me before they post their question.  When we first started this activity, students were including the answer IN their question, but with more practice, they are now thinking more about writing the word problem with the answer in mind.  To extend this activity, students can choose a word problem written by a peer, and work to solve it - PROVING the answer is correct.

5.  Work Together - 

Students can be a GREAT reference for each other.  I love letting my students explore word problems together.  They often have different ideas and strategies for how to solve a problem, which leads to awesome conversations about justifying their answers.  I always make them responsible for completing their own pages, but they can work together on the solutions.  My favorite resource for this is my Stick-It-Together Math Resources.  These resources have students working together in groups of 4.  Each students is responsible for solving the problem independently first (on a sticky note), then working together with the success criteria to build the best response they can from each other's responses.  I hear the BEST math talk when using this resource, and I watch them go back into their notes to help them with their solutions, which makes my teacher heart smile.  Each of these resources also contains an editable template, so you can add in any word problem you want to work on - better yet, give your students the opportunity to come up with the problem themselves.

You can also have them work together on large chart paper, or just give them some clip boards for their paper and let them work anywhere they wish within the room.  This is my students' favorite way to problem solve.

6.  Let them be the experts - 

Lastly, let your students be the experts.  Allow them plenty of time to see other solutions, and comment on the work (their own, and other's work).  I do a lot of peer and self-evaluation with problem-solving.  After problem solving activities, we do gallery walks - where students' work is displayed, and the students are asked to go around and view all the work, giving "stars and wishes" to their peers.  This makes them really think about what is needed in the solution to fully answer the problem.

I also like to post all the responses on the board, and allow students time to present their solutions - explaining what they did and how they know they are correct, or what they would do differently next time.

To help them differentiate between explaining HOW they solved the problem, and WHY they chose the steps they did, I also like to do an activity where I give the students two different colors of stickers or sticky notes.  They walk around the gallery of math, examining all the solutions, and place the one color where they see students explaining HOW, and the other color for WHY.  This really helps them see the difference between the two.  I then allow them time to go back to their own solutions, to see if they have completed both steps, and improve upon their communication.

These are just a few ideas.  I'd love to hear some of your fabulous ideas for helping your students dig deeper into word problems - just leave a comment below to share your ideas.

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My Favorite No

My Favorite No is my favorite YES in my math class.  I seriously LOVE this.  I fell in love with this idea when I first saw it just over two years ago (you can read a previous blog post HERE).  I started in the classroom then, and it has stayed a favorite.

The first 10-15 minutes of my math block is always reserved for some sort of review or spiral activity.  This year, from September to December we dove into Number Talks, from January to March Break we did Math Meetings (4th Grade Frolics has a great post about it HERE), and from now until the end of the year we'll be switching between Math Meetings on Tuesdays and Thursdays, and My Favorite No on Mondays, Wednesdays, and Fridays.  Perfection.

If My Favorite No is new to you, you have to start out by watching THIS VIDEO (then come right back here so I can share more with you).  ;)

My Favorite No is the perfect companion to all the Growth Mindset teaching I've been doing in the classroom this year - teaching that the best learning opportunities come from making mistakes.  This is EXACTLY the message I want my students to take away from all of this.

OK ... so here's how it goes.  I pose a problem.  In the past I've done more skill (knowledge) based questions, but this year I'm focusing on word problems because my data is showing my students need this.  I write the problem on a small cue card and project it for the students to see.

We then go through and mark the question together using CUBES.  (You can read more about CUBES in a blog post I have HERE).

Then, my students get 5 minutes to independently solve the problem (I set a timer and everything because they love the timer).  They are also writing on the small cue cards (I'll post a link to the cue cards I bought at the bottom of the page. I buy them in bulk, but any method you choose will work fine).

When they are finished, before they hand their cards in, they have to do a traffic light comprehension dot in the corner (green for no problems, yellow for a little difficulty, and red for a lot of difficulty).  I have a poster hanging in my class explaining the Traffic Light Comprehension Dots (I had it made at Vistaprint) - but you can grab a free copy of the pdf I made HERE.  We use traffic light comprehension dots on everything!

The students also color code their answers according to our Math Response Learning Goals.  These learning goals are EVERYTHING in my classroom.  They are seriously one of my most favorite resources.  They are from my Building Better Math Responses resource and they have truly made our responses better.  You can take a peek at them HERE.

Students don't write their names on the front of the card before they hand it in, instead, I just have them initial the back - that way, if their card is chosen as "My Favorite No" it still has a little anonymity to it.  When they hand the cards in, I quickly sort them into 3 piles - 1) correct, 2) correct strategy but a small error somewhere, and 3) incorrect.  

I then sort through the "correct strategy but a small error somewhere" pile and choose "My Favorite No".  This particular card will have a lot of wonderful thinking shown, but contains a small error or misconception somewhere that probably more than a few students in the class are making.  I put this card up under the projector for all to see (remember, it's anonymous) and talk about all the wonderful things I love about the answer given.  I then give students one to two minutes to do some error analysis with their partner or elbow buddy.  After this time, together we fix up the problem and (hopefully) clear up any misconceptions.

So ... what do I do with the cards when we're done?  Well, remember the three piles I sort the cards into?  I record this data into my grade book with easy colored dots - green for correct, yellow for correct strategy but small error, and red for incorrect.  I write the math concept for the word problem at the top of the page so I can go back and see where we need to do a little extra review.  These circles will also make it very easy to see which students need a little extra help with some small group sessions during math stations.  We've done two of these activities this week, so this is what my grade book looks like so far.  (See all those yellows and reds ... we've got some work to do during this last term ... but we're up for it).

I put a little sticker in the corner of the correct cards, and then I file all the cards in little library card pockets for each student.  When the pockets get too full, we simply clip the cards together and file them in our portfolios, and start filling them again.  The students crowd around me as I'm putting the cards in the pockets to see if they got a sticker on their work - it's kind of fun.  :) 

Now, if you're thinking this will embarrass the students, it won't.  I promise.  In all the time I've done this in my class, I've never ever ever had a student upset by it.  There is a solid rule in my room that we don't say whose card is My Favorite No, and their names aren't on the cards.  But ... even so, often the student who was chosen is really excited they had the chosen card, and will want people to know (even though I still don't let them say anything in the classroom).  When you teach your students how important mistakes are to the learning process, they are not embarrassed by their mistakes, they look forward to growing from them.  And that, my friends, is a beautiful thing.

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