Developing Multidigit Number Sense

Cn you find a reasonable solution for the question mark?
                         Find a reasonable solution for the question mark

Lately I’ve been reading David Sousa’s book and have come across some interesting (at least to me) observations.  Humans have an innate ability to subitize.  That subitizing can lead to estimation and through appropriate practices this leads to bettering number sense skills.

After speaking with a few middle school teachers this summer I’m finding that one area that tends to need strengthening is found in the overarching umbrella of math skills: number sense.  Without adequate number sense skills, students flounder when asked to complete higher level math concepts.  Year after year I find some students have a decent understanding of procedures, but fall apart when it comes to explaining their reasoning for completing particular math problems.  Students are able to identify and repeat processes (usually copying the teacher), but not understand why they’re using that strategy.  I find this particularly a concern when students don’t question the reasonableness of an answer.  In my mind, I think finding a reasonable answer or estimation shows a form of number sense.  This isn’t always the case, but I tend to find it when students blindly follow only a set of procedures. I believe that this can lead to more problems down the line.

This is especially a problem with larger numbers. Researchers Carmel Diezmann and Lyn English found that the activities below seem to help students develop multidigit number sense.  All of these activities can be used at the elementary or middle school levels.  The headings in bold are found in David’s book and highlighted in Diezmann and English’s research.  My narrative is below each heading.

Reading Large Numbers

Placing an emphasis on place value when reading large numbers is important.  Being able to identify and see the value of each digit can help students read large numbers more accurately.  I find that students in kindergarten and even first grade start to combine place values when speaking of large numbers (e.g. one hundred million thousands).  Giving students opportunities to take apart these large number by digit value can help reduce this issue.  I find comparing standard and written forms of numbers can also help students start to recognize the place value of large numbers.

Develop Physical Examples of Large Numbers

Visually observing 100 and 1,000 dots can show students the physical difference between large numbers. There’s so much math literature that can help with this. 100 Angry Ants, Really Big Numbers and How Much is a Million can be used with manipulatives to show examples of large numbers. If you teach in an elementary school, I recommend adding these books to your math library. Using unifix cubes, base-ten blocks, or replicas can also be used to show a physical example of large numbers.  Having the physical example can benefits students as they start to question if their solution to abstract problems are reasonable.  A number line with 1:1 correspondent is also one way to showcase large numbers.

Appreciating Large Numbers in Money

Kids tend to like to talk about money.  Showing how $1 compares to 100 $1 bills can show students a visual scale between the amounts.  Visual representations of money in dollar and coin forms can lend itself to having students become more aware of how place value impacts the value.  A problem that tends to always get students curious relates to how much money will fit in a briefcase.  Will $10,000 in $5 bills fit in a 20″ x 18″ briefcase?  These types of questions can have students start visualizing money and the reasonable of their answers.

Appreciating Large Number in Distance

Maps can be useful here.  I remember having students use Google Maps to calculate the distance from one particular destination to another.  Also looking at the distance from one continent to another, or even from Earth to another planet.  I find that a good amount of scaffolding is needed to help students experience large numbers in distance/measurement.  Comparing skyscrapers to distances can also play a role with this as well. If you stacked one Willis Tower on top of another, how many would it take to reach the moon?

They’re many ways to have students observe and interact with large numbers.  I’d like to add appreciating distance in relation to time to the the list.  Time can also be used highlight and compare large numbers. I’m thinking of the dates in history and the Science involved in evolution. Many of these activities can be interdisciplinary as connections between curriculum content exist. Digital and physical forms play a role in having students conceptualize an understanding of large numbers.  Students should be given opportunities to recognize large numbers in a  variety of contexts.  By doing so, I believe students should be able to better question whether their answers are reasonable or not.

By the way, the answer to the top image is 1,000 dots.

Making Math Meaning Through Writing

Yesterday I was able to dive deeper in my summer reading.  I’ve been reading David Sousa’s book on how the brain learns mathematics.  I’m finding the chapter related to making meaning interesting.  David says that there are basically two questions that determine whether an item in the memory is saved or deleted.

Does this make sense?

Does it have meaning?

I believe students ask these questions on a daily basis. Some of the asking is mumbled under their breath, while other students will down-right ask the teacher.  I find myself asking these questions as I sit in professional development sessions.  Students want to know how this new learning applies to their life.  Students are better able to retain what they’re learning when it makes sense and can be connected to past experiences.  Those past experiences can develop into having meaning for students.  What’s also interesting is that experiences that have an emotional component present have meaning for students. Past experiences that are clear in my memory are often related to some type of emotional component. I feel like this is similar with students.  Those experiences are more likely to be stored in long-term memory.

This chapter in particular emphasizes the need to spend more time creating opportunities for students to develop meaning.  Without meaning, students often use formulas to compute numbers.  Their confidence falls on the formula and the student doesn’t necessarily understand the concept. Students eventually become so skilled at computing numbers that they find answers without thinking of the context. Most teachers have had conversations with students about their answers and if they make numerical sense?  In those cases students understanding the procedural aspect (formula) but it’s not in relation to the context (meaning).  In order to create meaning, students need time to connect and personalize the content.  In addition, they need time to explore, reflect and practice.  Writing in math class is one way for students to practice and create meaning.

I’ve been a long time advocate for using writing in math class.  My students in 2-5th grade have used math journals in the past.  They reflect on their past performance and set goals moving forward.  Writing in math class gives students time to process information.  That processing can lead to personal meaning.  Writing in math class can take many different forms. I believe interactive notebooks and foldables can also provide opportunities for student to process and make meaning.  By writing, students are required to organize their thoughts and find sense and meaning in their learning. Using math notebooks/journals can assist in giving students a way to also communicate their current understanding of the material.  Having a component where the teacher responds to the students’ writing can also provide another opportunity for feedback. Regularly writing in math class can also provide students with an outlet to create a record that they can look back at to review their growth.  Something that I need to keep in mind is that the math writing doesn’t have to be on paper.  Writing through a math blog or in some other digital format can also play a role in making meaning.

How do you use writing in math class?

Patterns and Pre-Algebra

Yesterday I was able to get outside and walk around a local park.  While soaking up the sun I started to notice a variety of patterns on the sides of the path.  The patterns changed depending on the vegetation and location.  As I searched for additional patterns I started to find more and and then looked for consistency among the sequences. I took out my phone and started taking pictures of the patterns that I saw thinking that I might use them next school year. After collecting a few I started thinking about how this connects to the math strand of algebra.

What patterns exist?  What about lines of symmetry?
 How does symmetry play a role in the pattern?

Taking the pictures had me thinking of a class I had a few years ago.  I remember reading a district-adopted fourth grade text that introduced pre-algebra to students as patterns and solving for the unknown.  This simple kid-friendly definition was explained to elementary students in a short paragraph. After thoroughly discussing the definition of a pattern (yes, that took time), students took that definition and ran with it.  They started to find patterns (number and otherwise) in and outside of the classroom. If a pattern didn’t seem to exist, students would make a prediction based on the prior sequence.  A completed pattern seemed to make sense and an uncompleted sequence didn’t have meaning.  Students started to put on their “pattern glasses” to identify sequences.  Students would argue whether something was a pattern or not.  I distinctly remember one student saying that to complete the pattern you need to find the missing puzzle piece. These discussions were interesting to observe as students were developing their own rules to the patterns and offering their suggestions to others.

Additional pictures and questions:

What type of pattern exists?
             What type of pattern exists?  Do multiple patterns exist if you zoom in on the picture?
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               Would you consider this a pattern?

After uploading the pictures from the walk I started to think of how students make meaning out of patterns.  This past year my students were able to find patterns in nature, use Which One Doesn’t Belong, and then transition that idea to Visual Patterns.  Understanding the rule or rules behind the pattern can lead to different levels of pre-algebra moving forward. It’s amazing when students start to realize that there can be more than one rule to a pattern or question.  Simple patterns can allow students multiple entry point to access pre-algebra concepts.  Before the school year starts I’ll be pondering the question below.

How do students identify patterns and does that help them become better problem solvers?

I’ll leave you with one more picture:

Does this qualify as a pattern?
Does this qualify as a pattern?

Using Excel to Explore Rates and Proportions

My fifth graders are currently studying rates and proportions. Earlier in the week they explored rates by looking at unit prices and solving problems with some type of cross-multiplication strategy.  Although they’ve made progress I still feel as some many still need to cement their understanding of a ratio and proportion. So it was time to switch up the instruction model.

I decided to go with using a spreadsheet. In this case, the spreadsheet would be in the form of an Excel document. Each student grabbed a laptop and opened up Excel. The students used Excel earlier in the year so they were familiar with some of the basic functions.

After entering a few text cells, students were asked to put a random number above zero in cells B4 and C4. Then the class discussed what GCD stood for. Most of the students said “greatest common denominator.” That response made sense because that’s heavily emphasized in fourth grade as students add and subtract fractions. In this case, GCD means greatest common divisor. The class then discussed what that meant when comparing two numbers and the helpfulness in finding the GCD when exploring equivalent fractions. The discussion then transitioned from equivalent fractions to finding ratios.

Students entered in the formula =GCD(b4,c4) to find the GCD of the two different numbers. Students observed how the GCD changed as they updated their numbers.

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The next part was a bit tricky. I asked the students to write a formula to express the ratio in simplest form. The class used the GCD and trial and error to come up with the ratio formula. Once students wrote the formula and placed it in E4.  Students then explored how the ratio changed when their numbers were updated.

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The class then reviewed why the formula actually worked.  The class discussed that basically the formula took each number and divided it by the GCD of both numbers. What was great was that students were starting to connect the reasoning behind the creation of a ratio. Instead of just cross-multiplying, students are starting to show a deeper understanding of how ratios are constructed and the process used to simplify. The students were able to save and print out their spreadsheets for later review.


Excel Template

Example for Class Use


Math Menu Boards


My fifth grade group has been learning about probability for the past few weeks. Our class discussion have revolved around probability trees and likelihood concepts. The summative assessment on probability is coming up around the corner so last week I was scouring my resources to find a way to review some of the concepts taught earlier in the unit. One of my colleagues and I had a conversation about the idea of using a menu board. I heard of using them through #msmathchat but haven’t used them much. I’ve always thought that giving students a choice in their assignments matters. I feel like an assignment menu encourages student choice and often increases engagement.

So I found a probability menu resource and decided to use it with my fifth grade crew. I added a rubric and a few other options to Yuliana’s template.  Here is the template that I adapted and used for this project.

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After explaining the directions I fielded a few different student questions. During the question time some students needed more clarification than others. A group of students were confused to what the expectations were.  Many of them are used to playing school and expect the teacher to tell them what assignment or what to do to get all of the points on an assignment.  I feel like menu boards, to a small extent, help students become more self-directed in their learning journey. It was encouraging to see some of the students take the reigns and be assertive in deciding which menu option to complete.   After all the questions were answered I gave students time to complete the project.  Students completed the work in just over two class sessions.  After reviewing all of the projects I decided to reflect on the entire process. Here’s what I need to remind myself the next time I have the students create a menu board project:

  • Students need time to brainstorm before creating. I had a few students that immediately started working on their project just to throw it out five minutes later. These particular students didn’t brainstorm or organize their ideas before starting a final copy. On the opposite end, I had students that took out scratch paper and started to write out a few ideas before carefully crafting their project.
  • Students need checkpoints along the way. Throughout the project I had to remind students to check the rubric and generally check-in with students to answer questions and provide feedback. During this time I also had to ensure that I had the technology in my classroom ie. iPads and computers. Next time I assign a similar project I’m thinking of having students fill out a work log to help keep us all on time.
  • Students need time. They need time to put together their thoughts, create and produce a product that follows the minimum guidelines. Some of the students took around two class periods while others took longer. Ensuring that other assignments are in place after the project is important. Having additional work afterwards is important. It also helps eliminate the dreaded “what do I do next?” questions.
  • Review the projects. I reviewed each project with the students. I tried to limit my own talking, which was difficult, and let the students explain their project. During that time I filled out the rubric with the student. The time spent discussing the student project was vital. Students came ready to speak to me on what they created and what they thought was important. Some of the student projects were amazing and other projects needed a bit more work. The majority of students put a decent amount of effort into the project and met the minimum criteria.

This project took a good amount of time and had students create a product that was aligned with different probability standards. I thought it was worth the time and I’d like to bring out the project at some point next year.

Exploring Subtraction Computation Strategies

During the past few weeks my second grade class has been taking apart and putting together two and three-digit numbers. In the process students have been developing a better understanding of numbers.  They’ve been exposed to using a variety of computation strategies to find the sum and differences of numbers.  Through all of this I’m finding that the students are becoming more confident in their ability to use these different computation strategies more fluently.   Although they’re confident they tend to gravitate towards using one specific strategy for computation.  The traditional algorithm is usually the primary method that they use.  Even though students can add/subtract using that method I found that they weren’t expanding their understanding of other computation strategies.  This was a bit of an issue for me because students started to look at computation as the shortcut and not delve into the understanding of why it works.

After speaking with a few other teachers I decided to use a math task found in this book.  I briefly reviewed the different strategies that we’ve learned this year and gave the students this prompt.

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I wanted to make sure that students showed two different strategies and provided some type of written explanation.  The template I copied also had fields for a number model and explanation boxes.

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The bottom of the sheet was designed for students to be able to check their work using addition.

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I gave the students about 10-15 minutes to complete the formative assessment.  Most of the students tried out the standard subtraction algorithm but had a bit of trouble with the second strategy.  After a few moments students started to dig deep and think of how to take apart numbers using different strategies.  Some of the  students truly had trouble using a different strategy and this was evident in what they produced.  I was impressed with some of the different strategies that students used.

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I wrote feedback on the papers and handed them back to the students the next day. Afterwards, I removed the names off of the papers and shared some of the results with the class.  As a class we decided on the following:

  • Students remembered many of the different computations strategies that were discussed earlier in the year
  • Some of the students invented their own strategies on this particular sheet
  • Students need to strengthen their written explanations
  • Students had some trouble explaining what regrouping means

Next week the class will be setting goals in improving our written responses.  Overall, I feel like this activity helped showcase different computations strategies while bringing awareness to areas the need improvement.  I’d like to use this template with a few other classes later in the year.  Feel free to download and edit this file for your own classroom.

Exploring Discounts and Amazon Prices

Exploring Discounts, Percentages and Amazon
Exploring Discounts, Percentages, and Amazon

Today one of my classes explored discounts and percentages.  This particular class reviewed how to convert fractions and decimals yesterday.  Today’s step was to introduce students to the idea of taking a percentage off of a set number.

So I dug through some of my resource from the past and came across a sheet asking students to go on a shopping spree.  Yes, that caught my attention.  A shopping spree not only sounds fun but could be a great way to connect discounts and percentages.  From there I edited the document and decided that the students will be given a specific amount of money to spend and a site to visit to find the items. wasn’t blocked by my school district so I went with that store.  Students were also required to use coupons (10%, 20%, 25% …) to purchase the items.  The winner of the contest will be the student that has a sum closest to $500.

Click for file
Click for file

Students could buy whatever items they wanted.  I’m sure this could be repurposed and have the students buy items for a specific reason.  After I explained the directions each student was given an iPad or computer and asked to visit and find five items.

Students initially started looking at whatever caught their interests.

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Some students looked at shoes while others were finding flying drones.  Yes … I said drones. Thankfully all the searches came up without being blocked by the firewall.  Students then found the original price and calculated the discount.  Their sale price was documented and students went to the next item.

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I found that students started to have a challenging time with the last couple of items.  They had to carefully consider the coupon before writing down their options.  You could tell that they were trying to account for the discount. Understanding the magnitude of the discount started to take priority in the students’ minds.

Not all the students finished, but they will tomorrow.  I’m looking forward to comparing the total dollar amounts tomorrow and see who’s closest to $500.  Overall, this activity helped students see discounts from a different perspective.  This may be an activity that I’d like to edit and use with my other classes.

The idea in this post was adapted from this product.  Feel free to download and use for your own classroom.