## Estimating as Part of the Process

My fourth and fifth grade classes explored fraction models this week.  I enjoy teaching about the concept at both of these levels concurrently.   I can see the linear progression of skills associated with fractions and the different perceptions of fractions.  My fourth grade crew is finding equivalent fractions while my fifth graders are multiplying/dividing fractions.   Both groups are finding success, but I’m also seeing similar struggles.  Students are fairly consistent with being able to convert mixed numbers to fractions and combine fractions. Issues still exist in being able to estimate fraction computation problems and determining which operation to use while completing word problems

This year I’ve been focusing in on making sure students are using estimation strategies.  This is especially important when dealing with fractions and eventually decimals.  Unfortunately, I tend to find that time spent on the process (algorithm) trumps the reasonableness (estimate) from time to time.  Part of this is due to past math experiences and time management.  After the last assessment on fractions, I started to look for additional ways to incorporate estimation within my fraction unit.  I came across Open Middle last year and I’m finding their fraction resources to be a great addition.  Both, my fourth and fifth graders completed a few different Open Middle fraction problems this week.

I’m finding that students are estimating a lot more when they are involved in these types of activities.  The tasks I use from OpenMiddle emphasize the need to estimate first and calculate second.  These types of puzzles are interesting for students.  They are low-risk, but yet have a high ceiling.  I also found this to be evident with an activity that I found out of this book. I can’t say enough good things about the ideas and resources found within that resource.

Students had to find the missing numerator, denominator or variable.  In both, the Open Middle and Make it True activity, student worked in groups of 2-3.   I gave them about 10-15 minutes to collaborate.  The sheet below was adapted from the book above.

They shared ideas, estimated and came to a consensus on what the solution should be. I had the student groups write their answers on the board and the class discussed all the different solutions afterwards.  The class conversation incorporated a decent amount of review and also gave an opportunity for students to ask for clarification.  I’m looking forward to having more classes like this. The class conversation component that occurs after a collaborative effort is starting to become an even more valuable piece of my math instruction.

## Developing Multidigit Number Sense

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.

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.