Making equal parts

For this post, we're back to fractions....
The third grade classes are beginning to explore fractions.  To introduce the idea of equal parts (an essential foundation for fractions), we worked on equal parts situations on-line using ST Math.

Students had to figure out which object to choose and then we discussed why our choice was correct.  (If you choose correctly, the penguin is able to walk across the screen.)  After conversation about what made the parts equal, students were given geoboards so they could explore equal parts with a partner.

These girls created parts that were not equal.  They shared with me that the parts were not equal because they did not cover the same area.  To find the area, the girls counted the "squares" between the pegs.  (The third grade classes had just finished a study of area - so this was a great connection!)

These boys created parts that were equal.  They described the parts as being the "same".  The boys in this group are learning English as a second language and this activity gave them a great opportunity to explore examples and nonexamples of equal parts.

After having time to explore with their partner, we came back together as a whole group.  Students were asked to share one of their representations and explain whether the parts were equal or not - and how they knew.  This group described knowing the parts were equal because they divided the square into four equal parts (smaller squares).  Then they divided each square in half.  Such great conversation and thinking from these third grade students!

As a challenge to the class, I left them with three representations on geoboards.  Each geoboard was divided into fourths - one with squares, one with rectangles, and one with triangles.  Students agreed that each of the geoboards was divided into equal parts.  I then asked whether the triangle was equal to the square - or if the rectangle was equal to the square.  Students were intrigued by this idea - intuitively they thought yes but they weren't certain.  I left the problem as an open question and asked them to think about it further.  I can't wait to hear their ideas!

#GETMath: March 2014

I have posted a new #GETMath task for March 2014.  http://bit.ly/1fA0GGf  This task is based on Goldbach's Conjecture.  I selected this task because there are some similarities to the Sums of Consecutive Numbers task from February.  It might be interesting to discuss the ways in which the two problems are alike and how they differ.

This task also has multiple entry points.  Students are adding two numbers - which students begin to do in grade 1.  For students who might struggle with computation, you can give them a calculator.  For students who might not know the prime numbers, you can give them a list of the prime numbers.

For students who need an extension, they can explore finding prime numbers and how one might generate that list.  These students could also work to generalize or write a rule for the patterns.

Remember that the purpose of these tasks is not necessarily to focus on content.  While mathematical content is important, these tasks lend themselves to the Standards for Mathematical Practice - an essential part of mathematical learning.  Use these as an opportunity to engage your students in mathematics and get them excited about mathematics so they are eager to learn additional content!

I hope that you will share your thinking and pictures on Twitter!  I look forward to seeing what you have to share!

Chatting about Sums of Consecutive Numbers

It has been so much fun to connect with classes about the Sums of Consecutive Numbers problem!  I have been so impressed with the thinking and problem solving skills of students!  A 1st grade class had the fantastic idea to use a measurement tool as a number line!

A 4th grade class at my school used video chat to connect with a 4th grade class in Dublin, Ohio.  Before our chat, students explored the problem and discussed questions with their homeroom classes.  Some of the questions they explored included:

• Were there any numbers that couldn't be made by the sum of consecutive numbers?
• What happened if you added 3 numbers?  Or 4 numbers? 
• Could you predict whether the sum would be even or odd based on the number of addends?
• Is there a pattern?  If so, how might you describe it?
• Is there anything special about the prime numbers in comparison to the composite numbers?  What do you notice?

Last week, the classes were able to connect and share their observations:

Some of the observations students made included:

1. When you have 2 consecutive addends, the sum is an odd number.  For example, 3 + 4 = 7.
2. When you have 3 consecutive addends, the sum is a multiple of 3.  For example, 1 + 2 + 3 = 6 and 6 is a multiple of 3.
3. When you have 4 consecutive addends, the sum is NOT a multiple of 4.  They are curious why the pattern that seems to work for 3 addends does not work with 4 addends.  For example, 1 + 2 + 3 + 4 = 10 and 10 is not a multiple of 4.
4. When you have 5 consecutive addends, the sum is a multiple of 5.  For example, 1 + 2 + 3 + 4 + 5 = 15 and 15 is a multiple of 5.

The conversation generated further questions.  We're hoping that students can explore these questions in groups and then we can have small groups video chat rather than a whole group video chat.

Using technology to connect with others and share our mathematical thinking is a great way to take our learning beyond the physical walls of our school building!

#GETMath: Launching the Problem for Sums of Consecutive Numbers

Have you had a chance to try the Sums of Consecutive Numbers problem with your students?  If not, you might be wondering how you might launch the problem.  I did not give specific ways to launch the problem in the video because of the wide range of learners that might be participating.  After further reflection, I thought it might be worthwhile to share how I involved students in the problem.

I started the problem with a grade 3 class late last week.  We began by talking about the Global Read Aloud project and what they enjoyed about it.  Students said what they enjoyed most was talking with others about the book.  After explaining that I wanted to do something similar with a math problem, the class was eager to get started!  I shared the title of the problem "Sums of Consecutive Numbers" and asked students to tell me what they knew about "sums".  The class quickly agreed that sums were answers to addition problems.  Defining "consecutive" was a little more challenging but with the help of a dictionary, students soon agreed that consecutive numbers would be numbers that follow one another in sequence, such as on a number line.

 After drawing a number line to use as a reference, we began to brainstorm possible answers.  On a large chart, students were able to write their answers once they shared their idea and the class was in agreement that the example fit the rules of the problem.

After we brainstormed a few examples and I felt that students had a strong understanding of the problem, I encouraged students to explore the numbers on their own.  Students received a chart for them to record their thinking individually.  To begin, students were encouraged to copy the examples from the class chart.

While students were working, I circulated the room and asked questions about their strategies.  Some students immediately saw a pattern with the odd numbers - noticing that any odd number could be made as the sum of two consecutive numbers.  Other students tried using three or four addends and writing down the sum they discovered.

Students worked for about 15 minutes and then I stopped the class.  We discussed our observations about the patterns and wrote them on a chart.  I encouraged students to continue working on the problem and told them that they should not be limited to the 50 numbers on the chart.  I asked students to continue adding to the chart as they noticed other patterns.  I can't wait to check in this week to see what else they have added to the chart!

If you have been waiting for the moment to introduce this problem, I hope this helps you get started!  The students were so engaged with this problem that they continued working during their indoor recess!  Please ask questions or share how things are going on Twitter at #GETMath!  I look forward to hearing from you!

Using #GETMath to teach the Standards for Mathematical Practice

Some of you may have thought about using the Sums of Consecutive Numbers and joining in the #GETMath conversation but you're feeling stuck because you don't feel like the problem clearly connects to the content for your grade level.  (If you haven't heard about #GETMath, learn more here: http://bit.ly/1eybFKx)

This problem does not need to be (nor should it necessarily be) completed in one or two class periods.  Students can play with the problem during the month of February and share their observations once or twice each week.

This is an opportunity emphasize the Standards for Mathematical Practice!  

1. Make sense of problems and persevere in solving them.
• Sums of Consecutive Numbers has many different aspects to explore.  In introducing the problem to a 4th grade class, some students immediately thought to use more than two addends.  Another student made a conjecture that no even number could be made by only two addends.  The students were ready to get started and excited to look for patterns!  We created a chart where they can confirm or revise their findings and add additional observations.
2. Reason abstractly and quantitatively.
• Because this standard focuses on connecting real situations with symbolic representations, it is not really a standard that could be emphasized during this exploration.  
3. Construct viable arguments and critique the reasoning of others.
• As noted above, students were making conjectures about the patterns and working to prove or disprove their thinking.  As students continue working through the problem, they can add to the class chart.  We even plan to video chat with another 4th grade class - what a great opportunity to share our arguments and discuss our thinking about the problem!
4. Model with mathematics.
• As with SMP #2, this would not be the best standard to apply with the Sums of Consecutive Numbers problem.  Students might use a table to organize their thinking but they do not apply math to solve a real-world problem.
5. Use appropriate tools strategically.
• As with SMP #2 and #4, this would not be a strong standard to emphasize with this particular problem.  While the table or organizer used by students might be considered a tool, it would be a weak connection to this standard.
6. Attend to precision.
• In the Sums of Consecutive Numbers problem, students need to be sure that the numbers they use are indeed consecutive numbers and check to be sure the sum they find is accurate.  They also need to be able to communicate their thinking in a way that is clearly understood by others.  This would be a strong standard to emphasize with this problem!
7. Look for and make use of structure.
• The Sums of Consecutive Numbers problem provides students with an opportunity to notice and discuss the structure of mathematics.  For example, why can the odd numbers always be made by the sums of two consecutive numbers? 
8. Look for and express regularity in repeated reasoning.
• The Sums of Consecutive Numbers problem provides students with an opportunity to observe patterns and reason about generalizations.  For example, what numbers cannot be made by the sums of consecutive numbers?  Could you predict the next number that cannot be made as a sum of consecutive numbers?  Can you write a rule that will help you find all of the numbers that cannot be made by the sum of consecutive numbers?

The Sums of Consecutive Numbers problem has clear connections to five of the eight Standards for Mathematical Practice.  The CCSS emphasizes that the Standards for Mathematical Practice are as essential as the mathematical content.  While it would be difficult to emphasize five Mathematical Practices within your lesson, there are many opportunities for you to engage students in the Standards for Mathematical Practice through this problem. 

I hope you will join us and share your thinking at #GETMath!

#GETMath

Since the beginning of this school year, I have wanted to collaborate with teachers in my building for some type of problem solving "club" where we can work together on a mathematical task.  My original plan was to have this during lunch once or twice per month and invite teachers to discuss a problem we could work on throughout the month.  As so often happens in our school lives, schedules interfered and I realized it would be difficult to schedule these meetings during lunch.  I tried to think of a way to involve as many teachers in our building as possible without requiring additional meetings.

As I've been engaging in some productive struggle of my own about this situation, I have also been working to plan the elementary school methods for mathematics course that I begin teaching this week. I thought how wonderful it would be if the students in my class could discuss some of these rich mathematical tasks with others outside of our small classroom.

Finally - all of these thoughts merged together!  Many of the teachers in our building participated in the Global Read Aloud that was started by Pernille Ripp.  Wouldn't it be amazing to do something similar - but this time in mathematics?!?

I am inviting you to participate in our Global Engagement Task in Math (GETMath, for short).  For the month of February, I have selected a problem from Marilyn Burns about sums of consecutive numbers. The object of the task is to find patterns when adding consecutive numbers.  Think about what patterns you find, what surprises you, and anything else you notice!

I encourage you to begin the problem on your own - then talk with others about it!  Share your thinking and your findings.  Compare your results with what others have found.  Engage in some productive struggle!

You can also share your thoughts  on Twitter at #GETMath.  I hope you will join me in this experiment!  You can learn more here: http://bit.ly/1eybFKx

Productive Struggle

When working with students in math, it is important to engage them in productive struggle.  Through productive struggle, students feel that learning goals are attainable and the effort necessary to attain these learning goals is worthwhile.  It creates a sense of hope and students feel empowered with an increased sense of efficacy.

I love this quote about productive struggle: "Basically, academic rigor is helping kids learn to think for themselves."

If we use the above quote to frame our thinking about how students might approach a math problem, it is important that the teacher not make assumptions about the student's understanding.  Using questions to prompt student thinking is a powerful way to have students use metacognition - thinking about their own thinking.

Some of the questions I frequently use include:

• How might you begin?
• What do you already know?
• What do you need to do next?
• Why did that happen?
• What have you tried?  What happened?
• Show me how this will work on the next problem.
• Why did you ______?
• What do you think will happen?
• What might you try?
• How did you decide this answer is correct?

Answering a student's question with a question of your own allows you - the teacher - to help the student explain his/her thinking and develop his/her understanding of mathematics.