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!