Glimpse into Lesson Studies in the region

5^th Grade Lesson Study – Introductory Lesson on Division with Decimals

December 7, 2016

Research Question:

What are the essential understandings that students need when dividing decimals, and what can we do to ensure all students learn something new towards 5NBT.7?

By providing students with opportunities to start the problem-solving process on their own without a teacher-prescribed strategy, represent their thinking, share their thinking with their peers, and listen to their peers’ way of thinking, students’ current understandings are illuminated in a way that drives decision-making throughout the lesson.  In this lesson, 19 out of 23 students were able to provide some representation of their thinking and/or problem-solving strategy on a concept that they’ve had no previous experience with. 

Developing expertise in estimating with decimals, paired with students’ reasoning behind their estimates, prior to experiences using computation with decimals strongly supports understanding.  Estimation supports reasoning, elicits evidence of students’ place value understanding and relative size of decimals, which were found to be 2 of 3 key ingredients in developing understanding of decimal computations (per van de Walle).  Given significant evidence that students rarely refer back to their estimate after determining a more precise answer (in this lesson, and based on the experiences of all teachers in the group), there is a need to support students in increasing their buy-in of estimation as a tool for problem-solving.  Also important to initially developing understanding of division with decimals, understanding the relationship between multiplication and division can serve as an entry point for students, and simultaneously supports understanding of both models of division (partitive and quotitive).

A third essential understanding of dividing with decimals lies in the concept of equivalence.  It seemed that in working towards developing generalizable methods for dividing with decimals, equivalence will become increasingly important.  Understanding that 1.1 could also be called 11 tenths will support the generalized method of re-writing the expression that involves decimals (3.6 divided by 0.4) to work with an equivalent expression that does not (36 divided by 4). 

Therefore, there seems to be a potential instructional sequence that involves significant work with estimation of decimal values to determine relative size of decimals (individually and with computation), place value understanding and equivalence, with equivalence having the closest connection to the more common generalizable methods for dividing with decimals.  What is more certain is that introducing procedures or algorithms, such as the long-division algorithm, before these understandings are in place can inhibit students’ reasoning, intuitive thinking and repertoire of problem-solving strategies.

Ideas for further study…

Why are students not transferring use of strategies/models to new situations?

How do we know when students are making meaning of models in a way that they will choose to use it in another situation?

In the Library

What's Math Got To Do With It?

A recent assessment of mathematics performance around the world ranked the United States thirty-sixth out of sixty-four countries in the study. When the level of spending was taken into account, we sank to the very bottom of the list. We are falling rapidly behind the rest of the developed world when it comes to math education- and the consequences are dire.

In this straight forward and inspiring book, Jo Boaler named by the BBC as one of eight people who "are changing the future of education," outlines concrete solutions that can transform students' math experiences, including classroom approaches, essential strategies for students, and advice for parent. Now updated to address the controversial Common Core, this is a must read for anyone who is interested in the future of our children and our country!

Contact the Regional Math Center if you would like to borrow this or any of our other resources

http://www.lcsc.edu/irmc/professional-development-resources/books-to-loan/

Peanuts- doing math in your sleep

Elementary Conundrum

3 x 3 cube problem

Challenge:

Twenty-seven identical cubes have two opposite faces in black with the remaining four faces white; these are used to make a 3x3x3 cube. What is the greatest fraction of surface area that may appear black?

Given Answer:

Imagine a large 3x3x3 cube made out of 27 unit cubes. Of the 27 unit cubes, 1 is completely hidden and cannot be seen, and each of the remaining 26 cubes has one, two, or three of its faces exposed. Three faces are exposed on 8 of the 27 unit cubes (at the 8 corners of the large cube), 12 of the 27 unit cubes have two faces exposed (on the edges of the large cube), and 6 of the 27 cubes have 1 face exposed (in the middle of each face of the large cube). Each of the 26 unit cubes that have one or more faces exposed has at most one black face exposed. A large 3 Å~ 3 Å~ 3 cube made out of 27 unit cubes has a surface area of 54 unit squares. Therefore, the answer to our question is 26/54 = 13/27.

Credit:

NCTM Mathematics Teacher, October 2016, Vol. 110, Issue 3,  Calendar and Solutions

Did you know NASCO has free lesson plans?

Adding and Subtracting Integers

Find this and other free lessons at www.enasco.com

• Put an equation on the board, but note that this time they will be subtracting. Subtraction can also be called take away or the opposite of. 
• Set up this example: -8 – (-4) = 
• You will have one pile of 8 red counters and another of 4 red counters. When you subtract, you will turn over the counter pile that is directly after the minus sign because subtract means to do the opposite. Therefore, you will now have 8 red counters and 4 yellow counters. You now have an addition problem and the answer is -4. 
• When subtracting integers, you make two moves: one is to change the subtraction sign to addition and the other is to change the sign of the number directly following the subtraction sign.

Workshop Reviews

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Football Math

In this interactive game from Idaho Public Television, students add and subtract plays on a football field to practice working with negative and positive integers on the number line.
http://idahoptv.pbslearningmedia.org/resource/mket-math-ns-ratnumb/football/

Each set of randomized drives consists of seven to nine plays and ends with a touchdown. Plays are represented with equations. When students place the football at the correct point for each play, the answer to the corresponding equation is automatically filled in. The accompanying activity suggests ways that teachers can integrate the interactive into the classroom setting.