# When Will We Reach One Half?

### How many digits are needed before the hundreds chart is half covered?

First grade students became deeply engaged in this open problem.  They explored patterns, made conjectures (predictions) as to what digit they would be on when half the 100 board would be covered. They discussed and defined what one half would mean on the hundred chart.  In the process of solving this interesting problem, they discovered other numeration concepts along the way. They learned the meaning of a “digit” versus a number. They also gained reinforcement and practice in recognizing, sequencing, writing, and saying numbers up to 100.

# Is It Still a Triangle if it’s Not Pointing Upward?

When young children are first introduced to shape categories, like the triangle, they are presented with symmetrical models such as an equilateral triangle (all sides and angles equal in measure). Children are usually shown the equilateral triangle positioned only one way- with one vertex pointed upward, and a horizontal base.

However, research suggests that as soon as these typical examples are introduced, a variety of different positions and sizes should also be shown to children so that their notions of triangles don’t become rigid and limited to only one type of triangle, or a single example of a shape. “Children two to three years of age are not too young for this type of learning” (NCTM, 2010).

## “There’s lots of ways you can make a triangle”

Examining various triangles and determining whether or not they are, in fact triangles, leads children to grapple about specific attributes that define what a triangle is (three sides, three corners, straight lines, etc.). As children experience a broader variety of each shape, they begin to build more accurate geometric concepts and ideas. Through this investigation process, the Fours are laying the foundation for geometric discussions that will take place in later grades, such as transformations (rotations, flips, slides), symmetry, and angles.

# Teachers posed this question to first graders during their study of geometry.

First graders went beyond naming basic two-dimensional shapes to exploring specific characteristics, or attributes, that define them. Their conversations evolved from seeing a shape as a “whole” such as a triangle, to analyzing and deciding the specific features that prove that it is a triangle. Students decided that a shape can only be called a triangle if it has three straight lines that make up the sides, three corners, or vertices, and does not have any openings (it needs to be closed).  Through this specific definition, they discovered that there are many types of triangles. They used this critical thinking foundation to explore a variety of two-dimensional shapes.

# “Does a Circle Have Sides?”

This was an interesting question to ponder. First grade teachers, Sarah and Ariane, posed it to their students to see if they could apply the skills they had learned about defining geometric attributes to this question. It turns out that it wasn’t an easy question to answer! Continue reading

## Area, Perimeter & Little Red Square Park

### Third Graders Explore Area and Perimeter by Measuring a “Pocket Park”

As part of their study of area and perimeter, third graders in Elaine and Jessie’s class measured the perimeter of “Little Red Square”, the small pocket park that lies just in front of LREI on Sixth Avenue. Each class divided into small groups and used trundle wheels to measure the four sides of the park. Then they calculated the perimeter by adding up the side dimensions. When the class looked at the set of data, they realized that their perimeter data varied, and they attributed this to the inexactness of using the trundle wheel. They decided to use the middle number of the data set (the median) as their “working” perimeter for the park. Continue reading

## Apollo Theater Problem

### How many New York City blocks is it to the Apollo Theater from LREI?

(The Apollo Theater is located at 125th Street in Harlem)

This problem seemed easy enough until Tasha’s second grade realized that the West Village, where LREI is located, isn’t laid out in an organized city grid system, like the rest of Manhattan is. An interesting math problem ensued, and the class enlisted Nick, LREI’s resident historian to help us understand why the streets in the West Village are so confusing! Continue reading

## Tasha’s second grade class watched the video below and came up with several mathematical questions to investigate:

1. How many steps could there be on the stairs if Robin is on the middle step?
2. Is the step Robin is on odd or even?
3. Can I count by “twos” and land on the middle step?
4. What step could Robin be on?
5. What step could Robin not be on?

## Student-to-Student Mathematical Discussion in the Classroom

Young mathematicians need to be able to “Construct viable arguments and critique the reasoning of others”, according to the National Council of Teachers of Mathematics. This philosophy aligns with LREI’s progressive educational goals of placing an emphasis on student voice, and creating a classroom culture of engaging student-to-student discussions. Students take on the role of leaders who believe that they can actively defend their own mathematical ideas, and help shape the ideas of their colleagues in a supportive, nurturing environment. Continue reading

## A rich student-to-student discussion by first graders on the meaning of the equal symbol.

What does: 1 + 7 = ___ + 6

have to do with: 3x + 9 = 5x + 5

…and why are first graders arguing with each other over the meaning of the equal symbol?