How many Ways Can You Make 121 Using a Combination of These Blocks?
Students in second grade worked on this challenging open problem for several days using base ten blocks.
To understand place value deeply, students need to know that numbers can be represented more than one way. Through this problem, students explored the 10:1 base ten relationship and became flexible with decomposing hundreds, tens and ones many different ways. For example, in addition to representing 121 with 1 hundred, 2 tens, and 1 unit, students realized that 121 can be further decomposed into 12 tens and 1 unit cube, or 9 tens and 31 unit cubes.
This work provides a foundation for student strategies of breaking apart numbers by place to solve addition and subtraction problems, leading to a better understanding of the U.S. algorithm.
Student Examples- Can you find a Pattern?
As students worked on this problem, teachers challenged them to see if they could discover a pattern, and a systematic method of finding all of the possible ways to make 121.
Taking the Problem Further:
The openness of this problem allows students to work on many different levels and some may notice that there is a pattern with the number of blocks used (the number increases by 9), as tens are systematically decomposed, as well as an alternating odd and even number of blocks.
When Will We Reach One Half?
Working with a hundreds chart, how many numbers can be covered that contain only the digit 1? If we used the digits 1 and 2, how many numbers could be covered that contain only the digits 1 or 2, or both? By following this pattern:
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 A Pattern?”
Students in the Fours constructed their own rules for what makes a pattern, and then used those rules to create patterns with different attributes of color, size, and direction. The students determined that “With a pattern, you always know what comes next”, and that patterns “Repeat again and again”.
Where’s the Differentiation?
A Size Pattern Was Posed As An Open Problem
Students created their own patterns independently, and then tested their pattern against the rules. Does my pattern repeat? Do I know what comes next? This open problem resulted in a range of work that served as a formative assessment for how well children understood the concept of patterns. This approach was also a useful way to differentiate the needs of the math learners in the class. Teachers could observe students as they worked, and support or challenge them by asking targeted questions based on the individual patterns the children were creating.
Thinking of Numbers to Represent a Size Pattern:
“Two big squares, two medium squares, two big, two medium…”
(Click picture to view video)
Thinking of Letters to Represent a Size Pattern
“B, B, A… B, B, A… B, B, A…”
(Click picture to view video)
The Squares and Vertices Problem
Students in third grade used one-inch square tiles to create a growing pattern by adding one additional tile to each consecutive pattern configuration. Tiles were placed vertex-to-vertex. Then students created a function table to record and analyze the relationship between the number of tiles and the number of vertices in their growing pattern. For example, in one pattern, for 2 tiles, there were 7 vertices, and for 3 tiles, there were 10 vertices, and so on. Students realized that once they recognized the abstract numerical pattern in their function charts, they could predict the growing pattern for any number of tiles without having to actually create further concrete models.
Functional thinking is an important entry point into algebra, and allows students to generalize and express relationships quantitatively. Natural language can be used to describe and express these relationships. Students were asked, “How would you describe the relationship between the number of squares and the number of vertices in your repeating pattern diagram?” Students compared various ways of describing their numerical relationships. We also analyzed the statement: “The number of vertices is equal to three times the number of squares, plus one” v = 3s + 1
Fourth Graders Participated in a Political Art Installation, and then Posed Mathematical Questions About It.
“It’s my first political performance and I just felt I had to do it now. I wanted to show that any wall is dismantlable. We, the public, can tear down walls when society gets together. It could be a mental, physical, or political wall – the point is, it’s ephemeral.”–Bosco Sodi, Artist
Photos by Robert Banat: RobertBanat.com RobertBanat@gmail.com
Here Are the Range of Math Questions They Asked:
How Can We Use the Size of One Brick to Determine the Height and Length of the Wall?
How Many Inches Taller Is the Wall Height Compared to the Average Height of a 4th Grader?
How Many Bricks Are in Each Layer?
Does Each Row Have the Same Number of Bricks?
If Muro is Dismantled in 4 Hours, How Many Bricks Are Taken Down in 15 Minutes?
If There Were 20 People on Each Side of the Wall Taking Down 1 Brick Each, How Many Bricks Would Each Person Take Down?
How Many Bricks Could 8 People Remove During Each Hour If They Worked All Three Hours?
How Many Bricks In One Row?
How Many Bricks Need to be Removed Every 15 Minutes to Take Down Muro in Four Hours?
How Can Two Lines of People Dismantle 1,600 Bricks?
How Many Bricks Are Going Horizontally?
1,644 Bricks Were Dismantled in 4 Hours. How Many Bricks Were Dismantled in One Hour?
How Many Bricks Are There In Every Other Row Starting From the Top?
How Tall Are We Compared To The Muro Wall?
How Can 80 Volunteers Dismantle Muro?
Six Scenarios To Take Down Muro in Four Hours
There Are 1,644 Bricks. How Many Bricks Would Each Person Remove?
Blog entry by Ariane Stern & Julie Kim
First Graders Explore Area and Perimeter
Earlier in the year, we introduced the concept of area—that you can measure how small or large a space is. After measuring various shapes with non-standard units (e.g. tiles, beans, paperclips), first graders developed an understanding that with bigger units like popsicle sticks, you’d need fewer of them to cover a certain area and that you would need many more small units like beans to fill that same area. Later in the year, we began a unit on linear measurement. Again, we used manipulatives to measure the length of objects. This time, we focused on ways to accurately measure length, such as figuring out which side is the length, and measuring by starting at the edge. After learning how to measure length accurately with one unit, we moved again to the idea of measuring the same object with different sized units. Once again, it takes less popsicle sticks to measure the length of a book, and more cubes, if you were measuring the same length.
This year, to culminate our study on linear measurement, we challenged first graders to measure how long each Lower School classroom’s block area is. In small groups, first graders went to each classroom, calculated which side was the longest side and measured it with string. They brought their strings back to our classroom and proceeded to measure how long they were with popsicle sticks, tiles, and double unit wooden blocks. Afterwards, we collected each group’s data and compared the number of double unit blocks it took to measure the length of each string. We put the lengths in order to see which block area was the longest.
Along the way, we ran into some problems, just as mathematicians do in real life! What would we do if the double units were too long or too short for the end of the string? That led us into a conversation about fractions. We looked at halves, quarters, and thirds of rectangles and also looked at circles to deepen our understanding of fractions. This helped some groups to make more accurate measurements.
Then, Julie showed the class some pictures. Tasha’s block area was the longest, but it was very narrow and skinny. Diane’s block area was the shortest, but it was very wide. We wondered, just because a block area is the longest, does that mean it’s the biggest?
What a big question!
How would we figure that out? We knew what all the lengths of the block areas were, but how would we figure out how big they were? What were we even looking for? One student shouted, “The area! How were we going to figure out the area? We drew a rectangle on the board, showing how we knew the lengths. What else would we need to know to find the area? This was a puzzle. But calling upon what we had already learned about area, we realized that we needed to know the width of each block area. Wow!
Each group then went back to their assigned classroom, this time measuring the width of each block area. When they came back, they measured their new strings again with double unit wooden blocks. Then they used graph paper to replicate their block area. Each square of the graph paper represented one double unit block. Then they counted all the squares to figure out the area. This was hard work! Many of the shapes they drew had over 100 squares! Each group had to count the squares in their shape multiple times to try and get the most accurate count.
It wasn’t the longest block area!
Estimating Protest Crowd Size Using the Jacob’s Method of Crowd Counting.
“For many events, especially political rallies or protests, the number of people in a crowd carries political significance and count results are controversial”- Wikipedia
“Almost everyone who has tried to make a crowd estimate has a vested interest in what the outcome of the estimate is”– Charles Seife, Professor of Journalism and Mathematician at NYU.
Fourth graders applied the Jacob’s method of crowd counting to estimate the protest crowd that gathered in Washington Square Park on Wednesday, January 25th, 2017. The emergency rally was organized in response to President Trump’s executive order implementing a ban on immigrants entering the country from large Muslim populations, including Iran, Iraq, Libya, Somalia, Sudan, Syria, and Yemen.
Getting the Numbers Right: The Jacobs Method of Crowd Counting
Herbert Jacobs, a University of California journalism professor in the 1960’s, devised a basic density rule that has been widely accepted. Watching students protesting the Vietnam War from his office window, Jacobs saw that had gathered on a plaza that was arranged in a grid. He counted those in a few squares to get an average number per square and multiplied that by the total number of squares. He also came up with a basic density rule that states a “light crowd” has one person per square meter, and doubled that for a “dense crowd”. A “heavy crowd” would have as many as four people per square meter, according to this method.
Using People Per Square Meter as Density Factors to Determine Crowd Size.
Students Estimated Densities by Using an Aerial View of the Protest Crowd, a Scale Diagram of the Park, and Multiplication.
Washington Square Park Diagram Showing Density Arrays.
Kindergartners Use Dramatic Play to Model the Structure of Addition and Subtraction Problems.
Kindergartners related math stories to birds, their current social studies and science topic, by using dramatic play to act out addition and subtraction problems. Through these stories, students explored the action in different structures of problems (see chart below). Some examples of addition structure bird stories that students acted out were result unknown, change unknown, and both addends unknown. “Students who are taught to approach problems by looking at their structures through the use of a visual model are more likely to perform better than students who do not” (NCTM, 2014, pg. 50). This approach helps students make better sense of the action that is happening in problems and rely less on using keywords that can often be interpreted in many different ways (Karp, 2014, pg. 21).
An Example of Student Work Showing Change Unknown Structure Along With a Structure Diagram:
“Six pigeons were pecking on the sidewalk on Bleecker Street. Some more pigeons came to see what they were pecking at. Then there were ten pigeons.” (CHANGE UNKNOWN/ADD TO).
Addition/Subtraction Problem Structures:
2014 Karp, Karen S., Teaching Children Mathematics. National Council of Teachers of Mathematics
2014 NCTM. Putting Essential Understanding of Addition and Subtraction into Practice
Laying the Foundation for Addition by Exploring Parts of Six.
Kindergarten students decomposed the number six by placing square inch tiles into different arrangements and then recorded how they saw the arrangements using numbers. One of the goals of this problem was to help students understand that numbers can be decomposed (broken apart) in many ways. Developing an understanding of part-part-whole relationships is the first critical step to grasping the concepts of addition and subtraction, properties, and algorithms (NCTM, 2014, pg. 11). For example, in later years, decomposition (part-part-total understanding) of multi-digit numbers are used in place value concepts and the operations of addition/subtraction, multiplication/division by applying the various properties of commutative, associative, and distributive.
Differentiating Through an Open Problem:
How Many Ways Can Six Tiles Be Arranged?
Each child arranged tiles in as many ways as they could. This open problem was accessible to every student in the class, and at the same time provided differentiation because of the challenging open-ended nature of the problem (see below, the same problem given at the college level). The class came together and examined all of the possible ways of arranging six tiles “side-to-side, and corner-to-corner”. Through this exploration, the geometric concepts of congruence and transformations (flips, slides, turns), were introduced. The students found 20 ways to arrange six tiles.
Six Tiles Problem by Professor David Meredith:
Below is the same six tiles problem given to mathematics students at San Francisco State University by Professor Meredith, of the mathematics department. Prior to San Francisco State Univ., David was a professor at MIT, and a Woodrow Wilson Fellow. Meredith’s students found 35 distinct non-congruent ways to arrange six tiles.
2014 National Council of Teachers of Mathematics, Putting Essential Understanding of Addition and Subtraction into Practice.