Differentiating Mathematics Through Open Problems

Lower School
Guest post from Lower School Math Coordinator Debra Rawlins:
LREI Lower School students are engaging with rich, open-ended problems that allow them to be real mathematicians. These interesting problems are often messy and have no obvious pathway to a solution pathway. They require our students to use creativity and critical thinking in the process of solving them. Worthwhile problems can be solved on a variety of levels, with every child being able to individually tackle the problems in their own way. A good problem has multiple entry points, and therefore offers both accessibility and challenge for all students in a class.

You have 30 cans.
How many ways can you arrange the cans
to make them easier to count?

This is the open-ended problem that students worked on in second grade while collecting cans for Saint John’s Food Pantry. They discovered many layers of mathematics through this investigation.
The teachers differentiated and deepened the mathematics by asking targeted questions to individual students based on the approach each student took.
For example, students who counted the cans by ones were asked:
“Is there a way to group 30 cans that would allow you to count them more efficiently?”
In second grade, students need to develop the capacity to move from counting objects by ones to understanding and counting sets. For example, when counting by fives, the child comprehends that each number in the count represents five objects, and as the child says each number in the skip-count sequence, they are adding another group of five. This takes place on a conceptual level and is different from merely rote skip counting. This experience with equal groups of objects lays the foundation for multiplication.
For students who arranged 30 cans into equal groups of five, teachers asked, “Have you
found all the possible equal groups that you can make with 30 cans?”  As students worked through this question, they discovered all the factors of 30, and also found that repeated addition was one way to record equal groups. 
Some students placed the cans into six equal rows of five cans each. This represented a visual array model of multiplication. The array is used in later grades to help students understand the U.S. multi-digit multiplication algorithm. A question for these students would be, “Is there another way to record this can arrangement?” This would help the students to notice the spatial-structuring nature of the array (If it can be recorded as 5 + 5 + 5 + 5 + 5 + 5 = 30, it can also be recorded as 6 + 6 + 6 + 6 + 6 = 30)
Another student created equal groups of five, and also equal rows of five. The student was asked: “On the left, you have five cans in each group, and you have six groups. On the right, you have five cans in each row. Is six represented anywhere in your diagram on the right?” This helped the student relate the six equal groups to the number of rows in the array.
“The rows turned into columns, and the columns turned into rows.”
This is what one student noticed while looking at the three rows of ten cans from another point of view. The teacher inquired, “How can you record what you discovered?” The student showed the commutative property of multiplication by drawing the array in the rotated position.
How Many Different Nets Can You Find That can Be Folded To Make A Cube?
As third graders worked on this problem, several questions were individually posed to help students delve deeper into the mathematics. “If you flip the net over, does that count as another way?” This question introduced the concept of congruence, as well as transformations. Additional questions meant to encourage further mathematical explorations included: “How can you prove that you have found all of the ways?” “Why is it that the 11 nets are distinctly different, but all of them have an area of 6 square units?” “If all nets for a cube have the same area, will they also all have the same perimeter?” “How can you test this idea?
What Can a Student in the Fours Do When They Run Out of a Certain Size Block?
Fours’ teachers used this question as an opportunity for a rich open-ended math problem for their students. The Fours used classroom blocks to estimate, and then test, how many of each of the same size smaller blocks it would take to cover one double unit block. A poster was created of their mathematical conversions to use as a tool. The students have found this conversion chart, or “recipe,” useful to refer to when they run out of a certain block size because they can now create any size they need by combining other blocks.
Some of the targeted questions teachers asked students in order to differentiate this activity included: “Can you predict how many of this shape (unit square) it will take to cover the double unit block?” This question explored the concept of area, as well as measurement. “If it takes two unit squares to cover half the double unit block, how many would it take to cover the whole double unit block?” This question explored fractional parts, as well as addition concepts.
The openness of these problems, along with carefully posed, individualized questioning by teachers, guide our students along their journey toward meaningful mathematical thought and exploration. Rich problems such as these also provide further differentiation within a classroom because they provide multiple points of entry while allowing students to deepen their mathematical thinking.
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Lower School Conference Childcare 
The Lower School will be closed on both conference days (April 13 and 17). Childcare will be available from 8 a.m. to 4 p.m. for Lower School students. This service is complimentary on a drop-in basis during your conference, and $20 for the whole day. Whole day childcare is limited to 20 slots per day so please be sure to sign up below. Lunch and activities will be provided. Please email kmateos@lrei.org for any questions.
Sign up for full day childcare on April 13th here
Sign up for full day childcare on April 17th here