Squirreling it Away
In the problem Squirreling it Away, students use number sense, comparison subtraction, division, factors and divisibility, counting principles, systemic charting, and closed-form equations to solve problems. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.
In this task, the students use a set of acorns or cubes to answer questions related to subtraction. The teacher poses questions about how many acorns a squirrel took and how many they have left.
In this level, students are presented with a situation that involves making sense of totals and comparison differences. The task involves making a number story about giving acorns to two different-sized groups of squirrels and then determining how many were left over from the original total acorns.
This level addresses Common Core Standard 2.OA.1 by using subtraction to solve a two-step word problem involving taking from. Students are asked to use drawings to reason abstractly and quantitatively (MP.2).
In this level, students start to examine how the acorns can be partitioned into different sets. The students are told that different squirrels can carry different amounts of acorns on given trips. Students are asked to find the number of trips it takes for each type of squirrel to carry the acorns.
This level addresses Common Core Standard 4.OA.3 by using multiplication and division to solve a multi-step word problem involving remainders that must be interpreted. Students must reason abstractly and quantitatively (MP.2) in order to interpret the remainders.
In this level, students are asked how many ways three squirrels can carry away 24 acorns, if each can carry a different number of acorns.
This level connects to Common Core Standard 7.SP.8b as students can find all the possibilities—an important precursor to generating a sample space—using an organized strategy, such as lists, tables, or tree diagrams. High school students might tackle the problem by modeling the situation with a series of linear equations in two variables, each of which has a graph whose solutions can be interpreted in the context of the problem situation (A-CED.2, A-REI.10). Students can choose from a variety of tools (MP.5) to model the situation (MP.4), including lists, tables, or algebraic methods . They can also look for and express regularity in repeated reasoning in order to help them determine all the solutions (MP.8).
In this level, students determine the number of ways 24 acorns can be divided among three different squirrels, subject to given constraints.
This level can support 8-F.1, as students may choose to approach this level by beginning with a simpler problem (a smaller number of acorns) and then generating a table of values in which each value is the number of ways for larger and larger numbers of acorns. At the high school level, this problem can be modeled with an arithmetic sequence that can be represented recursively or with an explicit formula (F.LE.2, F-BF.2). Students will need to demonstrate regularity in repeated reasoning in order to help them determine all the possibilities (MP.8).
In this level, students are asked to find and justify a closed-form equation that will determine the number of ways that n acorns can be divided among three squirrels. Students may benefit from engaging in Levels D and E to help them make sense of this level.
This level addresses Common Core Standards A-CED.2 and F-BF.1a by having students generalize and write expressions or equations that describe the relationships between the total number of acorns and the number of acorns 3 squirrels could carry. Some strategies students use to find the equation might lead them to fit a quadratic function to data generated in an earlier level (S-ID.6a) or create and solve a system of 3 equations in 3 unknowns (A-CED.3, A-REI.6, A-REI.8). Students will model the situation (MP.4) and will look for regularity in repeated reasoning in order to help them determine their generalizations (MP.8).
PROBLEM OF THE MONTH
Download the complete packet of Squirreling it Away Levels A-E Here.
You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide. Download the guide as iBook with embedded videos or Download as PDF without embedded videos.
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