Essential Questions: How do I determine the most efficient way to represent a number (pictorial, symbolic, with objects) for a given situation?
In what ways can items be grouped to make exchanges for unit(s) of higher value?
How does the position of a digit in a number affect its value?
In what ways can numbers be composed and decomposed?
How are place value patterns repeated in numbers?
How can place value properties aid computation?
How does using the base ten system make it easier for me to count?
How does the place value system work?
How do I know which computational method (mental math, estimation, paper and pencil, or calculator) to use?
How do you solve problems using addition, subtraction, multiplication, or division in real world situations?
What computation tools are best suited to which circumstances?
Lesson Plan B.5: Using the Distributive Property to Multiply
Lesson Seed B.6: Using the Understanding of Place Value to Divide
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Content Emphasis By Clusters in Grade 4
Progressions from Common Core State Standards in Mathematics
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Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
In this unit, students invent and use strategies while also working towards understanding general methods and becoming fluent in using the standard addition and subtraction algorithms. These algorithms rely on adding or subtracting like base-ten units (ones with ones, tens with tens, hundreds with hundreds, and so on) and composing or decomposing base-ten units as needed (such as composing 10 ones to make 1 ten or decomposing 1 hundred to make 10 tens). In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable. Students in Grade 4 also compute products of one-digit numbers and multi-digit numbers (up to four digits) and products of two two-digit numbers. They divide multi-digit numbers (up to four digits) by one-digit numbers. As with addition and subtraction, students should use methods they understand and can explain. Visual representations such as area and array models that students draw or build and then connect to equations and other written numerical work are useful for this purpose. By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities..
At the completion of the unit on Arithmetic with multi-digit numbers, the student will understand that:
Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):
Possible Student Outcomes:
Evidence of Student Learning:
The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.
Fluency Expectations and Examples of Culminating Standards:
Sample Assessment Items:
Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)
Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.
Part I – Focus Cluster
Use Properties of Operations to Generate Equivalent Expressions
equation: is a number sentence stating that the expressions on either side of the equal sign are, in fact, equal.
factor pairs: two numbers that when multiplied equal a product. Examples of factor pairs for the number 12 are 2 and 6; 3 and 4; 1 and 12.
multiple: is the product of a whole number and any other whole number. Example: 20 is a multiple of 5 because 4 × 5 = 20.
place value: the value of a digit as determined by its position in a number. Example: in the number “101” the one is worth either 100 or 1, depending upon its position.)
base ten numerals: a base of a numeration system is the number that is raised to various powers to generate the place values of that system. In the base ten numeration system the base is ten. The first place is 100 or 1 (the units place), the second is 101 or 10 (the tens place, the third is 102 or 100 (the hundreds place), etc. This determines the place value of the different positions in a number. Example: One number written as a base ten numeral would be 6,427.
number names: base ten numerals written in word form. Example: 6,427 would be written as six thousand four hundred twenty-seven.
expanded form: a numeral expressed as a sum of the products of each digit and its place value. Example: 6,427 written in expanded form would be 6,000 + 400 + 20 + 7.
multi-digit whole numbers: a whole number comprised of more than one digit. Example: 27 and 246,910 are both multi-digit whole numbers.
standard algorithm: an algorithm is a systematic scheme for performing computations, consisting of a set of rules or steps.
properties of operations:
Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.
rectangular arrays: the arrangement of counters, blocks, or graph paper squares in rows and columns to represent a multiplication or division equation. Example:
Part II – Instructional Connections outside the Focus Cluster
multiplicative comparison: comparing the size of a product to the size of one factor on the basis of the size of the other factor.
Example: If 2 bags of candy cost $4.00 then
Then 8 bags of candy would cost $16.00
decompose : breaking a number into two or more parts to make it easier with which to work.
Example: When combining a set of 5 and a set of 8, a student might decompose 8 into a set of 3 and a set of 5, making it easier to see that the two sets of 5 make 10 and then there are 3 more for a total of 13.
Decompose the number 4; 4 = 1+3; 4 = 3+1; 4 = 2+2
Math Related Literature: