These are problems that have been investigated in the classroom, and teacher annotated. We decided to organize the problems by "Big Ideas," taken from the research of Marian Small (Good Questions: Great Ways to Differentiate Mathematics Instruction), and they are those enduring understandings that kids need to have if they really understand a concept. Students need many opportunities to explore and investigate concepts if we want them to have more than a rote or memorized understanding of them. So we have tried to label the important math, make explicit those big ideas we think kids really need to "get", and show you a range of strategies that you may see students using, so that you know where to push them next. Even though the students are investigating and building their understanding through working collaboratively with peers, the teacher is key in monitoring their strategies, extending and challenging their thinking with probing and prompting questions, and then intentionally orchestrating the debrief/consolidation of the problem to bring out those big ideas and that learning goal. The debrief is the most important part of the process, that is where the teaching happens. Marian Small speaks about the importance of consolidation: http://www.curriculum.org/k-12/en/videos/instructional-ideas Math Study Groups: Analyzing and Interpreting Students' Thinking, LNS webcast: http://www.curriculum.org/k-12/en/videos/analyzing-and-interpreting-students-thinking
Grades 4-6
Number Sense and Numeration
General Big Ideas: There are many ways to represent numbers Numbers tell how many or how much Number benchmarks are useful for relating numbers and estimating amounts By classifying numbers, conclusions can be drawn about them The patterns in the place value system can make it easier to interpret and operate with numbers It is important to recognize when each operation (addition, subtraction, multiplication, or division) is appropriate to use There are many different ways to add, subtract, multiply or divide numbers
Multiplication and Division: Multiplication and division undo each other, they are related inverse operations You can multiply numbers in any order (the Commutative Property) To multiply two numbers, you can divide one factor and multiply the other by the same amount without changing the product (variation of the Associative Property, for example, 8x3= (8/2) x (3x2)= 4x6) To divide two numbers, you can multiply both numbers by the same amount without changing the quotient (for example, 15/3= 30/6, because (15x2)/(3x2)) You can multiply in parts (the Distributive Property)
You can multiply in parts by breaking up the multiplier (for example, 6x5= 2x3x5) You can divide in parts by splitting the dividend into parts (the Distributive Property), but not the divisor (for example, 48/8=32/8+16/8. It is not equal to 48/4+48/4) You can divide by breaking up the divisor (for example, 36/6= 36/3/2) When you multiply by 0, the product is 0 When you divide 0 by any number but 0, the quotient is 0 You cannot divide by 0 When you multiply or divide a number by 1, the answer is the number you started with
Fractions can represent parts of regions, parts of sets, parts of measures, division or ratio. A fraction is not meaningful without knowing what the whole is.
There are multiple models and/or procedures for comparing and computing with fractions (decimals, percents), just as there are with whole numbers. Operations with fractions (decimals, percents) have the same meanings as operations with whole numbers, even though the algorithms differ.
The same object can be described by using different measurements The numerical value attached to a measurement is relative to the measurement unit Units of different sizes and tools of different types allow us to measure with different levels of precision The use of standard measurement units simplifies communication about the size of objects Knowledge of the size of benchmarks assists in measuring Measurement formulas allow us to rely on measurements that are simpler to access in order to calculate other more complicated measurements
There are always many representations of a given shape New shapes can be created by either combining or dissecting existing shapes Shapes can be located in space and relocated by using mathematical processes
A group of items form a pattern only if there is an element of repetition or regularity, that can be described with a pattern rule Any pattern, algebraic expression, relationship, or equation can be represented in many ways Patterns are all around us I the everyday world Many number, geometry, and measurement ideas are based on patterns Arranging information in charts and tables can make patterns easier to see Variables can be used to describe relationships
Data Management and Probability
Many data collection activities are based on the sorting of information into meaningful categories
This Marian Small problem, from Eyes on Math, was used for the lessons in the accompanying pdfs:
To collect useful data, it is necessary to decide, in advance, what source or collection method is appropriate and how to use that source or method effectively Sometimes a large set of data can be usefully described by using a summary statistic, which is a single meaningful number that describes the entire set of data. The number might describe the values of individual pieces of data or how the data are distributed or spread out Graphs are powerful data displays because they quickly reveal a great deal of information
An experimental probability is based on past events, and a theoretical probability is based on analyzing what could happen. An experimental probability approaches a theoretical one when enough random samples are used In probability situations, one can never be sure what will happen next. This is different from most other mathematical situations. Sometimes a probability can be estimated by using an appropriate model and conducting an experiment
Teaching Through Problem Solving
These are problems that have been investigated in the classroom, and teacher annotated. We decided to organize the problems by "Big Ideas," taken from the research of Marian Small (Good Questions: Great Ways to Differentiate Mathematics Instruction), and they are those enduring understandings that kids need to have if they really understand a concept. Students need many opportunities to explore and investigate concepts if we want them to have more than a rote or memorized understanding of them. So we have tried to label the important math, make explicit those big ideas we think kids really need to "get", and show you a range of strategies that you may see students using, so that you know where to push them next. Even though the students are investigating and building their understanding through working collaboratively with peers, the teacher is key in monitoring their strategies, extending and challenging their thinking with probing and prompting questions, and then intentionally orchestrating the debrief/consolidation of the problem to bring out those big ideas and that learning goal. The debrief is the most important part of the process, that is where the teaching happens.Marian Small speaks about the importance of consolidation: http://www.curriculum.org/k-12/en/videos/instructional-ideas
Math Study Groups: Analyzing and Interpreting Students' Thinking, LNS webcast:
http://www.curriculum.org/k-12/en/videos/analyzing-and-interpreting-students-thinking
Grades 4-6
Number Sense and Numeration
General Big Ideas:There are many ways to represent numbers
Numbers tell how many or how much
Number benchmarks are useful for relating numbers and estimating amounts
By classifying numbers, conclusions can be drawn about them
The patterns in the place value system can make it easier to interpret and operate with numbers
It is important to recognize when each operation (addition, subtraction, multiplication, or division) is appropriate to use
There are many different ways to add, subtract, multiply or divide numbers
Multiplication and Division:
Multiplication and division undo each other, they are related inverse operations
You can multiply numbers in any order (the Commutative Property)
To multiply two numbers, you can divide one factor and multiply the other by the same amount without changing the product (variation of the Associative Property, for example, 8x3= (8/2) x (3x2)= 4x6)
To divide two numbers, you can multiply both numbers by the same amount without changing the quotient (for example, 15/3= 30/6, because (15x2)/(3x2))
You can multiply in parts (the Distributive Property)
You can multiply in parts by breaking up the multiplier (for example, 6x5= 2x3x5)
You can divide in parts by splitting the dividend into parts (the Distributive Property), but not the divisor (for example, 48/8=32/8+16/8. It is not equal to 48/4+48/4)
You can divide by breaking up the divisor (for example, 36/6= 36/3/2)
When you multiply by 0, the product is 0
When you divide 0 by any number but 0, the quotient is 0
You cannot divide by 0
When you multiply or divide a number by 1, the answer is the number you started with
Fractions (Decimals, Percents):
Fractions can represent parts of regions, parts of sets, parts of measures, division or ratio.
A fraction is not meaningful without knowing what the whole is.
Renaming fractions is often the key to comparing them or computing with them. Every fraction can be renamed in an infinite number of ways.
There are multiple models and/or procedures for comparing and computing with fractions (decimals, percents), just as there are with whole numbers.
Operations with fractions (decimals, percents) have the same meanings as operations with whole numbers, even though the algorithms differ.
Measurement
A measurement is a comparison of the size of one object with the size of anotherThe same object can be described by using different measurements
The numerical value attached to a measurement is relative to the measurement unit
Units of different sizes and tools of different types allow us to measure with different levels of precision
The use of standard measurement units simplifies communication about the size of objects
Knowledge of the size of benchmarks assists in measuring
Measurement formulas allow us to rely on measurements that are simpler to access in order to calculate other more complicated
measurements
Geometry and Spatial Sense
Shapes of different dimensions and their properties can be described mathematicallyThere are always many representations of a given shape
New shapes can be created by either combining or dissecting existing shapes
Shapes can be located in space and relocated by using mathematical processes
Algebra and Patterning
A group of items form a pattern only if there is an element of repetition or regularity, that can be described with a pattern ruleAny pattern, algebraic expression, relationship, or equation can be represented in many ways
Patterns are all around us I the everyday world
Many number, geometry, and measurement ideas are based on patterns
Arranging information in charts and tables can make patterns easier to see
Variables can be used to describe relationships
Data Management and Probability
Many data collection activities are based on the sorting of information into meaningful categories
This Marian Small problem, from Eyes on Math, was used for the lessons in the accompanying pdfs:
To collect useful data, it is necessary to decide, in advance, what source or collection method is appropriate and how to use that source or method effectively
Sometimes a large set of data can be usefully described by using a summary statistic, which is a single meaningful number that describes the entire set of data. The number might describe the values of individual pieces of data or how the data are distributed or spread out
Graphs are powerful data displays because they quickly reveal a great deal of information
An experimental probability is based on past events, and a theoretical probability is based on analyzing what could happen. An experimental probability approaches a theoretical one when enough random samples are used
In probability situations, one can never be sure what will happen next. This is different from most other mathematical situations.
Sometimes a probability can be estimated by using an appropriate model and conducting an experiment