Podcast
Questions and Answers
What principle is demonstrated when a value is subtracted from one addend and added to another in an additive expression without changing the total?
What principle is demonstrated when a value is subtracted from one addend and added to another in an additive expression without changing the total?
- Commutative Property
- Compensation (correct)
- Associative Property
- Distributive Property
The order of apples in a box (e.g., 'red, green, red') determines whether an arrangement of apples is unique.
The order of apples in a box (e.g., 'red, green, red') determines whether an arrangement of apples is unique.
False (B)
How many unique additive combinations can be made with the number 7?
How many unique additive combinations can be made with the number 7?
8
When comparing 5 + 3 and 4 + 4 using compensation, you can take 1 from 5 and add it to 3 to create the expression ____ + ____.
When comparing 5 + 3 and 4 + 4 using compensation, you can take 1 from 5 and add it to 3 to create the expression ____ + ____.
Match the following scenarios with the mathematical principle they demonstrate:
Match the following scenarios with the mathematical principle they demonstrate:
Which of the following expressions is equivalent to 7 + 5, using the compensation strategy?
Which of the following expressions is equivalent to 7 + 5, using the compensation strategy?
According to hierarchical inclusion, smaller numbers do not 'nest' inside bigger numbers.
According to hierarchical inclusion, smaller numbers do not 'nest' inside bigger numbers.
In the context of apples, what determines a 'unique arrangement'?
In the context of apples, what determines a 'unique arrangement'?
The number of unique arrangements of a number is always _____ more than the number itself.
The number of unique arrangements of a number is always _____ more than the number itself.
Match the following terms with their descriptions:
Match the following terms with their descriptions:
Which of these is an example of compensation being used to create additive equivalence?
Which of these is an example of compensation being used to create additive equivalence?
Riemann sums in calculus relate to the concepts discussed in this unit because they both involve understanding digit place value.
Riemann sums in calculus relate to the concepts discussed in this unit because they both involve understanding digit place value.
What two contexts are being used to provide an understanding of additive equivalence and part whole relationships?
What two contexts are being used to provide an understanding of additive equivalence and part whole relationships?
The number 10 has _____ unique additive combinations.
The number 10 has _____ unique additive combinations.
Match the concept presented with its description:
Match the concept presented with its description:
When comparing the expressions 8 + 5 and 7 + 6 using compensation, what is the new, equivalent expression you can create from 8 + 5?
When comparing the expressions 8 + 5 and 7 + 6 using compensation, what is the new, equivalent expression you can create from 8 + 5?
Moving objects between groups will change the total number of objects.
Moving objects between groups will change the total number of objects.
What property states that changing the order of addends does not affect the sum?
What property states that changing the order of addends does not affect the sum?
In a two-digit number, the leftmost digit represents ___, while the rightmost digit represents the number of loose ones.
In a two-digit number, the leftmost digit represents ___, while the rightmost digit represents the number of loose ones.
Match the following course sections with their descriptions:
Match the following course sections with their descriptions:
Flashcards
Compensation in Math
Compensation in Math
Using compensation to manipulate an additive expression while maintaining equivalence.
Additive Compensation
Additive Compensation
Moving a value from one addend to another without changing the total sum.
Commutative Property of Addition
Commutative Property of Addition
The principle that changing the order of addends does not affect the sum.
Hierarchical Inclusion
Hierarchical Inclusion
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Equivalent Expressions
Equivalent Expressions
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Compensation Definition
Compensation Definition
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Unique Arrangement
Unique Arrangement
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Compensation for Arrangements
Compensation for Arrangements
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Arrangement Conjecture
Arrangement Conjecture
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Additive Combination
Additive Combination
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Whole
Whole
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Study Notes
- This procedural sequence is about using compensation to create new expressions and finding all unique arrangements of a number.
- The context of bunk beds and apple boxes helps to understand additive equivalence and part-whole relationships.
Bunk Beds
- Compensation is used to manipulate an additive expression.
- Removing a value from one addend and adding it to the other maintains equivalence.
- Example: 4 + 6 = 3 + 7 = 10
Apple Boxes
- The number of green and red apples determines if the arrangement is unique, not the order of the apples.
- Compensation is used to find all unique arrangements of 5 and 10 systematically.
- The number of unique additive combinations of any number is one more than the value of that number (e.g., 10 has 11 unique combinations).
Intellectual Progression
- Scholars already know: switching addends doesn't change the sum (commutative property of addition) and combinations of 5 and 10.
- Numbers grow by one when counting (hierarchical inclusion).
- Objects can be moved between groups without changing the total.
- In this Sequence: scholars will develop an understanding of how compensation can be used to create equivalent expressions, and they will start to be able to manipulate additive expressions flexibly
- The number of unique additive combinations of a number is one more than the number itself.
- Through compensation, understanding of part-whole relationships is deepened.
Later in the Course
- Part-whole relationships are explored through place value.
- Digits in a two-digit number represent tens and ones.
After the Course
- Riemann sums in calculus approximate the area under a curve.
- The area under the curve is sectioned into different parts, which when added together, produce a finite whole.
- The unit sets the foundation for understanding part-whole relationships at a more complex level.
Essential Questions
- How can we prove that two additive expressions are equivalent?
- Can we find all unique additive combinations of a specific number?
- How do you know you have exhausted all combinations?
Content Overview
- Compensation is used to create equivalent expressions by taking from one group and giving it to another.
- Equivalence is maintained because the total number of objects doesn't change.
Example: Bunk Beds
- Moving kids between bunks mimics compensation.
- 4 kids on top, 4 on the bottom (4 + 4).
- One kid moves, resulting in 3 + 5.
- Both expressions equal 8.
Comparing Expressions
- Compensation compares expressions without solving.
- Comparing 2 + 6 and 3 + 4: changing 2 + 6 to 3 + 5 demonstrates that 2 + 6 must be greater than 3 + 4.
Unique Arrangements
- The number of possible arrangements is 1 more than the number itself.
- A unique arrangement is when no other arrangements have the same quantities of objects.
- The quantity of objects determines uniqueness, not the arrangement.
- Example: "red, red, green, red, green" is the same as "green, green, red, red, red" given they have 2 green apples and 3 red apples each.
Finding Arrangements of 5 Apples
- There are 6 unique arrangements of 5 apples.
- 11 unique arrangements can be made with 10 apples.
Unique Arrangements of 10
- The unique arrangements of 10 total as follows:
- 10 + 0
- 9 + 1
- 8 + 2
- 7 + 3
- 6 + 4
- 5 + 5
- 4 + 6
- 3 + 7
- 2 + 8
- 1 + 9
- 0 + 10
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