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Prove That the Average Really Lies Between Two Numbers

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The World of Numbers · Chapter challenge problems · 3:18

You'll learn: Justify both rationality of the average and its strict location between distinct endpoints.

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Prove That the Average Really Lies Between Two Numbers

3:18

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Prove That the Average Really Lies Between Two Numbers

You'll learn: Justify both rationality of the average and its strict location between distinct endpoints.

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Chapter challenge problems

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Find More Reciprocals with Cyclic Repeating Blocks
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Find More Reciprocals with Cyclic Repeating Blocks
You'll learn: Test reciprocal decimals and compare rotations of their repeating blocks.
What Does the Fourth Decimal Place Tell Us?
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What Does the Fourth Decimal Place Tell Us?
You'll learn: Relate a last non-zero fourth decimal digit to a numerator over 10⁴ and examine reduction.
Make Enough Space Between Fraction Numerators
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Make Enough Space Between Fraction Numerators
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Can Three Rational Numbers Satisfy Both Zero Conditions?
You'll learn: Combine the sum and pairwise-product conditions to obtain a sum of squares equal to zero.

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Understanding and proving identities

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Three Consecutive Squares Always Give Two
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Three Consecutive Squares Always Give Two
You'll learn: Explore the opening square-number calculation and formulate the observed pattern.
An Identity Is an Equation True for Every Allowed Value
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An Identity Is an Equation True for Every Allowed Value
You'll learn: Distinguish an identity from an equation holding only for certain inputs.
Build the Square of a Sum from Four Pieces
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Build the Square of a Sum from Four Pieces
You'll learn: Partition a square of side a+b into a², b² and two ab rectangles.
Why the Square Identity Also Works for Negative Numbers
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When Is the Square of a Sum Bigger Than the Sum of Squares?
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When Is the Square of a Sum Bigger Than the Sum of Squares?
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Square a Binomial with Coefficients and Fractions
You'll learn: Identify the complete expressions playing the roles of a and b and expand correctly.
Use Nearby Round Numbers to Calculate Squares
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Use Nearby Round Numbers to Calculate Squares
You'll learn: Choose a convenient additive split and evaluate with (a+b)².

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Perfect-square factorisation

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Read a Perfect Square Backwards
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Read a Perfect Square Backwards
You'll learn: Recognise a²+2ab+b² and write its repeated factor.
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Take Out a Common Factor Before Looking for a Square
You'll learn: Extract a common factor to expose a simpler perfect-square expression.
Derive the Square of a Difference by Substitution
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Derive the Square of a Difference by Substitution
You'll learn: Replace b with −b in the square-of-a-sum identity.
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See the Square of a Difference as an Area
You'll learn: Remove the two appropriate rectangles from the larger square.