How many Squares are there on a Checkerboard?

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We may see this problem as somewhat easy, counting just the 64 small (1x1) squares.

However, the board also has 2 x 2 squares, 3 x 3 squares, and so on. Watch this video to see one way to think about the other sized squares:




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A systematic way of counting the squares can be helpful:

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What did you realize?How many 3 x3 Squares are there on a Checkerboard?What is the total number of squares are there on a Checkerboard?

Extension

How many cubes are there in an 8x8x8 cube such as this?

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AuthorEda Aydemir

Target Grades3 – 12

TagsStudent Exploration, Algebra, Geometry, Number Theory

Common Core Standards6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers. (a) Write expressions that record operations with numbers and with letters standing for numbers. (b) Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient). View one or more parts of an expression as a single entity. (c) Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole- number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). • 7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. (a) Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. (b) Solve word problems leading to inequalities of the form px + q > r or px + q • 8.EE.A.2Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number.

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Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.