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Week 4: Algebra - Patterns


Arranging Toothpicks (Printable PDF)

Toothpicks are used to build a rectangular grid that is 20 toothpicks long and 10 toothpicks wide. The grid is filled with squares that have 1 toothpick on each side. What is the total number of toothpicks used?
If a represents the number of toothpicks in the length of a grid and b represents the number of toothpicks in the width of a grid (again, the grid is filled with squares that have 1 toothpick on each side), write an expression representing the total number of toothpicks in any rectangular grid of this sort.

 

GPS As seen in Problem Exploration

M6A2: Students will consider relationships between varying quantities.

a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.

 

Using a spreadsheet program, the student can set up two columns: one representing the number of toothpicks for the width and one for the height. Then they can test different hypotheses.

The number of rectangles and number of toothpicks are related and can be expressed as varying quantities in an equation. A pattern exists when you extend the rectangle. For rectangles that are 1 toothpick high, you will need 3 more toothpicks to extend the rectangle. For rectangles that are 2 toothpicks high, you will need 5 toothpicks to extend the rectangle. Etc.

M7A1. Students will represent and evaluate quantities using algebraic expressions.

a. Translate verbal phrases to algebraic expressions.

b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.
 

The number of toothpicks can be represented as an algebraic expression: a+2ab+b. One way to see this is to let a = number of rows and b = number of columns in the rectangular grid. Then the number of toothpicks needed is a+1 for one column, so one needs b(a+1) toothpicks for all the columns and similarly one needs a(b+1) toothpicks for the columns.

Adding these two together: b(a+1) + a(b+1) = 2ab+a+b.
 

M8A1. Students will use algebra to represent, analyze, and solve problems.

a. Represent a given situation using algebraic expressions or equations in one variable.

b. Simplify and evaluate algebraic expressions.

 

Same as M7A1.