Activity Set 2.1PREP PPTX

Visual Algebra for Teachers

Chapter 2

LINEAR EXPRESSIONS, EQUATIONS AND GRAPHS

Visual Algebra for Teachers

Activity Set 2.1

Introduction to Toothpick Figure 

Sequences

Visual Algebra for Teachers

PURPOSETo learn how to analyze sequences of toothpickfigures:

NumericallyIn words andSymbolically

to see and define an underlying algebraic pattern andto use that algebraic pattern to answer questionsabout the sequence of toothpick figures.

Toothpicks

MATERIALS

INTRODUCTION

Figure NumbersFigure Numbers, indexed here by the counting numbers;n = 1, 2, 3 ... indicate the figure positions in a sequenceof figures.

Counting Using LoopsWhen counting the number of shapes in a figure, we canuse Looping to mark off portions of the figure.

A Loop is simply a shape drawn around a portion of afigure as illustrated here.

Counting Using Loops

Notice that the number of picks looped and the shapesof the loops can change; for looping, there are manycorrect choices.

Counting Using LoopsWhen counting the number of toothpicks in a figureusing loops; mark your counts on the figure and writeout the complete numerical expression.

Don’t just simplify, for example; “13” is not as helpful forseeing this pattern as the addition expression 4 + 5 + 4.

Describing Looped DiagramsWhen describing looping in words, try to use simplephrases.Words and phrases should be clear and allow anotherperson to see your counting technique.Common types of phrases you might use while loopinginclude:

• 1 more (or less) than the figure• Half of 1 more (or less) than the figure• 1 more (or less) than twice the figure• Half of 1 more (or less) than twice the figure• Half of the figure• Three times a figure

PREP Question #1 (numerically)Write in your coursepack, Activity Set 2.1 #1 (ONE)

Consider the following sequence of toothpick figures,Triangles. Build the figures with toothpicks if you can.

The FIRST STEP is to look at the figures NumericallyUse looping and count the total number of toothpicks in each figure without simply counting each individual toothpick. Mark your individual counts by your loops and give each toothpick count as an addition expression (as illustrated in the Introduction).

PREP Question #2 (words)Write in your coursepack, Activity Set 2.1 #2 (ONE)

The SECOND STEP is to describe your previous methodsin Words.

Your words should describe the general technique youused for each toothpick figure in the set.

Another person should be able to recreate loops byreading your words.

Use the diagram for #2 ONE to remark loops and write words.

PREP Question #3 (use method)Write in your coursepack, Activity Set 2.1 #3

Use your looping ideas and words to answer:

“How many toothpicks will there be in the 5th Trianglesfigure?”

Sketch or build the figure to double check your method.

PREP Question #4 (symbols)See your coursepack, Activity Set 2.1 #6 (not #4!)

The THIRD STEP for analyzing the sequence oftoothpick figures is to convert your words intoSymbols.

Use the symbol n for the toothpick figure number andthe symbol T for the total number of toothpicks.

Go back to activity 2, and below your Words, try towrite each word expression as a symbolic equationT = an expression involving n and numbers, such asT = 7n + 2 + 2 (not a valid answer for this sequence).

You are now ready for:

PREP QUIZ 2.1

See Moodle

Visual Algebra for Teachers