Lesson 3-5 page 20

Tiles

Lesson 3-6 page 20

Tiles

Lesson 3-7 page 22

Tiles

Lesson 3-1 page 17 Lesson 3-2 page 18 Lesson 3-3 Lesson 3-4Tiles

Copy and extend teachermade patterns.

Extend teacher made patternsthat change directions.

Students create and recordtile patterns.

Students extend patterns

created by other students.

Lesson 3-8 page 22Cubes Cubes Cubes Cubes

Copy and extend teachermade patterns.

Extend teacher made patternsthat use color.

Students create patterns which

other students extend.Students create and record

color patterns.

Lesson 3-9 page 22

CubesExamine recorded patterns

for the answers to specificquestions.

Prerequisite chapters:None

MATERIALS

For overhead projector:Transparencies graph paper, 2.5 cm squares Worksheet 4Clear acetate squares lettered A through E Materials chapter, page 295Tiles Materials chapter, page 294Cubes Materials chapter, page 295

If no overhead projector is available:Make charts in place of transparencies Materials chapter, page 294Square shapes lettered A through E Materials chapter, page 295Square shapes Materials chapter, page 294

Student materials:Dittos Graph paper, 2.5 cm squares

Graph paper, 1.7 cm squares Worksheet 5TilesCubesCrayonsUnlined paper

/

PATTERNS WITH TILES AND CUBES 16

If there are students who are unable to foresee futuresteps of a pattern, the teacher asks those students who dosee the pattern to explain why a certain number of tilesshould be added to each step of the pattern.

The patterns are kept very simple at the beginning.Examples of the kinds of patterns used may be found inthe figure below. As the students' skill increases, morecomplicated patterns may be attempted.

oo

CD

EE

PATTERNS WITH TILES

PURPOSE:

To copy teacher-generated patterns thatchange directions, and predict additionalsteps in each sequence

MATERIALS:

1. If no overhead projector is available,square sha~es

2. Tiles

The patterns in Lesson 3-1 made each new step largerthan the one before. It is not necessary, however, for apattern to continually grow. It may change direction, itmay grow to a certain height then decrease in size in anequally predictable manner. This figure represents stepsin a pattern that changes direction.

Students can learn to predict next steps for change-of-direction patterns, too. The only difficulty comes whenthey guess a next step that is larger, while the next rowplaced on the overhead is smaller. The teacher then tellsthe students:

I see from your predictions you are thinking of a differentpattern than I am using this time. Your predictionswould have been correct if I were using that pattern.Now that you see what my pattern gives for my nextrow, what row am I going to add next?

There is more than one possibility for a next row thatwill fit into a predictable sequence. The only waystudents can know what comes next is to consider allthat has gone before then assume that what is addedfollows from what they think they have already seen.

This may seem arbitrary, but most of Science and, forthat matter, most learning consists of observing closelywhat has already happened, and from that trying toproject what will happen. The more reliable the patterns,the better the predictions. More difficult or less regularpatterns lead to more tentative predictions.

Chemists mix two chemicals in known proportions,and feel they can predict with a high degree of accuracywhat will happen next. Meteorologists, on the otherhand, can give only an approximate weather forecastfrom their knowledge of weather patterns. After thefront has passed, they can explain it. Although weatherpredictors know the logical consequence of any particularpattern they cannot guarantee which pattern will be ineffect at any given moment.

Change·of-direction patterns lead to an interesting ques-tion: when a pattern has gone up and back down again whatwill the next row look like?

.--

I II I J J

PATTERNS WITH TILES AND CUBES 18