GEOMETRY: SPATIAL SENSE AND MEASUREMENT
For many adults, geometry is one math topic that immediately makes sense to them and gives them confidence in their ability to learn. It is also true, however, that many adults associate geometry, like algebra, with failure. Measurement, a foundation skill for geometry, is also an essential life skill, one that adults use in many different but familiar contexts.
Measurement is not an end in itself. It is a tool used in many contexts; home, work, and community. We measure many different attributes of physical objects and time in many different ways in many different contexts. Measurement is essential to our sense of ourselves and our orientation to the world. Because measurement is used so often and in so many contexts, many learners have great confidence in their measurement skills. For ESL learners, teaching measurement is very important as a cross-cultural component of mathematics and second language learning, since many of these learners have use the metric measurement system much more than the U.S. system. Measurement skills can be critically important. Time management is another critical measurement skill.
Some adult learners identify geometry with failure. Other learners recognize their excellent everyday skills in geometry, although they may or may not use the term "geometry" in relation to these skills. Some adult learners don't see geometry as useful, however, geometry is and can be related to all aspects of life: home, school, work and community. Geometry and spatial sense can be used to describe the physical world.
IMPLICATIONS FOR LEARNING AND TEACHING
We must use exact and estimated measurements to describe and compare phenomena to increase the understanding of the structure, concepts and process of measurement. Despite the fact that competence in measurement is vital, some adult learners have difficulty selecting and determining appropriate units of measure as well as appropriate tools of measurement. Concrete activities with non-standard and standards units help learners develop an understanding of the many measurable attributes of physical objects (length, time, temperature, capacity, weight, mass, area, volume, and angle). This is the natural way of building a vocabulary for measurement, and for comprehension of what it means to measure.
We must address the impact of measurement skills on self-efficacy and
self-reliance. Math is everywhere and to be independent and survive on a limited budget one needs to be able to do things oneself and find the best values along the way.
Measurement skills should be extended to concepts areas such as volume, proportion, and problem solving.
We must increase the awareness of acceptable tolerances (margins and upper and lower limits) and the consequences of being within and outside of these tolerances. In the workplace, everything is measured. Someone has to understand what upper and lower limits are, and how to input data. Much is computerized but the results are only as good as the information inputted.
The place to start is the learner's strengths; instruction must be practical and useful for learners to overcome their fears regarding geometry. Opposites must be provided for learners to make connections between instruction and real-life situations common to their lives.
Finally, it is necessary to focus on hands-on problem solving and to give special attention to developing spatial sense in order for learners to develop an understanding of geometric principles. Spatial reasoning includes not only geometry, but measurement and the ability to visualize. It is often the visual and concrete models that can help people understand and learn what we want to teach about number and statistics. We must realize that this kind of reasoning, this part of mathematics, often helps students who have talents in this direction realize and accept that they do have. mathematical potential.
Title: Celsius Temperature
Materials:
Celsius thermometers, worksheet (attached).
Target Audience:
Type of Setting:
Small or large group, individual.
Activity Description:
1 . Instructor develops the concept of Celsius temperature (e.g. O*C is the freezing point of water and 100*C is the boiling point of water).
2. Students estimate Celsius temperatures of locations listed on worksheet.
3. Students place a thermometer at each location and leave it there for about 1 hour.
4. Students record and compare actual Celsius temperatures.
Evaluation: This hands-on activity will help students develop an understanding of the Celsius temperature scale.
Modifications:
Other temperature-sensitive locations can be substituted depending on your particular classroom setting.
Celsius Temperature Worksheet
Directions:
Estimate each temperature in Celsius degrees. Place a thermometer in each location, leave it there for about an hour, then check and record actual temperature.
| Estimated
Temperature | Actual
Temperature |
| 1. On a tabletop | | |
| 2. On a windowsill | | |
| 3. In the refrigerator | | |
| 4. In the freezer | | |
| 5. In your shirt pocket | | |
| 6. Outside | | |
| 7. On the wall | | |
| 8. Near a heat source | | |
Title:
Materials:
Metric measuring cups or graduated cylinders; sand; an
assortment of household containers such as coffee cup, drinking
glass, soft drink bottle, gallon jug.
Target Audience:
Type of Setting:
Small group, one-on-one, or individual.
Activity Description:
1. Students first estimate the capacity of the container.
2. Students fill the container with sand and note actual capacity.
3. Students calculate the difference between their estimate and the actual capacity.
Evaluation:
This activity helps students develop an understanding of metric units of capacity.
Modifications:
This activity could become a competition; the student (or team) whose estimates are most accurate is the winner. Also, this concept could be modified for measurements of length or weight.
Title:
Materials:
1 000 ml graduated cylinder of water and an apple for each group.
Target Audience:
Intermediate to advanced math skills.
Type of Setting:
Activity Description:
Volume:
1 . Instructor introduces the concept of "displacement" and how the volume of water displaced by the apple is equal to the volume of the apple,
2. Students explore the concept using the attached worksheet.
Weight:
1 . Instructor develops the concept that an object that floats in a liquid displaces an amount of liquid equal to the weight of the .Object. Tell students that they will be measuring the weight of the apple without using a scale.
2. Students explore the concept using the attached worksheet.
Evaluation:
This activity explores two important measurement topics: volume of an object by water displacement; and determination of weight by volume of water displaced. It is important for the instructor to develop these ideas for the class.
Resources:
Let's Measure Metric, Ohio Dept. of Education, 1975
Modifications:
This activity can be expanded to include other objects which will float in water.
An Apple for the Teacher Worksheet
Volume
1. Fill the beaker about 2/3 full of water. Record the level of the water.
2. Place the apple in the beaker, taking care not to spill any water. Using a pencil, hold the apple just below the surface of the water. Record the water level.
3. Calculate the volume of the apple by subtracting water level 1 from
Water level 2 - water level 1 = ml = volume of the apple
4. Explain why the difference between the two water levels is the same as the volume of the apple.
Weight
1 .Estimate the weight of the apple.
2. Fill the beaker about 213 full of water. Record the level of the water.
3. Place the apple in the beaker, taking care not to spill any water. Let the apple float on the surface of the water. Record the water level.
4. Calculate the volume of water displaced.
Water level 2 - water level 1 = ml
5. Given that 1 ml of water weighs 1 gram, what is the weight of the apple? How does this compare with your estimate?
Title:
Materials:
Metric measuring cups, small mixing bowl, trail mix ingredients.
Target Audience:
Type of Setting:
Small group or individual.
Activity Description:
Students make individual servings of trail mix using the following recipe:
125 ml shelled peanuts 60 ml M&M's 80 ml raisins 30 ml coconut
Thoroughly mix ingredients in a small bowl. Enjoy!
Evaluation:
This hands-on activity helps the student develop familiarity with metric units of capacity. Students with very basic reading and math skills will be able to experience success in a practical classroom activity.
Modifications:
Other recipes which lend themselves to classroom use may be substituted. If an oven is available, baking temperatures should be given in Celsius degrees.
Title:
Materials:
Masking tape strip about 2.7 meters long, meter stick, decimeter square, worksheet (attached).
Target Audience:
Beginning to intermediate math skills.
Type of Setting:
.
Activity Description:
1. Instructor attaches masking tape strip to classroom wall or table.
2. Students cut out the decimeter square.
3. Students work in pairs or very small groups following the directions given on worksheet.
Evaluation:
This activity demonstrates the need for standard units of measurement. Students also learn that accuracy is increased by use of smaller standardized units which are more suited to the particular measurement task.
Resources:
Let's Measure Metric, Ohio Dept. of Education, 1975
Hands Down Worksheet
Directions:
1. On the wall or table, you will find a long strip of masking tape. Find the length of this strip by measuring it with your hands.
2. Compare your answer with your partners. Are the answers the same? If not, why not?
3. Now use the meter stick to measure the strip. It is called a meter stick because it is 1 meter long.
4. Compare your answer with your partners. Are the answers the same? Why or why not?
5. Now use the decimeter square to measure the strip. Each side is 1 decimeter long.
Strip length = decimeters
6. Compare your answer with your partner's. Are the answers the same? Why or why not?
7. Compare your answers for the strip length in questions 3 and 5.
8. How many decimeters are the same as one meter?
Title:
How much time is it? Developing the concept of time.
Materials:
An alarm clock or kitchen timer.
Audience Targeted:
Type of setting:
Classroom, small group or one-on-one
Activity:
1. Discuss how long various times are: second, minute, hour, day.
2. Tell students you are going to measure various lengths of time. Set the
alarm clock for a period of time. Tell the students how much time it was set for. Go on with your lesson.
3. When the alarm rings, discuss what occurred in the time frame. Make estimates, how long does it take to do an activity.
Evaluation:
This is a good activity to get a feel for different time periods.
Recommended resources:
Measurement Investigations, by Tamara J. Dream and Randall J. Souviney.
Modifications:
1. As a pre-activity have pairs of tasks such as tie your shoe and write your name written on 3x5 cards. Have students guess which takes longer. Then one student does one task and the other does the other task to find out which one takes longer. Do the talks again but measure how much time each takes with a watch.
2. Set the alarm clock for an amount of time but don't tell students how long. When the alarm rings students guess how much time passed.
3. Tell students to close their eyes for a specified amount of time, one minute, five minutes. Have students raise their hand when they think the amount of time is up. Record whose hand goes up when. When all hands are up have the class open their eyes. Show them the guesses. Practice this to get an internal feel for time.
Title:
Snake and Ball: Developing the concept of mass.
Materials:
Audience Targeted:
Type of setting:
Class, small group and one-on-one.
Activity:
1. Divide clay into equal pieces. Use the balance to ensure each is equal to the rest.
2. Give each student two pieces of clay. Let them balance them on the balance to ensure the mass is equal.
3. Have students roll one piece into a ball and the other piece into a snake.
4. Ask if the two pieces weigh the same or if one is now heavier.
5. Use balance to ensure they are still the same.
6. Have students experiment rolling clay into different shapes and weighing on
the balance.
Evaluation:
Good tactile, kinesthetic activity. Helps students understand that things can be different shapes and have the same mass.
Recommended resources:
Measurement Investigations, by Tamara J. Dream and Randall J. Souviney.
Modifications:
Try weighing different density objects that are the same size.
Title:
Straight line, crooked line: Developing the concept or length,
Materials:
Sidewalk chalk, clothes line, yard or meter sticks.
Audience Targeted:
Type of setting:
Classroom, small group, one-on-one
Activity:
1. Draw curved and crooked lines of various lengths on the sidewalk. Different shapes may also be drawn.
2. Have students lay clothes line along a figure, marking the starting and ending points.
3. Measure clothes line with yard stick. Record.
4. Have students guess if other lines are longer or shorter than the first one.
Record guesses.
5. Use the same method to measure the other lines. Determine which are longer/shorter than first line.
6. Rank lines from longest to shortest
Evaluation:
Great kinesthetic activity when students need to move to learn, or have extra energy. Fun treat to go outside on a nice day. Shows how length can be the same even if.. lines are not straight.
Recommended resources:
Measurement Investigations, by Tamara J. Dream and Randall J. Souviney.
Modifications:
Can be done on a smaller scale indoors with paper and rulers.
Title:
Materials:
One inch squares of cardboard or some other heavy material (twelve per student). Twelve inch strips of cardboard and markers
Audience Targeted:
Type of setting:
Classroom, small group or one-on-one.
Activity:
1. Have students place squares side by side on objects to be measured.
Books are a good size.
2. Have students count the number of squares it take to be the same length as
the object.
3. Have students measure the cardboard strip using the squares.
4. Discuss how one can use blocks or use a stick with blocks marked off on it
to measure.
5. Have students mark off blocks on cardboard strip. Have students color each block an the cardboard strip a different color. Or, paste blocks on to strip if they are of different colors.
6. Students need only to count the blocks that line-up with the object they are measuring to give the inches.
Evaluation:
This "easy to read ruler" gets rid of all the extra lines which can be confusing to someone learning to use a ruler. The colored in blocks also help the student understand she is counting units when measuring, not just looking at the lines. This ruler is very useful for people with poor eye sight or visual dyslexia.
Recommended resources:
Modifications:
If students are more advanced, or begin asking how to measure if the object being measured falls in the middle of a block, the concepts of half an inch and quarter of an inch may be introduced.
Title:
Measurement Box: Developing the Concept of Length
Materials: Large cardboard box filled with materials of various lengths: pencil, tie, rope, straw, paper clips, ribbons, belt, extension cord, etc. Standard ruler, yard stick, metric ruler, meter stick (depending upon what units you are working on). 3x5 cards
.
Audience Targeted:
Type of setting: Classroom, small group or one-on-one.
Activity:
1. Let students explore the objects, ask them to describe objects to classmates.
2. Ask students to compare the lengths of objects. (which is shorter?)
3. Ask students to put objects in order from shortest to longest.
4. Ask students to compare using another item as a nonstandard measure.
(How many straws does it take to make an extension cord?)
5. Ask students to measure items using a ruler.
6. These and other questions can be written on cards so students can explore
items at their own pace.
Evaluation:
This is a good hands on activity. It encourages students to understand the concept of length before measuring with a ruler. It allows the tutor to observe the students current skills and knowledge before beginning to teach.
Recommended resources:
Up the Math Ladder, Lynn Molyneux.
Modifications:
This same idea can be used to create a weight box, volume box or other measurement box
TITLE:
Water, Water, Everywhere!
OBJECTIVES:
Students will be able to use various liquid measures into other measures.
MATERIALS LIST:
Empty plastic gallon milk jugs, empty plastic quart milk jugs, empty plastic pint milk jugs, plastic 8oz. Cup measure, funnels, water, paper, pencil, markers.
TARGET AUDIENCE:
ABLE INSTRUCTIONAL GROUP SIZE:
ESTIMATED TIME:
ACTIVITY DESCRIPTION:
Have each container marked A, B, C, D. Introduce container and have students estimate how many of the small containers of water (A), it would take to fill container. Place estimate on graph on board (see attached sheet).Ask how many of the containers marked B it would each take to fill the container marked C, and put the guess on the graph. Repeat this process with each of the containers.
Next have the students divide into groups and have them fill container A (8oz. Cup) with water and pour the water into container B (pint). Record the number of cups it takes to fill the container on the graph on the board. Repeat the process with each container. Conclude activity with discussion of the graph and comparison of results with estimates.
Liquid Measure Graph
| Pint | Quart | Gallon |
| Estimate Actual | Estimate Actual | Estimate Actual |
| Cups | | | |
| Pints | | | |
| Quarts | | | |
| Gallons | | | |
TITLE:
OBJECTIVES:
Students will be able to use a tape measure to accurately
make sure the length and width of a room. Students will be able to use graph paper to show relationships between an actual square foot and a square on the graph paper.
MATERIALS LIST:
Tape measure(s), paper, overhead projector, graph paper, pencils, graph transparency.
TARGET AUDIENCE:
INSTRUCTIONAL GROUP SIZE:
ESTIMATED TIME:
DESCRIPTION:
Teams of three students are given a tape measure and asked
to measure the length and width of the room. One person acts as the recorder. When measuring is completed, graph paper is given to each student The teacher will work on the overhead transparency as students work on theirs to draw a floor plan of the classroom. More ABLE students may use the tape to locate the placement of file cabinets, tables, etc. These would be included in the scale drawing. The graphs can then be used to answer questions about perimeter and square feet.
RESOURCES./SOURCES USED:
Measure Up, by Charles H. Kahn, Sylvia K. Herring,
and Robert Tong. Fearon/Janus/Quercus, Pub. Contemporary's Critical Thinking With Math Contemporary Books.
QUESTIONS FOR "INCHES, FEET, YARDS"
1. Acme Super Store has vinyl floor covering on sale this week. The covering comes in the same width as the room. How many yards of covering will you need to cover the floor of the classroom?
2. Home Corp has carpet advertised for $5.99 a square yard. If 9 square feet make one square yard, how many square yards of carpet will you need to purchase to cover the classroom floor?
3. What will be the cost of the carpet in question #2?
4. After the new floor is laid, we are going to put molding around the room. Using your graph paper drawing, determine how many feet of molding will be needed. Molding sells for 69t a foot. How much will the molding cost?
5. What will be the total cost of the carpet in #3 and the molding in #4?
TITLE:
How Does Your Garden Grow?
OBJECTIVES:
Students will demonstrate the correct use of a tape measure.
Students will use graph paper to draw a diagram of their yard. Students will compute area and perimeter accurately.
MATERIALS LIST:
Graph paper, tag board, tape measure, string, paper, pencils
.
TARGET AUDIENCE:
INSTRUCTIONAL GROUP SIZE:
ESTIMATED TIME:
Two or more class periods, could be used after "Inches, Feet Yards.
ACTIVITY DESCRIPTION:
In class have students make a yard stick out of tag board or string if students do not have access to a tape measure.
measure their yards at home and bring the measurements to class. Use the graph paper to draw a scale model of their yards. Have students use the graph of their yard to answer the questions on their attached worksheet.
RESOURCES/SOURCES USED:
Consumer Math Success Kit, J. Weston Walch, Pub.
ACTIVITY QUESTIONS FOR "HOW DOES YOUR GARDEN GROW?
1. ACME PLANT STORE HAS GRASS SEED ON SALE FOR $2.99 A BOX. ONE BOX WILL COVER 400 SQUARE FEET OF LAWN. USING YOUR GRAPH, HOW MANY SQUARE FEET OF LAWN DO YOU HAVE IN YOUR YARD? HOW MANY BOXES OF GRASS SEED WILL YOU NEED TO PLANT YOUR LAWN?
2. YOU WISH TO PUT A FENCE AROUND YOUR BACK YARD TO KEEP IN YOUR DOG. HOW MANY FEET OF CHAIN LINK FENCE DO YOU NEED TO BUY?
3. CHAIN LINK FENCE SELLS FOR $1.99 A YARD. HOW MANY YARDS OF FENCING WILL YOU NEED? HOW MUCH WILL IT COST TO FENCE YOUR YARD?
4. YOU WANT TO PUT A FLOWER BED IN THE FRONT YARD OF YOUR HOUSE.
THE BED WILL BE 8 FEET BY 10 FEET. YOU WANT TO SURROUND THE BED WITH EDGING. THE EDGING COMES IN ROLLS 10 FEET LONG. WILL 3 ROLLS BE ENOUGH?
5. IF THE EDGING COSTS $.98 A ROLL, HOW MUCH WILL YOU SPEND FOR THE EDGING?
6. YOU NEED TO KNOW HOW MUCH TO BUDGET FOR YOUR YARD. IF YOU SPEND $20. FORFLOWERS FOR THE FLOWER BED, HOW MUCH WILL YOU HAVE SPENT FOR ALL YOUR YARD IMPROVEMENTS? (GRASS SEED, FENCING, EDGING, AND FLOWERS)
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