Resource Overview Quantile® Measure: 1010Q Skill or Concept: Use grids to develop the relationship between the total numbers of square units in a rectangle and the length and width of the rectangle (l x w). (QT‐M‐191) Use models to develop formulas for finding areas of triangles, parallelograms, trapezoids, and circles. (QT‐M‐256) Excerpted from: Gourmet Learning 1937 IH 35 North Suite 105 New Braunfels, TX 78130 www.gourmetlearning.com © Gourmet Learning This resource may be available in other Quantile utilities. For full access to these free utilities, visit www.quantiles.com/tools.aspx.
The Quantile® Framework for Mathematics, developed by educational measurement and research organization MetaMetrics®, comprises more than 500 skills and concepts (called QTaxons) taught from kindergarten through high school. The Quantile Framework depicts the developmental nature of mathematics and the connections between mathematics content across the strands. By matching a student’s Quantile measure with the Quantile measure of a mathematical skill or concept, you can determine if the student is ready to learn that skill, needs to learn supporting concepts first, or has already learned it. For more information and to use free Quantile utilities, visit www.Quantiles.com. 1000 Park Forty Plaza Drive, Suite 120, Durham, North Carolina 27713 METAMETRICS®, the METAMETRICS® logo and tagline, QUANTILE®, QUANTILE FRAMEWORK® and the QUANTILE® logo are trademarks of MetaMetrics, Inc., and are
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3rd Grade
Measurement
Student Expectation: Students will determine the area of two-dimensional surfaces
using concrete models or pictures
Unit 1 – Lesson 3
The student directly compares the attributes of area, and uses comparative language to
solve problems and answer questions. The student selects and uses standard units to
describe area. The student is expected to use concrete and pictorial models of square units
to determine the area of two-dimensional surfaces.
Study the TEKS . . .
Prior Knowledge
In 2nd grade, students were first introduced to the
concept of area using concrete, square tiles only.
Next Steps
3rd
In 4th grade, students will stop using concrete
models and pictures and begin using tools and
formulas to find the areas of figures.
Grade
In third grade . . .
Teachers will expand the students’ knowledge of area by continuing the use of concrete
models and moving beyond to pictorial representation of square units to determine area.
This is NOT the year to jump to formulas to solve area problems. However, since next
year the students will be solving these problems with measuring tools and formulas, this
is the year to finalize the concrete models and pictorial methods. We have provided lots
of materials and options to insure students understand this concept.
Gourmet Curriculum Press, Inc.©
1
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use a parachute to visualize the concept of area
versus perimeter
Focus Activity
Finding Area with Concrete/Pictorial Models
C
Teacher note: This Focus Activity is designed to make the link between perimeter and
area. Just as perimeter was introduced as a “fence” to form a visual image in the students’
minds, area will be shown as the enclosed portion inside the fence. This will be done with
a parachute.
Group size: whole class
Materials: parachute - requested from the gym teacher - If a parachute is not available a
large sheet or tarp can be used, but consider that it needs to be large enough for the entire
class to hold on to it.
Before class: Find a large area for this activity. If it is a nice day, it can be done outside; if
not, a gymnasium or cafeteria will be needed for the entire class to participate.
Directions:
• Have the students hold hands and create a large circle in the middle of the area you
have chosen.
• Spread the parachute open in the middle of the circle.
• Have each student grab the part of the parachute nearest him/her and stretch it out to
the edge of the circle. Students will lay it on the ground and hold hands again.
Questioning Technique
Instructional Strategy
Ask: What shape have we formed as we stand holding hands? (a circle) What is it that
our bodies and arms and hands have made for this circle? (perimeter)
Ask: How do we know this? (Answers will vary, but from our definition of “perimeter,”
it is the “fence” surrounding a space, the distance around a space. It is the length of the
outside rim of the parachute.)
Ask: Is the parachute part of our perimeter? (No, it fills the space inside the perimeter; it
is not the perimeter.)
Ask: Does anyone know what we would call the parachute? (The students can brainstorm
at this point. They should have been exposed to the idea of area before, so lead them to the
idea that the parachute is the area.)
Ask: Will the area always be a circle? (No.)
Say: As a group, work together to create a perimeter that has an area with a shape other
than a circle. (The students can stand in a square, triangle, or any other shape. You can
try to adjust the parachute to fill this space, or simply discuss the shape they have created.
Challenge them to find other areas.)
Ask: Will we measure area in the same way that we measured perimeter? (Answers will
vary, but lead the students to understand that perimeter is a length - one dimension. Therefore,
they will need another way to measure area. That is what they will learn in this lesson!)
2
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will learn vocabulary related to measurement of area
Initial Instruction—Part I—Vocabulary
Finding Area with Concrete/Pictorial Models
K
Definitions:
area: the number of square units needed to cover a surface
square units: A square is the same length on all 4 sides.
Square “units” is a generic term. Usually, area is measured
using specific square units, such as square inches.
Customary Square Units
square inch: a square with 1 inch on all four sides
square foot: a square with 1 foot on all four sides
Metric Square Units
square centimeter: a square with 1 cm on all four sides
square meter: a square with 1 meter on all four sides
Example: Square A has an area of 4 square units; square B
has an area of 9 square units.
Square A
continued on page 4
Square B
Gourmet Curriculum Press, Inc.©
3(T)
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will learn vocabulary related to measurement of area
Initial Instruction—Part I—Vocabulary
Finding Area with Concrete/Pictorial Models
Definitions:
scale drawing: a drawing that represents a real object—The
drawing is the same shape as the actual object, but the size
of the drawing may be larger or smaller than the object.
2 cm
Example 1: school library
5 cm
Each cm = 10 ft
1 in
Example 2: bolt
2 in
Each in = 1 cm
key: used to show a chosen item represents something else—
These are commonly used in statistics on pictographs or bartype graphs.
Example: Each
4(T)
represents 15 minutes.
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will create square units most commonly used in this
lesson
Initial Instruction—Part II
Finding Area with Concrete/Pictorial Models
K
Teacher note: Students frequently have a hard time picturing the different square units
because they have always used linear measurement. In this first part of our Initial
Instruction, groups of students will create some of the most commonly used square units,
so they can manipulate and see the units with which they will be working.
Group size: no more than three students
Materials: instructions, transparency page 6; poster boards or construction paper;
scissors; meter sticks and yardsticks; pencils; pieces of cardstock for the corners; rulers
with customary and metric units
Before class: Make one set of the square units, see page 6. This will help you easily
evaluate the ones the students create.
Directions:
• Explain to the students that they will create some “square units” of measure to use in
this lesson to measure area.
Questioning Technique
Instructional Strategy
Ask: If we are creating a square, what must be true in order for us to be sure it is a square?
(All sides must be the same length. There have to be 90 degree angles at the corners.
Although students might not be familiar with the actual degrees, it is important they
realize that there is a specific corner that must be created.)
Ask: How can we be sure that we have the right angles? (Answers may vary, but suggest
using the corners of the sheet of cardstock to be sure).
Ask: How can we be sure that we have the same side lengths? (Use a ruler or meter stick
to measure. Caution them that we do NOT just estimate the lengths.)
• Distribute poster boards or construction paper to each group.
• Place transparency page 6 on the overhead.
• Direct the students to create the square units listed.
• As the groups think they have created one of the squares, test their creations by placing
the ones you created on top of the ones they have created. You may want to do this
BEFORE they cut them out.
When everyone has completed all the square units, ask the students for things in the room
(or common items) that might be measured with each; e.g., the area of a book’s front cover
might be measured in square inches. Also ask what might be measured in square miles.
(a large ranch, a city, a state, etc.)
Gourmet Curriculum Press, Inc.©
5
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will create square units most commonly used in this
lesson
Initial Instruction—Part II—Instructions
Finding Area with Concrete/Pictorial Models
Create the following square units . . .
1 square centimeter
1 square inch
1 square foot
1 square yard
1 square meter
6(T)
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III
Finding Area with Concrete/Pictorial Models
K
Teacher note: This standard asks students to use both physical and pictorial models to
find area. In this portion of the Initial Instruction, students will use inch and centimeter
graph paper to estimate the area of some figures. Through questioning, students will
extend using true-sized squares to a scale model with a square representing a larger area
(for example 1 square inch could represent 1 square mile). In the next section of the Initial
Instruction, pictorial models will be addressed.
Group size: no more than four students
Materials: Instructional Strategy, page 8; example, transparency page 9; shapes, pages
10-11; inch and centimeter graph paper, pages 12-13; cardstock; scissors, tape or glue sticks;
pencils; colored pencils
Before class: Copy the shapes for each group onto cardstock. To save time, you may opt
to cut these out before class begins. Copy enough graph paper sheets so that each group
has 3 inch sheets and 3 centimeter sheets.
Teacher note: Save the copies of pages 12-13 for use in several other parts of this lesson.
Directions:
• Students will cut out the shapes from page 10. Instruct the students to cut very neatly
in the middle of the line and to watch the corners.
• Each group will start with the rectangle.
• Have the students take a sheet of graph paper that matches the units written on the
figure. For example, for the rectangle, use the centimeter paper. Students will trace the
figure onto the page twice. (Tell them to line up the figure with one of the squares so
that they don’t have too many fractional squares.)
• Students will then cut out one of the traced figures and glue it onto the shape so that
the lines for the graph paper can be seen. (Show the example on page 9. It is not one
of the problems from page 10 or 11.)
• Have the students shade in the other traced figure with a colored pencil. (Show the
example on page 9.)
• On BOTH of the figures, students should use a pencil to count the squares to find the
area. They should write their answers on the figure. Remind them to include “sq cm”
or “sq in.” (Show the example on page 9; the answer for their rectangle should be 40
square centimeters.)
• Direct the students to continue this process with the other 2 figures on page 10. When
all the groups are finished, continue with the questions on page 8 before moving on to
the shapes on page 11.
Gourmet Curriculum Press, Inc.©
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Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III
Finding Area with Concrete/Pictorial Models
Questioning Technique
Instructional Strategy
Ask: What is the area of each of these figures? (The rectangle is 40 square cm. The square
is 16 square inches. The triangle is 4 ! square inches.)
Ask: Which one was the most difficult to figure out? Why? (Most will agree that the
triangle was the most difficult, since there were halves along the diagonal.)
Ask: How did you find the area of a triangle? (Most groups will have taken 3 full inch
squares + 2 half squares (which equals a whole) + 1 extra half square to get 4 ! square
inches. Some, however, could have realized that this triangle is half of a square, found
the area of a 3 x 3 square, and halved it. Allow students to share their methods.)
• Explain to the students that the next page is a little more challenging. Remember the
strategies we just shared with the triangle, and look for patterns you can use.
• Hand out the second page of shapes, and have the students cut them out and find the
area the same way they did on the first page.
• When all the groups have finished, continue with the following questioning strategy.
Ask: What areas did you get for these three figures? (The pentagon is 45 square
centimeters. The trapezoid is 21 square centimeters. The rectangle with the missing
piece is 16 square inches.)
Ask: What are some of the methods you used to find the areas? (Allow students a chance to
share methods. Accept all reasonable responses. For example, in the last one, some students
may have found the area of the entire rectangle (20) and subtracted the square (4).)
Ask: Is it possible to draw a rectangular area that is 21 square miles? (No, to draw it to
actual size, you would need miles and miles of paper.)
Ask: How might someone show a plot of land on a piece of paper that represents 21 square
miles? (It would be drawn smaller, and each square centimeter or square inch would
represent a square mile as indicated in a key.)
Say: This would be called a scale drawing. For example, our trapezoid was 21 square
centimeters. If I told you when we started that each square centimeter represented a square
mile, then this trapezoid (show the trapezoid to the class) would represent 21 square miles.
Ask: If every square inch represents a square kilometer, then how big of an area does this
square (hold up the square) represent? (16 square kilometers)
If time permits, allow the students to create their own shapes using cardstock, straight
edges, and centimeter or inch graph paper. Have the groups determine the area of their own
figures first, and then trade with another group to see if they come up with the same areas.
8
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III—Example
Finding Area with Concrete/Pictorial Models
Example:
If we started with this figure:
cm
Cut out one set of square centimeters, glue them, and count:
1
2
3
4
5
6
7
8
9
10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
Trace one set of square centimeters, shade them, and count:
1
2
3
4
5
6
7
8
9
10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
Answer: area = 36 square centimeters
Gourmet Curriculum Press, Inc.©
9(T)
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III—Shapes
Finding Area with Concrete/Pictorial Models
cm
in
in
10
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III—Shapes
Finding Area with Concrete/Pictorial Models
Use inches to find
the area of the
shaded part of this
picture.
cm
cm
Gourmet Curriculum Press, Inc.©
11
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III—Inch Graph Paper
Finding Area with Concrete/Pictorial Models
12
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use inch and centimeter graph paper squares to
estimate the area of figures
Initial Instruction—Part III—Centimeter Graph Paper
Finding Area with Concrete/Pictorial Models
Gourmet Curriculum Press, Inc.©
13
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use pictorial representations to determine area
Initial Instruction—Part IV
Finding Area with Concrete/Pictorial Models
K
Teacher note: This standard asks students to use physical and pictorial models to find
area. In this portion of the Initial Instruction, students will use pictures with square units
already marked on them to determine area.
Group size: whole-class discussion
Materials: examples, transparency pages 16-17; overhead markers
Before class: Gather materials.
Directions:
• Use the Instructional Strategy below to guide students through this lesson.
Questioning Technique
Instructional Strategy
• Explain to the class that many times on tests and activities, area will already be marked
with squares in a picture form. Sometimes the pictures will be drawn to size. Many
times, however, the pictures will actually be smaller than the real object, and the
squares will represent a square unit much larger than the actual picture. This is called
a scale drawing, which we discussed in the last part of the Initial Instruction. For
example, if a square is said to be 1 mile on each side, then the answer will be in square
miles.
• Place transparency page 16 on the overhead. Cover everything except example 1.
Use with example 1.
Ask: How is this different from the previous problems we completed? (The squares are
already drawn for us.)
Say: Step one is to look at the key. We must always look at the key to find out what each
square represents.
Ask: What does each square represent on this figure? (Each square is 1 square inch.
Although each square is 1 inch in your materials, each square will be larger than 1 inch on
the overhead because the overhead magnifies the objects.)
Say: I need one student to come up and count the squares. (Have the student use the
overhead marker to write the numbers on the squares as he/she did earlier in the Initial
Instruction (show the example on page 9). This is to prevent missing a square or counting
a square more than once.)
• Once students have completed counting the squares:
Ask: What is the area of this figure? (21 square inches—Be sure the students use the
units when stating the area. 21 is not the same as 21 inches, which is not the same as 21
square inches. They must get used to using square units when measuring area and linear
units when measuring length.)
14
• Use the same process to determine the area of example 2.
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use pictorial representations to determine area
Initial Instruction—Part IV
Finding Area with Concrete/Pictorial Models
Questioning Technique
Instructional Strategy
• Place transparency page 17 on the overhead. Cover example 4.
Use with example 3.
Ask: What does the key say that each square represents here? (1 square yard)
Ask: Is this 1 square yard in “real life?” (No, it is much smaller.)
Ask: Why would we use something this small and say that it is 1 square yard? (Answers
will vary but could include something like the fact that we couldn’t draw something that
large on a sheet of paper.)
Say: This is called a scale drawing. It means that the picture is just a scale, or representation
of the real drawing. In this case, the drawing is smaller than the actual figure.
Ask: Have you ever used something that included a scale?
responses, which could include a map, a globe, or blueprints.)
(Accept all reasonable
• Continue going through examples 3 and 4 using the same process to find area that
you did for examples 1 and 2. Only once you have counted the squares, be sure the
students are able to tell the units that are represented with the figure shown.
Answers:
Example 2: 80 square centimeters
Example 3: 56 square yards
Example 4: 8 square kilometers
Gourmet Curriculum Press, Inc.©
15
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use pictorial representations to determine area
Initial Instruction—Part IV—Examples
Finding Area with Concrete/Pictorial Models
Example 1:
= 1 sq in
Example 2:
= 1 sq cm
16 ( T )
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use pictorial representations to determine area
Initial Instruction—Part IV—Examples
Finding Area with Concrete/Pictorial Models
Example 3:
= 1 sq yd
Example 4:
= 1 sq km
Gourmet Curriculum Press, Inc.©
17 ( T )
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use children’s literature to examine the concept
of area
Initial Instruction—Part V
Finding Area with Concrete/Pictorial Models
Optional Reading Activity
K
C
Teacher note: In this activity, students will read the book Bigger, Better, Best! by Stuart J.
Murphy and discuss area.
Group size: whole class, then groups of three to four students
Materials: Instructional Strategy, pages 18-19; recording table, page 20; square foot tiles—
If you do not have square foot tiles in your classroom to count, then you will need to make
square foot pieces of paper to use; masking tape; copy of the book Bigger, Better, Best!
Before class: Cut out square foot pieces of paper. (You might want to make some stencils
and have students cut out squares throughout the week, since there are so many. How
many? Select an area to be measured—a section of the hallway or your classroom or the
whole classroom—and either measure or estimate the area in square feet.) If you are
going to use a section of hallway or classroom, mark it off with masking tape on the floor.
Copy the recording table, page 20, for each small group of three to four students.
Directions:
• Before reading the book, ask the students who has the “best” bedroom? (Most students
will either think THEY have the best bedroom or know someone who they think has
a great bedroom.)
Questioning Technique
Instructional Strategy
Ask: What makes a bedroom “best”? (Answers will vary and could include anything
from video games, to the size, to the color, to the furniture, etc.)
Ask: Would you rather have a large room or a small room? (Answers will vary; most
would like a large room for all of their stuff plus room to move and play, but if they are
thinking how much needs to be cleaned, they may opt for a smaller one.)
• Read the book to the class.
After reading the book:
Ask: What did these kids think that having the “best” room meant? (It had a bigger
window and more floor space.)
Ask: Who had the best room? (They were tied with the same area window and same area
floor.)
Ask: Did they have the same shape of windows and floor? (No, the area was the same,
but the dimensions were different.)
Say: Estimate how big you think our floor is in square feet. (Record all answers on the
front board.)
18
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use children’s literature to examine the concept
of area
Initial Instruction—Part V
Finding Area with Concrete/Pictorial Models
Optional Reading Activity
Questioning Technique
Instructional Strategy
• Have the students begin by carefully lining up the paper squares along one side. Since
we are only estimating, if they do not reach exactly, use the closest number. Tape each
down with a small piece of tape.
• When they finish one line, start the next. DO NOT start from different parts of the
room, as they might not come together as you anticipate. It is better to start on the
same side and work in one direction.
Teacher note: If your classroom has 1-square-foot tiles, these can be used instead.
Ask: What other things can we measure in this room using our square feet? (Answers may
vary, but could include a wall, a window, etc.)
• Divide the students into groups of 3 or 4, and give each a stack of square feet and a
copy of the recording table, page 20.
• Each group will use its square feet to find the area of at least 3 things in the room
(or school) and then sketch (make a scale drawing) and record their results on the
recording table, page 20.
• Determine if groups included a key for their scale drawings.
• Compare and contrast groups’ results. Which is “the best” and why?
Gourmet Curriculum Press, Inc.©
19
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will use children’s literature to examine the concept
of area
Initial Instruction—Part V—Recording Table
Finding Area with Concrete/Pictorial Models
Optional Reading Activity
20
Object 1: Sketch with squares
Object 1: Area
Object 2: Sketch with squares
Object 2: Area
Object 3: Sketch with squares
Object 3: Area
Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will practice newly-established methods to find area
Initial Instruction—Guided Practice
Finding Area with Concrete/Pictorial Models
K
C
Teacher note: Students have just had explicit instruction in using manipulatives and
counting squares to find the area of objects. The following are practice problems and
questions to informally assess the students’ comprehension and abilities. It is the teacher’s
discretion to use or not use this section.
Group size: pairs
Materials: inch and centimeter graph paper, pages 12-13; transparency, page 22; copies of
the figures below; answer key, page 57
Before class: Make copies of pages 12-13 for each pair on cardstock and laminate (or
use saved copies). Students may place shapes over the squares to find area or cut out
the squares to place on top of shapes. It would be a good idea to cut out the squares in
advance if possible. The squares are often altered with elementary cutting skills. Make
copies of the bottom half of this page for each pair.
Directions:
• Instruct the students to find the area of each of the figures below using their cm or inch
squares (as indicated on each figure).
• Place transparency page 22 on the overhead, and have the students determine the area
of each of the figures. Be sure they pay attention to the key used for each.
Example #1
Measure me in
square inches.
Example #2
Measure me in
square centimeters.
Gourmet Curriculum Press, Inc.©
21
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will practice newly-established methods to find area
Initial Instruction—Guided Practice
Finding Area with Concrete/Pictorial Models
Example #3: Find the area of the sidewalk around the
park. Next, find the area of the park.
This is
the park
area.
= 1 sq yd
Example #4: Find the area of the stained glass windows
shown here.
= 1 sq dm
Bonus: What type of drawings are examples #3 and #4?
22 ( T )
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