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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.

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Resource Overview

Quantile® Measure: 400Q

Skill or Concept:

Measure lengths in inches/centimeters using appropriate tools and units. (QT‐M‐99) Determine perimeter using concrete models, nonstandard units, and standard units. (QT‐M‐146)

Excerpted from:

Gourmet Learning 1937 IH 35 North Suite 105 New Braunfels, TX 78130 www.gourmetlearning.com © Gourmet Learning

3rd Grade Measurement

Gourmet Curriculum Press, Inc.© 1

Unit 1 – Lesson 2

The student directly compares the attributes of length, [area, weight/mass, and capacity] and uses comparative language to solve problems and answer questions. The student selects and uses standard units to describe length [area, capacity/volume, and weight/mass]. The student is expected to use standard units to find the perimeter of a shape.

Study the TEKS . . .

Prior Knowledge

In 2nd grade, students used measurement in relationship to length (as we did in Unit 1 Lesson 1), but they have never made the connection to perimeter (the distance around an object).

Next Steps

In 4th grade, students will continue to find the perimeter of objects as they did in 3rd grade, but in addition to measuring perimeter, they will learn how to “estimate and use.” Perimeter is also linked with other aspects of length, so at this point, it will be assumed that the concept of perimeter is already learned and ready to be used as a tool.

3rd

Grade

Student Expectation: Students will use standard units to find the perimeter of a shape

In third grade . . .

This is the year that students are introduced to the concept of perimeter. Tactile representations will help cement the idea (e.g., a fence). Finding perimeter is a skill that students will use often, so it is important that they understand what perimeter is and how to find it.

Unit 1 – Lesson 2 Measurement

Gourmet Curriculum Press, Inc.©2

Focus Activity

Find Perimeter

Teacher note: By linking the concept of perimeter to a fence around a piece of land, the students will remember the definition more easily. This Focus Activity is designed to introduce the concept of perimeter before teaching the formal definition.

Group size: whole class

Materials: 20 or more rulers—If you don’t have enough rulers, you can cut strips of cardstock or poster board in 12-inch strips.

Before class: Gather rulers, or create the 12-inch strips.

Directions: Follow the Instructional Strategy provided.

Ask: Have you ever ridden through the country and seen animals in fields? Most of the animals we see are kept inside fences. (Permit students to share some stories of where they have seen animals in fences - on the side of the road, at the zoo, in a kennel, at the vet.)

Ask: How do the farmers know how much fencing they need to buy? (Accept reasonable answers. Some may say things like they buy enough to cover all the land, or that the cows have enough grazing room but not too much so that you have to go find them.)

• Have a few students come to the front of the room and stand together.

• Have other students use the rulers or 12-inch strips to create a “fence” around the students.

Ask: How much fencing was needed in this case? (If you have enough of the strips, ask the students to figure out how much fencing they’d need to put around the “entire field,” i.e., the classroom. Have them count the strips and rulers.)

• Explain to the class that they have just found the “perimeter.” This is the focus of this lesson.

Ask: Are there any other situations for which you might have to find the perimeter of an item? (Answers will vary, but some might include the frame on a picture, trim on the cuffs of a shirt, the track around a field—It doesn’t all have to be fences!)

Student Expectation: Students will create fences as an introduction to perimeter

K

Questioning Technique

Instructional Strategy

Unit 1 – Lesson 2

Gourmet Curriculum Press, Inc.© 3 ( T )

Measurement

Student Expectation: Students will learn vocabulary related to measurement of perimeter

KInitial Instruction—Part I—Vocabulary

Find Perimeter

Definitions:

perimeter: the length or distance around the outside of a figure

Example:

The perimeter can be found in any customary units (inches, feet, yard, etc.) or metric units (millimeters, centimeters, meters, etc.).

length: the straight distance measured between two ends

regular polygon: a closed plane figure with straight sides all of equal length

Examples:

Whether around the shape or pulled out to a line, the sum of the lengths is the perimeter.

a

b

c

d

e

f

a b c d e f

continued on page 4


Gourmet Curriculum Press, Inc.©4 ( T )

Student Expectation: Students will learn vocabulary related to measurement of perimeter

Initial Instruction—Part I—Vocabulary

Find Perimeter

2 cm

5 cmEach cm = 10 ft

Example 2: bolt

1 in

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 bar-type graphs.

Example: Each represents 15 minutes.

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.

Example 1: school library

Unit 1 – Lesson 2


Measurement

Initial Instruction—Part II

Find Perimeter

Teacher note: In this section of the Initial Instruction, students will actually have a figure in front of them, and they will measure it to find the perimeter. They will use inches, millimeters, and centimeters for this activity.

Group size: three students

Materials: Instructional Strategy, below; figures, pages 6-8; metric and customary rulers (copies of rulers are available in Unit 1 Lesson 1 in Objective 4)

Before class: Make copies of pages 6-8 for each group. Gather both types of rulers for each group.

Directions:

• Hand out page 6 and rulers to each group.

Ask: What figure is this? (rectangle)

Ask: How do you know? (Opposite sides are the same length, and the corners are all the same (90º). Opposite sides are also parallel.)

Say: Take one finger and trace around the perimeter of the rectangle. (Have each student run his/her finger around the rim of the shape. Verify that they understand the definition of perimeter.)

Ask: What do we have to do to find the perimeter of this figure? (Answers will vary, but students should realize that they will have to find the measurement of each side and add them all together.)

• Instruct the students to take turns measuring the sides. The measuring unit they are supposed to use is written in the center of the figure. Students will record the measurements on the paper. (Two sides are 6 inches, and two sides are 5 inches.)

• Instruct the students to add the measurements together to find the perimeter of the object. (5 + 5 + 6 + 6 = 22 inches)

• Compare the results from each group. (It is plausible that the measurements might be off an infinitesimal amount, but considering the fact that they have only measured to the nearest whole or half number, they should be the same. Discuss the concept of “human error” in measurements, if there are any considerably off.)

Repeat this procedure for each of the figures on pages 7-8. Students practice measuring in inches, centimeters, and millimeters.

hexagon (centimeters): 7 cm + 6 cm + 6 cm + 7 cm + 6 cm + 6 cm = 38 cm

triangle (millimeters): 90 mm + 120 mm + 150 mm = 360 mm

Student Expectation: Students will measure around the sides of figures to find the perimeters



K



Initial Instruction—Part II—Figure 1

Find Perimeter


Measure in:

INCHES

Unit 1 – Lesson 2


Measurement


Find Perimeter


Measure in:

CENTIMETERS




Find Perimeter


Measure in:

MILLIMETERS

Unit 1 – Lesson 2


Measurement

Initial Instruction—Part III

Find Perimeter

Teacher note: Often on tests or activities, geometric figures are drawn and measurements given on the figures for students to use. Almost always, the measurements are not accurate as they are shown. An example would be a picture of a football field with the length shown as 100 yards. After using real measurements, sometimes students have trouble using given numbers. This part of the Initial Instruction will address these types of problems.

Group size: three students

Materials: Instructional Strategy, pages 9-11; examples, transparency page 12; group sheet, page 13; colored pencils; overhead markers

Before class: Make copies of page 13 for each group; sharpen colored pencils.

Directions:

• Distribute a copy of page 13 and colored pencils to each group.

• Go through the first two examples together using the Instructional Strategy below.

• Place transparency page 12 on the overhead. Direct the students to see that the figure in example 1 on the overhead is the first figure on their group sheet.

Ask: If we measured this figure with rulers, would we get the same numbers that are shown here? (No. The lines aren’t as long as stated.)

Ask: Why would we use something this small and say that it is bigger? (Answers will vary but could include something like the fact that we couldn’t draw something that big on a sheet of paper, or tests don’t have enough space to draw the real pictures.)

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? (Accept all reasonable responses, which could include a map, a globe, blueprints, etc.)

Say: Many times on tests, you will see problems like these.

Ask: Has the definition of “perimeter” changed, since we don’t actually have to measure anything? (No. It is still the distance around the object.)

Ask: So what do we need to do? (Add the numbers all around the figure.)

Say: To be sure that we are adding every side once and only once, we are going to use colored pencils as we do our work. Pick out enough colors so that you have a different color for each side of example 1. How many colors will we need? (6)

Student Expectation: Students will use given scaled measurements on sides of figures to find the perimeters

K






Find Perimeter

Example 1:

• Choose a student to pick any side on which to start.

• Highlight the side chosen with one color, and direct the students to do the same at their table. (The color does not matter.)

• Record the measurement associated with this side next to it.

• Continue clockwise until all sides are highlighted with different colors and all corresponding lengths are recorded. (ex: 4, 3, 1, 4, 5, 3)

Ask: What do we do with these numbers to find the perimeter? (add them)

Ask: What is the perimeter of this figure? (20 inches - Be sure that the students use the label inches. “20” is not a complete answer.)

• Direct the students to see that the figure in example two on the overhead is the second figure on their group sheet.

Ask: What is the same between this figure and the last one? (Answers will vary, but the students should realize that it is a geometric figure that has measurements on it that do not match true lengths of the sides.)

Ask: What are some differences between this figure and the last one? (Answers will vary, but the students might mention that the sides of this figure are being measured in yards instead of inches, that there are only 2 measurements given, that there are only 4 sides, and that it is a rectangle instead of a strange figure.)

Ask: Just because our picture changed, did our definition of perimeter change? (No, it is still the total length of all the sides.)

Ask: Do we have enough information in this picture to find the perimeter? (Yes, we know it is a rectangle, and the opposite sides of a rectangle are the same length. That is why they only had to show us the length of 2 of the sides.)




1 in

4 in

4 in

3 in

5 in

3 in

Unit 1 – Lesson 2


Measurement


Find Perimeter

Example 2:

• Direct the students to highlight one of the long sides with a colored pencil and record the length next to it. (While they are doing it on their paper, use an overhead pen to do the same on the transparency.)

• Direct the students to highlight one of the short sides with another colored pencil and record the length with the first side.

• Reinforce that it is a rectangle by using the first color a second time when you get to the other long side. This will visually show the students that it is the opposite sides that are congruent. Record the length with the first two sides.

• Reinforce the concept of a rectangle by using the second color a second time when you get to the other short side. This will visually show the students that it is the opposite sides that are congruent. Record the length with the first three.

Ask: What is the perimeter of this rectangle? (40 yards: 12 + 8 + 12 + 8 = 40. Be sure to use the label “yards.”)

Ask: Can you think of other figures that you do not have to label every side in order to find the perimeter? (Answers may vary, but it includes every regular polygon or other figures, such as a rhombus, parallelogram, etc., with some equal sides.)

Ask: How many sides would have to be labeled with a measurement if the figure was a square? (Only 1 - all four sides are the same length.)

• Have the students find the perimeters of the figures in examples 3 through 6.

• When they’re finished, select students to come to the overhead and explain their strategies.




8 yd

12 yd



Initial Instruction—Part III—Examples

Find Perimeter


Example 1:

8 yd

12 yd

Example 2:

1 in

4 in

5 in 4 in

3 in3 in

Unit 1 – Lesson 2


Measurement

Initial Instruction—Part III—Group Sheet

Find Perimeter


1 in

4 in

3 in

4 in3 in

5 in

2 ft

13 ft

5 m

6 mi

5 mi

5 mi

3 mi4 mi

4 mi

3 mi

9 cm

8 yd

12 yd

Example 1: Example 2:





Initial Instruction—Part IV

Find Perimeter

Optional Reading Activity

Teacher note: In Spaghetti and Meatballs for All, students will analyze how changing the arrangement of tables during a dinner changes the number of people who can sit at the table(s). These are examples of the concept of perimeter.

Group size: four students

Materials: Instructional Strategy, pages 14-15; table square sheet, page 16; people circle sheet, page 17; large sheet butcher paper; markers; masking tape; copy of book Spaghetti and Meatballs for All by Marilyn Burns

Before class: Copy pages 16-17 so that each group has 20 squares and 32 circles. Laminate and cut out the pieces for each group (or use other plastic counters).

Directions:

• Before reading the book, ask the students if they have ever been to a big family gathering such as a reunion or wedding. Discuss what math goes into a big event like this. (Answers will vary, but could include things like how much food to buy, how many tables and chairs they need, how many invitations to buy, etc.)

• Read aloud the book Spaghetti and Meatballs for All, or allow the students to read it.

Ask: What does Mrs. Comfort’s arrangement of her guests have to do with perimeter? (Instead of feet or meters, we are measuring in number of guests that can be seated. If each side of a table is 1 person, then the number of people that can sit around the table is the perimeter of the arrangement . . . This isn’t in units we are used to using.)

• Distribute 20 squares from page 16 and 32 people circles from page 17 to each group.

Ask: What does this represent? (20 tables and 32 people) So, this is a scale representation.

Say: Display Mrs. Comfort’s original seating arrangement. (Watch that the students set out 8 squares with 4 people around each.)

Ask: What is the perimeter of each table if we are measuring in people seated and each person sits at one side of a table? (4 people seated)

Say: Combine the tables so that there are 4 double tables (still using 8 total tables). Place the people around the tables.

Ask: What is the perimeter of each table if we are measuring in people seated and each person sits at one side of one table? (6 people served) How many people can sit down with this arrangement of 4 double tables? (24 people)

Student Expectation: Students will use the concept of perimeter while enjoying children’s literature



CAn

Unit 1 – Lesson 2


Measurement

Initial Instruction—Part IV

Find Perimeter


Say: Combine the tables so that there are 2 groups of 4 tables (still using 8 total tables). Place the people around the tables. (Please note that there are various ways to put 4 tables together. Let the students choose how they want to arrange them. Allow time to show the different configurations and discuss the similarities and differences in the arrangements.)

Ask: What is the perimeter of each table if we are measuring in people served and each person sits at one side of one table? (Answers will vary depending on how they connected the 4 tables. See each of the possibilities below.)

seats 8 seats 10 seats 10 seats 10

Ask: What is the maximum number of guests Mrs. Comfort could seat using 2 groups of 4 tables? (The maximum number is 20. The first arrangement only seats 8. With 2 of those groups, you could seat 16, but with 2 of any of the other configurations, 20 people could be seated.)

Ask: Did this book show the only ways that we could set up the tables to seat 32? (No.)

• Take the students through an example showing an additional arrangement for the tables. (Some rectangular examples would be an 8 x 8 or a 2 x 14, although they don’t have to be rectangular. Challenge the students to seat the guests in any way that 32 will fit.)

• After the students design a layout that will seat 32 guests, transfer their layout onto a large sheet of butcher paper. (If necessary, demonstrate this using your example in the step prior to this one.)

• Under their drawings, have the students list benefits of setting the tables up the way they did. (These answers will vary, but could include close proximity so guests can talk or that they needed less tables; accept all reasonable responses.)

• When all the groups are finished, post the pictures around the room. Allow the students to look at all the different layouts.

Ask: What is the same in all of these pictures? (They all have the same number of people seated; the perimeter is the same.)

Ask: What is different in these pictures? (They have different numbers of tables; they are different arrangements/shapes.)

Optional extension: Give students 8 more circles and the scenario that Mrs. Comfort’s old 3rd grade teacher, her husband and 6 kids decided to join them. Go through the process again from the beginning.






Initial Instruction—Part IV—Table Square Sheet

Find Perimeter



Unit 1 – Lesson 2


Measurement

Initial Instruction—Part IV—People Circle Sheet

Find Perimeter





Initial Instruction—Guided Practice

Find Perimeter

Teacher note: This Guided Practice will provide opportunities for students to practice finding perimeter by measuring items and adding the lengths of the sides.

Group size: pairs

Materials: problems, pages 18-19; rulers (both customary and metric) (optional); pencils or colored pencils; answer key, page 49

Before class: Copy page 18 for each pair, and cut the pages along the dotted line to separate the student portion.

Directions:

• Distribute page 18 to each pair, and have students complete problems 1-4.

• Discuss the answers found with the whole group.

• Place page 19 on the overhead, and have the students complete problems 5-8.

• Discuss the answers with the whole group.

Problem 1: Measure the following perimeter using inches.

Problem 2: Measure the following perimeter using centimeters.

Problem 3: On the back of this page, draw a figure with a perimeter of 12 inches.

Problem 4: On the back of this page, draw a figure with a perimeter of 48 centimeters.

Student Expectation: Students will find perimeters by measuring and adding given lengths

Unit 1 – Lesson 2


Measurement

Initial Instruction—Guided Practice

Find Perimeter

Find the perimeter:

Problem 5: Problem 6:

Sketch a figure for each of the following perimeters:

Problem 7: 20 Km

Problem 8: 75 mi

Student Expectation: Students will find perimeters by measuring and adding given lengths

3 m

11 m

4 yd

4 yd

7 yd 2 yd


Unit 1 – Practice #2 Measurement

Problem #1

A cabinet door in the shape of a rectangle is 60 centimeters long and 30 centimeters wide.

What is the perimeter of the cabinet door?

A 90 cm

B 120 cm

C 180 cm

D 240 cm

Problem #2

Allison had a square swimming pool. One side of the pool was 4 feet. What is the perimeter of Allison’s pool?

F 4 ft G 8 ft

H 12 ft J 16 ft

30 cm

60 cm

4 feet

Problem #3

Sean got a watch for his birthday. The face of the watch is shown below. Use a ruler to measure the perimeter of this watch face in inches.

What is the perimeter of Sean’s watch to the nearest inch?

A 5 in B 6 in

C 8 in D 4 in

Problem #4

Charlie laid down some toothpicks and made the following drawing.

Each toothpick is 3 inches. What is the perimeter of this picture?

F 12 inches G 15 inches

H 18 inches J 24 inches

Directions: Read each problem carefully. Decide which answer best completes the question. Show your work.

Teacher note: Students will need a ruler with customary and metric units to complete this assessment. Reproducible rulers are provided in Objective 4 Unit 1 Lesson 1, pages 14-15.

12

6

9 3



Problem #5

Lupe wanted to enter a kite in the San Marcos Kite Festival. In order to participate, the kite’s perimeter had to be 30 inches.

Which of the following kites could Lupe enter?

A B

C D

Problem #6

Phillipe sketched his trip to the store and back home again.

What is the perimeter of Phillipe’s entire trip?

F 13 blocks G 22 blocks

H 27 blocks J 37 blocks

10 in10 in

20 in20 i

n

8 in

8 in

9 in

9 in

6 in6 in

12 in

12 i

n

5 in 5 in

10 in

10 in

10 blocks

12 blocks

2 blocks

3 blocks

Problem #7

Use a ruler to measure the perimeter of the stop sign to the nearest cm.

A 6 cm

B 16 cm

C 18 cm

D 30 cm

Problem #8

Each year, school photo companies distribute “wallet-sized” pictures to students. Penny’s photo is shown below. Use a ruler to measure its perimeter in inches.

F 3 inches G 5 inches

H 6 inches J 8 inches




Problem #9

Farmer Bullock had 20 feet of fencing to make a pen for his dog in the backyard. Which of the following designs would not work with Farmer Bullock’s fencing?

A

B

C

D

12 ft

10 ft6 ft

3 ft

7 ft

5 ft 5 ft

4 ft4 ft

4 ft4 ft

4 ft

2 ft

2 ft

8 ft8 ft

Problem #10

Use a ruler to draw a figure with an actual perimeter of 21 cm. Label each side. Show the calculations to prove it is 21 cm.

Problem #11

Darlene has a square piece of chocolate that has a perimeter of 8 inches. How long is each side of the square?

A 1 inch

B 2 inches

C 4 inches

D 8 inches



Unit 1 – Application #2 Measurement

Problem #1

Renee had a pipe cleaner that was 12 inches long. Which polygon below has a perimeter that Renee could have made with her pipe cleaner?

Problem #2

Gloria’s sandbox is in the shape of a triangle. Each side has the same length. If one side of the sandbox is 6 feet, what is the perimeter of Gloria’s box?

6 ft

12 ft

18 ft

24 ft

4 in

4 in

4 in 4 in

4 in3 in

3 in

3 in3 in

3 in

6 in4 in

5 in

5 in1 in1 in

Problem #3

Katie walked the path around the playground. The path formed a triangle. What was the perimeter of Katie’s triangle?

12 yds

13 yds

18 yds

25 yds

Problem #4

A diving board is in the shape of a rectangle. Its length is 80 inches and its width is 25 inches.

What is the perimeter of the diving board?

105 in

185 in

210 in

825 in

7 yds

5 yds

6 yds

80 in25 in


Teacher note: Students will need a ruler with customary and metric units to complete this assessment. Reproducible rulers are provided in Objective 4 Unit 1 Lesson 1, pages 14-15.



Problem #5

Use a ruler to measure the perimeter of the star below in inches.

8 in 10 in

12 in 14 in

Problem #6

A stop sign is an octagon. It has 8 equal sides. Robby has a stop sign hanging in his room that has a perimeter of 56 centimeters. How long is each side of his stop sign?

5 cm

6 cm

7 cm

8 cm

Problem #7

Use a ruler to measure the perimeter of this polygon. Measure to the nearest centimeter.

14 cm 18 cm

22 cm 26 cm

Problem #8

At Leslie’s new school, students change classrooms and teachers for each major class (Language Arts, Science, Math and Social Studies). One day for fun, she counted how many steps it took her to get from one class to another. The diagram of her route is below.

What is the perimeter between all of her classes?

How did you determine your answer? Show your work.

What unit did you use? Why?


12 steps

8 steps

6 steps

19 steps

Ms. SmithSocial Studies

Mr. BrownScience

Mrs. BakerLanguage Arts

Mr. JamesonMath



Problem #9

The zoo just got a new baby kangaroo. The keepers bought 40 ft of fencing to make a home for the new animal. Which of the following designs would not use all of the zoo’s fencing?

8 ft8 ft

8 ft8 ft

8 ft

5 ft5 ft

15 ft

15 ft

5 ft5 ft

5 ft5 ft

5 ft5 ft

10 ft

10 ft

10 ft 10 ft

Problem #11

Yani’s name begins with a Y. He made a Y like the following using toothpicks.

What is the perimeter of this letter?

33 cm 60 cm

66 cm 55 cm

Bonus

Pappi had a small picture album that he carried with pictures of his grandson. His album is shown below. Use a ruler to measure the perimeter of the top cover in centimeters.

What is the perimeter of Pappi’s book to the nearest centimeter?

12 cm 16 cm

20 cm 24 cm

Each toothpick is 6 cm long.


Problem #10

A hexagon has 6 sides. Use a ruler to draw a hexagon with a perimeter of 36 cm on the back of this page.

Under the picture, show your calculations that prove the perimeter is 36 cm.

Pappi’s

Picture

Album

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