6th grade unit 1 lesson 07.notebook...6th grade unit 1 lesson 07.notebook 9 7.2 activity 7.2...

6th Grade Unit 1 Lesson 07.notebook

1MATERIALS

Lesson 7FROM PARALLELOGRAMS TO

TRIANGLES

Materials

needed

for this

lesson

• rulers• preprinted slips, cut

from copies of the blackline master

• geometry toolkits

UNIT 1

link to blackline masterPrint pairs of triangles from the blackline master for A Tale of Two Triangles (Part 2). If students are cutting out the triangles, use the first page only. If the triangles are to be precut by the teacher, print the second and third pages. Prepare enough sets so that each group of 3–4 students has a complete set (2 copies each of triangles P–U).

For classes using the digital version of the activity, an applet is provided that can be used in place of, or in addition to, the cut out triangles.

crudiselText Boxteacher page


2LESSON TITLE

UNIT 1 Lesson 7FROM

PARALLELOGRAMS TO TRIANGLES


3OBJECTIVE

UNIT 1 Lesson 7

Objective: • I can explain the special relationship

between a pair of identical triangles and a parallelogram.

Let’s compare parallelograms and triangles.

FROM PARALLELOGRAMS

TO TRIANGLES


4WARMUP

Warmup

link to worksheet Cont...


5WARMUP

Warmup7.1: Same Parallelograms,

different bases

2 min quiet time

2 min partner talk

partner

Cont...

Here are two copies of a parallelogram. Each copy has one side labeled as the base and a segmentdrawn for its corresponding height and labeled .

1. The base of the parallelogram on the left is 2.4 centimeters; its corresponding height is 1 centimeter. Find its area in square centimeters.

2. The height of the parallelogram on the right is 2 centimeters. How long is the base of that parallelogram? Explain your reasoning.

link to worksheet


6WARMUP

Warmup

Can we use any side of a parallelogram as a base?

7.1: Same Parallelograms, different bases

Once we have identified a base, how do we identify a height?

Is the height always the length of one of the

sides of the parallelogram?

How did you find the base of the second parallelogram?

Can a height segment be drawn outside of a parallelogram?


77.2 ACTIVITY

Activity 7.2



87.2 ACTIVITY

7.2 Activity partner

Cont...link to applet

7.2: A Tale of Two Triangles

Two polygons are identical if they match up exactly when placed one on top of the other.

Draw one line to decompose each of the following polygons into two identical triangles, if possible. Use a straightedge to draw your line.Which quadrilaterals can be decomposed into two identical triangles?

34 students

2 min quiet time

5 min partner talk

http://m.openup.org/1/6-1-7-2


97.2 ACTIVITY

7.2 Activity 2 min quiet time2 min partner talk

partner




Study the quadrilaterals that can, in fact, be decomposed into two identical triangles. What do you notice about them? Write a couple of observations about what these quadrilaterals have in common.



10ACTIVITY SYNTHESIS

For a quadrilateral to be decomposable into two identical triangles it must be (or have) . . .

Complete this statement.

Activity Synthesis




Could each quadrilateral be divided into two identical triangles?What do quadrilaterals A, B and D have that C and E do not?



11ARE YOU READY FOR MORE

Activity 7.2

Are You Ready For More



127.3 ACTIVITY

Activity 7.3

link to worksheet Cont...link to blackline master


137.3 ACTIVITY

PQ

R

R

U U

T T

S S

P Q

7.3 Activity partner


7.3: A Tale of Two Triangles PART 2

Which pairs of identical triangles can be composed into a rectangle? What do they have in common?

34 students

2 min quiet time

5 min partner talk

Manipulative

Which pairs of identical triangles can be composed into a parallelogram?



14ACTIVITY SYNTHESIS

P

QR

RUU

TT

S S

P

Q


7.3: A Tale of Two Triangles PART 2

How many different parallelograms can be created with any two copies of a triangle? Why?

ManipulativeHere is one way of composing triangles S into a parallelogram.

Did anyone obtain a parallelogram a different way?

S S

Activity Synthesis



15LESSON SYNTHESIS

Lesson Synthesis

First, we tried to decompose or break apart quadrilaterals into two identical triangles.Then, we explored the relationship between triangles and quadrilaterals the other way around. We tried to compose or create quadrilaterals from pairs of identical triangles.

Which types of quadrilaterals could always be decomposed into two identical triangles?

Can quadrilaterals that are not parallelograms be decomposed into triangles?

What types of quadrilaterals were you able to compose with a pair of identical triangles?

Does it matter what type of triangles was used?

Was there a particular side along which the two triangles must be joined to form a parallelogram?

Cont...1 of 2


16LESSON SYNTHESIS

Lesson Synthesis

Two identical copies of a

triangle can be combined to make a

parallelogram.

Any parallelogram can be split into two identical triangles.

This is True

We will use this idea to find the area of any triangle in upcoming

lessons.2 of 2


17LESSON SUMMARY

Lesson SummaryRead through your Lesson Summary...

link to worksheet


18COOL DOWN

Cool Downcomplete your "cool down"

link to worksheet


19PRACTICE PROBLEMS

PRACTICE PROBLEMS


Page 1


20PRACTICE PROBLEMS

PRACTICE PROBLEMSPage 2



21PRACTICE SOLUTIONS

PRACTICE PROBLEMS SOLUTIONS

Cont...

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Cont...

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g, h


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Attachments

6173blackline_master.pdf

617a student_task_statements.pdf

617b student_cool_down.pdf

617c practice_problems.pdf

Use this blackline master if triangles are to be cut by students.

Blackline Master for Classroom Activity 6.1.7.3: A Tale of Two Triangles (Part 2)

Use this blackline master if triangles are pre-cut by teacher.

Use this blackline master if triangles are pre-cut by teacher.

Blank Page

SMART Notebook

Unit 1, Lesson 7: From Parallelograms to TrianglesLet’s compare parallelograms and triangles.

7.1: Same Parallelograms, Different Bases

Here are two copies of a parallelogram. Each copy has one side labeled as the base and a segmentdrawn for its corresponding height and labeled .

1. The base of the parallelogram on the left is 2.4 centimeters; its corresponding height is 1 centimeter.Find its area in square centimeters.

2. The height of the parallelogram on the right is 2 centimeters. How long is the base of thatparallelogram? Explain your reasoning.

GRADE 6 MATHEMATICS BY

NAME DATE PERIOD

Student Task Statement

m.openup.org/1/6-1-7-27.2: A Tale of Two Triangles (Part 1)

Two polygons are identical if they match up exactly when placed one on top of theother.

1. Draw one line to decompose each of the following polygons into two identical triangles, if possible.Use a straightedge to draw your line.

2. Which quadrilaterals can be decomposed into two identical triangles?

Pause here for a small-group discussion.

3. Study the quadrilaterals that can, in fact, be decomposed into two identical triangles. What do younotice about them? Write a couple of observations about what these quadrilaterals have in common.


NAME DATE PERIOD

m.openup.org/1/6-1-7-3

Are you ready for more?

On the grid, draw some other types of quadrilaterals that are not already shown. Try to decompose theminto two identical triangles. Can you do it?

Come up with a rule about what must be true about a quadrilateral for it to be decomposed into twoidentical triangles.

7.3: A Tale of Two Triangles (Part 2)

Your teacher will give your group several pairs of triangles. Each group member shouldtake 1–2 pairs.

1. a. Which pair(s) of triangles do you have?

b. Can each pair be composed into a rectangle? A parallelogram?

2. Discuss with your group your responses to the first question. Then, complete each of the followingstatements with all, some, or none. Sketch 1–2 examples to illustrate each completed statement.

a. ________________ of these pairs of identicaltriangles can be composed into a rectangle.

b. ________________ of these pairs of identicaltriangles can be composed into a parallelogram.


NAME DATE PERIOD

Lesson 7 Summary

A parallelogram can always be decomposed into two identical triangles by a segment that connectsopposite vertices.

Going the other way around, two identical copies of a triangle can always be arranged to form aparallelogram, regardless of the type of triangle being used.

To produce a parallelogram, we can join a triangle and its copy along any of the three sides, so the samepair of triangles can make different parallelograms.

Here are examples of how two copies of both Triangle A and Triangle F can be composed into threedifferent parallelograms.


NAME DATE PERIOD

This special relationship between triangles and parallelograms can help us reason about the area of anytriangle.


NAME DATE PERIOD

Unit 1: Area and Surface Area Lesson 7: From Parallelograms to Triangles 5

Unit 1, Lesson 7: From Parallelograms to Triangles

7.1: Same Parallelograms, Different Bases


Are you ready for more?


Lesson 7 Summary

Blank Page

SMART Notebook


1. Here are some quadrilaterals.

a. Circle all quadrilaterals that you think can be decomposed into two identical triangles using onlyone line.

b. What characteristics do the quadrilaterals that you circled have in common?

2. Here is a right triangle. Show or briefly describe how two copiesof it can be composed into a parallelogram.


NAME DATE PERIOD


COOL DOWN


Blank Page

SMART Notebook

Unit 1, Lesson 7: From Parallelograms to Triangles1. To decompose a quadrilateral into two identical shapes, Clare drew a dashed line as shown in the

diagram.

b. Did Clare partition the figure into two identical shapes? Explain your reasoning.

2. Triangle R is a right triangle. Can we use two copies of Triangle R to compose a parallelogram that isnot a square?

If so, explain how or sketch a solution. If not, explain why not.

3.Two copies of this triangle are used to compose a parallelogram. Which parallelogram cannot be aresult of the composition? If you get stuck, consider using tracing paper.

a. She said the that two resulting shapes have the same area. Doyou agree? Explain your reasoning.


NAME DATE PERIOD


Practice Problems

4. a. On the grid, draw at least three different quadrilaterals that can each be decomposed into twoidentical triangles with a single cut (show the cut line). One or more of the quadrilaterals should havenon-right angles.

5. a. A parallelogram has a base of 9 units and a corresponding height of units. What is its area?

b. A parallelogram has a base of 9 units and an area of 12 square units. What is the correspondingheight for that base?

c. A parallelogram has an area of 7 square units. If the height that corresponds to a base is unit,

what is the base?

(from Unit 1, Lesson 6)

6. Select all segments that could represent a corresponding height if the side is the base.

(from Unit 1, Lesson 5)

b. Identify the type of eachquadrilateral.


NAME DATE PERIOD



SMART Notebook

Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 21Page 22Page 23Attachments Page 1

6th grade unit 1 lesson 07.notebook...6th grade unit 1 lesson 07.notebook 9 7.2 activity 7.2...

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