Appointment Sheet

Transparency / Handout 6A-1

Appointment Sheet

1st Appointment 2nd Appointment 3rd Appointment 4th Appointment 5th Appointment 6th Appointment

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section A Clock Arithmetic Page 9


Clock Arithmetic

mod12

mod10

mod8

mod5


Handout 6A-4

Number Line

To make the centimeter Number Line: 1. Cut on the dotted lines 2. Overlap the ends, putting the 20 from the first strip on top of the 20 on the second strip 3. Continue overlapping in the same way, putting the 40 on the 40, the 60 on the 60, and the 80 on

the 80 4. Make sure the centimeters are accurate at the overlapped edges 5. Tape the strips together


Handout 6A-4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140



Clock Arithmetic Recording Sheet 1

Which clock are you using to solve these problems? Make sure you label your clock answers with the clock’s name. For example, on the mod12 clock, 8 + 5 = 1 mod12. The same problem on the mod10 clock would be 8 + 5 = 3 mod10.

Problem Number Line Solution Clock Solution 4 + 3 2 + 2

8 + 10 11 + 2 15 + 4 3 + 0 5 + 3 2 + 8

3 + 10 3 + 14 12 + 2 2 + 10 3 + 8 5 + 2 5 + 4

What patterns do you see in your Clock Solutions? If you know the Number Line Solution, how could you figure out the Clock Solution without the clock?



Clock Arithmetic Recording Sheet 2

Problem Number line

Solution

mod12 Clock

Solution

mod10 Clock

Solution

mod8 Clock

Solution

mod5 Clock

Solution 4 + 3 2 + 2

8 + 10 11 + 2 15 + 4 3 + 0 5 + 3 2 + 8

3 + 10 3 + 14 12 + 2 2 + 10 3 + 8 5 + 2 5 + 4

Look at the solutions for each of the clocks and the number line. What patterns do you see? What generalizations can you write for using the different clocks to solve addition problems? What patterns do you think you would find if you used the clocks for the other operations?


Handout 6B-1

The Lesson Planning Process

The Lesson Planning Process is used to set learning expectations and instructional goals to create a rigorous mathematics classroom that results in the development of mathematically powerful students. 1. Big Ideas 2. Evidence of Understanding 3. Orchestrating for Rigorous Learning

Framing the Experience Developing Deep and Complex Knowledge

4. Communication to Support Learning

Incorporating Inquiry Supporting Reflection

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 23

Handout 6B-2

--1-- Big Ideas

What is a Big Idea? The purpose of a Big Idea is to provide a focus for instructional planning. A Big Idea should represent a connected set of related knowledge and skills that contributes to our ability to use mathematics in our daily lives. Although Big Ideas can be worded in a variety of ways, they should all have the following characteristics:

A Big Idea should be mathematically important. A Big Idea should be valuable in terms of the instructional goals. A Big Idea should encompass a variety of levels of understanding and application.

The teacher must use the TEKS to inform the development of Big Ideas and can use the Big Ideas to plan mathematically worthwhile tasks that are organized to meet students’ needs. Questions to consider in developing Big Ideas:

How can I create a Big Idea to make connected expectations from the TEKS the basis of my instruction?

How is this Big Idea mathematically important? (Does it have a mathematical “feel”?)

How is this Big Idea valuable in terms of the instructional goals? (Does it have an instructional “feel”?)

How does this Big Idea encompass a variety of levels of understanding and application? (Does it have a developmental “feel”?)

References: Wiggins, G. & McTighe, J. (1998). Understanding by Design. Alexandria, VA: ASCD.


Handout 6B-3

--2-- Evidence of Understanding

What is Evidence of Understanding? Evidence of Understanding is criteria for quality work based on how students show their ability to apply mathematics knowledge and skills. Students use Evidence of Understanding to examine their own work and the work of others for the purpose of improving the quality of their work. Evidence of Understanding statements are most useful when they come from the students and are worded in terms that students can comprehend. In order to help students meet learning expectations, teachers must ensure that students are familiar with and attend to Evidence of Understanding as they work. Questions to consider in identifying Evidence of Understanding:

How can I use the TEKS to help students determine criteria for quality work? How will I help students become familiar with and use Evidence of Understanding in

their work? How will Evidence of Understanding help me assess students’ progress in meeting

learning expectations? How will Evidence of Understanding be made public to students, families,

administrators, and others? References: Wiggins, G. & McTighe, J. (1998). Understanding by Design. Alexandria, VA: ASCD.


Handout 6B-4

Sample Big Ideas and Evidences of Understanding for Number and Operations (from Week 1)

Using number and operations to solve problems I can recognize uses of numbers and operations in the world around me. I can use what I know about numbers and operations to decide if my answer makes

sense. I can talk about and write about the numbers and operations I used to solve a

problem. Using representations and models to show whole numbers

I can use objects to show a whole number. I can draw pictures to show a whole number. I can read and write the number.

Using whole numbers to describe quantities

I can count to tell how many are in a set. I can use a whole number to describe how many. I can talk about what a number means. I can write about what a number means. I can pick a reasonable number to describe something.

Using place value to describe a quantity

I can use 10s and 1s to describe a whole number. I can talk about the patterns I see in numbers.

Describing relationships between whole numbers

I can compare whole numbers using sets of objects. I can draw pictures to compare whole numbers. I can use symbols (and place value) to compare whole numbers. I can talk about the relationships between whole numbers. I can write about the relationships between whole numbers.

Using representations and models to show rational numbers

I can tell what the whole is. I can use different wholes to show a fraction or decimal. I can use a set of objects to show a fraction or decimal. I can draw pictures to show a fraction or decimal. I can read the number. I can write the number.


Handout 6B-4

Using rational numbers to describe a quantity in relation to a whole I can use a fraction or decimal to describe how much of something. I can talk about what a fraction or decimal means. I can write about what a fraction or decimal means. I can explain (talk about, write about) what the numerator means. I can explain (talk about, write about) what the denominator means. I can pick a reasonable fraction or decimal to describe how much of something.

Identifying equivalent symbolic representations for rational numbers

I can write different fraction names for the same rational number. I can write different decimal names for the same rational number. I can write the same rational number either as a fraction or a decimal.

Extending place value to represent numbers less than one

I can use tenths and hundredths to describe a decimal. I can talk about patterns I see in decimals.

Describing relationships between rational numbers

I can compare rational numbers using objects. I can draw pictures to compare fractions and decimals. I can use symbols (and place value) to compare fractions (and decimals). I can talk about the relationships between fractions and decimals. I can write about the relationships between fractions and decimals.

Applying meanings and properties of operations

I can use objects to show the addition/subtraction/multiplication/division. I can draw pictures to show addition/subtraction/multiplication/division. I can talk about why I chose addition/subtraction/multiplication/division. I can add/multiply in any order (commutativity). I can add/multiply any two numbers first (associativity). I can read an addition/subtraction/multiplication/division number sentence.

Using procedures for finding sums/ differences

I can count on to find a sum. I can count back to find a difference. I can use landmark numbers (like 5 and 10) to find sums/differences. I can use patterns to find sums/differences. I can use basic facts to find sums/differences. I can write a number sentence to describe the sum/difference. I can pick a reasonable number for the sum/difference. I can estimate the sum/difference. I can use tens and ones to find a sum/difference. I can write about the different ways I found the sum/difference.


Handout 6B-4

Connecting addition and subtraction I can check my subtraction with addition. I can use addition/subtraction fact families to solve problems.

Using procedures for finding products/ quotients

I can skip count to find a product or quotient. I can use equal groups to find a product or quotient. I can use factors and multiples to find products/quotients. I can use patterns to find products/quotients. I can use basic facts to find products/quotients. I can write a number sentence to describe the product/quotient. I can pick a reasonable number for the product/quotient. I can estimate the product/quotient. I can use tens and ones to find a product/quotient. I can write about the different ways I found the product/quotient.

Connecting multiplication and division

I can check my division with multiplication. I can use multiplication/division fact families to solve problems.


Handout 6B-5

--3-- Orchestrating for Rigorous Learning

Framing the Experience How do I help students understand the reasons why this

mathematics is important (e.g. use of real-world contexts, connections to other mathematical ideas, etc.)?

How can I use my assessment of students’ prior knowledge to design the learning experience so that students can apply that knowledge to rigorous new mathematics learning?

How can I provide multiple entry points for student engagement?

How will I enable students to access multiple strategies and tools without being too prescriptive?

Developing Deep and Complex Knowledge How can I organize the learning experience around

interrelated concepts? How can I design the learning experience so that students find

the work to be personally challenging? What opportunities have I built in for students to develop

flexible mathematical thinking (e.g. examining and using multiple representations of a mathematical idea)?

How can I guide students to become familiar with and demonstrate Evidence of Understanding?

--4-- Communication to Support Learning

Incorporating Inquiry How can I design the learning experience so that students

have the opportunity to explain and justify their mathematical thinking?

What strategies can I use to facilitate collaboration and effective learning discourse among students?

How can I use questioning strategies to ensure that students think and communicate about their thinking?

How can I use questions to encourage students to talk about their work in terms of the Evidence of Understanding?

Encouraging Reflection How can I implement the learning experience in a way that

values and supports the development of students’ metacognitive strategies?

How can I provide students the opportunity to think about how they felt when engaged in rigorous mathematics learning?

How can I help students use the Evidence of Understanding to reflect on their work?

How can I use student reflection to assess what students have learned and plan further instruction?


Handout 6B-6

The Aquarium Problem Set

The First Aquarium The Problem: Eric works for a large pet store. Part of his job responsibilities is filling new aquariums. The store is adding several aquariums that Eric needs to fill. He decides to fill the first one and keep a record of how long it takes. Each aquarium contains 40 gallons of water. The dimensions of the aquarium are 2

136 inches long, 2

112 inches wide and 2123 inches tall. Eric decides to put a tape measure along the

height of the aquarium and time how long it takes to fill it up, inch by inch, It took 55 seconds to reach the first inch, 1 minute 50 seconds to reach the second inch, 2 minutes 45 seconds to reach the third inch, 3 minutes 40 seconds to reach the fourth inch, 4 minutes 35 seconds to reach the fifth inch, and 5 minutes 30 seconds to reach the sixth inch. This rate continued until the tank reached the 22-inch mark. The Eric turned the water off, knowing he would add a little more water when he added the fish. Your Task: Make a T-chart that shows how long it took the water to reach each inch from 0 to 22.

The Second Aquarium The Problem: Eric filled the second aquarium the same way. This aquarium had the same dimensions and Eric used the same water source. This time, Eric had to stop the water when it reached the 10-inch mark so that he could help a customer. It took Eric 10 minutes to help the customer. Your Task: Make a written record that shows how long it took to fill the second aquarium. Try to think of a unique display for the information.


Handout 6B-6

The Third Aquarium The Problem: The third aquarium was a problem for Eric. It was the same size as the other two and he started filling it with the same hose. Unfortunately, when the water level reached 11 inches, the aquarium started to leak. Eric turned off the water and started to siphon the water back out of the aquarium. It took Eric 3 minutes to get the siphon started. The siphon hose was much smaller than the water hose, and it took 2 minutes for the water to drop to 10 inches. The siphon continued at this rate until the aquarium was completely empty. Your Task: With your group, decide on the display you will use to show the solution to the problem. Make sure you choose the display you think best shows how this problem is different from the first two problems. Be prepared to share your thinking, your solution, and your choice of display with another group. Debriefing Questions:

Does the answer make sense? How do you know? How does the display clearly show the answer? How does the display show the differences between this problem and the first two

problems? Why did your group choose the display you used?

The Next Step The Problem: Ms. McNemar, Eric’s boss, thought it was great that Eric was keeping written records of how he filled the aquariums. She decided she wanted some kind of graph that showed the same information for the three aquariums Eric had filled (or tried to fill). She wants to be able to compare how the aquariums were filled. Your Task: Create one graph that shows all three aquariums. Make sure Ms. McNemar can compare the way the three aquariums were filled.


Handout 6C-1

Sample Big Ideas and Evidences of Understanding for Patterns and Algebraic Thinking

Using patterns and relationships to solve problems I can recognize patterns and relationships in the world around me. I can use what I know about patterns and relationships to decide if my answer makes

sense. I can talk about and write about the patterns and relationships I used to solve a

problem. Identifying and creating patterns and relationships

I can make a sound or action pattern. I can continue a sound or action pattern. I can use objects to make a pattern. I can draw pictures to make a pattern. I can draw which object comes next in a pattern.

Describing and labeling patterns and relationships

I can describe a pattern I see or hear. I can tell how two patterns are the same or different. I can make the same pattern in more than one way. I can make a record of my pattern. I can use symbols (such as letters, numbers, shapes, or pictures) to describe my

pattern. I can tell about the patterns I see on a calendar. I can tell about the patterns I see on a number line. I can tell about the patterns I see on a hundred chart. I can use symbols to describe relationships (e.g. =, >, <).

Using patterns to understand numbers

I can make a number pattern. I can make a number pattern longer. I can tell how a number pattern works. I can tell the missing number(s) in a pattern. I can use patterns to count by ones. I can use patterns to skip count (such as by twos, fives, or tens). I can tell how I know a number is odd or even. I can use patterns to read and write numbers. I can find patterns in the hundred chart. I can use place value (such as hundreds, tens, and ones) to put numbers in order. I can use patterns in place value (such as hundreds, tens, ones, tenths, hundredths) to

tell which number is least or greatest. I can use place value to describe a number in more than one way (e.g. 1 ten and 4

ones or 14 ones.)

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Patterns, Relationships, and Algebraic Thinking Sampler Page 36

Handout 6C-1

I can use patterns to find equivalent fractions. I can use models to tell how I know a number is prime or composite. I can use patterns to tell how I know a number is prime or composite.

Using patterns to understand operations

I can use patterns to remember facts when I add. I can use patterns to remember facts when I multiply. I can use fact families to subtract. I can use fact families to divide. I can use patterns to multiply by 10 or 100. I can use multiplication to count possible combinations. I can use patterns to find how many combinations I can make.

Using patterns and relationships to make predictions

I can tell which object comes next in a pattern. I can draw which picture comes next in a pattern. I can tell which number comes next in a pattern. I can tell what is missing in a pattern. I can predict what is coming later in a pattern. I can use cause-and-effect patterns to make predictions.

Using representations to make generalizations about patterns and relationships

I can organize information in different ways to look for a pattern. I can make an organized list of related numbers to describe a situation. I can use a list of related pairs of numbers to look for a relationship. I can explain why I think a pattern or relationship happens. I can select a diagram to represent a situation. I can use objects or pictures to help me explain why I can use multiplication to count

possible combinations. I can pick a number sentence to represent a situation. I can use patterns to find how many combinations I can make.


Transparency / Handout 6C-2

Lesson Planning Process Chart for the Patterns, Relationships, and Algebraic Thinking Sampler

Step 1 Big Ideas

Step 2 Evidence of Understanding

Step 3 Orchestrating for Rigorous Learning

Step 4 Communication to Support Learning



Lesson Planning Process Chart for the Patterns, Relationships, and Algebraic Thinking Sampler

Step 1 Big Ideas





10 x 22 centimeter grid Transparency / Handout 6C-4

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Name Scarves Page 45

Centimeter Graph Paper Transparency / Handout 6C-5

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Name Scarves Page 46


Two of Everything Problems to Solve

Directions: Each problem should be solved in two different ways. Students may use numbers, words, charts or pictures to explain their thinking. 1. If Mr. Haktak took $5.00 out of a pot that doubles its contents,

what did Mr. Haktak drop into the pot? 2. How many gold coins would you have to put into the pot that

doubles its contents in order to take 100 gold coins out of the pot?

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Two of Everything Page 49


3. The following year, Mr. Haktak found another brass pot. When he dropped one coin into the pot, he got 5 coins out. When he dropped 2 coins into the pot, he got 10 coins out. If this pattern continues, how many coins would Mr. Haktak have if he put 3 coins into the pot? What would happen if he put 5 coins into the pot?

4. Write your own problem about a magic pot.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Two of Everything Page 50


Hundred Chart

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 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Odd and Even Page 54

1 Inch Graph Paper Handout 6C-8



Odd and Even Recording Sheet

What is the total number of tiles you

picked up?

Do you and your partner have the same amount?

Do the tiles make a rectangle that is two

tiles wide?

Is the total number of tiles odd or even? How do you know?



Number Machines Recording Sheet

This is a +5 Number Machine. Record the missing numbers on the T-chart.

Add Five Machine +5

In Out 3 _____

1 _____

9 _____

_____ 15

_____ 7

_____ 9 Look at this T-chart. What would you name this machine?

__________ Machine +5

In Out 4 11

7 14

15 22

19 26 What is the mathematical rule that explains what this machine does to the “In” numbers?

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Number Machines Page 62


What would you name this machine? Complete this T-chart.

__________ Machine +5

In Out 19 15

9 5

6 2

8 _____

12 _____

_____ 9 Create your own Number Machine. Design a T-chart with “In” numbers and “Out” numbers that fit your secret rule. On the back of this piece of paper copy your T-chart leaving some numbers blank. Let your partner try to fill in the blanks and name your secret rule.

__________ Machine +5

In Out _____ _____

_____ _____

_____ _____

_____ _____

_____ _____

_____ _____

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Number Machines Page 63


Multiplication Chart

x 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Fraction Frame Game Page 67


Just Being Crafty Kits

The Just Being Crafty Company makes Crafty Kits. The Crafty Kit of Pattern Blocks includes:

hexagons trapezoids blue rhombi squares triangles white rhombi

You are the order manager. As orders come in, you tell the production manager how many of each shape you need. Here are today’s orders:

Order 1097 needs 5 Crafty Kits of Pattern Blocks. Order 1098 needs 2 Crafty Kits of Pattern Blocks. Order 1099 needs 12 Crafty Kits of Pattern Blocks.

The chart below can help you organize the information. Don’t forget to total the amounts before sending instructions to the production manager.

Hexagons Trapezoids Blue rhombi Squares Triangles White

rhombi One Kit 3 4 5 6 7 8

Order 1097

Order 1098

Order 1099

TOTAL

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Just Being Crafty Page 70


Just Being Crafty Kits Production Manager Sheet

The Production Manager has been given the totals for today’s orders. When he cuts the shapes, they are cut from sheets. Each shape is printed on a separate sheet. The list below tells how many of each shape is on a sheet.

6 hexagons 9 trapezoids 12 blue rhombi 12 squares 12 triangles 12 white rhombi

Create a chart that shows the total number of each shape needed for today’s orders, the number of shapes per sheet, the number of sheets needed for today’s orders, and the number of each shape left over.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C Just Being Crafty Page 71


99,999 Recording Sheet

A) B)

C) D)

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 6: Section C 99,999 Puzzle Page 73


The Stars Problem

Vincent is going to put glow-in-the-dark stars on the wall of his bedroom. His mother said he could as long as he could tell her exactly where each star would go. He drew this diagram.

.
8 ft
12 ft. Scale:

21 in. = 1 ft.

After Vincent drew the diagram, he remembered using ordered pairs in mathematics class. He thought ordered pairs might help his mother know where the stars would go. Draw a diagram on graph paper that shows how ordered pairs can be used to place the stars.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section A The Stars Problem Page 7


Part 2 After Vincent finished decorating one wall, he asked his mother if he could decorate another wall. She agreed, as long as Vincent could make the design symmetrical. Vincent chose a wall that was perpendicular to the first wall. The second wall joined the first wall on the left side. Vincent thought that ordered pairs might help again, but he wasn’t sure. Use grid paper and help Vincent draw the plan for the second wall.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section A The Stars Problem Page 8


Clock Arithmetic

mod12

mod10

mod8

mod5

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section B The Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 16

Transparency / Handout 7B-1

Lesson Planning Process Chart

Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 17


Sample Big Ideas and Evidences of Understanding for Patterns and Algebraic Thinking

Using patterns and relationships to solve problems I can recognize patterns and relationships in the world around me. I can use what I know about patterns and relationships to decide if my answer makes

sense. I can talk about and write about the patterns and relationships I used to solve a

problem. Identifying and creating patterns and relationships

I can make a sound or action pattern. I can continue a sound or action pattern. I can use objects to make a pattern. I can draw pictures to make a pattern. I can draw which object comes next in a pattern.

Describing and labeling patterns and relationships

I can describe a pattern I see or hear. I can tell how two patterns are the same or different. I can make the same pattern in more than one way. I can make a record of my pattern. I can use symbols (such as letters, numbers, shapes, or pictures) to describe my

pattern. I can tell about the patterns I see on a calendar. I can tell about the patterns I see on a number line. I can tell about the patterns I see on a hundred chart. I can use symbols to describe relationships (e.g. =, >, <).

Using patterns to understand numbers

I can make a number pattern. I can make a number pattern longer. I can tell how a number pattern works. I can tell the missing number(s) in a pattern. I can use patterns to count by ones. I can use patterns to skip count (such as by twos, fives, or tens). I can tell how I know a number is odd or even. I can use patterns to read and write numbers. I can find patterns in the hundred chart. I can use place value (such as hundreds, tens, and ones) to put numbers in order. I can use patterns in place value (such as hundreds, tens, ones, tenths, hundredths) to

tell which number is least or greatest. I can use place value to describe a number in more than one way (e.g. 1 ten and 4

ones or 14 ones.)



I can use patterns to find equivalent fractions. I can use models to tell how I know a number is prime or composite. I can use patterns to tell how I know a number is prime or composite.

Using patterns to understand operations

I can use patterns to remember facts when I add. I can use patterns to remember facts when I multiply. I can use fact families to subtract. I can use fact families to divide. I can use patterns to multiply by 10 or 100. I can use multiplication to count possible combinations. I can use patterns to find how many combinations I can make.

Using patterns and relationships to make predictions

I can tell which object comes next in a pattern. I can draw which picture comes next in a pattern. I can tell which number comes next in a pattern. I can tell what is missing in a pattern. I can predict what is coming later in a pattern. I can use cause-and-effect patterns to make predictions.

Using representations to make generalizations about patterns and relationships

I can organize information in different ways to look for a pattern. I can make an organized list of related numbers to describe a situation. I can use a list of related pairs of numbers to look for a relationship. I can explain why I think a pattern or relationship happens. I can select a diagram to represent a situation. I can use objects or pictures to help me explain why I can use multiplication to count

possible combinations. I can pick a number sentence to represent a situation. I can use patterns to find how many combinations I can make.



Sample Big Ideas and Evidences of Understanding for Geometry

Using geometry to solve problems I can recognize geometry in the world around me. I can use what I know about geometry to decide if my answer makes sense. I can use geometry words to talk and write about solving problems.

Identifying and comparing shapes and solids

I can sort a set of objects into groups. I can tell how I sorted a set of objects into groups. I can describe an object by telling about its shape. I can tell how shapes are the same or different.

Using formal geometric vocabulary to identify and define

I can name two-dimensional shapes. I can name three-dimensional objects. I can name the shapes I see on three-dimensional objects. I can talk about shapes I see in real objects. I can use the right math words to tell about an object. I can identify different kinds of angles. (right, acute, obtuse) I can identify parallel lines and tell how I know they are parallel. I can identify perpendicular lines and tell how I know they are perpendicular. I can tell about the vertices, edges, and faces on an object. I can define a shape or solid by telling about its attributes.

Combining and dividing shapes and objects

I can put shapes together to make new shapes. I can cut a shape apart and tell what new shapes I make.

Modeling and describing transformations

I can use objects to show translations. I can use objects to show reflections I can use objects to show rotations. I can draw what happens when I translate a shape. I can draw what happens when I reflect a shape. I can draw what happens when I rotate a shape.



Recognizing congruence and symmetry I can identify congruent shapes. I can make shapes that are symmetrical. I can find the line of symmetry on a shape. I can verify that a shape is symmetrical by reflecting it. I can show how to translate, reflect, or rotate a shape onto another to show they are

congruent. Using numbers to describe location

I can use numbers to name points on a line. I can use fractions to name points on a line. I can use decimals to name points on a line. I can locate points on a grid from ordered pairs.



Modular Art Mod5 Number Patterns

Mod5 Addition Chart

+ 0 1 2 3 4

0

1

2

3

4

Mod5 Multiplication Chart

x 0 1 2 3 4

0

1

2

3

4



Modular Art Quilt Tile Square Patterns

Mod5 Quilt Pattern



Modular Art Mod5 Quilt Patterns Recording Sheet


Transparency / Handout 7B-7a

Modular Art Translations Recording Sheet

7

6

5

4

3

2

1

0

-1 -7 -6 -5 -4 -3 -2

-1

1 2 3 4 5 6 7

-2

-3

-4

-5

-6

-7


Transparency / Handout 7B-7b

Modular Art Recording Sheet

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14



Modular Art Rotations Recording Sheet

7

6

5

4

3

2

1

0

-1 -7 -6 -5 -4 -3 -2

-1

1 2 3 4 5 6 7

-2

-3

-4

-5

-6

-7



Modular Art Reflections Recording Sheet

7

6

5

4

3

2

1

0

-1 -7 -6 -5 -4 -3 -2

-1

1 2 3 4 5 6 7

-2

-3

-4

-5

-6

-7



Modular Art Where Are We?

Look at your original Mod5 quilt pattern as it was copied onto the three coordinate grids. Each of the nine quilt tile squares in your Mod5 quilt pattern has a center point that can be identified by an ordered pair. Find the center point for each of the five squares shaded in the diagram to the right. List the points next to the correct number in each quadrant of the coordinate grids. (Quadrant I is the upper right hand quadrant. Quadrant II is the upper left-hand quadrant. Quadrant III is the lower left-hand quadrant. Quadrant IV is the lower right hand quadrant.)

B E

C

A D

Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4

A B C D

Translations

E A B C D

Rotations

E A B C D

Reflections

E


Handout 7B-11

--3-- Orchestrating for Rigorous Learning

Framing the Experience What does it mean to frame a learning experience? The frame is used at the beginning of the learning experience to set the stage. It should be carefully designed to invite students into the experience. The teacher must use knowledge about students’ interests and abilities to create a frame that results in meaningful student engagement. Questions to consider in planning the frame for the learning experience:

How do I help students understand the reasons why this mathematics is important (e.g. use of real-world contexts, connections to other mathematical ideas, etc.)?

How can I use my assessment of students’ prior knowledge to design the learning

experience so that students can apply that knowledge to rigorous new mathematics learning?

How can I provide multiple entry points for student engagement?

How will I enable students to access multiple strategies and tools without being too

prescriptive? References Allen, R. (2001). Train Smart: Perfect Trainings Every Time. San Diego, CA: Brain Store.



Lesson Planning Process Chart for the Connecting Algebraic Thinking and Spatial Reasoning Sampler

Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Connecting Algebraic Thinking and Spatial Reasoning Sampler Page 35


Lesson Planning Process Chart for the Connecting Algebraic Thinking and Spatial Reasoning Sampler

Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Connecting Algebraic Thinking and Spatial Reasoning Sampler Page 36


A Mysterious Tadpole

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C The Mysterious Tadpole Page 43


The Mysterious Tadpole Recording Sheet

1. Draw a picture or write a description of how you determined how much longer the large tadpole is than the small tadpole.

2. Draw a picture or write a description of how you

determined how many times as long as the small tadpole is the large tadpole.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C The Mysterious Tadpole Page 44

Making Rectangles Grid Transparency / Handout 7C-6

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Making Rectangles Page 48


Making Rectangles Recording Sheet

Rectangle Number Number of square

centimeters 1 2 3

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Making Rectangles Page 49

Centimeter Graph Paper Handout 7C-8

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Measuring Area with Rectangles Page 51


Measuring Area with Rectangles Recording Sheet

Problem: How many 2x3 rectangles does it take to make each similar rectangle? To do this activity you need to have completed the activity “Making Rectangles.” Cut a 2-cm by 3-cm rectangle from a sheet of centimeter graph paper. Use this 2x3 rectangle to count how many rectangles it takes to cover each of the rectangles you drew on the Making Rectangles Grid. This number is the “Area in 2x3 Rectangles.” Fill out the information on the chart below.

Rectangle Number Area in Squares Area in 2x3 Rectangles 1 6 1

What patterns in the numbers do you see? Predict how many rectangles and how many squares would be in the 10th, 11th, and 12th rectangles in the sequence. Use centimeter graph paper to check your predictions. Bonus: How many more rectangles does it take to make the next larger rectangle? Is there a pattern in those numbers?

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Measuring Area with Rectangles Page 52


Similar Quadrilaterals: Rectangles Recording Sheet

Problem: Is there a relationship between the corresponding sides of similar quadrilaterals? A ratio expresses a relationship. The two different sides of the small rectangle used in Making Rectangles were 2 cm and 3 cm. The

relationship can be written as 2 cm to 3 cm, or 2:3, or as a fraction 32 .

For this activity, use the fraction form of the ratio. First, fill in the unshaded parts of the chart below. Write the rectangle number in the first AND last columns. Then write the length of the short side of each rectangle. Finally, write the length of the long side of the rectangle. Data for the first two rectangles are already filled in.

Rectangle Number

Measure of Short side

Numerator Simplified

Denominator Simplified

Measure of Long Side

Rectangle Number

1 2 3 1 2 4 6

2

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Similar Quadrilaterals Page 55


After you finish filling out the unshaded part of the chart, use the fraction calculator. Enter the short side/long side. Then simplify the fraction. You may need to simplify it more than once. Make sure the fraction is in simplest terms. When it is in simplest terms, nothing will change in the display. Use the keystroke chart below for your calculator.

Casio Fraction Mate TI Math Explorer Or Explorer Plus

TI 15

2 b/c 3 SIMP 2 / 3 SIMP = 2n 3d SIMP = To simplify again, press

SIMP To simplify again, press

SIMP = To simplify again, press

SIMP = When you have the fraction in lowest terms, enter the numerator and the denominator in the correct shaded columns. Answer these questions. What happened when you simplified the fractions? Look at the first three columns. What relationship is there between

the numbers in each row? For example, what is the relationship between the numbers 2, 4, and 2 (the numbers in the second row?)

Did any of the fractions simplify to the same fraction? What do you suppose that means?

What does it mean when two fractions simplify to the same fraction?

What does it mean if the ratio of two sides of a rectangle and the ratio of the corresponding two sides of another rectangle each simplifies to the same fraction?



Similar Quadrilaterals: Trapezoids

Problem: Is there a relationship between the sides of similar quadrilaterals? Similar rectangles are not the only kind of quadrilateral that could have a relationship between corresponding sides. Use the trapezoids from the Pattern Blocks to build similar trapezoids. As you build them, enter the length of the long side and the length of one of the shorter sides in the unshaded parts of the chart. Use the calculator to simplify the fractions like you did with the rectangles. (For this problem, we will consider the short side of the trapezoid pattern block to be 1 unit long.)

Trapezoid Number Short side Numerator

Simplified Denominator

Simplified Long Side Trapezoid Number

1 1 2 1 2 2 4 2

After you finished filling out the white part of the chart, use the fraction calculator. Enter the short side/long side. Then simplify the fraction. You may need to simplify it more than once. Make sure the fraction is in simplest terms. When it is in simplest terms, nothing will change in the display. Use the keystroke chart below for your calculator.



Casio Fraction Mate TI Math Explorer

Or Explorer Plus TI 15

1 b/c 2 SIMP 1 / 2 SIMP = 2n 3d SIMP = To simplify again, press

SIMP To simplify again, press

SIMP = To simplify again, press

SIMP = When you have the fraction in lowest terms, enter the numerator and the denominator in the correct shaded columns. Answer these questions. What happened when you simplified the fractions? Look at the first three columns. What relationship is there between

the numbers in each row? Did any of the fractions simplify to the same fraction? What do

you suppose that means? What does it mean when two fractions simplify to the same

fraction? What does it mean if the ratio of two sides of one trapezoid and the

ratio of the corresponding two sides of another trapezoid each simplifies to the same fraction?



Examples of figures that are connected:

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Balloon Geometry Page 64


Examples of figures that are not connected:



Balloon Geometry Recording Sheet

1. Can a triangle be made to look like a closed curve? 2. Can an empty triangle be made to look like a filled in or

blackened triangle? 3. Can the letter F be made to look like the letter H? 4. Can the letter F be made to look like the letter Y? 5. Can = be made to look like ( )? 6. Can = be made to look like x? 7. Can a rectangle be made to look like a hexagon? 8. Can a rectangle be made to look like a five-pointed star?



Shadow Geometry Recording Sheet

Draw the different shadows you can make with each shape.

Circle Square Triangle

Which of the following shapes could be a shadow of O? Explain your choices.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Shadow Geometry Page 70


Comparing K Recording Sheet

Can the K be made to look like each of the given shapes if K is:

on a balloon? a wire making a shadow? on a card?

1.

2.

3.

4.

5.

6.

7.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 7: Section C Comparing K Page 73



Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 26

Handout 8B-2

Sample Big Ideas and Evidence of Understanding for Geometry

Using geometry to solve problems I can recognize geometry in the world around me. I can use what I know about geometry to decide if my answer makes sense. I can use geometry words to talk and write about solving problems.

Identifying and comparing shapes and solids

I can sort a set of objects into groups. I can tell how I sorted a set of objects into groups. I can describe an object by telling about its shape. I can tell how shapes are the same or different.

Using formal geometric vocabulary to identify and define

I can name two-dimensional shapes. I can name three-dimensional objects. I can name the shapes I see on three-dimensional objects. I can talk about shapes I see in real objects. I can use the right math words to tell about an object. I can identify different kinds of angles. (right, acute, obtuse) I can identify parallel lines and tell how I know they are parallel. I can identify perpendicular lines and tell how I know they are perpendicular. I can tell about the vertices, edges, and faces on an object. I can define a shape or solid by telling about its attributes.

Combining and dividing shapes and objects

I can put shapes together to make new shapes. I can cut a shape apart and tell what new shapes I make.

Modeling and describing transformations

I can use objects to show translations. I can use objects to show reflections I can use objects to show rotations. I can draw what happens when I translate a shape. I can draw what happens when I reflect a shape. I can draw what happens when I rotate a shape.


Handout 8B-2

Recognizing congruence and symmetry I can identify congruent shapes. I can make shapes that are symmetrical. I can find the line of symmetry on a shape. I can verify that a shape is symmetrical by reflecting it. I can show how to translate, reflect, or rotate a shape onto another to show they are

congruent. Using numbers to describe location

I can use numbers to name points on a line. I can use fractions to name points on a line. I can use decimals to name points on a line. I can locate points on a grid from ordered pairs.


Handout 8B-3

--3-- Orchestrating for Rigorous Learning (cont.)

Developing Deep and Complex Knowledge What does it mean to develop deep and complex knowledge in a learning experience? Students need deep and complex mathematical knowledge in order to apply skills and concepts to solve problems and make decisions. Students develop deep knowledge when they pursue a concept from multiple perspectives in order to understand it more fully; they develop complex knowledge when they consciously make connections between concepts. A learning experience must be carefully designed to engage students in mathematically rigorous thinking. The teacher must create learning experiences that provide the opportunity for students to purposefully and thoughtfully process ideas in order to strengthen their mathematical understanding. Questions to consider in planning for the development of deep and complex knowledge:

How can I organize the learning experience around interrelated concepts? How can I design the learning experience so that students find the work to be

personally challenging? What opportunities have I built in for students to develop flexible mathematical

thinking (e.g. examining and using multiple representations of a mathematical idea)? How can I guide students to become familiar with and demonstrate Evidence of

Understanding? References National Research Council. (2002). Helping Children Learn Mathematics. Mathematics Learning Study Committee, J. Kilpatrick and J. Swafford, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Strong, R. W., Silver, H. E., & Perini, M. J. (2001). Teaching what matters most: Standards and strategies for raising student achievement. Alexandria, VA: ASCD.


Handout 8B-4

Sorting Cards Set A

1 2 3

4 5 6

7 8 9


Handout 8B-5

Sorting Cards Set B

10 11 12

13 14 15

16 17 18


Handout 8B-6

Sorting Cards Set C

19 20 21

22 23 24

25 26 27


Handout 8B-7

Sorting Cards Set D

28 29 30

31 32 33

34 35 36


Handout 8B-8

Sorting Cards Set E

37 38 39

40 41 42

43 44 45


Handout 8B-9

Sorting Cards Set F

46 47 48

49 50 51

52 53 54



How We Sorted Our Shape Cards

Sorting # 1 Cards we used: How we sorted:





How We Sorted Our Shape Cards (continued)






Sorting Polygons

Write the numbers of the shapes that fit in each.

Equilateral Equiangular

Part 2: Quadrilaterals

Equilateral Equiangular


Handout 8C-1

How to Make A Tangram

(Note: Directions to read aloud are in bold. Statements about geometric vocabulary are in italics with the vocabulary words to emphasize underlined.) A tangram can be cut from a square of any size or material. 1. Fold the square in half to bring diagonal corners together. Cut along the diagonal

line. We have cut the original square into two congruent pieces. How can we tell the two pieces are congruent? Since we have two congruent pieces cut from the original square, how much of the original square is each piece? These pieces are right triangles and each of them is one-half of the original square.

2. Take one of the right triangles. Hold it so that its longest side, called the hypotenuse,

is on the bottom. Bring the end points of the hypotenuse together and fold to divide the hypotenuse in half. Cut along this line. When you cut along this line, you will be dividing the right triangle into two congruent pieces. Since that right triangle was one-half of the original square and we have divided that in half, how much of the original square is each of these pieces? These pieces are also right triangles and each of them is one-fourth of the original square. These two pieces go together at right angles to form one-half of the original square. For the purpose of making our discussion about the actual tangram easier, we will call these two pieces the large triangles. Put these down for later use.

3. Take the other right triangle that we cut at first. The other five tangram pieces will

be cut from this half of the original tangram square. On the hypotenuse this time, you will find the midpoint by bringing the ends together and pinching (not folding) in the middle. Now bring the corner that is NOT on the hypotenuse down to touch the midpoint of the hypotenuse and carefully fold. Cut off the triangle that is folded down. When you bring the vertex of the right angle to meet the midpoint of the hypotenuse and fold, you will have a right triangle and a trapezoid. The right triangle that you cut formed one of the right triangles of the original square. We will refer to this triangle as the medium triangle.

4. Take the trapezoid and fold it in half along its longest side where you had already

found the midpoint. Cut along this fold. You have now divided the trapezoid into two congruent pieces. They are still trapezoids, since they are quadrilaterals with exactly one pair of parallel sides, but they are right angle trapezoids. They look like pointy shoes.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 8: Section C Geometry Sampler Page 43

Handout 8C-1

5. Hold one of the right angle trapezoids so its longest side is on the bottom, as you

would imagine a shoe sitting on the ground. Try to picture this shape as a shoe with a pointy toe, a heel, the laces, and the place where the foot slides in. You will be bringing diagonal corners together when you bring the toe back to touch the heel and fold. Cut on this fold. When you cut, you will have two more pieces of the tangram, the only square and one of the two small triangles. Remember how the small triangle met at right angles with the square to make the right trapezoid and this will be a good clue to putting the pieces back together to make the original square.

6. With this last right trapezoid, you are again going to think about it as a pointy shoe,

but this time you will bring the heel up to touch the laces. Fold and cut on this line. When you cut, you will have the final two pieces of the tangram, the other parallelogram (remember that the square is also a parallelogram) and the other one of the two small triangles. Remember how the small triangle joined with the parallelogram on their longest sides to make the right trapezoid and this will be a good clue to putting the pieces back together to make the original square. Another hint to putting the seven tangram pieces together is that the two small triangles join at right angles with the square to make a trapezoid.

7. Let's check our pieces. Notice how none of the original square we started with was

lost when we cut. There are seven pieces: two large triangles, one medium triangle, two small triangles, one square and one parallelogram. The experience of making a tangram should assist you in using the pieces to make shapes and pictures. Now you are ready to explore the fascinating relationships between pieces in the tangram!



Geometry Sampler Reflection Sheet

Task: Evidence of Understanding: Ways in which this task addressed developing deep and complex knowledge: Task: Evidence of Understanding: Ways in which this task addressed developing deep and complex knowledge: Task: Evidence of Understanding: Ways in which this task addressed developing deep and complex knowledge:



Lesson Planning Process Chart for the Geometry Sampler

Step 1 Big Ideas






Lesson Planning Process Chart for the Geometry Sampler

Step 1 Big Ideas





Transparency / Task Card 8C-4

Sorting Task A: Sorting Shapes

You will work with a partner or small group using the sorting cards and 2 yarn loops for this task. The 2 yarn loops should overlap so there is an intersection between the loops. 1. The first person will choose an attribute for

each of the loops and place sorting cards so that there are at least 3 sorting cards in each loop outside the intersection and at least 2 sorting cards in the intersection.

2. The second person will examine the cards to determine what attributes were chosen and to verify that the sorting cards in the intersection do indeed have both attributes.

3. Trade roles so that the second person above now places the cards and the first person determines the attributes.

4. As you work, write the two attributes chosen for each sort. Continue sorting until each person has placed sorting cards at least 3 times (or at least 6 sorts).

Questions to Ponder

What do you notice as you examine the pairs of attributes you have chosen? Are there any patterns in the pairs?

Do some attributes make it more difficult to find shapes for this intersection? Why do you think this is so?

What new shapes might you want to add to the sorting cards for this task? Why would you want to add those shapes?



Sorting Task B: Missing Shapes

This task requires 4 people: 2 people for each team. Each team will need a set of sorting cards and 2 yarn loops for this task. The 2 yarn loops should overlap so there is an intersection between the loops. 1. Each team chooses an attribute for each of the loops and places

sorting cards so that there are at least 3 sorting cards in each loop outside the intersection and at least 2 sorting cards in the intersection.

2. Once the sorting has been created, the team removes all of the cards from one loop (that are NOT in the intersection) and stacks them face down on the table. Cards that remain on view are all in 1 loop, including the intersection, and can now be shown to the other team.

3. The teams exchange places to examine each other's display. The task for the viewing team is to determine the attribute of the missing cards. The viewing team can either choose or draw sorting cards that fit in the loop where the cards are missing.

4. When both teams are done, the creating team examines the cards added by the viewing team to determine if they fit.

Questions to Ponder

Could the viewing team find a different attribute for the missing cards than the creating team? Think of an example for one of your sorts.

Is there a way to make sure the other team can determine the attributes your team selected? Would it help to have more sorting cards on display?



Sorting Task C: Three Attributes

Here is a challenge for you to work on as a team: Use 3 (three!) yarn loops and the sorting cards to create a display to show how three attributes can be connected. Your display should look something like the diagram to the right. Try to place at least one sorting card in each space, as pictured in the diagram. You may draw new sorting cards if you need them. Before you begin working, talk with your team members about what this diagram means and which attributes the cards you choose for each of the spaces representing intersections must have.

Questions to Ponder What was the most challenging part of doing this task? Why do you think

this was challenging? What do you notice about the three attributes you selected? Is there a

relationship between the attributes? Would this relationship work to sort with 3 attributes again?

How could you use crayons or designs to explain the intersections in the diagram above? (Hint: yellow + blue = green)



Sorting Task D: Sorting by Attributes

Here is a challenge for you to work on as a team: As a team, create the table below and write the numbers of any sorting cards that fit in each cell on the table. Is ONLY

Equilateral Is ONLY Equiangular

Is Regular: Equilateral AND Equiangular

Is NEITHER Equilateral NOR Equiangular

3 sides 4 sides 5 sides 6 sides 7 sides 8 sides Before you begin working, talk with your team members about how information is organized on this table and what cards you can choose for each of the cells.


Questions to Ponder What was the most challenging part of doing this task? Why do you think

this was challenging? Are any of the cells empty? Why do you think this is so? Could you draw

shapes to fit in any empty cells? How could sorting information in this way be helpful? In what other ways

could you sort the information?


Sorting Task E: Sorting Containers

You will work with a partner or small group using the containers and 2 yarn loops for this task. The 2 yarn loops should overlap so there is an intersection between the loops. 1. The first person will choose an attribute for

each of the loops and place containers so that there are at least 3 containers in each loop outside the intersection and at least 2 containers in the intersection.

2. The second person will examine the containers to determine what attributes were chosen and to verify that the containers in the intersection do indeed have both attributes.

3. Trade roles so that the second person above now places the containers and the first person determines the attributes.

4. As you work, write the two attributes chosen for each sort. Continue sorting until each person has placed containers at least 3 times (or at least 6 sorts).

Questions to Ponder

What do you notice as you examine the pairs of attributes you have chosen? Are there any patterns in the pairs?

Do some attributes make it more difficult to find containers for this intersection? Why do you think this is so?

What new containers might you want to add to those available for this task? Why would you want to add those containers?



Sorting Task F: Sorting Objects

Each member of your team should select 5 to 8 small objects from your desk, backpack, or classroom. You will also use 2 yarn loops that should overlap so there is an intersection between the loops. 1. Examine your objects and make a list of attributes that could be

used to describe them. 2. Choose one attribute and place objects so that there are at least 3

objects in one loop outside the intersection. (If you don't have at least 3 objects, try another attribute). Write this attribute on a card.

3. Now choose a second attribute and place objects so that there are at least 3 objects in the other loop outside the intersection. (If less than 3 objects, try another attribute). Write this attribute on a card.

4. Examine the remaining objects. Are there any objects that match BOTH attributes? If so, place them in the intersection.

5. If you did not have any objects that matched BOTH attributes, then switch one or both of the attributes you chose and try again. Keep working until you have at least one object that goes in the intersection because it matches BOTH attributes. When you have done this, draw a picture and write the attributes to show your work.

Question to Ponder

What things do you need to consider as you are working on this task?



Tangram Task A: Tangram Area

One of the fascinating things about the tangram pieces is the way they relate to each other in size. Use the table below to help you explore this relationship. Given the area of one of the tangram pieces, find the relative areas of the other pieces.

then what is the area of each of these? If this piece has an area of 1 unit,

small triangle

medium triangle

large triangle

square parallel-ogram

whole tangram

small triangle 1

medium triangle

large triangle

square

parallelogram

whole tangram


Questions to Ponder What patterns do you notice when you examine areas of the tangram pieces? How does this table help you recognize the patterns? How could you use this information to work with tangrams in other ways?


Tangram Task B: Measuring Tangrams

Tangram pieces have interesting relationships in the lengths of their sides. Use the lengths given below for the small triangle to help you compare the other pieces and the whole tangram.

1. Trace each of the pieces on a sheet of paper. 2. Mark all the sides that are congruent to A or B with those letters. 3. Which pieces have sides that are not marked with A or B? 4. Can you relate those sides to length A or B in some way? 5. Now trace the original tangram square. What is the length of its

sides?


Questions to Ponder What patterns do you notice when you examine lengths of the tangram

pieces? How does your marking of lengths help you recognize the patterns? How could you use this information to work with tangrams in other ways?


Tangram Task C: Buying Tangrams

A tangram can be cut from any square. Since tangrams are such interesting and entertaining puzzles, people make them from different types of materials. For each of the materials listed below, a base price is given in terms of the cost of a small triangle. Determine the cost for each piece and for the tangram as a whole. For example, if the small triangle cost 10¢, then a square would cost 20¢ to produce since a square is equal in size to 2 small triangles.

Materials Plastic Magnetic Wood Marble small triangle 8¢ 25¢ 60¢ $2.00 small triangle

medium triangle

large triangle

large triangle

square

parallelogram

TOTAL for tangram


Questions to Ponder What patterns do you notice when you examine costs of the tangram pieces? Which total cost was easiest to determine? Why do you think this is so? Which of these materials would you choose for a tangram puzzle? Why?


Tangram Task D: Tangram Angles

Tangram pieces have interesting relationships in the size of their angles. Use the wedge protractor below to help you measure and compare angles on the other pieces.

1. Trace each of the pieces on a sheet of paper. 2. Mark all the angles to label the measure of each in wedges. 3. Now trace the original tangram square. What is the measure of its

angles?


Questions to Ponder What patterns do you notice when you examine angle measures of the

pieces? How does your marking of angle measures help you recognize the patterns? How could you use this information to work with tangrams in other ways?


Tangram Task E: Building With Tangrams

For this task, you will work with one partner. Each of you should have one complete set of tagboard tangram pieces. In consideration of the attractiveness of your product, you may wish to choose two different colors of tangram pieces. Using only the 2 sets of 7 tangram pieces, you will construct as a team one three-dimensional object. You will do this by carefully joining the edges of tangram pieces with tape. Here are things to consider when you are working. Use all seven pieces of both tangrams in your construction. The tape should join tangram pieces only on their edges. The tangram pieces themselves should not be cut or bent but

should remain flat. Your construction should be able to stand on its own as a three-

dimensional object. You should name your construction to help others understand its

design or function.


Questions to Ponder What characteristics of the sides and/or angles of the tangram pieces

influenced your decisions as a team? What challenges would an artist or architect face in using only these pieces? How could you adapt the directions to make this task better?


Tangram Task F: Shapes With Tangrams

For this task, you will work with one partner. The two of you will need one set of 7 tangram pieces. Choose one of the following sets of pieces. 3 smaller triangles (2 small and 1 medium) all 7 pieces 5 smallest pieces (NOT the 2 large triangles) 4 OR 5 of the triangles

With each of these sets, you can build the following shapes: square rectangle (that is not a square) parallelogram trapezoid triangle 1. Use one of the sets of tangram pieces described above to build the

5 shapes listed above. As you build each shape, draw a picture to record how the tangram pieces fit together to build that shape.

2. Choose a second set of tangram pieces from the list above. Build and record the way pieces fit together to form each shape.


Questions to Ponder How could your work with the first set of tangram pieces help the work you

did with the second set of tangram pieces? Did you notice any patterns? Do you think any of the 4 sets listed could be more challenging in building

the shapes? Why do you think this? Which tangram piece(s) might be more challenging to work with? Why?


Pattern Block Task A: Floor Tiles

You have been assigned to design a pattern for a kitchen floor. Choose one of the floor spaces (Pattern Block Spaces 1 to 4). Create a design for the kitchen floor using any of the following pattern block shapes: hexagon, trapezoid, parallelogram, and triangle. When you have completed a pleasing floor design, use crayons to record the design by matching shapes and colors with the pattern blocks used. Now that you have your design, determine the total cost for your floor. 1. Find out how many of each block type you used. (ex: 3 hexagons) 2. Use the prices below to determine the costs for the number of tiles

needed. hexagon = $6 trapezoid = $3 parallelogram = $2 triangle = $1

(ex: need 3 hexagons x $6 = $18 worth of hexagons) 3. Add the costs for each block to find the total cost for the floor. Now create a different design on the same floor space so that the owner has a choice between two designs. Follow the same steps to record and find the cost for the second design.


Questions to Ponder Why did you make the choices you made for the design of the floor? How do your two designs compare in cost? Why do you think this is so? Which design would you select for your home? Why would you choose this?


Pattern Block Task B: Patio Stones

You have been asked to design a pattern for a patio floor. Choose one of the floor spaces (Pattern Block Spaces 1 to 4). Create a design for the patio floor using any of the following pattern block shapes: hexagon, trapezoid, parallelogram, and triangle. The floor will be made from flagstone. Since larger pieces of stone are harder to cut and move, they cost more. After you create the design of your floor with the shape of stones you want, use the table below to determine the cost of your design. The costs for each piece of stone are: hexagon = $ 25 trapezoid = $ 10 parallelogram = $ 5 triangle = $ 1

stone shape cost per stone shape

number of stones needed

total cost per shape

hexagon $ 25

trapezoid $ 10

parallelogram $ 5

triangle $ 1

TOTAL - -


Questions to Ponder Why did you make the choices you made for the design of the patio floor? How could you arrange the same stones to create a different design? Do you prefer large stones or small stones? Why is this?


Pattern Block Task C: Measuring the Lawn

The owners of a new home have asked for a bid on the cost of putting grass in and a fence around their backyard. Choose one of the lawn spaces (Pattern Block Spaces 1 to 4). In order to determine the cost, you will need to know the area of the space for the grass and the perimeter of the area for the fence. For this plan you will use the triangle as your unit of measure. One length of fence is equal to the length of one side of the

triangle. One length of fence costs $4. One plot of grass is equal to the area of the triangle. One plot of

grass costs $3. You will need to find: The number of lengths of fence needed to go around the perimeter. The cost for this number of lengths of fence for the backyard. The number of plots of grass needed to fill the area completely. The cost for this number of plots of grass for the backyard. The total cost of both fence and grass for the backyard.


Questions to Ponder How could you have used different pattern blocks to measure the area? Is there something that could have been done to make this task easier? How could you alter this plan while working with the same space?


Pattern Block Task D: Build It Bigger

1. Choose any three of the shapes below. hexagon trapezoid parallelogram rhombus triangle square rectangle 2. Use at least 4 pattern blocks to build these shapes.

You may use any of the 6 different pattern block pieces. 3. Draw and label each of the 4 pattern block shapes on your

paper showing what you used to build each shape. Challenge: Build a larger shape with 4 or more pattern blocks using ONLY blocks of that type. Explain your thinking. (Ex: Build a large triangle with 4 or more triangle blocks.)


Questions to Ponder Are some of the shapes harder to build larger? Why do you think this is so? Did you use pattern blocks of all the same shape? Why did you do this? Could you build these shapes with more than 9 pattern blocks? How do you

know?


Pattern Block Task E-1: Hexagon Fractions

Use pattern blocks to fill each of the hexagons below. The fractions near each hexagon tell how much of each hexagon should be a specific color.


Challenge: Create your own design and write the fractional parts that describe it.


Pattern Block Task E-2: Hexagon Fractions

Use pattern blocks to fill each of the hexagons below. The fractions near each hexagon tell how much of each hexagon should be a specific color.


Challenge: Create your own design and write the fractional parts that describe it.


Pattern Block Task F: Symmetry

Fold a regular sheet of paper in half. The fold on the paper will serve as your line of symmetry. Use pattern blocks to construct a symmetrical picture or design. Whatever piece you place on one side of the line must be reflected onto the other side. Here is an example of how this might look:

1. Use your pencil and crayons to trace and record your symmetrical design. 2. Make a list of how many pieces of each pattern block shape you used. 3. We could now reflect this design onto another sheet of paper. 4. How many of each type of pattern block would you need in all?

Questions to Ponder Are some of the shapes easier to use to create symmetry? Why do you think

this? Did you see a pattern in your design? How would you describe the pattern?


Handout 8C-23

Pattern Block Space 1


Handout 8C-24



Handout 8C-25



Handout 8C-26




3-D Task A: Boxes Inside Out

Work with one or two partners on this activity. 1. Choose one of the boxes on display. 2. Cut along one of the edges of the box so that the box is opened flat.

Discuss which one edge you should cut on and why before you cut. 3. After you have opened the box, place it flat on your table so you

are looking at the inside. Talk with your partners about what shapes you see and how many shapes you see. Talk about how these shapes are connected.

4. Fold the box back into its original shape, but this time fold it so the decorated or printed faces are on the inside. It will look like the box is inside out. Do not tape the box together yet.

5. Examine the box. If this box was NOT a container for food, what might it hold? Decide with your partners what you think the box could hold and how much or how many of this thing it could hold.

6. Open the box flat again with the inside part showing. Use crayons or markers to create a new box. Be sure to label the box to tell what is in it and how much it holds. When you have finished decorating, put the box back together with your new decoration showing on the outside. Tape the edge that you cut so that your new box can be put on display.


Questions to Ponder Why is your box a good choice to hold the object(s) you selected? How did you figure out how much of the object your box would hold?


3-D Task B: Shrinking Boxes

Work with one or two partners on this activity. 1. Choose one of the boxes on display. 2. Cut along each of the edges of the boxes to separate the 3-D box

into 2-D shapes. How many shapes and what shapes do you see? 3. With a ruler, draw a line down the middle of each shape parallel to

its longest side. The line you draw should divide each shape in half.

4. Cut along the lines you drew to divide each shape in half. Put one half of each shape in one pile and one in a separate pile.

5. Take one pile (half of each shape). Discuss how you could tape the edges together to form a new box. Do you need to make any additional cuts? How does the size of this new box compare to the box you started with? If this box was NOT a container for food, what might it hold? Decide with your partners what you think the box could hold and how much or how many of this thing it could hold.

6. Use crayons or markers to create a new box. Be sure to label the box to tell what is in it and how much it holds. When you have finished decorating, put the box back together with your new decoration showing on the outside. Tape all of the edges that you cut so that your new box can be put on display.

Questions to Ponder

How does your new box compare in size to the original box? Why do you think this is so?

What did you need to consider as you put the shapes together to make the new box?



3-D Task C: Building Boxes

In this task, you will work with one or two partners to design a new box. 1. Examine the available shapes cut from poster board. Choose the

right number of pieces of the right size and shape to build a new box. At least 2 of your shapes should NOT be rectangles.

2. Discuss how you think your box will look once you have put it together. Draw a picture to show your prediction of how the completed box will look.

3. Use tape to join the shapes you have chosen to make a box. 4. Discuss with your partners what this box could be used to hold. On

your drawing paper, give your box a name, tell what it could hold and how much or how many of this thing you think the box could hold.

5. If you wish, you may use crayons or markers to decorate your new box. Now your new box can be put on display.

Questions to Ponder

How did you decide to choose the size and shape of your pieces? How did you know how many of each size and shape to choose?

Do we often see boxes like yours? Why do you think this is so? If you were to choose shapes for a second box, what would you choose?

Why?



3-D Task D: Bag-Net

You have been asked to create a new design for a small paper lunch bag. Your new bag should look different from regular paper bags. You will work with one or two partners on this design. 1. Examine a small lunch bag. Determine what shapes compose its

faces and how the flaps overlap to make the container. You may wish to deconstruct the lunch bag to see how it is put together.

2. Discuss the shapes and numbers of faces your new bag will have. Consider the length of the sides of each of these faces.

3. Draw the design for your new bag on a piece of paper. When you draw, be sure the faces are connected to show the “net” of the design. You probably want to include some overlapping flaps that will enable you to join the faces.

4. Cut out the “net” and fold along the edges to test the construction of your paper bag design. Discuss with your partners what this bag could be used to hold. On your drawing paper, give your bag a name, tell what it would hold and how much or how many of this thing you think the bag could hold.

5. If you wish, you may use crayons or markers to decorate your new paper bag before you put it on display.

Questions to Ponder

How did you decide to choose the size and shape of your faces? How did you know how many of each size and shape to choose? What did you need to consider as you were drawing the “net” of your paper

bag?



3-D Task E: Roll With It

You will work with a partner on this task. 1. Select one of the empty paper towel or toilet paper tubes. 2. Examine the way this tube was constructed. What 2-D shape do

you think was used to make this tube? 3. You will use scissors to make one cut in a straight line from one

open end to the other. BEFORE you cut, talk with your partners about what 2-D shape you think you will have when you cut.

4. What 2-D shape do you have? Use your ruler to measure the sides. What do you notice about the lengths of the sides?

5. Select a second tube. This time, you will cut in a straight line at a different angle than the cut on the first tube. Before you cut, predict the shape you will get. Why do you predict this?

6. What 2-D shape do you have? Use your ruler to measure the sides. Are you noticing any patterns? What would you predict would happen if you cut a third tube in a straight line at a different angle?

7. Based on your observations and any patterns you see, use your ruler to draw a 2-D shape that you think will make a tube. Test out the shape you cut. Does it make a tube? Why do you think this happened?

Questions to Ponder

What did your work help you understand about making a tube? What objects have you seen that use tubes as one of their parts?



3-D Task F: Cones

You will work with a partner on this task. 1. Think of something that is the shape of a cone. Did you think of an

ice cream cone? What does an ice cream cone do? It holds the ice cream, so it acts as a container. What else might a cone hold?

2. Draw a circle or trace around something round on paper. Cut out your paper circle. Fold the circle in half. Then fold it in half again.

3. Open up the circle. You should see two intersecting lines that go from one side of the circle to the other, like cutting a round pizza or cake into four equal parts. Take your pencil and put a dot on the place where these two lines cross. This is called the center of your circle.

4. Use scissors to cut on one of the lines from the outside to the center.

5. On the line where you cut, you now have 2 edges. Bring those edges together so you have a small overlap. What do you have? Make a larger overlap. What happens to the size of the cone when you change the amount of overlap?

6. Try this with circles of different sizes. What patterns do you notice as you are working with the overlap of paper? How can you turn a circle into a cone that will hold the most?

Questions to Ponder

What did your work help you understand about making a cone? As part of what other objects do we see cones?


Transparency / Handout 8D-1


Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 8: Section D Orchestrating to Develop Deep and Complex Knowledge Page 82


Attributes That Can Be Measured

Attribute Unit of Measure

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 9: Section A Defining Measurement Page 13

Handout 9A-2

How Bouncy Is a Ball?

Objective: The student will explore ways to quantify and record the bounciness of various balls in order to compare them. Materials: a variety of balls of different sizes and compositions; a variety of measuring tools for use with both customary and metric units (such as balances and masses, cm and customary tape measures, meter sticks, yardsticks, stop watches, string, adding machine tape, scissors, inch2/cm2 graph paper or chart paper) Procedure:

Introduction: Show students several balls and ask them to suggest ways they could compare them. Ask students to brainstorm ways they could measure the bounciness of balls in order

to compare them. Have each group of students devise an investigation to measure the bounciness of a

ball, make and record their measurements, and present their results to the class. Discuss the various methods of measurement used and the groups’ results.

Exploration:

How bouncy do you think your ball is? What information are you using to make your estimate?

Can different balls have different bounciness? Can you get different measures of bounciness with the same ball?

What measuring tools could you use to measure the bounciness of your ball? What makes a good unit of bounciness? Why? What else might make a good unit of bounciness? Why? What would NOT make a good unit of bounciness? Why? If you test the bounciness of a ball several times, do you get the same result for each

test? Why or why not? How are you using your unit to find the bounciness of the ball you are measuring? How will you report the bounciness of your ball to someone else? How close is the actual measure to your estimate? Did you over-estimate or under-

estimate? How do you know? How did your group decide to record your data? What conclusions can you make from the data? Does the size of the ball affect its bounciness? How do you know? Does the composition of the ball affect its bounciness? How do you know? Do two balls that look exactly alike have the same bounciness? How do you know?


Handout 9A-2

Extension: Design a display of your results (a poster, demonstration, etc.). Design and conduct another investigation to measure the bounciness of a ball.

Compare the results to the measurements from your first investigation. Change something about your ball (such as covering it with tape or wrapping it in

paper). How does that change the ball’s bounciness?

Summary: What techniques did your group use to measure the bounciness of your ball? How did your group decide what tools to use to measure? How did your group decide how to gather and represent your data? What conclusions did your group make from your results? How were your group’s results different from or the same as other groups’ results?

What do you think caused the differences? How could we determine the bounciest ball in the room? The least bouncy ball? Do

we have enough information to order all of the balls from least bouncy to most bouncy?

How or when could this information be useful to you? If you investigated bounciness again, would you use one of your investigations or

another group’s investigation? Why? How did mathematics help you in your exploration of the bounciness of a ball?

Assessment: Questions:

See summary questions. Observations:

Were students making measurements as accurately as possible? Did students attempt to account for errors in measurement? Did students select appropriate units of measure? Were students discussing how to use their units of measure to determine the

bounciness of a ball? Did students try several different methods? Or did they keep repeating and refining a

single method? Task:

In your journal, describe how your group decided to measure the bounciness of a ball and why you decided to measure it in that way.

Write a story about a situation in which this data would be useful to someone.


1 Inch Graph Paper Handout 9A-3


Centimeter Graph Paper Handout 9A-4



Measurement Is _________




Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 9: Section B Measurement and Inquiry Page 23

Handout 9B-2


Sample Big Ideas and Evidence of Understanding for Measurement

Using measurement to solve problems I can recognize ways measurement is used in the world around me. I can use what I know about measurement to decide if my answer makes sense. I can use measurement words to talk and write about solving problems.

Comparing and ordering objects based on a common attribute

I can tell if an object is shorter or longer than or about the same as another. I can tell if a container holds more or less than or about the same as another. I can tell if an object is heavier or lighter than or about the same as another. I can tell if something takes more or less time than or about the same as another. I can tell if something is hotter or colder than or about the same as something else. I can put things in order from shortest to longest. I can put containers in order from holds most to holds least. I can put objects in order from heaviest to lightest. I can order things by how much time they take. I can put a set of events in the right order. I can put objects in order from coldest to hottest.

Applying the process of measuring an attribute such as length (including perimeter) weight/mass, time, temperature, area, capacity, or volume

I can tell what kind of measuring I need to do. I can choose the right kind of unit. I can tell why I picked a certain unit to use. I can choose the right kind of measuring tool. I can tell why I picked a certain measuring tool. I can use the unit/tool correctly to measure. (Example: I can use a clock to tell how

long something takes.) I can explain how I am using the unit/tool to measure. (Example: I can explain how I

use a ruler to measure length.) I can use numbers and units to tell the measurement. (Example: I can use a balance

and gram masses to tell how many grams something weighs.) Reading measurement tools

I can tell what kind of measure a tool will give me. I can use the measuring tool to tell a measure. (Example: I can read a thermometer to

tell temperature.) I can tell the right unit when I read a measuring tool. I can write the right unit when I record the measure. I can read a calendar. I can read a clock to tell what time it is.

Handout 9B-2

Estimating measurements I can name something that is about the same as (some standard unit). (Example: I can

name something that holds about as much as a liter.) I can use other things I’ve measured to help me predict the measurement of

something else. Understanding the nature of measurement

I can describe what happens to my measurement when I use a smaller or larger unit. I can describe the relationship between related units of measure. (Example: I can tell

how inches and feet are related.) I can use numbers to describe the relationship between two measures. I can tell the measurement to the nearest unit. I can explain the error in a measurement.


Handout 9B-3


displace 6 mL.

How Dense Are You?

Objective: The student will use measurement tools to determine the density of a variety of samples. They will then use that information to confirm or refute their estimates for the density of a golf ball. Materials:

Samples (one kit per group): Glass (5 marbles) Paraffin (1 medium candle) Copper (10 pennies) Plastic (5 dice) Wood (3 two cm wooden cubes)

Measurement tools (one set per group): Spring scales (1g x 250g) Snack-size zipper bag with hole punched in an upper corner (These will be filled

with samples and attached to the hook on the spring scale to weigh the sample.) Graduated cylinders (1 x 100mL) 10 cm cubes Graduated beakers

Water Cm measuring tapes Sticky notes, chart paper Golf balls (one per group) How Dense Are You? recording sheet Calculators

Procedure: Pre-assessment: Ask each student to “guess” the density of the golf ball by recording on a sticky note and placing randomly on chart paper. Begin asking questions about data on the chart. Make the point that data must be organized to be useful for asking and answering questions. Ask volunteers to organize sticky notes into a graph. Retrieve golf balls to use later. Introduction:

1. Give students an opportunity to practice using a spring scale to determine the weight/mass of a variety of objects. It will be important for them to remember to “zero” the scale each time they use it to weigh samples.

2. Have students use graduated cylinders to measure the amount of water displaced by cm cubes. Ask, “What do you notice about the difference in the level of the water when you place 2, 4, or 6 cubes in the graduated cylinder?” Students will notice that 2 cm cubes displace 2 milliliters of water; 4 cubes displace 4 mL of water; 6 cubes

Handout 9B-3

3. Have students use the procedure below to determine the density of the samples: Use the spring scale to determine the weight/mass of each sample. Record. Use the displacement method or other method(s) to determine the volume of each

sample. Record. Students will use this information to determine the density of each sample.

Exploration: While students use measurement tools to find the mass and volume of each sample, ask questions such as:

Which tool are you using to determine the weight/mass of your sample? What unit of measure are you using?

How accurate do you think your measurement is? What could affect the accuracy? How could you make your measurements more accurate?

How are you finding the volume of your samples? Could another method be used? Explain. What unit of measure are you using?

What if your sample floats? How could that affect the accuracy of your measure? What is the relationship between the number of milliliters displaced and the volume

of your sample? After students have found the weight/mass and volume of each of their samples, discuss how to use the relationship between weight/mass and volume to find the density of each sample. Introduce the process with a triangular flash card:

If you know the product (56) and one of the factors (7), you divide to find the other factor (8).

If you know both factors (7 and 8), you multiply to find the product (56).

Dialogue Box:

Remind students that the triangle represents the relationship between the product and the factors. Use another triangle to represent other relationships: miles per gallon, miles per hour, etc.

If you know the number of miles and the number of gallons, you divide to find miles per gallon.

If you know the miles driven and the miles per gallon, you divide to find the gallons used.

If you know the number of gallons used and the miles per gallon, you multiply to find the number of miles.

Dialogue Box:


Handout 9B-3

Use the triangle to represent the relationship between mass, volume and density.

If you know the mass and the volume, you divide to find the density.

If you know the density and the mass, you divide to find the volume.

If you know the volume and the density, you multiply to find the mass.

Dialogue Box:

Summary: After students have investigated the density of each sample, ask questions such as:

What relationships did you use to determine the density of your samples? (the relationship between the mass and the volume)

What is the unit of measure for density? (mass per unit volume or g/cm

Did any of you come up with a formula for finding density? What variables did you use and what do they represent?

What is density? Did anyone describe density in a different way? Assessment:

Ask students to revisit their estimate of the density of the golf ball, and predict where they think the golf ball belongs in the “least dense” to “most dense” order.

Does a golf ball float? If not, how does knowing that information affect your estimate of its density?

How can you find the density of your golf ball? Where does the golf ball fit in the order of least to most dense samples? How does your measure of the density compare to where you placed the golf ball in

our density sequence? Did everyone get the same density for their golf ball? How can you explain that?

Continuing the Investigation:

Ask students to bring in additional samples from home, predict their densities by placing the samples in order from least dense to most dense, and then take measurements to determine how valid their predictions were.

3) Let’s order our samples from least to most dense.



How Dense Are You? Recording Sheet

Density relationship:

Paraffin:

Plastic:

Copper:

Wood:

Glass:

Golf Ball:

Density is:


Handout 9B-5

--4-- Communication to Support Learning

Incorporating Inquiry What does it mean to incorporate inquiry in a learning experience? Both the teacher and the students can be involved in inquiry to collaboratively build understanding of concepts. Inquiry is supported by the use of effective questioning strategies. Teachers need to plan and use questions that promote discussion, to provide time for discussion, and to attend to what occurs during the experience to facilitate continued discussion. Questions to consider in planning ways to incorporate inquiry:

How can I design the learning experience so that students have the opportunity to explain and justify their mathematical thinking?

What strategies can I use to facilitate collaboration and effective learning discourse

among students? How can I use questioning strategies to ensure that students think and communicate

about their thinking? How can I use questions to encourage students to talk about their work in terms of the

Evidence of Understanding? References Beck, I. L., McKeown, M.G., Hamilton, R. L. & Kucan, L. (1997). Questioning the Author. Newark, DE: International Reading Association. Burns, M. (February, 1985). The role of questioning. Arithmetic Teacher, 14-16. Dacey, L. S., & Eston, R. (1999). Growing Mathematical Ideas in Kindergarten. Parsippany, NJ: Pearson Learning. Schielack, J. F., Chancellor, D., & Childs, K. M. (2000). Designing questions to encourage children’s mathematical thinking. Teaching Children Mathematics, 6, 398-402.


Handout 9C-1

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 9: Section C Measurement Sampler Page 35

of Shaq’s foot?

When is a Foot NOT a Foot?

Objective: Students will measure a variety of lengths/distances using their cut-out foot, Shaquille O’Neal’s cut-out foot and non-standard units in order to develop

An understanding of the need for standard units; and The relationship between the size of a unit and the number of units in the

measurement. Materials: scissors, paper, linking cubes, linear measuring tools (rulers, tape measures, string, etc.) calculators, When Is a Foot NOT a Foot? recording sheet, cut out of Shaq’s foot Procedure: Introduction:

Have students work with a partner to draw around their foot and cut out the drawing. Ask students to write their names on their cut-out feet.

Ask each student to use his/her cut-out foot to measure and record a variety of lengths/distances: e.g. the distance from their desk to the chalkboard, the length of the chalk tray, the distance from the classroom door to the office, the height of the door.

Next, students will think about using Shaq’s foot as a unit of measure. Ask them to estimate the number of Shaq’s feet it will take to measure and record the same lengths/distances they measured with their cut-out feet. Talk about how they estimated: Did they just guess? Or did they use some information to help them?

Have students use Shaq’s foot to measure the same lengths/distances they measured with their cut-out foot. Compare the measurements made using the children’s feet and Shaq’s foot. Discuss the differences.

Have students use another non-standard unit (snap cubes) to estimate and measure their foot and Shaq’s foot. Each student will make a train of the unit cubes equal to the length of their foot and a train equal to the length of Shaq’s foot. Remind them to keep these trains to use later.

Ask them to estimate and then find their lengths/distances in cubes. Exploration: Questions to ask students as they do the activity for the purpose of focusing student attention on the math in the activity—

How are you using your cut-out foot as a tool of measurement? Why is it important to use it in that way?

What information are you using to predict how many of Shaq’s cut-out foot it will take to measure your lengths/distances?

What information are you using to estimate the number of snap cubes it will take to equal the length of your foot and Shaq’s foot?

How are you using the snap cubes to measure the length of your foot and the length

Handout 9C-1

How are you grouping your cubes to count them? Did anyone use a different way to group? Did you come up with the same total?

How are you using your trains to find the length/distances you measured with the cut-out feet?

Summary: Questions to include in a follow-up discussion—

How many of your feet did it take to measure (the chalk tray)? How many of Shaq’s feet did it take? What do you notice about the two measurements? How can you explain the difference?

Explain how you made your estimates. Did you guess or did you use information to help you? Did anyone think of this in another way? Explain your thinking.

How long is your foot in cubes? How long is Shaq’s foot in cubes? How did you use this information to find out how long (the chalk tray) is in cubes? How did your cube measurements compare? How can you explain this?

Extension: Bring from home a cut-out foot from a family member or a pet. Use it as your unit of measure. Estimate first and then find the actual measurements of a variety of lengths/distances. Compare these to your original measurements.


Handout 9C-2

Shaq’s Foot


Handout 9C-2


Handout 9C-3

When is a Foot NOT a Foot?

Length/distance measured

It took:

of my feet. of Shaq’s feet.

My foot =

snap cubes.

Shaq’s foot =

snap cubes.Length/distance in cubes:

Length/distance in cubes:

How we figured this out: What we discovered:



Lesson Planning Process Chart for the Measurement Sampler

Step 1 Big Ideas







Lesson Planning Process Chart for the Measurement Sampler

Step 1 Big Ideas






Area with Tiles

You need: color tiles or 1 inch ceramic tiles, colored paper cut in a variety of rectangular shapes (labeled for identification), small cups (for grouping tiles to count) Choose a rectangle. How many tiles do you think it will take to cover the rectangle? Record your estimate.

Use tiles to find out. Group your tiles in tens and extras. Record the actual count in square units. Choose 3 other rectangles. Estimate first and then find out how many tiles will cover each rectangle.

If you group your tiles in a different way (such as by fives or twos) will your total change? Try it to see.

Write about what you find out. Use pictures, words and numbers.



Area with Unit Cubes

You need: white cubes from Base-10 blocks, colored paper cut in a variety of rectangular shapes (labeled for identification), small cups (for grouping cubes to count) Choose a rectangle. How many cubes do you think it will take to cover the rectangle? Record your estimate.

Use cubes to find out. Group your cubes in tens and extras. Record the actual count in square units. Choose 3 other rectangles. Estimate first and then find out how many cubes will cover each rectangle.

If you group your cubes in a different way (such as by fives or by twos) will your total change? Try it to see.

Write about what you find out. Use pictures, words and numbers.



Area in Square Meters

You need: newspaper, masking tape, scissors, meter stick, several areas outlined in tape on the floor Tape together newspaper to make a square meter.

Use your square meter to measure areas on the floor outlined with tape.

Write about it. Use pictures, words, and numbers.



Linear Units

You need: Popsicle sticks, classroom objects, small rubber bands or small cups (for bundling Popsicle sticks and toothpicks), toothpicks Choose 5 things to measure. Estimate their lengths in Popsicle sticks. Record your estimates. Use Popsicle sticks to measure. Bundle your Popsicle sticks in tens and extras to count them.

Record the actual measurements. Repeat the activity using toothpicks as the unit of measure.

Write about how your numbers changed when you used toothpicks to measure.

Explain why the numbers changed.



Body Measures

Estimate first. Record your estimates. Measure & record the actual measurement.

Use baby steps to measure the distance between your desk and the door.

Stretch your arms wide. This is your span. How many spans wide is the chalkboard?

How many of your feet does it take to equal the width of the door?

Take a normal step. The distance between your feet is a pace. How many paces wide is your classroom?

How many palms wide is your desk?

Check with your classmates. Did you all get the same answers?

Why or why not?



Make an Inch Ruler

You need: a length of adding machine tape, scissors, one color tile Make a ruler using one color tile and adding machine tape.

Where should the tic marks be? How far apart are your tic marks? Number your tic marks to make it easier to read your ruler.

What number should be next to the first tic mark? Why?

Use your new ruler to measure 3 things in your classroom.




Make a Cm Ruler

You need: a length of adding machine tape, scissors, one white cube from Base-10 blocks Make a ruler using one white cube and adding machine tape.

Where should the tic marks be? How far apart are your tic marks? Number your tic marks to make it easier to read your ruler.

What number should be next to the first tic mark? Why?

Use your new ruler to measure 3 things in your classroom.




What Could It Be?

You need: a variety of lengths of string, classroom objects Choose a length of string. Your string is the perimeter of objects in our classroom.

What might some of the objects be? Find several objects in the classroom with the same perimeter as the length of your string.

Make a list of the objects you found.



What’s the Perimeter?

The area of your garden is 12 square yards. What might its perimeter be? Use words, numbers and pictures to record your solution.



Change the Mass

You need: pan balance, a variety of objects (such as scissors, block, golf ball, markers, eraser, crayons), non-standard units (pennies, marbles, color tiles, cubes, 2-color counters, etc.) Choose an object. Estimate its mass. Record your estimate.

Put your object in one pan and your units in the other pan until they balance.

Count your units. Record. Keep the same object, but change your unit. How many of your new unit do you think it will take to balance your object? Record your estimate. Check to see.

Do this twice more. Talk about it.



Change the Object

You need: pan balance, a variety of objects (such as scissors, block, golf ball, markers, eraser, crayons), non-standard units (pennies, marbles, color tiles, cubes, 2-color counters, etc.) Choose an object. Estimate its mass. Record your estimate.

Put your object in one pan & your units of mass in the other pan until they balance.

Count your units. Record. Keep the same unit of mass, but change your object.

How many of your units do you think it will take to balance your new object? Record your estimate. Check to see. Do this twice more. Talk about it.



How Many Grams?

You need: pan balance, gram masses, non-standard units (such as large and small paperclips, 2-color counters, pennies, color tiles, cubes, teddy bear counters) How many grams do you think will balance one cube? Two cubes? Three cubes? Record your estimates.

Check and see. Choose 3 more non-standard units. Estimate first, then find out how many grams will balance one unit, two units, three units.

Do any of your non-standard units weigh less than one gram each? How do you know? Write about it.



How Many Ounces?

You need: pan balance, customary masses, non-standard units (such as large and small paperclips, 2-color counters, pennies, color tiles, cubes, teddy bear counters) How many ounces do you think will balance one cube? Two cubes? Three cubes? Record your estimates.

Check and see. Choose 3 more non-standard units. Estimate first, then find out how many ounces will balance one unit, two units, three units.

Do any of your non-standard units weigh less than one ounce each? How do you know? Write about it.



Order! Order!

You need: pan balance, gram and customary masses, a variety of objects to weigh Choose 5 objects. Estimate their masses and order the objects from heaviest to lightest.

How will you know how close your estimates are?

Write about it. Use words, pictures and numbers.



How Much Is a Kilogram?

You need: a plastic grocery bag Work as a group. Fill your bag with items until you think it holds 1 kg.

Do not weigh your bag until later. Be ready to describe the process you used to fill your bag.



The Time of Our Lives!

You need: chart paper or butcher paper, markers, scissors, glue, magazines and newspapers Make 3 posters. Label your posters: “Things That Take One Second” “Things That Take One Minute” “Things That Take One Hour”

Collect pictures or make a list of things that belong on each poster.

How many things can you find? How can you check to make sure every event really belongs on each poster?



How Long Is a Minute?

You need: a way to time one minute, a partner How long do you think a minute is? Close your eyes. Tell your partner when to start timing. Raise your hand when you think a minute has passed.

Open your eyes and check the timer. Do you think you can get closer to one minute?

Try it again to see.



What Time?

The hands of a clock make an angle that is less than a right angle.

What time might it be?

Use words, numbers and pictures to record your solution.



How Much Rice Does It Hold?

You need: small paper bathroom cup, larger containers of a variety of sizes, rice, tray or tub to catch spills How many cups do you think it will take to fill the container? Record your estimate.

Check to see. Choose a different container. How many cups do you think it will take to fill the new container? Record your estimate.

Check to see. Repeat with a third container. Order your containers: holds most to holds least. Explain how you did it.



Same Blocks, Different Boxes

You need: linking cubes, a variety of boxes of different sizes Choose a box. How many linking cubes do you think it will take to fill it? Record your estimate.

Try it to see. Choose a different box. How many linking cubes do you think it will take to fill THIS box?

Record your estimate. Try it to see. How did you use the information from your first experiment to help you make a better estimate for the second experiment?



Same Box, Different Blocks

You need: snap cubes, cm cubes, a box How many linking cubes do you think it will take to fill the box? Record your estimate.

Try it to see. Using the cm cubes, how many do you think it will take to fill the same box?

Record your estimate. Try it to see. How did you use the information from your first experiment to help you make a better estimate for the second experiment?



Bottle Cap Count

You need: a variety of screw-on caps, juice glass, container of water, rice or sand, tray or tub to catch spills Estimate how many capfuls of water it will take to fill the juice glass. Record your estimate.

Count as you fill the glass with capfuls of water. How many did it take?

Find a different cap. Using this cap, do you think it will take more, fewer, or the same number of capfuls of water to fill the juice glass? Why?

Try it to see.



How Much Is a Cup?

You need: measuring cup, stuff to measure (rice, beans, sand, or water), a variety of different-sized containers (labeled for identification), tray or tub to catch spills Choose a container you think may hold about a cup.

Fill it with “stuff” and pour it into the measuring cup to check.

Choose 2 other containers to check. Write about what you discover. Use pictures, words, and numbers.



How Much Does It Hold?

You need: a container (such as a quart jar), scoop, rice and beans, rubber band, tray or tub to catch spills How many scoops of rice do you think will fill the jar? Record your estimate.

Put 3 scoops of rice in the jar. Put a rubber band on the jar to show the level of the rice.

Change your estimate if you wish. Continue until the jar is filled. Record the actual number of scoops it took.

How many scoops of beans do you think will fill the jar? Write about what you

think and why. Try it to see.


Cubes from Cubes

You need: snap cubes, paper/pencil for table If the first cube is 1 unit on an edge, it takes 1 unit cube to build it.

The next larger cube is 2 units on an edge. It takes 8 unit cubes to build it.

Build the next 3 cubes. Make a table. How many units are on an edge? How many unit cubes does it take to build each of the larger cubes?

Describe the pattern(s) you see. Use numbers to show how your pattern would continue if you kept building larger cubes.

If you know the number of unitsof a cube,the number of unit cubes it would take to build the larger cube.

on the edge explain how you could find out

If you know the number of unit cubes it takes to build a larger cube, can you figure out the units on a side? Explain.




Prisms from Cubes

You need: snap cubes, paper/pencil for table Use linking cubes to build a variety of rectangular prisms.

For each prism, record: the dimensions of your prism (Example: 2 x 3 x 4)

the number of cubes it took to build it.

Organize your data into a class chart. Describe the patterns you see in your data. If you know the dimensions of your prism, can you figure out how many cubes it takes to build it? Explain.

If you know the number of cubes it takes to build a prism, can you figure out its dimensions? Explain.



Hidden Hexahedra

You need: linking cubes, paper/pencil for recording Use linking cubes to build a cube with 3 units on a side. How many cubes does it take? Do you notice that one cube is hidden by the other cubes no matter how hard you look for it?

How many inside cubes are hidden from view in a cube that is 4 units on a side? Record your estimate first, then build the cube and find out.

How many cubes are hidden inside a cube that is 5 units on a side?

Keep investigating the hidden cubes inside a variety of different-sized cubes recording your estimate first.

Organize your information. Do you notice any patterns? Describe them.

Do your patterns help you to predict the number of hidden cubes in larger cubes of any size? How?

Transparency / Task Card 9D-1

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 9: Section D Incorporating Inquiry Page 72

Potato, Po-tah-to

What can you find that is bigger than a potato but lighter than it is?


Measurement Scavenger Hunt I

You need: A variety of measuring tools for length, area, volume, mass, temperature Look around the room, find something you estimate has a measurement that is very close to the Target Measurement on the Recording Sheet, and write its name in the Estimate column.

Measure the attribute of the object and record the actual measurement.

Record the difference between your actual measurement and the Target, which you used as your estimate.

Add the differences to find your total score.

The smallest Total Score WINS!


Repeat for each of the Target Measurements.


Measurement Scavenger Hunt I Recording Sheet

Target Measurement

Object of Estimation

Actual Measurement

Score (difference between Target and Actual)

72º F

60 square cm

112 inches

2 meters

84 square inches

5 yards

150 feet

Total Score:



Measurement Scavenger Hunt II

You need: A variety of measuring tools for length, area, volume, mass, temperature Look around the room and write down your estimates for the measurements of the things on your recording sheet.

Find and record their actual measures. Record the


differences between the actual measurements and your estimates.

Add the differences to find your Total Score.

The smallest score WINS!


Measurement Scavenger Hunt II Recording Sheet

Attribute To Be Measured

Estimate (unit)

Actual (unit)

Score (difference between

Estimate and Actual)

The temperature of the room

º F º F º F

The area of the top of my desk cm cm cm22 2

The perimeter of the bulletin board

inches inches inches

The height of the chalk board meters meters meters

The temperature outside

º F º F º F

The distance between our classroom & the library

yards yards yards

The distance around our trash can

cm cm cm

Total Score:



Measurement Scavenger Hunt III Recording Sheet

Attribute To Be Measured

Estimate (unit)

Actual (unit)

Score (difference between

Estimate and Actual)

Total Score:


Handout 9D-7


Step 1 Big Ideas






What’s In the Bag?

Each group has a bag with two different colors of cubes in it. Without looking in the bag, draw a cube and record its color on the recording sheet. Return the cube to the bag and repeat the process until you are confident you can draw a conclusion and describe the contents of your bag. Make a graph of the data on your recording sheet. Base your prediction of the contents of your bag on a total number of cubes fewer than 25. For example, your prediction might be: If there are 12 cubes in our bag, we predict that 4 are blue and 8 are red; or 4 of 12 are blue and 8 of 12 are red. Once you are confident about your prediction, select or design a spinner that represents the contents of your bag. Use the spinner to simulate drawing cubes from the bag (one spin represents one draw). Record and graph the data on your recording sheet. Compare the data set created by spinning the spinner to the data set created by drawing cubes. At least one other group has a bag with the same probabilities as your bag. Find a group with a matching spinner. Discuss why you think your bags contain contents with equal probabilities. “Dump” your bags to determine how close the probabilities of your bags actually are.

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 10: Section A What’s In the Bag? Page 12



What’s In the Bag? Recording Sheet

IA. Drawing Cubes Frequency Table

Red Blue

IB. Drawing Cubes Graph

R B Sketch or describe the spinner you selected or designed:

IIA. Spinner Simulation Frequency Table

Red Blue

IIB. Spinner Simulation Graph

R B















Step 1 Big Ideas




TEXTEAMS Rethinking Elementary School Mathematics Part II Day 10: Section B Fair or Not? Reflecting on Data Representation and Analysis Page 27

Handout 10B-2

Sample Big Ideas and Evidence of Understanding for Data Representation and Analysis

Using data to solve problems I can recognize uses of data in the world around me. I can use what I know about collecting and organizing data to decide if my answer

makes sense. I can talk about and write about the data I used to solve a problem.

Organizing and representing data to answer questions

I can collect data. I can select which data I need. I can sort data. I can choose a good way to show data. I can build a real graph to show data in an organized way. I can draw a picture graph to show data in an organized way. I can make a pictograph where one picture stands for more than one thing. I can make a bar graph to show data in an organized way. I can write a table to show data in an organized way. I can make a line graph from a table of number pairs. I can represent data in more than one way. I can decide what kind of graph is best for my data.

Describing data

I can answer questions about information in graphs and tables. I can describe data presented in tables and graphs. I can determine the spread of the data. I can determine the middle number of a set of data. I can use fractions to describe the results of an experiment.

Interpreting data

I can draw conclusions about information in graphs. I can tell if something is certain or impossible. I can tell how likely something is to happen. I can tell if things are equally likely to occur. I can list all possible outcomes of an experiment. I can use the shape of the data to answer questions about the results of an experiment. I can use numbers to answer questions about the results of an experiment. I can use experimental results to make predictions.


Handout 10B-2

Making predictions without data I can list all the possible outcomes of an experiment. I can determine how many possible outcomes there are. I can use all the possible outcomes to tell how likely something is. I can use numbers to compare one outcome to all possible outcomes. I can use fractions to describe the likelihood of an event.


Handout 10B-3

--4-- Communication to Support Learning (cont.)

Encouraging Reflection What does it mean to support reflection in a learning experience? The purpose of reflection is to look back at an experience in order to improve understanding. Tools such as journals and graphic organizers, where thoughts are recorded and shared, can enable students to reflect upon their learning. Teachers must model reflective processes, schedule time for student reflection, and offer instructive feedback to connect student reflection to the learning expectations. Questions to consider in planning ways to support reflection:

How can I implement the learning experience in a way that values and supports the development of students’ metacognitive strategies?

How can I provide students the opportunity to think about how they felt when

engaged in rigorous mathematics learning? How can I help students use the Evidence of Understanding to reflect on their work?

How can I use student reflection to assess what students have learned and plan further

instruction?

References National Research Council (2000). How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington, DC: National Academy Press. Strong, R. W., Silver, H. E., & Perini, M. J. (2001). Teaching what matters most: Standards and strategies for raising student achievement. Alexandria, VA: ASCD.



Fair or Not? Game One

Players 1 and 2 each toss a six-sided die. Player 1 wins if the faces differ by 0, 1, or 2. Player 2 wins if the faces differ by 3, 4, or 5. Predict: Is this a fair game? Roll the dice 20 times. Record the number of times that the difference is 0, the number of times the difference is 1, and so on. Also record the number of times each player wins. Is this a fair game? Pool your data with other players’ data. Do you see any patterns?



Data Representation for Game One

Difference of 0

Difference of 1

Difference of 2

Difference of 3

Difference of 4

Difference of 5

Player 1 wins

Player 2 wins



Fair or Not? Game Two

Each player tosses a six-sided die. Player 1 wins if the maximum face of the two tosses is a 1, 2, 3, or 5. Player 2 wins if the maximum face is a 4 or a 6. Predict: Is this a fair game? Roll the dice 20 times. Record the number of times that the maximum is 1, the number of times the maximum is 2, and so on. Also record the number of times each player wins. Is this a fair game? Pool data with other players. Do you see any patterns?



Data Representation for Game Two

Maximum of 1

Maximum of 2

Maximum of 3

Maximum of 4

Maximum of 5

Maximum of 6

Player 1 wins

Player 2 wins



TEXTEAMS Rethinking Elementary School Mathematics Part II Day 10: Section C Data Representation and Analysis Sampler Page 40

Lesson Planning Process Chart for the Data Representation and Analysis Sampler

Step 1 Big Ideas






Lesson Planning Process Chart for the Data Representation and Analysis Sampler

Step 1 Big Ideas






Peek Box

Materials: Small box with 10 marbles of various colors in it (taped shut with

one corner cut out just enough to view one marble) Paper and pencil for recording results

Design and carry out a plan to determine how many marbles of each color are in the box. (There are a total of 10 marbles in the box.) Write a description of your plan. Find a partner. Compare your plans.



Toss that Cube!

Think about tossing a number cube 30 times. Predict how many times the cube will land on 1, 2, 3, 4, 5, and 6. Write down your predictions.

Now, toss a number cube 30 times and record the result of each toss – 1, 2, 3, 4, 5, or 6. Graph your data in two different ways. Combine your data with the data from at least three other groups. Do the results change? If so, how?



Heads or Tails?

Materials: Coin or two-colored counter Paper and pencil for recording

Think about flipping a coin 25 times. Predict how many times heads will come up. Write down your prediction. ______________ Now, flip a coin 25 times and record each result – heads or tails. Graph your data in two different ways. Combine your data with the data from three other groups. Do the results change? If so, how?



Two-Dice Game

Materials: Game board for each player Two dice (1-6) 11 chips or counters for each player Paper and pencil for recording results

Think about rolling two dice and adding the two numbers that show. Predict which sums you think you will roll by placing your 11 chips on the totals on your game board that you think you will roll. (You may put as many of your chips as you like on any given total.) Now, roll the two dice, record the two numbers that show, then add the two numbers. The sum will be between 2 and 12. If you have a chip on that sum, remove it. (You may remove only one chip at a time. If you do not have a chip on that sum, you do nothing that turn.) The game is over when all of your chips have been removed from the gameboard. Play the game several times. Did your results match your predictions? What patterns do you notice?

Handout 10C-6


Two-Dice Game Board

Place your eleven chips on the sums that you predict you will roll.

2 3 4 5 6 7 8 9 10 11 12



It’s in the Bag?

Place ten red cubes and ten blue cubes in a paper bag. Predict which color you are most likely to draw if you remove one cube from the bag without looking. Remove a cube from the bag without looking. Record the color on a chart. Put the cube back into the bag. Remove another cube, record the color and return the cube to the bag. Continue removing a cube, recording the color and returning the cube to the bag until you have twenty pieces of data. Was your prediction correct? Next put fifteen red cubes and five blue cubes in the bag. Predict which color you are most likely to draw. Will your results be different? Why? Repeat the experiment, and check your prediction.


What’s the Spin?

Imagine spinning the spinner below 25 times. Predict how many times the spinner will land on blue, red and yellow. Record your predictions.

Spin the spinner 25 times and record the result of each spin – red, blue, or yellow. Graph your data in two different ways. Combine your data with the data from at least three other groups. Do the results change? If so, how?




Add It Up

Write the numbers from 2 to 12 in a column. If you roll two six-sided dice and add the numbers, which sum do you think would be most likely to occur? _______ Roll the dice, add the numbers, and record the sum with a tally mark next to the matching number on your paper. Continue the experiment until one of the numbers has ten tally marks. Which numbers received most of the tally marks? Which number received the fewest? _____ Why? Was your prediction correct?



Evens-Odds Number Game

Materials: Game board for each player 16 chips for each player Two tetrahedral (four-sided) dice

Examine the dice and decide how to read them. For this game you will be rolling the dice and multiplying the face numbers the dice land on to get a product. Before you roll the dice, predict whether you think you will roll more even products, more odd products, or about the same amount of each. Place half of your chips on the odd side and half on the even side of your game board. Take turns rolling the dice. For each turn, find the product of the numbers on the faces. Remove an “even” chip from your game board if your product is even and an “odd” chip if your product is odd. The first player to remove all chips from both sides of the game board wins. If a player rolls and cannot remove a chip from the game board, the turn is lost. Play several games. Decide how many chips to place on the even and odd sides of the game board. What type of chip placement most often results in a win? How would you place your chips if you were given 12 chips instead of 16?

Handout 10C-11


Evens-Odds Number Game Board

Even Odd



Fair and Square

Materials: Three number cubes (1-6) Paper and pencil for recording

Play this game with a partner. Player 1 rolls a pair of number cubes and multiplies the numbers showing to get a product. Player 2 rolls one number cube and squares the number showing. For example, if Player 2 rolls a 3, he or she would multiply 3 x 3 to get 9. The winner is the player with the greater result. Who do you predict will win most often? ______________ Play twenty rounds and record who wins each round. Graph your data in two different ways. Combine your data with the data from three other groups. Do the results change? If so, how?



Fair and Square 2

Materials: Three number cubes (1-6) Paper and pencil for recording

Play this game with a partner. Player 1 rolls a pair of number cubes and adds the numbers showing on the faces to get a sum. Player 2 rolls one number cube and multiplies the number showing on the face by 2 to get a product. The winner is the player with the greater result. Who do you predict will win most often? ______________ Play 20 rounds and record who wins each round. Graph your data in two different ways. Combine your data with the data from three other groups. Do the results change? If so, how?



Fair Counters?

With a partner, take turns tossing two two-color counters. If the counters land with the same color showing, Player 1 scores a point. If the counters show different colors, Player 2 scores a point. What are the possible outcomes? Is the game fair? Why or why not? Play the game until each player has taken ten turns, recording the results of each toss. The ratio of the number of times the counters showed a specific outcome to the total number of tosses is called the experimental probability of that outcome. What experimental probability does your data show for the outcome that the two counters will show the same color? ______ different colors? ______ Combine your results with those of the other groups. How does the experimental probability change? Can you determine the theoretical probability of getting the same color? If so, how does it compare to the experimental probability your data shows?



Mix and Match

Materials: Paper bag Two candies (or chips) of one color One candy (or chip) of a second color Paper and pencil for recording results

In a bag, put two candies of one color and one of a different color. Draw two candies out of the bag ten times, replacing them after each draw. Will you get more draws that are two candies of different colors, more draws that are candies of the same colors, or an equal number of both types of draws? Combine your data with data from other groups. How does your group data compare with the combined data of all groups? What do you think would happen if you repeated the experiment with three candies of one color and one of a different color?



Take Two

Materials: Seven chips Paper and pencil for recording results

This game takes two players. To begin the game “Take Two,” place seven chips in a row. Decide which player takes the first turn by flipping a coin. The two players take turns, removing one or two chips from the row on each turn. The player to remove the last chip is the winner. Alternate version: The winner is the player who does not remove the last chip. Is “Take Two” a fair game? Why or why not? Does it make a difference who plays first? What was your strategy for winning the game? How would the game change if the game was played with a row of eight chips? Nine chips?



Spin 1

Materials: Spinner Paper and pencil for keeping score

Game Rules 1. The player whose first name has the most letters is Player S. The

other player is Player D. If both first names have the same number of letters, use last names. If there is still a tie, use middle names.

2. To play a round of the game, spin the spinner twice. Player S

scores a point if the spinner lands on the same letter twice. Player D scores a point if the spinner lands on different letters. Take turns spinning the spinner each round. Play and record 20 rounds.

3. The winner is the player with more points at the end of 20 rounds.

Play two or three games. Does each player have an equal chance to win the game? Does the same person win each time? Is the game a fair game? Why or why not?

A

B

A

B



Spin 2


Game Rules 1. The player whose first name has the most letters is Player S. The

other player is Player D. If both first names have the same number of letters, use last names. If there is still a tie, use middle names.

2. To play a round of the game, spin the spinner twice. Player S

scores a point if the spinner lands on the same letter twice. Player D scores a point if the spinner lands on different letters. Take turns spinning the spinner each round. Play and record 20 rounds.

3. The winner is the player with more points at the end of 20 rounds.

Play two or three games. Does each player have an equal chance to win the game? Does the same person win each time? Is the game a fair game? Why or why not?

A

B



Spin Again 1


Game Rules 1. The player whose first name begins with the letter closest to A is

Number 1. The other player is Number 2. 2. Each time the spinner lands on a player’s number, that player will

get the point. Take turns spinning the spinner and recording. Each player spins 25 times.

3. The winner is the player with the most points after 50 spins. 4. Decide before playing who you predict will win the game and why.

Now play the game and see what happens.

1

1

1

2

2

2



Spin Again 2


Game Rules 1. Each player chooses one of the letters, A or B. 2. Each time the spinner lands on a player’s letter, that player will get

the point. Take turns spinning the spinner and recording. Each player will spin 25 times.

3. The winner will be the player with the most points after 50 spins. 4. Decide before playing who you predict will win the game and why.

Now play and see what happens.

B

AB

A

B

A

B

A



Dice Fractions 1

Materials: Two six-sided dice (1-6) Paper and pencil for keeping score

Game Rules 1. Decide which player will be A and which will be B. 2. Each player has a die. Both players roll at the same time. Use the

two numbers showing to make a fraction less than or equal to 1. 3. If the fraction is not in simplest terms, player A scores a point.

Otherwise, player B scores a point. 4. Play 12 rounds. 5. The winner is the player with more points at the end of 12 rounds. 6. Play two or three games, then answer these questions:

Does each player have an equal chance to win the game? Does the same person win each time? Is the game a fair game? Why or why not?



Dice Fractions 2

Materials: Two six-sided dice (1-6) of different colors Paper and pencil for keeping score

Game Rules 1. Decide which player will be A and which will be B. Decide which

color die will be the numerator and which will be the denominator. _____________________________________

2. Each player has a die. Both players roll at the same time. Make a

fraction with the numerator and the denominator identified by the colors chosen before the game began.

3. If the fraction is greater than 1, Player A scores a point. If the

fraction is less than 1, Player B scores a point. If the fraction equals 1, each player scores a point.

4. Play 12 rounds. 5. The winner is the player with more points at the end of 12 rounds. 6. Play two or three games, then answer these questions:




Chips 1

Materials: 2 red chips, each with A on one side and B on the other 1 blue chip with A on one side and B on the other Paper and pencil for keeping score

Game Rules 1. Decide which player will be Player 1 and which will be Player 2. 2. For each round, all three chips will be flipped at the same time.

Player 2 scores a point if both red chips show A or the blue chip shows A or all three chips show A. Otherwise, Player 1 scores a point. Players take turns flipping the chips and recording each round. Each player flips 8 rounds.





Chips 2

Materials: 3 yellow chips, each with A on one side and B on the other 1 green chip with A on one side and B on the other Paper and pencil for keeping score

Game Rules 1. Decide which player will be Player 1 and which will be Player 2. 2. For each round all four chips will be flipped at the same time.

Player 1 scores a point if all three yellow chips show A or the green chip shows A or all four chips show A. Otherwise, Player 2 scores a point. Players take turns flipping the chips and recording each round. Each player flips 12 rounds.





Match or Not 1

Materials: One chip with X on both sides One chip with X on one side and Y on the other side Paper and pencil for keeping score

Game Rules 1. Decide which player will be MATCH and which will be NO

MATCH. 2. Each round both chips are flipped at the same time. Player

MATCH scores a point if the chips match. Player NO MATCH scores a point if the chips do not match. Take turns flipping and recording.

3. The winner is the first player to get 15 points. Play two or three

games, then answer these questions:




Match or Not 2

Materials: One chip with A on one side and B on the other side One chip with A on one side and C on the other side One chip with B on one side and C on the other side Paper and pencil for keeping score

Game Rules 1. Decide which player will be MATCH and which will be NO

MATCH. 2. Flip all three chips at the same time. Player MATCH scores a point

if two of the chips match. Player NO MATCH scores a point if all three chips are different.

3. The winner is the first player to get 15 points. Play two or three

games. Then answer these questions:




Dice, Dice 1

Materials: Three six-sided dice Paper and pencil for keeping score

Game Rules 1. Decide which player will be Player A and which will be Player B. 2. For a round, roll all three dice at the same time and find the sum

of the three numbers. 3. If the sum is even, Player A scores a point. If the sum is odd,

Player B scores a point. Players take turns rolling the three dice and recording. Each player will roll 10 rounds.





Dice, Dice 2

Materials: Three six-sided dice, one of one color and two of another color Paper and pencil for keeping score

Game Rules 1. Decide which player will be Player A and which will be Player B. 2. For a round, roll all three dice at the same time and find the sum of

the numbers on the two dice of the same color. Find the difference between that answer and the number on the third die.

3. If the difference is odd, Player A scores a point. If the difference is

even, Player B scores a point. Take turns rolling the dice and recording. Each player will roll 10 rounds.





Design a Spinner

Design a spinner that fits these criteria: Blue has a 50 % chance of winning. Red has one chance in four of winning. Yellow and green are equally likely to win. Orange cannot possibly win.

Compare your spinner with one made by another group. Are they the same or different?


Looking Back

Activity ______________________ 1. Can you determine the theoretical probability for this activity? If

so, what is it? 2. How would knowing the theoretical probability affect the way you

did the activity or played the game? 3. If you did the activity again, what techniques would you use to

make your predictions? 4. How could you make the activity more mathematically rigorous? 5. Does the activity promote inquiry? If so, how? If not, how can you

change the activity to promote inquiry? 6. In answering these questions and revisiting the activities, what

have you been engaging in?

TEXTEAMS Rethinking Elementary School Mathematics Part II Day 10: Section D Looking Back Page 76

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