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

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1 Transparency / Handout 6A-1 Appointment Sheet 1 st Appointment 2 nd Appointment 3 rd Appointment 4 th Appointment 5 th Appointment 6 th Appointment Day 6: Section A Clock Arithmetic Page 9

2 Transparency / Handout 6A-3 Clock Arithmetic mod12 mod10 mod8 mod5 Day 6: Section A Clock Arithmetic Page 11

3 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 Make sure the centimeters are accurate at the overlapped edges 5. Tape the strips together Day 6: Section A Clock Arithmetic Page 12

4 Handout 6A Day 6: Section A Clock Arithmetic Page 13

5 Transparency / Handout 6A-5 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, = 1 mod12. The same problem on the mod10 clock would be = 3 mod10. Problem Number Line Solution Clock Solution 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? Day 6: Section A Clock Arithmetic Page 14

6 Transparency / Handout 6A-6 Clock Arithmetic Recording Sheet 2 Problem Number line Solution mod12 Clock Solution mod10 Clock Solution mod8 Clock Solution mod5 Clock Solution 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? Day 6: Section A Clock Arithmetic Page 15

7 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 Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 23

8 Handout 6B 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 24

9 Handout 6B 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 25

10 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 26

11 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 27

12 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 28

13 Handout 6B 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? 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? Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 29

14 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. 1 Each aquarium contains 40 gallons of water. The dimensions of the aquarium are 36 inches inches wide and 23 inches tall. Eric decides to put a tape measure along the 2 1 long, 2 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 30

15 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. Day 6: Section B Investigating Algebraic Thinking and The Lesson Planning Process Page 31

16 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.) Day 6: Section C Patterns, Relationships, and Algebraic Thinking Sampler Page 36

17 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. Day 6: Section C Patterns, Relationships, and Algebraic Thinking Sampler Page 37

18 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 Day 6: Section C Patterns, Relationships, and Algebraic Thinking Sampler Page 38

19 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 Day 6: Section C Patterns, Relationships, and Algebraic Thinking Sampler Page 39

20 10 x 22 centimeter grid Transparency / Handout 6C-4 Day 6: Section C Name Scarves Page 45

21 Centimeter Graph Paper Transparency / Handout 6C-5 Day 6: Section C Name Scarves Page 46

22 Transparency / Handout 6C-6 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? Day 6: Section C Two of Everything Page 49

23 Transparency / Handout 6C-6 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. Day 6: Section C Two of Everything Page 50

24 Transparency / Handout 6C-7 Hundred Chart Day 6: Section C Odd and Even Page 54

25 1 Inch Graph Paper Handout 6C-8 Day 6: Section C Odd and Even Page 55

26 Transparency / Handout 6C-9 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? Day 6: Section C Odd and Even Page 56

27 Transparency / Handout 6C-12 Number Machines Recording Sheet This is a +5 Number Machine. Record the missing numbers on the T- chart. Add Five Machine +5 In Out Look at this T-chart. What would you name this machine? Machine +5 In Out What is the mathematical rule that explains what this machine does to the In numbers? Day 6: Section C Number Machines Page 62

28 Transparency / Handout 6C-12 What would you name this machine? Complete this T-chart. Machine +5 In Out 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 Day 6: Section C Number Machines Page 63

29 Transparency / Handout 6C-13 Multiplication Chart x Day 6: Section C Fraction Frame Game Page 67

30 Transparency / Handout 6C-14 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 Order 1097 Order 1098 Order 1099 TOTAL Day 6: Section C Just Being Crafty Page 70

31 Transparency / Handout 6C-15 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. Day 6: Section C Just Being Crafty Page 71

32 Transparency / Handout 6C-16 99,999 Recording Sheet A) B) C) D) Day 6: Section C 99,999 Puzzle Page 73

33 Transparency / Handout 7A-1 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. Scale: 2 1 in. = 1 ft. 12 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. Day 7: Section A The Stars Problem Page 7

34 Transparency / Handout 7A-1 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. Day 7: Section A The Stars Problem Page 8

35 Transparency / Handout 6A-3 Clock Arithmetic mod12 mod10 mod8 mod5 Day 7: Section B The Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 16

36 Transparency / Handout 7B-1 Lesson Planning Process Chart Step 1 Big Ideas Step 2 Evidence of Understanding Step 3 Orchestrating for Rigorous Learning Step 4 Communication to Support Learning Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 17

37 Transparency / Handout 7B-2 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.) Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 18

38 Transparency / Handout 7B-2 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. Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 19

39 Transparency / Handout 7B-3 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. Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 20

40 Transparency / Handout 7B-3 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. Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 21

41 Transparency / Handout 7B-4 Modular Art Mod5 Number Patterns Mod5 Addition Chart Mod5 Multiplication Chart x Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 22

42 Transparency / Handout 7B-5 Modular Art Quilt Tile Square Patterns Mod5 Quilt Pattern Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 23

43 Transparency / Handout 7B-6 Modular Art Mod5 Quilt Patterns Recording Sheet Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 24

44 Transparency / Handout 7B-7a Modular Art Translations Recording Sheet Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 25

45 Transparency / Handout 7B-7b Modular Art Recording Sheet Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 26

46 Transparency / Handout 7B-8 Modular Art Rotations Recording Sheet Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 27

47 Transparency / Handout 7B-9 Modular Art Reflections Recording Sheet Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 28

48 Transparency / Handout 7B-10 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 A C E D A Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4 B Translations C D E A B Rotations C D E A Reflections B C D E Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 29

49 Handout 7B 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. Day 7: Section B Modular Art: Framing Algebraic Thinking and Spatial Reasoning Experiences Page 30

50 Transparency / Handout 7C-1 Lesson Planning Process Chart for the Connecting Algebraic Thinking and Spatial Reasoning Sampler Step 1 Big Ideas Step 2 Evidence of Understanding Step 3 Orchestrating for Rigorous Learning Step 4 Communication to Support Learning Day 7: Section C Connecting Algebraic Thinking and Spatial Reasoning Sampler Page 35

51 Transparency / Handout 7C-1 Lesson Planning Process Chart for the Connecting Algebraic Thinking and Spatial Reasoning Sampler Step 1 Big Ideas Step 2 Evidence of Understanding Step 3 Orchestrating for Rigorous Learning Step 4 Communication to Support Learning Day 7: Section C Connecting Algebraic Thinking and Spatial Reasoning Sampler Page 36

52 Transparency / Handout 7C-3 A Mysterious Tadpole Day 7: Section C The Mysterious Tadpole Page 43

53 Transparency / Handout 7C-4 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. Day 7: Section C The Mysterious Tadpole Page 44

54 Making Rectangles Grid Transparency / Handout 7C-6 Day 7: Section C Making Rectangles Page 48

55 Transparency / Handout 7C-7 Making Rectangles Recording Sheet Rectangle Number Number of square centimeters Day 7: Section C Making Rectangles Page 49

56 Centimeter Graph Paper Handout 7C-8 Day 7: Section C Measuring Area with Rectangles Page 51

57 Transparency / Handout 7C-9 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 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? Day 7: Section C Measuring Area with Rectangles Page 52

58 Transparency / Handout 7C-10 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 3 2. 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 Day 7: Section C Similar Quadrilaterals Page 55

59 Transparency / Handout 7C-10 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? Day 7: Section C Similar Quadrilaterals Page 56

60 Transparency / Handout 7C-11 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 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. Day 7: Section C Similar Quadrilaterals Page 57

61 Transparency / Handout 7C-11 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? Day 7: Section C Similar Quadrilaterals Page 58

62 Transparency / Handout 7C-12 Examples of figures that are connected: Day 7: Section C Balloon Geometry Page 64

63 Transparency / Handout 7C-13 Examples of figures that are not connected: Day 7: Section C Balloon Geometry Page 65

64 Transparency / Handout 7C-14 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? Day 7: Section C Balloon Geometry Page 66

65 Transparency / Handout 7C-15 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. Day 7: Section C Shadow Geometry Page 70

66 Transparency / Handout 7C-16 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? Day 7: Section C Comparing K Page 73

67 Transparency / Handout 8B-1 Lesson Planning Process Chart Step 1 Big Ideas Step 2 Evidence of Understanding Step 3 Orchestrating for Rigorous Learning Step 4 Communication to Support Learning Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 26

68 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. Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 27

69 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. Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 28

70 Handout 8B 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. Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 29

71 Handout 8B-4 Sorting Cards Set A Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 30

72 Handout 8B-5 Sorting Cards Set B Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 31

73 Handout 8B-6 Sorting Cards Set C Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 32

74 Handout 8B-7 Sorting Cards Set D Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 33

75 Handout 8B-8 Sorting Cards Set E Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 34

76 Handout 8B-9 Sorting Cards Set F Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 35

77 Transparency / Handout 8B-10 How We Sorted Our Shape Cards Cards we used: How we sorted: Sorting # 1 Cards we used: How we sorted: Sorting # 2 Cards we used: How we sorted: Sorting # 3 Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 36

78 Transparency / Handout 8B-10 How We Sorted Our Shape Cards (continued) Cards we used: How we sorted: Sorting # 4 Cards we used: How we sorted: Sorting # 5 Cards we used: How we sorted: Sorting # 6 Day 8: Section B Developing Deep and Complex Geometric Knowledge Page 37