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2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5

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    1. 2010 Mathematics Institute Patterns, Functions, and Algebra Grades 3-5

    2. Agenda Introductions Big Topics Patterns Properties Lunch Equalities and Inequalities Explain parking lotExplain parking lot

    3. Big Topics Patterns Using Number Lines Properties Building Vocabulary Equations and Inequalities Keeping it Balanced

    4. Unpacking Patterns Open activity unpacking the Pattern SOL. Split everyone into 6 tables. Each table is assigned one of the patterns SOL: Give each person an unpacking form and a copy of the curriculum framework for that SOL. Each person fills out the unpacking form and then one person will be the recorder and write it on chart paper. verbs using curriculum framework done at each table with 6 sols on chart paper. Then, have each person number off KG - 5th grade. They will then go to the place on the wall where they numbered off. When they get to the SOL that their table created, they are the one in charge of that chart. The group will discuss what they notice about the progression of the SOL through the grade levels. A new person in each of these groups should be the recorder and record what the group discusses about the progression.Open activity unpacking the Pattern SOL. Split everyone into 6 tables. Each table is assigned one of the patterns SOL: Give each person an unpacking form and a copy of the curriculum framework for that SOL. Each person fills out the unpacking form and then one person will be the recorder and write it on chart paper. verbs using curriculum framework done at each table with 6 sols on chart paper. Then, have each person number off KG - 5th grade. They will then go to the place on the wall where they numbered off. When they get to the SOL that their table created, they are the one in charge of that chart. The group will discuss what they notice about the progression of the SOL through the grade levels. A new person in each of these groups should be the recorder and record what the group discusses about the progression.

    5. Multiplication shown on a Number Line 2009 SOL 3.6 5 Have participants use smart pal to write multiplication factHave participants use smart pal to write multiplication fact

    6. 6 Number lines can be used to discuss skip counting, but can also bridge skip counting to multiplication sentences. When looking at the first number line, Factor Frog has jumped part of the way, but hasnt made it the whole way. If Factor Frog has to jump four spaces, six times, where will he land? Help Factor Frog finish hopping. Record his hops. On the second number line, Factor Frog has jumped five spaces twice, but he needs to jump five spaces, five times. Help Factor Frog finish his hops. Where will he land? Record his hops. On the third number line, Factor Frog needs to hop 10 spaces, three times. Record his hops. Where will Factor Frog end on this number line?Number lines can be used to discuss skip counting, but can also bridge skip counting to multiplication sentences. When looking at the first number line, Factor Frog has jumped part of the way, but hasnt made it the whole way. If Factor Frog has to jump four spaces, six times, where will he land? Help Factor Frog finish hopping. Record his hops. On the second number line, Factor Frog has jumped five spaces twice, but he needs to jump five spaces, five times. Help Factor Frog finish his hops. Where will he land? Record his hops. On the third number line, Factor Frog needs to hop 10 spaces, three times. Record his hops. Where will Factor Frog end on this number line?

    7. 7 Factor Frog has hopped three spaces, five times, and while his friend, Multiple Mouse hopped on the number line below, six spaces, two times. What landing spots were the same or common? What was their first common landing spot? This is their Least Common Multiple. Our Frog ActivityFactor Frog has hopped three spaces, five times, and while his friend, Multiple Mouse hopped on the number line below, six spaces, two times. What landing spots were the same or common? What was their first common landing spot? This is their Least Common Multiple. Our Frog Activity

    8. 8 Now, we are going to take the number lines we have explored with and stack them in a number grid or hundreds chart to explore Prime and Composites. In this activity, we are the owners of a Bubble Gum Factory. When we work in the bubble gum factory, we get to chew gum. Now, we are going to take the number lines we have explored with and stack them in a number grid or hundreds chart to explore Prime and Composites. In this activity, we are the owners of a Bubble Gum Factory. When we work in the bubble gum factory, we get to chew gum.

    9. 9 The gum that we work with, comes to us in 1-inch pieces. It is our job to stretch the gum to different sizes for different companies.The gum that we work with, comes to us in 1-inch pieces. It is our job to stretch the gum to different sizes for different companies.

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    15. Primes and Composites 2009 SOL 5.3 15

    16. Primes and Composites 2009 SOL 5.3 16

    17. Primes and Composites 2009 SOL 5.3 17

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    19. Think/Pair/Share 19 What did you find useful in the activities? What connections did you see that are helpful from grade level to grade level?What did you find useful in the activities? What connections did you see that are helpful from grade level to grade level?

    20. Properties Vocabulary New Vocabulary will be introduced in third, fourth, and fifth grade. It is important for your students to connect these next activities to the new vocabulary they need to know. A great way to do this is for them to compare and contrast the characteristics of the vocabulary. This template was inspired by a Dan Mulligan activity. It would be important for students to discuss the characteristics of the properties and the non characteristics of the properties as well. Take time to discuss with a partner which property this is and why. New Vocabulary will be introduced in third, fourth, and fifth grade. It is important for your students to connect these next activities to the new vocabulary they need to know. A great way to do this is for them to compare and contrast the characteristics of the vocabulary. This template was inspired by a Dan Mulligan activity. It would be important for students to discuss the characteristics of the properties and the non characteristics of the properties as well. Take time to discuss with a partner which property this is and why.

    21. Properties Vocabulary Now it is your turn. Choose a property and write down the characteristics that would fit that property and characteristics that would NOT fit that property. Can someone at your table name your property? Now it is your turn. Choose a property and write down the characteristics that would fit that property and characteristics that would NOT fit that property. Can someone at your table name your property?

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    23. Commutative Property 2009 SOL 3.20 23

    24. Commutative Property 2009 SOL 3.20 24 PINK PAPERPINK PAPER

    25. Commutative Property 2009 SOL 3.20 25

    26. Commutative Property 2009 SOL 3.20 26 Build an area model with cm cubes to show cummutativity. Ask them to build a model for 24. Discuss at table and share out.Build an area model with cm cubes to show cummutativity. Ask them to build a model for 24. Discuss at table and share out.

    27. Automaticity 27

    28. Associative Property 2009 SOL 4.16b 28

    29. Associative Property 2009 SOL 4.16b 29 Students should have multiple opportunities to build cubes to understand the volume. Finding the volume of a cube is in the new 2009 SOL. This is new for fifth grade (solving volume) we have not had to calculate volume up to this point. This is a good lead in to the fifth grade standard for measurement.Students should have multiple opportunities to build cubes to understand the volume. Finding the volume of a cube is in the new 2009 SOL. This is new for fifth grade (solving volume) we have not had to calculate volume up to this point. This is a good lead in to the fifth grade standard for measurement.

    30. Distributive Property 2009 SOL 5.19 30

    31. Distributive Property 2009 SOL 5.19 31 Activity for audience. Give them a piece of graph paper. Ask them to cut out an array in which they may not know the answer. Have them cut the array into smaller arrays (either cutting vertically or horizontally, not both) in which they do know the answer. Record the answer to the smaller array and add them up to find the answer to the original problem. They can share some of their answers. Blue cm grid paper and white grid paper to show model..work in pairs, each cut out the same area model from the cm paper. MODEL FOR THEM FIRST!Activity for audience. Give them a piece of graph paper. Ask them to cut out an array in which they may not know the answer. Have them cut the array into smaller arrays (either cutting vertically or horizontally, not both) in which they do know the answer. Record the answer to the smaller array and add them up to find the answer to the original problem. They can share some of their answers. Blue cm grid paper and white grid paper to show model..work in pairs, each cut out the same area model from the cm paper. MODEL FOR THEM FIRST!

    32. Think/Pair/Share What did you find useful in the activities? What connections did you see that are helpful from grade level to grade level? What did you find useful in the activities? What connections did you see that are helpful from grade level to grade level?

    33. Equations and Inequalities Vertical Alignment Modeling one step linear equalitiesVertical Alignment Modeling one step linear equalities

    34. What does the equal sign mean? 34 Using ideas from the article Childrens Understanding of Equality: A Foundation For Algebra lead discussion on what students will often say when asked that question - it means the answer. How can we facilitate the understanding of the equal sign? Using ideas from the article Childrens Understanding of Equality: A Foundation For Algebra lead discussion on what students will often say when asked that question - it means the answer. How can we facilitate the understanding of the equal sign?

    35. Equalities 2009 SOL 4.16a 35 Show the blue number balancesparticipants write on the smart pals.Robin makes a model, Kim make a modelWho can come up with a third?Show the blue number balancesparticipants write on the smart pals.Robin makes a model, Kim make a modelWho can come up with a third?

    36. 36 Using the Commutative property, we can allow students more opportunities to explore equality. Representing this property in a variety of forms will provide them more opportunities to connect the characteristics of these properties to future mathematical endeavors.Using the Commutative property, we can allow students more opportunities to explore equality. Representing this property in a variety of forms will provide them more opportunities to connect the characteristics of these properties to future mathematical endeavors.

    37. 37 Using the Commutative property, we can allow students more opportunities to explore equality. Representing this property in a variety of forms will provide them more opportunities to connect the characteristics of these properties to future mathematical endeavors.Using the Commutative property, we can allow students more opportunities to explore equality. Representing this property in a variety of forms will provide them more opportunities to connect the characteristics of these properties to future mathematical endeavors.

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    39. Equalities 2009 SOL 4.16a 39 Will students agree that 8 = 1 + 7? Many times they think that it has to say 1 + 7 = 8. Use the balance to prove that they are equal. Continue with the other equations. Discuss commutative property? How many different ways can you show 9 = 9? Can you show 10 - 1?Will students agree that 8 = 1 + 7? Many times they think that it has to say 1 + 7 = 8. Use the balance to prove that they are equal. Continue with the other equations. Discuss commutative property? How many different ways can you show 9 = 9? Can you show 10 - 1?

    40. Equalities 2009 SOL 4.16a 40 Hand out sets of True/False cards to groups of 2 or 3. Ask them to sort them into True and False piles. Then lead a discussion as to why some of them are true and some of them are false encouraging the use of properties. Use Double Mat (front and back) and cut cards for sorting cardsHand out sets of True/False cards to groups of 2 or 3. Ask them to sort them into True and False piles. Then lead a discussion as to why some of them are true and some of them are false encouraging the use of properties. Use Double Mat (front and back) and cut cards for sorting cards

    41. 41 Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation? Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation?

    42. 42 Students can discover what the unknown variable is here without having to write an equation. Students can discover what the unknown variable is here without having to write an equation.

    43. 43 Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation? Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation?

    44. 44 Now you try. How might you write this equation? What is the value for J here? Now you try. How might you write this equation? What is the value for J here?

    45. 45 How might you write this equation? What is the value for B here? How might you write this equation? What is the value for B here?

    46. 46 Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation? Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation?

    47. 47 How might you write this equation? What is the value for B here? How might you write this equation? What is the value for B here?

    48. 48 Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation? Again, this is an equation using multiplication. This would NOT be tested in fifth grade, but could be used to differentiate for those students who are ready for that discussion. What is the value for B here? How might you write this equation?

    49. 49 Place one redsquare on the left pan and one bluecircle on the right pan. Are they equal in weight? If not, which weighs more? Try to make the pans balance by adding another shape to one pan. Which pan should get another weight? Why? Continue adding either redsquares or bluecircles to the pans. When the balance, an entry will be added to the table. Do you think you know the weight of the bluecircle? Check by clicking the Guess Weights button and entering your guess. Is there another combination of redsquares and bluecircles that will make the pans balance? Try to find at least one more. Conduct similar investigations to find out how redsquares and bluecircles are related to pinktriangles and yellowdiamonds. Place one redsquare on the left pan and one bluecircle on the right pan. Are they equal in weight? If not, which weighs more? Try to make the pans balance by adding another shape to one pan. Which pan should get another weight? Why? Continue adding either redsquares or bluecircles to the pans. When the balance, an entry will be added to the table. Do you think you know the weight of the bluecircle? Check by clicking the Guess Weights button and entering your guess. Is there another combination of redsquares and bluecircles that will make the pans balance? Try to find at least one more. Conduct similar investigations to find out how redsquares and bluecircles are related to pinktriangles and yellowdiamonds.

    50. Modeling One-step Linear Equations 2009 SOL 5.18c 50 This is what 5th grade is preparing students for in sixth grade.here is an example of what they will see and be doing in 6th grade beyond this 5th grade standard. Move the slider to adjust the value ofx. For what value ofx do the red and blue pans have equal values? What other observations can you make about the relationship between the values of the pans? Can you model this with your candy and erasers? Now you try! Separate your cards into two stacks. If you choose from the equation stack, then show your answer by modeling. If you choose from the modeling stack, then record your equation on the recording sheet.This is what 5th grade is preparing students for in sixth grade.here is an example of what they will see and be doing in 6th grade beyond this 5th grade standard. Move the slider to adjust the value ofx. For what value ofx do the red and blue pans have equal values? What other observations can you make about the relationship between the values of the pans? Can you model this with your candy and erasers? Now you try! Separate your cards into two stacks. If you choose from the equation stack, then show your answer by modeling. If you choose from the modeling stack, then record your equation on the recording sheet.

    51. Think/Pair/Share What did you find useful in the activities? What connections did you see that are helpful from grade level to grade level? What did you find useful in the activities? What connections did you see that are helpful from grade level to grade level?

    52. 52 Visit the Parking Lot What questions came up? Address questions or table them to send to Debbie or Michael. Thanks everyone! ? Visit the Parking Lot What questions came up? Address questions or table them to send to Debbie or Michael. Thanks everyone! ?