2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra

1. 2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra

2. 2

3. Supporting Implementation of 2009 Standards Highlight key curriculum changes. 3

4. Properties of Operations 4

5. 5 Properties of Operations: 2001 Standards 7.3 The student will identify and apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero. 8.1 The student will a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;

6. 6 Properties of Operations: 2009 Standards 2001.7.3 focused on identifying; that has now moved down to grades 3-6, and the 2009 SOL focus in middle school is on applying the properties. 8.15c represents a new (and appropriate) change that is representative of the overall changes in the 2009 Standards.2001.7.3 focused on identifying; that has now moved down to grades 3-6, and the 2009 SOL focus in middle school is on applying the properties. 8.15c represents a new (and appropriate) change that is representative of the overall changes in the 2009 Standards.

7. 7 3.20a&b: Identity Property for Multiplication A multiplication chart can be used for more than just finding products: It�s a natural extension of the 100�s chart that students use in grades K-2. It can help students understand the meaning of multiplication and its properties if teachers highlight its structure.A multiplication chart can be used for more than just finding products: It�s a natural extension of the 100�s chart that students use in grades K-2. It can help students understand the meaning of multiplication and its properties if teachers highlight its structure.

8. 8 3.20a&b: Commutative Property for Multiplication This slide is connected to the following slide. Might want to make sure that participants are in fact familiar/comfortable with this idea of the perfect squares as a symmetry line.This slide is connected to the following slide. Might want to make sure that participants are in fact familiar/comfortable with this idea of the perfect squares as a symmetry line.

9. 9 3.20a&b: Commutative Property for Multiplication If you imagine the multiplication as Geoboard, you could stretch a rubber band from the upper-left corner (0 x 0 = 0) to form rectangles of different areas. The number in the lower right-hand corner of each rectangle corresponds to the area of the rectangle. For any pair of factors, it doesn�t matter if you stretch horizontally first or vertically first � the area (product) will be the same, just as if you had rotated the rectangle the way we did with the base ten blocks.If you imagine the multiplication as Geoboard, you could stretch a rubber band from the upper-left corner (0 x 0 = 0) to form rectangles of different areas. The number in the lower right-hand corner of each rectangle corresponds to the area of the rectangle. For any pair of factors, it doesn�t matter if you stretch horizontally first or vertically first � the area (product) will be the same, just as if you had rotated the rectangle the way we did with the base ten blocks.

10. 10 6.19: Multiplicative Property of Zero Highlights the relationship between the C&E and Geometry strands: One-dimensional line segments have no area; two-dimensional rectangles define an area (a product) based on the lengths of their sides (their factors).Highlights the relationship between the C&E and Geometry strands: One-dimensional line segments have no area; two-dimensional rectangles define an area (a product) based on the lengths of their sides (their factors).

11. 11 Meanings of Multiplication Multiplication is typically introduced as repeated addition or �skip counting�, but students need to develop progressively more sophisticated understandings of multiplication as they move through elementary school and into the middle grades. �Groups-of� and �Area� conceptions of multiplication are flexible models that can be easily extended from whole numbers to rational numbers.Multiplication is typically introduced as repeated addition or �skip counting�, but students need to develop progressively more sophisticated understandings of multiplication as they move through elementary school and into the middle grades. �Groups-of� and �Area� conceptions of multiplication are flexible models that can be easily extended from whole numbers to rational numbers.

12. 12 3.6: Represent Multiplication Using an Area Model The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.

13. 13 3.6: Represent Multiplication Using an Area Model The area model provides a nice illustration of the commutative property: Changing the orientation of the rectangle doesn�t change the length of its sides (the factors) or its area (the product).The area model provides a nice illustration of the commutative property: Changing the orientation of the rectangle doesn�t change the length of its sides (the factors) or its area (the product).

14. 14 3.6: Represent Multiplication Using an Area Model The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.

15. 15 5.19: Distributive Property of Multiplication The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.

16. 16 5.19: Distributive Propertyof Multiplication Over Addition The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.The area model is closely related to the �groups of� and �array� meanings of multiplication (the applet even references �3 groups of 6�), but it has two important distinctions: It�s a measurement. It always uses square units.

17. 17 5.19: Distributive Property of Multiplication Over Addition Most participants will be familiar with only the traditional algorithm, so that point of this slide (and the next two) is to connect the area model to the traditional algorithm. *If you download the free trial version of the NLVM software (http://www.mattimath.com/), you can show other examples using this applet without having access to the Internet. Most participants will be familiar with only the traditional algorithm, so that point of this slide (and the next two) is to connect the area model to the traditional algorithm. *If you download the free trial version of the NLVM software (http://www.mattimath.com/), you can show other examples using this applet without having access to the Internet.

18. 18 5.19: Distributive Property of Multiplication Over Addition Connect to the previous slide. Specifically ask participants to discuss at their tables all of the connections that they can see. Have them share out with the whole group. Acknowledging the connection to FOIL when it comes up (it�s almost guaranteed that someone will bring it up, but we should be sure to if no one else does) is a powerful way to increase participant �buy in� for using the area model and partial products as a foundational instructional model.Connect to the previous slide. Specifically ask participants to discuss at their tables all of the connections that they can see. Have them share out with the whole group. Acknowledging the connection to FOIL when it comes up (it�s almost guaranteed that someone will bring it up, but we should be sure to if no one else does) is a powerful way to increase participant �buy in� for using the area model and partial products as a foundational instructional model.

19. 19 Strengths of the Area Model of Multiplication The last bullet is the key point.The last bullet is the key point.

20. 20 4.16b: Associative Property for Multiplication The associative property states that the grouping of the factors doesn�t affect the product. Possible confusion: There are actually three different possible orientations of this rectangular solid, but the associative property only deals with two of them at a time. We may need to address this by showing how it would work with the missing orientation from our sample problem. e.g., (3 x 2) x 4 and 3 x (2 x 4) *Participants may confuse this idea with the commutative property, but we are not talking here about 2 x 3 x 4 = 2 x 4 x 3.The associative property states that the grouping of the factors doesn�t affect the product. Possible confusion: There are actually three different possible orientations of this rectangular solid, but the associative property only deals with two of them at a time. We may need to address this by showing how it would work with the missing orientation from our sample problem. e.g., (3 x 2) x 4 and 3 x (2 x 4) *Participants may confuse this idea with the commutative property, but we are not talking here about 2 x 3 x 4 = 2 x 4 x 3.

21. Expressions and Equations

22. Tell participants that they will use the algebra tiles on their tables as we work with expressions and equations.Tell participants that they will use the algebra tiles on their tables as we work with expressions and equations.

23. What do these tiles represent? 23 Note to presenters: It would be good to mention that with an interactive white board works nicely with this type of exploration. It might also be good to tell the audience that no special program was used to create the representations of the tiles. We want to start by establishing the value of the algebra tiles. (~10 minutes)� Many students will relate these tile to the base ten blocks and want to name them very concretely with a number value. � We start with the yellow tile which does represent one unit. We declare that it is one unit tall and one unit wide, so it has an area of one square unit. � Next, we move to the green tile. It is one unit tall. However, its width can vary. The tiles shown above are about 2.5 units long. The manipulatives our participants will have are about 5.1 units long. The width is unknown, so we name it with a variable, x. The green tile is one unit by x units, so it has an area of x square units. (The Multiplicative Identity can be mentioned here.) � After this, we take a look at the blue tile. It is x units tall and x units wide. (Many students will still go back to the yellow tiles as a unit of measurement.) The blue tile has an area of x2 square units. � Finally, discuss that the red tiles are used to denote negative quantities. Note to presenters: It would be good to mention that with an interactive white board works nicely with this type of exploration. It might also be good to tell the audience that no special program was used to create the representations of the tiles. We want to start by establishing the value of the algebra tiles. (~10 minutes)� Many students will relate these tile to the base ten blocks and want to name them very concretely with a number value. � We start with the yellow tile which does represent one unit. We declare that it is one unit tall and one unit wide, so it has an area of one square unit. � Next, we move to the green tile. It is one unit tall. However, its width can vary. The tiles shown above are about 2.5 units long. The manipulatives our participants will have are about 5.1 units long. The width is unknown, so we name it with a variable, x. The green tile is one unit by x units, so it has an area of x square units. (The Multiplicative Identity can be mentioned here.) � After this, we take a look at the blue tile. It is x units tall and x units wide. (Many students will still go back to the yellow tiles as a unit of measurement.) The blue tile has an area of x2 square units. � Finally, discuss that the red tiles are used to denote negative quantities.

24. Modeling expressions 24 Our next idea is to focus us modeling algebraic expressions. (~20 minutes) Now that we have established the value of each tile, have participants model each expression using tiles. a) x + 5 (one green tile and 5 yellow tiles) b) 5 + x (5 yellow tiles and 1 green tile) These two expressions represent the same group of tiles� Commutative Property of Addition Our next idea is to focus us modeling algebraic expressions. (~20 minutes) Now that we have established the value of each tile, have participants model each expression using tiles. a) x + 5 (one green tile and 5 yellow tiles) b) 5 + x (5 yellow tiles and 1 green tile) These two expressions represent the same group of tiles� Commutative Property of Addition

25. Modeling expressions 25 c) x - 1 Start with x and take away one.� (Put up one green tile.) We don't have one to take away. We only have an x. This is another picture of x where we can take one away. (Still have one green tile. Add one yellow tile and one red tile.) We added a zero pair. (The red tile and the yellow tile combine to make the zero pair� Additive Inverse) After we take one away, or subtract one, this is what we are left with. (Remove the yellow tile representing one, leaving one green and one red tile.) This process with the tiles is helpful in building understanding that subtracting one is the same as adding negative one. c) x - 1 Start with x and take away one.� (Put up one green tile.) We don't have one to take away. We only have an x. This is another picture of x where we can take one away. (Still have one green tile. Add one yellow tile and one red tile.) We added a zero pair. (The red tile and the yellow tile combine to make the zero pair� Additive Inverse) After we take one away, or subtract one, this is what we are left with. (Remove the yellow tile representing one, leaving one green and one red tile.) This process with the tiles is helpful in building understanding that subtracting one is the same as adding negative one.

26. Modeling expressions 26 Compare the expressions . . . x + 2 and 2x (Have participants model each expression at their tables.) Should they look the same? x + 2 . . . is an x with 2 ones added onto the end. (one green tile and two yellow tiles) 2x . . . means you have an x two times. (two green tiles) Compare the expressions . . . x + 2 and 2x (Have participants model each expression at their tables.) Should they look the same? x + 2 . . . is an x with 2 ones added onto the end. (one green tile and two yellow tiles) 2x . . . means you have an x two times. (two green tiles)

27. Modeling expressions 27 Now, we are ready to move to simplifying algebraic expressions. (~20 minutes) (combining like terms and applying the Distributive Property) � a) Build x2 + 3x + 2 (one blue tile, three green tiles, and two yellow tiles) This expression has three terms and each term looks quite different. Now, we are ready to move to simplifying algebraic expressions. (~20 minutes) (combining like terms and applying the Distributive Property) � a) Build x2 + 3x + 2 (one blue tile, three green tiles, and two yellow tiles) This expression has three terms and each term looks quite different.

28. Simplifying expressions 28 Build x2 + x - 2x2 + 2x � 1 (From left to right, pull in one blue tile, one green, two red large square tiles, two green, and one small red square tile.) Some of these terms do look �like.� Let�s rearrange, using the Commutative Property of Addition. (Rearrange so like tiles are together.) We see one zero pair with our x2 and -x2 (Additive Inverse). We see three x tiles. It is easy to see that the expression simplifies to x2 + 3x - 1. Build x2 + x - 2x2 + 2x � 1 (From left to right, pull in one blue tile, one green, two red large square tiles, two green, and one small red square tile.) Some of these terms do look �like.� Let�s rearrange, using the Commutative Property of Addition. (Rearrange so like tiles are together.) We see one zero pair with our x2 and -x2 (Additive Inverse). We see three x tiles. It is easy to see that the expression simplifies to x2 + 3x - 1.

29. Simplifying expressions 29 What would the model of this expression look like? Invite a volunteer to come up and draw what they have modeled. Ask if anyone has a different model to share (and compare). It is likely that most participants will make two sets of tiles that each contain 2 greens and 3 yellows. It is possible that someone may think about the area model. This should lead nicely into the next slide. This expression is the first one we have seen with parentheses. One would use the Distributive Property to simplify the expression.What would the model of this expression look like? Invite a volunteer to come up and draw what they have modeled. Ask if anyone has a different model to share (and compare). It is likely that most participants will make two sets of tiles that each contain 2 greens and 3 yellows. It is possible that someone may think about the area model. This should lead nicely into the next slide. This expression is the first one we have seen with parentheses. One would use the Distributive Property to simplify the expression.

30. Two methods of illustrating the Distributive Property: 30 One way to model the Distributive Property is to copy and paste. The copy and paste method reinforces the idea that you will be getting the whole quantity within the parentheses multiple times. Another method to model the Distributive Property is to use an area model for multiplication and build an array. The area model connects to the earlier work we did with multi-digit multiplication, and it carries over into Algebra I when multiplying and factoring polynomials is introduced. � Both of these methods have value for the students. One way to model the Distributive Property is to copy and paste. The copy and paste method reinforces the idea that you will be getting the whole quantity within the parentheses multiple times. Another method to model the Distributive Property is to use an area model for multiplication and build an array. The area model connects to the earlier work we did with multi-digit multiplication, and it carries over into Algebra I when multiplying and factoring polynomials is introduced. � Both of these methods have value for the students.

31. Solving EquationsHow does this concept progress as we move through middle school? 31 Finally, we can work with equations. � 6th grade . . . one-step equations . . . . remember that these students have not learned about integer operations. � 7th grade . . . up to two-step equations . . . we can now apply operations with integers. � 8th grade . . . up to four-step equations . . . simplifying expressions within an equation and working with variables on both sides of the equation make their debut. �Finally, we can work with equations. � 6th grade . . . one-step equations . . . . remember that these students have not learned about integer operations. � 7th grade . . . up to two-step equations . . . we can now apply operations with integers. � 8th grade . . . up to four-step equations . . . simplifying expressions within an equation and working with variables on both sides of the equation make their debut. �

32. Solving Equations 32 We should start with a discussion of the word equation. It is a statement that two expressions are equal. This idea is mirrored in the equation mat. Each rectangle should show the model of an expression. The two expressions are equivalent. We should start with a discussion of the word equation. It is a statement that two expressions are equal. This idea is mirrored in the equation mat. Each rectangle should show the model of an expression. The two expressions are equivalent.

33. Solving Equations 33 Let�s start with some examples from 6th grade SOL. x + 3 = 5 What would that look like on the equation mat? (Pull in one green and three yellow on left and five yellow on the right.) We want to isolate the x term. How can we achieve that goal? Take three yellows off each side Add two negatives to each side (Note this is not a 6th grade response.) Let�s start with some examples from 6th grade SOL. x + 3 = 5 What would that look like on the equation mat? (Pull in one green and three yellow on left and five yellow on the right.) We want to isolate the x term. How can we achieve that goal? Take three yellows off each side Add two negatives to each side (Note this is not a 6th grade response.)

34. 34 Here we start with a pictorial representation of what we just built with the tiles. How can we solve for x ? You can take three away from each side. You can see a pictorial representation as well as the symbolic representation of that step. Finally, we are left with the green tile that represents x alone on the left and two yellow tiles on the right. The solution is x = 2. Note: the �condensed symbolic representation� column allows you to rework the problem in a more stand-alone symbolic version. Here we start with a pictorial representation of what we just built with the tiles. How can we solve for x ? You can take three away from each side. You can see a pictorial representation as well as the symbolic representation of that step. Finally, we are left with the green tile that represents x alone on the left and two yellow tiles on the right. The solution is x = 2. Note: the �condensed symbolic representation� column allows you to rework the problem in a more stand-alone symbolic version.

35. 35 Let�s try another 6th grade example, 2x = 8. Model this on your equation mats. What does it look like? We have two green tiles on the left side, and eight yellow tiles on the right. If we want to split the green tiles to determine what each one is worth individually, how would we separate the yellow tiles to distribute them fairly into 2 equal groups? (Pull the yellow tiles one at a time into two separate groups, distributing one at a time to show the fair sharing idea. Discuss that the picture still shows 2x = 8, it has just been reorganized.) How many x�s do we want? Can you tell me what 1x is worth? Delete all but one section by editing.Let�s try another 6th grade example, 2x = 8. Model this on your equation mats. What does it look like? We have two green tiles on the left side, and eight yellow tiles on the right. If we want to split the green tiles to determine what each one is worth individually, how would we separate the yellow tiles to distribute them fairly into 2 equal groups? (Pull the yellow tiles one at a time into two separate groups, distributing one at a time to show the fair sharing idea. Discuss that the picture still shows 2x = 8, it has just been reorganized.) How many x�s do we want? Can you tell me what 1x is worth? Delete all but one section by editing.

36. 36 3 = x - 1 (This is a 6th grade problem that would be challenging with the tiles. It would require a discussion of zero pairs. Although integer operations and the Additive Inverse Property are 7th grade curriculum, we could approach problems like this to help create a bridge to the 7th grade content.) To model with tiles: Put three yellow tiles on the left and one green tile on the right. We don�t have any unit tiles to remove (or subtract), but we could add a zero pair (one yellow and one red unit tile that combine to equal zero). Then, we could remove the yellow unit tile (take away one) and this would leave x and negative one, or x plus -1 on the right. This example would cause sixth grade students to consider what might happen and would be an early example for seventh graders as they formulate their own rules for operating with integers. 3 = x - 1 (This is a 6th grade problem that would be challenging with the tiles. It would require a discussion of zero pairs. Although integer operations and the Additive Inverse Property are 7th grade curriculum, we could approach problems like this to help create a bridge to the 7th grade content.) To model with tiles: Put three yellow tiles on the left and one green tile on the right. We don�t have any unit tiles to remove (or subtract), but we could add a zero pair (one yellow and one red unit tile that combine to equal zero). Then, we could remove the yellow unit tile (take away one) and this would leave x and negative one, or x plus -1 on the right. This example would cause sixth grade students to consider what might happen and would be an early example for seventh graders as they formulate their own rules for operating with integers.

37. 37 7th grade examples: 2x + 3 = 13 2x + 3 = 13 Let�s first model the equation on the equation mat. (Pull in two green tiles and three yellow tiles on the left, and thirteen yellow tiles on the right.) We know that the two boxes must be equivalent. What can you do to solve the equation? What would be your first step? (First step is to subtract three tiles from each side, leaving two greens on left and 10 yellows on right.) What would you do next? Have participants model and discuss at their tables. Go to next slide for whole group discussion.7th grade examples: 2x + 3 = 13 2x + 3 = 13 Let�s first model the equation on the equation mat. (Pull in two green tiles and three yellow tiles on the left, and thirteen yellow tiles on the right.) We know that the two boxes must be equivalent. What can you do to solve the equation? What would be your first step? (First step is to subtract three tiles from each side, leaving two greens on left and 10 yellows on right.) What would you do next? Have participants model and discuss at their tables. Go to next slide for whole group discussion.

38. 38 2x + 3 = 13 This chart shows the progression of steps we followed. On the left, you see a pictorial representation of what we did with the tiles. In the middle column, you see the more abstract, symbolic representation of the our moves with the tiles. On the right, you see the condensed version that one would expect students to work toward as they become efficient. To recap this solution, remove three from each side (second row of chart) and then practice fair sharing (third row).2x + 3 = 13 This chart shows the progression of steps we followed. On the left, you see a pictorial representation of what we did with the tiles. In the middle column, you see the more abstract, symbolic representation of the our moves with the tiles. On the right, you see the condensed version that one would expect students to work toward as they become efficient. To recap this solution, remove three from each side (second row of chart) and then practice fair sharing (third row).

39. 39 Another 7th grade example: 0 = 4 � 2x What would that model look like? Have participants model and solve, keeping track of the steps to solve. Use next slide for whole group discussion of solution. Invite volunteer to draw pictorial representation on chart paper before moving to next slide. Another 7th grade example: 0 = 4 � 2x What would that model look like? Have participants model and solve, keeping track of the steps to solve. Use next slide for whole group discussion of solution. Invite volunteer to draw pictorial representation on chart paper before moving to next slide.

40. 40 In the first row, we see the yellow tiles and representing 4 and the two red tiles representing -2x. Students could put up 2 zero pairs and take away two x�s, leaving the 2 negative x tiles. We want to isolate the x term, so the first step is to add four red tiles to each side. You can easily see the -4 on the left side of the equation mat. On the right side of the equation, this gives us 4 zero pairs and the -2x. In the third row, you see the separation into two equal groups. But we are interested in the value of one x, so we can remove all but one of those groupings. This leaves us with -2 = -x. We can think about it this way. If the opposite of two is equal to the opposite of x, then 2 is equal to x. In the third column, what looks different? (Division by negative 2) Presenters have the option of solving additional equations that fit within the realm of the 7th grade curriculum. These equations could be solved on chart paper, with participants recording their pictorial and symbolic representations to foster discussion: x + 2 = 1 (Students could add -2 to each side and then simplify each side OR add a zero pair to the right side in order to then take away two from each side) 3 = x � 4 (Students could add four to each side and then simplify each side) -3x = 9 (Fair sharing and opposites in either order) -9 = -5x +1 (Add -1 to each side, then fair sharing and opposites in either order OR add one zero pair to -9 in order to take away positive one from each side, then division and opposites or division by negative 5)In the first row, we see the yellow tiles and representing 4 and the two red tiles representing -2x. Students could put up 2 zero pairs and take away two x�s, leaving the 2 negative x tiles. We want to isolate the x term, so the first step is to add four red tiles to each side. You can easily see the -4 on the left side of the equation mat. On the right side of the equation, this gives us 4 zero pairs and the -2x. In the third row, you see the separation into two equal groups. But we are interested in the value of one x, so we can remove all but one of those groupings. This leaves us with -2 = -x. We can think about it this way. If the opposite of two is equal to the opposite of x, then 2 is equal to x. In the third column, what looks different? (Division by negative 2) Presenters have the option of solving additional equations that fit within the realm of the 7th grade curriculum. These equations could be solved on chart paper, with participants recording their pictorial and symbolic representations to foster discussion: x + 2 = 1 (Students could add -2 to each side and then simplify each side OR add a zero pair to the right side in order to then take away two from each side) 3 = x � 4 (Students could add four to each side and then simplify each side) -3x = 9 (Fair sharing and opposites in either order) -9 = -5x +1 (Add -1 to each side, then fair sharing and opposites in either order OR add one zero pair to -9 in order to take away positive one from each side, then division and opposites or division by negative 5)

41. 41 Now we�re ready to try some examples from 8th grade. Model 3x + 5 � x = 11 on your equation mat. What does that look like? What would you do first? Solve at your tables and keep track of your steps. Go to next slide for whole group discussion of solution.Now we�re ready to try some examples from 8th grade. Model 3x + 5 � x = 11 on your equation mat. What does that look like? What would you do first? Solve at your tables and keep track of your steps. Go to next slide for whole group discussion of solution.

42. 42 We see the first model has the red tile. Where does it go? (talk here about zero pair/ combining like terms) After we combine like terms, what remains is a two-step equation. First we subtract five from each side. In the third row of the chart, you see those five tiles have been pulled out of the boxes on the equation mat. In the second step, we divide both sides by two. We can see the solution is three. Again, the third column shows the more efficient method of recording work that students will work toward.We see the first model has the red tile. Where does it go? (talk here about zero pair/ combining like terms) After we combine like terms, what remains is a two-step equation. First we subtract five from each side. In the third row of the chart, you see those five tiles have been pulled out of the boxes on the equation mat. In the second step, we divide both sides by two. We can see the solution is three. Again, the third column shows the more efficient method of recording work that students will work toward.

43. 43 Another 8th grade example: x + 2 = 2(2x + 1) There is a lot going on here. What would we tackle first? (Simplify the expression on the right using the Distributive Property.) Have participants work at tables to go through the steps to find the solution. Use next slide for whole group discussion.Another 8th grade example: x + 2 = 2(2x + 1) There is a lot going on here. What would we tackle first? (Simplify the expression on the right using the Distributive Property.) Have participants work at tables to go through the steps to find the solution. Use next slide for whole group discussion.

44. 44 Things to bring out in the discussion: First step is application of the Distributive Property. Here we have modeled the �copy and paste� method. What would happen if the 4x were subtracted from both sides instead of the x? Presenters can continue with additional equations that fall in the realm of 8th grade curriculum on chart paper: 6 = 2(x - 3) + x (Distribute and combine like terms to create a two-step equation.) ** 3x - 1 = x + 7 (Variables on both sides of the equation. In this example you can simply remove an x from each side to create a two-step equation.) � ** 3x - 1 = -x + 7 (Now, we need to extend our operations with integers to the variables.) **We thought this might be a spot where the group could split in half. Each half could work through one of these two, slightly different equations. We could then talk about the similarities and differences as a whole group. � � Things to bring out in the discussion: First step is application of the Distributive Property. Here we have modeled the �copy and paste� method. What would happen if the 4x were subtracted from both sides instead of the x? Presenters can continue with additional equations that fall in the realm of 8th grade curriculum on chart paper: 6 = 2(x - 3) + x (Distribute and combine like terms to create a two-step equation.) ** 3x - 1 = x + 7 (Variables on both sides of the equation. In this example you can simply remove an x from each side to create a two-step equation.) � ** 3x - 1 = -x + 7 (Now, we need to extend our operations with integers to the variables.) **We thought this might be a spot where the group could split in half. Each half could work through one of these two, slightly different equations. We could then talk about the similarities and differences as a whole group. � �

45. Modeling Multiplication and Division of Fractions

46. So what�s new about fractions in Grades 6-8? 46

47. Thinking About Multiplication Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions. Repeated Addition: makes sense if fraction is multiplied by a whole number (2 � 1/3 = 1/3 + 1/3), but not for fraction times fraction Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3) Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row) Rate: �Dave walked for 2 hours at a rate of 1/3 mile per hour� or �Dave walked � hour at a rate of 1/3 mile per hour� Comparison: �Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie� Combinations: Area: �I need to cover a bulletin board that measures � yard long and 1/3 yard wide; how many square yards of fabric are needed?� Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions. Repeated Addition: makes sense if fraction is multiplied by a whole number (2 � 1/3 = 1/3 + 1/3), but not for fraction times fraction Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3) Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row) Rate: �Dave walked for 2 hours at a rate of 1/3 mile per hour� or �Dave walked � hour at a rate of 1/3 mile per hour� Comparison: �Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie� Combinations: Area: �I need to cover a bulletin board that measures � yard long and 1/3 yard wide; how many square yards of fabric are needed?�

48. Thinking About Multiplication 48 Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions. Repeated Addition: makes sense if fraction is multiplied by a whole number (2 � 1/3 = 1/3 + 1/3), but not for fraction times fraction Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3) Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row) Rate: �Dave walked for 2 hours at a rate of 1/3 mile per hour� or �Dave walked � hour at a rate of 1/3 mile per hour� Comparison: �Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie� Combinations: Area: �I need to cover a bulletin board that measures � yard long and 1/3 yard wide; how many square yards of fabric are needed?� Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions. Repeated Addition: makes sense if fraction is multiplied by a whole number (2 � 1/3 = 1/3 + 1/3), but not for fraction times fraction Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3) Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row) Rate: �Dave walked for 2 hours at a rate of 1/3 mile per hour� or �Dave walked � hour at a rate of 1/3 mile per hour� Comparison: �Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie� Combinations: Area: �I need to cover a bulletin board that measures � yard long and 1/3 yard wide; how many square yards of fabric are needed?�

49. Making sense of multiplication of fractions using paper folding and area models Enhanced Scope and Sequence, 2004, pages 22 - 24 49 Take participants through the enhanced scope and sequence activity. Spend no more than 15 minutes on this� Participants will need blank paper and something to write with.Take participants through the enhanced scope and sequence activity. Spend no more than 15 minutes on this� Participants will need blank paper and something to write with.



52. The Importance of Context Builds meaning for operations Develops understanding of and helps illustrate the relationships among operations Allows for a variety of approaches to solving a problem 52

53. Contexts for Modeling Multiplication of Fractions 53 Ask participants to draw a picture to solve the problem. One possible pictorial solution is on the following slide.Ask participants to draw a picture to solve the problem. One possible pictorial solution is on the following slide.

54. 54 We start with one-half of one pizza. That is, in effect, the �whole� amount we have to work with. Each boy gets one-third of that amount, so we divide the half into three equal portions (dividing into three equal shares is the same as finding one-third of the amount). Since the question refers to the share of the original pizza, we have to determine what each of those three pieces is in relation to the original whole. The dotted segments extend across to show that each boy had 1/6 of the original pizza.We start with one-half of one pizza. That is, in effect, the �whole� amount we have to work with. Each boy gets one-third of that amount, so we divide the half into three equal portions (dividing into three equal shares is the same as finding one-third of the amount). Since the question refers to the share of the original pizza, we have to determine what each of those three pieces is in relation to the original whole. The dotted segments extend across to show that each boy had 1/6 of the original pizza.

55. Andrea and Allison are partners in a relay race. Each girl will run half the total distance. On race day, Andrea stops for water after running of her half of the race. What portion of the race had Andrea run when she stopped for water? Another Context for Multiplication of Fractions 55

56. 56

57. Another Context for Multiplication of Fractions Mrs. Jones has 24 gold stickers that she bought to put on perfect test papers. She took of the stickers out of the package, and then she used of that half on the papers. What fraction of the 24 stickers did she use on the perfect test papers? 57

58. Problems involving discrete items may be represented with set models. 58

59. What�s the relationship between multiplying and dividing? This slide serves as a transition between multiplication and division, and should be reached by the 30 min. mark for this segment�This slide serves as a transition between multiplication and division, and should be reached by the 30 min. mark for this segment�

60. 60 Meanings of Division

61. Sometimes, Always, Never? When we multiply, the product is larger than the number we start with. When we divide, the quotient is smaller than the number we start with. These questions are posed to get participants to consider what really happens during the operations of multiplication and division, and to recognize that the ways the two numbers in each expression are used is different (one is the multiplier, the other is a quantity or amount).. These questions are posed to get participants to consider what really happens during the operations of multiplication and division, and to recognize that the ways the two numbers in each expression are used is different (one is the multiplier, the other is a quantity or amount)..

62. 62 �I thought times makes it bigger...�

63. 63 �Groups of� and �Measure Out�

64. Thinking About Division 64



67. Contexts for Division of Fractions 67

68. 68 If 1/3 pizza is one serving, how many servings are in the � pizza we are starting with in this problem? Since 1/3 is less than 1/2, our solution will be more than one serving. Since the � pizza we have is less than 2/3 (or two servings), the solution will be less than 2 servings. We can portion off one full serving, leaving 1/6 of the original pizza. That sixth is � of a serving. We have a total of 1 � servings in our half pizza, that are each 1/3 pizza in size. If 1/3 pizza is one serving, how many servings are in the � pizza we are starting with in this problem? Since 1/3 is less than 1/2, our solution will be more than one serving. Since the � pizza we have is less than 2/3 (or two servings), the solution will be less than 2 servings. We can portion off one full serving, leaving 1/6 of the original pizza. That sixth is � of a serving. We have a total of 1 � servings in our half pizza, that are each 1/3 pizza in size.

69. Another Context for Division of Fractions 69

70. 70 How many thirds are in one half? Measurement situations can be represented with a linear model, even if length isn�t being measured.How many thirds are in one half? Measurement situations can be represented with a linear model, even if length isn�t being measured.

71. 71

72. 72 The blue amount represents the leftover � sheet cake. The green shaded amount shows 1/3 of the original cake. You can see that one complete 1/3 cake portion exists and then � of another portion. Another way to think: common denominators. We know that � is 3-sixths and 1/3 is 2-sixths. How many 2/6 are in 3/6? There is one 2/6 and then � of another 2/6, for a total of 1 �. With this diagram you can see that the �little bit more� is really one sixth� so why isn�t �one sixth� a part of the answer? The answer is 1� � where is that one half in the model?The blue amount represents the leftover � sheet cake. The green shaded amount shows 1/3 of the original cake. You can see that one complete 1/3 cake portion exists and then � of another portion. Another way to think: common denominators. We know that � is 3-sixths and 1/3 is 2-sixths. How many 2/6 are in 3/6? There is one 2/6 and then � of another 2/6, for a total of 1 �. With this diagram you can see that the �little bit more� is really one sixth� so why isn�t �one sixth� a part of the answer? The answer is 1� � where is that one half in the model?

73. What does it look like numerically? 73 Connect the discussion regarding the pictorial representation to the numerical solution. Toggle back and forth between the two slides pointing out where the common denominator 6 comes from , as well as where the 1 � comes from.Connect the discussion regarding the pictorial representation to the numerical solution. Toggle back and forth between the two slides pointing out where the common denominator 6 comes from , as well as where the 1 � comes from.

74. What is the role of common denominators in dividing fractions? 74 Connect concrete to abstract by giving participants some division problems to solve using the common denominator method.Connect concrete to abstract by giving participants some division problems to solve using the common denominator method.

75. Invariably, the traditional algorithm will come up even if the teacher has not introduced it himself. Teachers need to be ready to discuss it. Remember that one focus of the 2009 standards is to model multiple representations for division of fractions. Time considering the algorithm is worthwhile but it�s not the only way to divide fractions. The next slides attempt to give teachers some ideas about how to help students think about that traditional algorithm.Invariably, the traditional algorithm will come up even if the teacher has not introduced it himself. Teachers need to be ready to discuss it. Remember that one focus of the 2009 standards is to model multiple representations for division of fractions. Time considering the algorithm is worthwhile but it�s not the only way to divide fractions. The next slides attempt to give teachers some ideas about how to help students think about that traditional algorithm.

77. This is a difficult idea to understand and could take several days in itself. The gist is that we should always be teaching for understanding, and we want students to understand the procedures they are using to solve problems. Multiplying by the reciprocal is not the only method for dividing fractions. This is a difficult idea to understand and could take several days in itself. The gist is that we should always be teaching for understanding, and we want students to understand the procedures they are using to solve problems. Multiplying by the reciprocal is not the only method for dividing fractions.

78. Multiple Representations 78 Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with. � �1/2 2/3 � � 5/6 � 2/6 2/3 � 1/6 1/6 � 3/6 12/3� 1/2 It is important that everyone come back together and have some debriefing. Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with. � �1/2 2/3 � � 5/6 � 2/6 2/3 � 1/6 1/6 � 3/6 12/3� 1/2 It is important that everyone come back together and have some debriefing.

79. Using multiple representations to express understanding 79 Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with. � �1/2 2/3 � � 5/6 � 2/6 2/3 � 1/6 1/6 � 3/6 12/3� 1/2 It is important that everyone come back together and have some debriefing. Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with. � �1/2 2/3 � � 5/6 � 2/6 2/3 � 1/6 1/6 � 3/6 12/3� 1/2 It is important that everyone come back together and have some debriefing.

81. Mean: Fair Share and Balance Point 81

82. Mean: Fair Share 82 The blue text represents additional changes in the Standard. The red text represents the change that we are addressing today. *This convention is also used on Slide 76 for SOL 2009.6.15. Understanding the Standard is copied directly from the 2009 Curriculum Framework. The blue text represents additional changes in the Standard. The red text represents the change that we are addressing today. *This convention is also used on Slide 76 for SOL 2009.6.15. Understanding the Standard is copied directly from the 2009 Curriculum Framework.

83. Understanding the Mean 83 We will be asking participants to engage in two separate activities � this one uses snap cubes to demonstrate mean as fair share, and the sticky note activity demonstrates mean as balance point. Michael has suggested that we should take the group up through Slide 79 (the example where the mean doesn�t work out neatly to a whole number) BEFORE we conduct the second activity. The instructions for that activity are in the notes for Slide 79. We will be asking participants to engage in two separate activities � this one uses snap cubes to demonstrate mean as fair share, and the sticky note activity demonstrates mean as balance point. Michael has suggested that we should take the group up through Slide 79 (the example where the mean doesn�t work out neatly to a whole number) BEFORE we conduct the second activity. The instructions for that activity are in the notes for Slide 79.

84. Understanding the Mean 84 �Teach� this lesson similarly to the way you would with students.�Teach� this lesson similarly to the way you would with students.

85. Understanding the Mean 85 Obviously, the answer represents the mean � but make sure they understand it represents a �fair share� as the underlying concept of the mean.Obviously, the answer represents the mean � but make sure they understand it represents a �fair share� as the underlying concept of the mean.

86. Understanding the Mean 86 Building a line plot for the activity at the end of Slide 79.Building a line plot for the activity at the end of Slide 79.

87. Understanding the Mean 87 From the Curriculum Framework.From the Curriculum Framework.

88. Understanding the Mean 88 �Teach� again to make sure they can find the range, mode, and median before moving to the next slide about using a line plot to find the mean (the balance point).�Teach� again to make sure they can find the range, mode, and median before moving to the next slide about using a line plot to find the mean (the balance point).

89. Mean: Balance Point 89 Understanding the Standard and Essential Knowledge & Skills info is taken directly from the Curriculum Framework. *For the record, the last bullet on this slide is not the only bullet in the Framework: � Find the mean for a set of data. � Describe the three measures of center and a situation in which each would best represent a set of data. � Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data.Understanding the Standard and Essential Knowledge & Skills info is taken directly from the Curriculum Framework. *For the record, the last bullet on this slide is not the only bullet in the Framework: � Find the mean for a set of data. � Describe the three measures of center and a situation in which each would best represent a set of data. � Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data.

90. 90 We can manipulate the data points to help us see where the balance point would be. Even though we don�t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point �moves�), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it�s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you�d want to during our presentation, you could also make moves away from the center without affecting the mean.We can manipulate the data points to help us see where the balance point would be. Even though we don�t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point �moves�), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it�s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you�d want to during our presentation, you could also make moves away from the center without affecting the mean.





95. 95 Now refer back to the original line plot. We know that 3 is the balance point or mean. Talk about the sum of the distances above the mean being the same as the sum of distances from the mean below the mean.Now refer back to the original line plot. We know that 3 is the balance point or mean. Talk about the sum of the distances above the mean being the same as the sum of distances from the mean below the mean.

96. We can manipulate the data points to help us see where the balance point would be. Even though we don�t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point �moves�), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it�s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you�d want to during our presentation, you could also make moves away from the center without affecting the mean.We can manipulate the data points to help us see where the balance point would be. Even though we don�t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point �moves�), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it�s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you�d want to during our presentation, you could also make moves away from the center without affecting the mean.

98. Sticky Note Activity: Work with the whole group to use this strategy to find the mean number of cubes in one handful based on our data set. If it doesn�t work out to be whole number, discuss how we could find the exact decimal value of the mean if we could �zoom in� and how we could estimate the mean based on the modified line plot. *You may want to have a calculator handy to find the actual mean of your whole group data set. Sticky Note Activity: Work with the whole group to use this strategy to find the mean number of cubes in one handful based on our data set. If it doesn�t work out to be whole number, discuss how we could find the exact decimal value of the mean if we could �zoom in� and how we could estimate the mean based on the modified line plot. *You may want to have a calculator handy to find the actual mean of your whole group data set.

99. Mean: Balance Point 99 By �intentionally� we mean that teachers should begin with data sets that have whole number means, only progressing to rational number means once students understand the concept and process. When introducing decimal means, teachers should begin with �neat� data sets. - For example: {2, 3, 4, 5, 6, 7} would have a mean of 4.5 (27/6 = 4.5), which would be easy to see once the line plot was transformed so that there were three 4�s and three 5�s. By �intentionally� we mean that teachers should begin with data sets that have whole number means, only progressing to rational number means once students understand the concept and process. When introducing decimal means, teachers should begin with �neat� data sets. - For example: {2, 3, 4, 5, 6, 7} would have a mean of 4.5 (27/6 = 4.5), which would be easy to see once the line plot was transformed so that there were three 4�s and three 5�s.

100. Assessing Higher-Level Thinking 100 Emphasize the shift to higher-order thinking from the 2001 to 2009 Standards.Emphasize the shift to higher-order thinking from the 2001 to 2009 Standards.

101. Mean:Fair Share & Balance Point 101 Connection between the new SOL and the NCTM Standards emphasizing that the teacher must provide learning experiences that will promote students� conceptual understanding. The days of �just add �em up and divide� are behind us :o) Connection between the new SOL and the NCTM Standards emphasizing that the teacher must provide learning experiences that will promote students� conceptual understanding. The days of �just add �em up and divide� are behind us :o)

102. Inequalities

103. Inequalities SOL 6.20 The student will graph inequalities on a number line. SOL 7.15 The student will a) solve one-step inequalities in one variable; and graph solutions to inequalities on the number line. SOL 8.15 The student will b) solve two-step linear inequalities and graph the results on a number line Why spend the time talking about inequalities at grade 6?Why spend the time talking about inequalities at grade 6?

104. Inequalities What does inequality mean in the world of mathematics? mathematical sentence comparing two unequal expressions How are they used in everyday life?

105. Equations vs. Inequalities x = 2 x > 2 How are they alike? How are they different? So, what about x > 2? Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?

106. Equations vs. Inequalities x = 2 x > 2 x > 2 Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?

107. Open or Closed? x > 16 -5 > y m > 12 n < 341 -3 < j How do you read these? What do they really mean in real life terms? It�s more than just putting dots and arrows, isn�t it?How do you read these? What do they really mean in real life terms? It�s more than just putting dots and arrows, isn�t it?

108. Equations vs. Inequalities x + 2 = 8 x + 2 < 8 How are they alike? How are they different? So, what about x + 2 < 8? Engage participants in dialogue comparing x +2 = 8 with x+2 > 8. What is the difference between using the = or the <? What is the difference between using the < and the <? How could we represent these in real life terms? (i.e. (x+2 =8) Antonio has $8 to ride go-karts and play games. If the go-karts cost $2, how much can he spend on games? (x+2<8) Antonio has $8 to ride go-karts and play games at the fair. If the go-kart costs $2, what is the most he can spend on games?) Have participants use white boards or paper to show graphical representations of the two. So, what about x+2>=8? Engage participants in dialogue comparing x +2 = 8 with x+2 > 8. What is the difference between using the = or the <? What is the difference between using the < and the <? How could we represent these in real life terms? (i.e. (x+2 =8) Antonio has $8 to ride go-karts and play games. If the go-karts cost $2, how much can he spend on games? (x+2<8) Antonio has $8 to ride go-karts and play games at the fair. If the go-kart costs $2, what is the most he can spend on games?) Have participants use white boards or paper to show graphical representations of the two. So, what about x+2>=8?

109. Equations vs. Inequalities x + 2 = 8 x + 2 < 8 How are they alike? Both statements include the terms: x, 2 and 8 The solution set for both statements involves 6. How are they different? The solution set for x + 2 = 8 only includes 6. The solution set for x + 2 < 8 does includes all real numbers less than 6. What about x + 2 < 8? Engage participants in dialogue comparing x +2 = 8 with x+2 > 8. What is the difference between using the = or the <? What is the difference between using the < and the <? How could we represent these in real life terms? (i.e. (x+2 =8) Antonio has $8 to ride go-karts and play games. If the go-karts cost $2, how much can he spend on games? (x+2<8) Antonio has $8 to ride go-karts and play games at the fair. If the go-kart costs $2, what is the most he can spend on games?) Have participants use white boards or paper to show graphical representations of the two. So, what about x+2>=8? Engage participants in dialogue comparing x +2 = 8 with x+2 > 8. What is the difference between using the = or the <? What is the difference between using the < and the <? How could we represent these in real life terms? (i.e. (x+2 =8) Antonio has $8 to ride go-karts and play games. If the go-karts cost $2, how much can he spend on games? (x+2<8) Antonio has $8 to ride go-karts and play games at the fair. If the go-kart costs $2, what is the most he can spend on games?) Have participants use white boards or paper to show graphical representations of the two. So, what about x+2>=8?

110. Equations vs. Inequalities x+ 2 = 8 x+ 2 < 8 x+ 2 < 8 Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?

111. Inequality Match With your tablemates, find as many matches as possible in the set of cards.

113. Operations with Integers

114. Operations with Integers This 2009 SOL has flown under the radar because at first glance the content doesn�t not appear to have changed � it�s still about operating on integers in 7th Grade. However, the new verb �model� should not be underestimated. This 2009 SOL has flown under the radar because at first glance the content doesn�t not appear to have changed � it�s still about operating on integers in 7th Grade. However, the new verb �model� should not be underestimated.

115. Assessing Higher-Level Thinking These �chip� models must be read from left to right: FOLLOW THE ARROWS. These �chip� models must be read from left to right: FOLLOW THE ARROWS.

116. Assessing Higher-Level Thinking This one doesn�t seem to �do� very much, but the arrow is showing a move from 3 groups of (-4) to a product of (-12). This one doesn�t seem to �do� very much, but the arrow is showing a move from 3 groups of (-4) to a product of (-12).

117. Assessing Higher-Level Thinking Most follow the arrows � begin at the zero. This model could be representing: 5 + (-17) = -12 OR 5 - 17 = -12 The slide animations will display both number sentences.Most follow the arrows � begin at the zero. This model could be representing: 5 + (-17) = -12 OR 5 - 17 = -12 The slide animations will display both number sentences.

118. Assessing Higher-Level Thinking Could also be shown using �hops�. Could also be shown using �hops�.

119. Another Example of Assessing Higher-Level Thinking Emphasize the vocabulary: �scale factor�.Emphasize the vocabulary: �scale factor�.

120. Tying it All Together Improved vertical alignment of content with increased cognitive demand. Key conceptual models can be extended across grade levels. Refer to the Curriculum Framework. Pay attention to the changes in the verbs.

121. Exit Slip 1. Aha... 2. Can�t wait to share� 3. HELP!

2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra

2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra

Presentation Transcript

Ms. Green's Jeopardy 7th Grade

Maggie P. Fay Ph.D. Candidate in Sociology, CUNY Graduate Center

THE ONTARIO CURRICULUM GRADES 1-8 (read p 1-9 of the mathematics curriculum )

Discrete mathematics Discrete i.e. no continuous Set theory, Combinatorics, Graphs, Modern AlgebraAbstract algebra, Alg

National Council of Teachers of Mathematics

Mathematics – Proposal

Reading & Mathematics Grades 3–8 Science Grades 5 & 8 2013

4-3 Patterns and Nonlinear Functions

Welcome to

The Algebra of Functions

THIS

GCSE Mathematics

Finding Focus for Mathematics Instruction Grades 6-8

Quick lesson in some Mathematics used in Managerial Economics

Mathematics Grades K-2

Welcome to Ms. Reed’s Mathematics Class

FOCUS GROUP WITH PARENTS OF CHILDREN IN GRADES 8, 9 OR 10 February, 2010

Core-Plus Mathematics

Common Core State Standards for Mathematics: Illustrating the 3 Shifts

STAR 2010

THIS

Ms. Katie Runkle Sun Valley Mathematics X Algebra II and Honors Algebra II

2010 VDOE Mathematics Institute Grades 6-8 Focus: Patterns, Functions, and Algebra