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# K-2 Learning Trajectories for Mathematics

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Tweet## Amanda James

on 7 April 2011#### Transcript of K-2 Learning Trajectories for Mathematics

K-2 Learning Trajectories for Mathematics

Quad 2 Geometry K 1st 2nd Reference Big Idea Big Idea Big Idea Big Idea Big Idea Big Idea Describe objects in the environment using shapes. (NGA Center & CCSSO, 2010) Correctly name shapes regardless of their orientations or overall size. – Students will state the name of the shapes no matter how big or small and the way the shapes are positioned. (NGA Center & CCSSO, 2010) National Governors Association Center for Best Practices & The Council of Chief State School Officers.(2010). Common Core Standards for Mathematics. Retrieved from www.corestandards.org. Task Introduce the name of the shapes and characteristics in a creative way like a poem or song. Then place the shapes on the board or overhead and have the students name off the characteristics. This is a great chance to introduce all the vocabulary for this lesson. After all the shapes have been introduced have the students take out white boards, then say the characteristics out loud and have the students draw the shape it goes too. Finally after getting the identification steps are done play “I Spy” game with the shapes in the classroom. (Indiana standards and resources) Describe the relative positions of objects using terms such as above, below, beside, in front of, behind, and next to. - Students will state where an object is using the terms above. (NGA Center & CCSSO, 2010) Task Tape off a large area on the classroom floor and collect pictures of people and animals in different locations. The taped off area will be for the game “Simon Says” tell the students the rules using the vocabulary words such as inside, outside, between, above, and below. Use phrases like “Simon Says stand inside the space.” Then after the game show the students the pictures and have them use the vocabulary words in a sentence. (Indiana standards and resources) Identify shapes as two-dimensional like shapes on a flat plane. Students need to be able to understand the shapes that are flat or 2-D. (NGA Center & CCSSO, 2010) Identify shapes as three-dimensional like solid shapes. Students need to be able to understand the shapes that are solid and tangible are 3-D. (NGA Center & CCSSO, 2010) Analyze two- and three- dimensional shapes, in different sizes and orientations. Students should be able to discuss in detail about two- and three- dimensional shapes regardless of their size or position. (NGA Center & CCSSO, 2010) Compare two- and three- dimensional shapes, in different sizes and orientations. Students should be able to discuss how two- and three-dimensional are different in size and position. (NGA Center & CCSSO, 2010) Using informal language to describe two- and three-dimensional similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other attributes (e.g., having sides of equal length). Students should be able to describe every part of two- and three-dimensional shapes. (NGA Center & CCSSO, 2010) Task When exploring two- and three- dimensional students will be given materials to make a flower out of paper. When the students are finished making there paper flower (2D) then real flowers (3D) will be placed in the classroom for students to observe. The students will need to compare and analyze the similarities and differences between the paper flower and real flower. Then the students will make a venn-diagram, draw pictures, or write about the flowers using the vocabulary and characteristics of 2D and 3D shapes. Model shapes in the world by building shapes from components (like sticks and clay balls).(NGA Center & CCSSO, 2010) Drawing shapes in the world.(NGA Center & CCSSO, 2010) Task Collect many materials that could be made into a shape for example, Popsicle sticks, clay, paper, yarn, ect. Then ask the students how many shapes can they made out of these materials. After the class has discuss a few ways of making shapes have the students explore making the shapes by prompting them to find as many as possible. After a while come back together and discuss using the vocabulary. Compose simple shapes to form larger shapes. (NGA Center & CCSSO, 2010) Task Burns, M. (1994) The Greedy Triangle. New York, NY: Scholastic. Big Idea Big Idea Distinguish defining attributes- The students define the characteristics that make up the shape, for example a triangle’s defining attributes are it is closed and has 3 sides and 3 angles. (NGA Center & CCSSO, 2010)

Distinguish non-defining attributes- The students say characteristics that are not important to defining a triangle, for example size and color. (NGA Center & CCSSO, 2010)

Draw shapes to possess defining attributes- drawing the shape to show the characteristics that define the shape.(NGA Center & CCSSO, 2010) Read the Greedy Triangle (Burns, 1994) and discuss what they think a quadrilateral is. Focusing on rectangles and squares, have the students explore different sized squares and rectangles and ask them questions like “How are these shapes different?” and “Can you prove it?” After exploring these shapes have the students tell you what defines a square and a rectangle. Afterward they made their own quadrilateral book to put their definition in and draw shapes to go along with it. Compose 2 dimensional or 3 dimensional shapes to create a composite shape- putting together shapes that are either two-dimensional or three- dimensional to make a common shape like triangle, rectangle, and square. For example putting together two triangles would make a square. (NGA Center & CCSSO, 2010)

Compose new shapes from the composite-Using common shapes, such as triangles, squares, and rectangles, to make new shapes like diamonds or pentagon. (NGA Center & CCSSO, 2010) To assist in learning about 2D and 3D shapes, the students will make two snowmen (or something to go along with the theme of that week). One of these snowmen will be made out of paper. The second will be made out of Styrofoam balls and other objects. The students will then compare the two snowmen to notice the differences. The goal is to notice that the snowman made out of paper is flat and the one made out of Styrofoam has “depth”. (Indiana standards and resources) Partition circles into two equal shares- splitting a circle into two equal pieces(NGA Center & CCSSO, 2010)

Partition circles into four equal shares- splitting a circle into four equal pieces(NGA Center & CCSSO, 2010)

Describe shares using halves- describe that a rectangle or circle is divided into halves when it is split in half.(NGA Center & CCSSO, 2010)

Describe shares using fourths- describe that a rectangle or circle is divided into fourths when it is split into four equal pieces.(NGA Center & CCSSO, 2010)

Describe shares using quarters- Describe that a rectangle or circle is divided into quarters when it is split into four equal pieces(NGA Center & CCSSO, 2010)

Describe the whole as two of- describe the cookie, for example, as being two of the one half pieces put together to make one whole cookie.(NGA Center & CCSSO, 2010)

Describe the whole as four of- describe the cake, for example, as being four of the one-fourth pieces put together to make one whole cake.(NGA Center & CCSSO, 2010) In order to teach halves fourths and quarters, we are going to use examples like cakes, brownies and cookies. Start by asking the students a question like “ I have one cake but I want to share it with three of my friends, how could we each get an equal size piece?” We would ask similar questions for getting halves and with using a circle shape like a cookie or pie. Then follow up with “One of my friends decided she didn’t want a piece so I ate her piece, how much of the cake did I eat? How many fourths or quarters?” The cake was divided into fourths. If we each just ate one piece and there were only three of us how much of the cake would be left? Big Idea Indiana standards and resources. Indiana department of education. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.(NGA Center & CCSSO, 2010) Big Idea Big Idea Big Idea Big Idea Big Idea Big Idea Recognize and draw triangles, quadrilaterals, pentagons, hexagons and cubes (NGA Center & CCSSO, 2010). Here students will be able to look at a triangle, quadrilateral, pentagon, hexagon or cube and be able to tell his/her teacher what that shape is by name. They will also be able to draw each shape when instructed by just hearing the name of the shape. Gumdrop Geometry: In this activity “students will construct various geometric shapes and describe the shapes according to the number of faces, edges, and vertices (corners)” (NGA Center & CCSSO, 2010). The students would first construct these shapes out of construction paper and then out of gum drops and toothpicks. Recognize the number of angles or equal faces for triangles, quadrilaterals, pentagons, hexagons and cubes (NGA Center & CCSSO, 2010). Students can do this by “finding measures of the interior and exterior angles of polygons, justifying the method used” (website). They can also do this for the other shapes. Activity: An activity I could do here is give each student a piece of paper and ask them to draw a triangle, quadrilateral, pentagon and hexagon then count the number of angles each one has. I could then ask the students to see if they can find a relationship between the shapes using the angles as a clue. I would ask them in they see a pattern and to redraw the shapes in a pattern they see and then justify why they thought that pattern worked. “Partition circles and rectangles into two, three, or four equal shares” (NGA Center & CCSSO, 2010). Student are going to need to “recognize...shapes that have symmetry” (National Standards). They can then use circles and rectangles to create these equal shares.

Example:

Activity: In this activity I could bring in some small pies and give each group of four students a pie. I could ask them how they think the pie needs to be cut so that everyone has an equal share. I could then go around to each table and cut the pies for them in the way they tell me. We could do the same thing with a rectangular candy bar. After doing this the students would then be asked to draw a picture on a piece of paper of how the pie/candy was cut up. I would then ask them to prove to me why each piece was equal. “Describe shares of circles and rectangles using the words halves, thirds, half of, a third of, etc, and describe the whole as two halves, three thirds, four fourths” (NGA Center & CCSSO, 2010). To accomplish this students could “use a variety of problem-solving strategies, such as drawing a diagram, making a chart” (website) and draw out real life situations to learn about halves, thirds and so on. These real life situations could be things like drawing a present and then drawing a ribbon around the present cutting it into four equal parts. Activity: I could divide the class in groups of four. I could then write half, third, half of the group, a third of the group, two halves, three thirds, four fourths up on the wipe off board. I would then ask the groups to figure out how to cut their group in half and one third and so on to make these words come to life. I would assess their work by having them write how they cut their groups up by drawing stick figures on a piece of paper. “Recognize that equal shares of identical wholes need not have the same shape” (NGA Center & CCSSO, 2010).

This best way to understand this big idea is by looking at an example. These identical squares are divided into equal shares and yet the divided shapes on the right square are different than the divided shapes on left square. The square on the left is broken up into four equal squares while the square on the right is broken up into four equal triangles.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.(NGA Center & CCSSO, 2010) I will use a math work page from the book Indiana Mathematics grade K(2005, p. 209). The children will be given simple shapes and an outline of a biger shape. The children will uses the manupulative shapes to see how many of one particular shape it will take to reconstruct the bigger shape. They will then write how many smaller shapes it took to do it.

Now if there are some students who need a little harder problem. I will give them the next page from the book Indiana Mathematics grade K(2005, p. 210). In this page the students will be given different shapes to make a bigger shape. Foresman, S., Addison, W. (2005) Indiana Mathematics Grade K. Glenview, Il: Person Education, Inc. I got this idea from the book Indiana Mathematics Grade 1 Teachers Edition (2005, P. 165B). This page shows mulitple intelegence, and how to teach for each one. One lesson would be to take simple shapes and make different animals out of them, another would be to cut out the simple shapes with construction paper and then give the students clay and some toothpicks. The students would then take the small balls of clay and make the shape of the construction paper with the toothpicks. I would have a couple of different options for kids to play around with this core standard. I would have a regualar page with differnt squars to have kids count, then i would have a station where there is blocks put together and the kids could then try to count with the blocks together, or if they needed to they could take the blocks apart to count or even to check their answers. I would also have a more advanced station, I got this from the book Indiana Mathematics Grade 2 Teachers Edition (2005, P. 251). This would be about netting, where instead of just counting the blocks they could see that the blocks put in specific locations could be brought together and make a 3D shape. Foresman, S., Addison, W. (2005) Indiana Mathematics Grade 2 Teachers Edition. Glenview, Il: Person Education, Inc. Foresman, S., Addison, W. (2005) Indiana Mathematics Grade 1 Teachers Edition. Glenview, Il: Person Education, Inc. Task Task Task Task Task Activity: I could give each student two or three pieces of square paper, two or three pieces of circular paper, two or three pieces of pentagonal paper, two or three pieces of hexagonal paper and two or three pieces of rectangular paper. I would then ask each student to try to divide these papers into three or four equal shares. They could do this by drawing on the paper or folding it. I would tell them to try to figure out different ways to divide them. Task Task Task Task Task By:

Kayla Cole

Brittany Brugh

Melissa Dean

Amanda James

Full transcriptQuad 2 Geometry K 1st 2nd Reference Big Idea Big Idea Big Idea Big Idea Big Idea Big Idea Describe objects in the environment using shapes. (NGA Center & CCSSO, 2010) Correctly name shapes regardless of their orientations or overall size. – Students will state the name of the shapes no matter how big or small and the way the shapes are positioned. (NGA Center & CCSSO, 2010) National Governors Association Center for Best Practices & The Council of Chief State School Officers.(2010). Common Core Standards for Mathematics. Retrieved from www.corestandards.org. Task Introduce the name of the shapes and characteristics in a creative way like a poem or song. Then place the shapes on the board or overhead and have the students name off the characteristics. This is a great chance to introduce all the vocabulary for this lesson. After all the shapes have been introduced have the students take out white boards, then say the characteristics out loud and have the students draw the shape it goes too. Finally after getting the identification steps are done play “I Spy” game with the shapes in the classroom. (Indiana standards and resources) Describe the relative positions of objects using terms such as above, below, beside, in front of, behind, and next to. - Students will state where an object is using the terms above. (NGA Center & CCSSO, 2010) Task Tape off a large area on the classroom floor and collect pictures of people and animals in different locations. The taped off area will be for the game “Simon Says” tell the students the rules using the vocabulary words such as inside, outside, between, above, and below. Use phrases like “Simon Says stand inside the space.” Then after the game show the students the pictures and have them use the vocabulary words in a sentence. (Indiana standards and resources) Identify shapes as two-dimensional like shapes on a flat plane. Students need to be able to understand the shapes that are flat or 2-D. (NGA Center & CCSSO, 2010) Identify shapes as three-dimensional like solid shapes. Students need to be able to understand the shapes that are solid and tangible are 3-D. (NGA Center & CCSSO, 2010) Analyze two- and three- dimensional shapes, in different sizes and orientations. Students should be able to discuss in detail about two- and three- dimensional shapes regardless of their size or position. (NGA Center & CCSSO, 2010) Compare two- and three- dimensional shapes, in different sizes and orientations. Students should be able to discuss how two- and three-dimensional are different in size and position. (NGA Center & CCSSO, 2010) Using informal language to describe two- and three-dimensional similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other attributes (e.g., having sides of equal length). Students should be able to describe every part of two- and three-dimensional shapes. (NGA Center & CCSSO, 2010) Task When exploring two- and three- dimensional students will be given materials to make a flower out of paper. When the students are finished making there paper flower (2D) then real flowers (3D) will be placed in the classroom for students to observe. The students will need to compare and analyze the similarities and differences between the paper flower and real flower. Then the students will make a venn-diagram, draw pictures, or write about the flowers using the vocabulary and characteristics of 2D and 3D shapes. Model shapes in the world by building shapes from components (like sticks and clay balls).(NGA Center & CCSSO, 2010) Drawing shapes in the world.(NGA Center & CCSSO, 2010) Task Collect many materials that could be made into a shape for example, Popsicle sticks, clay, paper, yarn, ect. Then ask the students how many shapes can they made out of these materials. After the class has discuss a few ways of making shapes have the students explore making the shapes by prompting them to find as many as possible. After a while come back together and discuss using the vocabulary. Compose simple shapes to form larger shapes. (NGA Center & CCSSO, 2010) Task Burns, M. (1994) The Greedy Triangle. New York, NY: Scholastic. Big Idea Big Idea Distinguish defining attributes- The students define the characteristics that make up the shape, for example a triangle’s defining attributes are it is closed and has 3 sides and 3 angles. (NGA Center & CCSSO, 2010)

Distinguish non-defining attributes- The students say characteristics that are not important to defining a triangle, for example size and color. (NGA Center & CCSSO, 2010)

Draw shapes to possess defining attributes- drawing the shape to show the characteristics that define the shape.(NGA Center & CCSSO, 2010) Read the Greedy Triangle (Burns, 1994) and discuss what they think a quadrilateral is. Focusing on rectangles and squares, have the students explore different sized squares and rectangles and ask them questions like “How are these shapes different?” and “Can you prove it?” After exploring these shapes have the students tell you what defines a square and a rectangle. Afterward they made their own quadrilateral book to put their definition in and draw shapes to go along with it. Compose 2 dimensional or 3 dimensional shapes to create a composite shape- putting together shapes that are either two-dimensional or three- dimensional to make a common shape like triangle, rectangle, and square. For example putting together two triangles would make a square. (NGA Center & CCSSO, 2010)

Compose new shapes from the composite-Using common shapes, such as triangles, squares, and rectangles, to make new shapes like diamonds or pentagon. (NGA Center & CCSSO, 2010) To assist in learning about 2D and 3D shapes, the students will make two snowmen (or something to go along with the theme of that week). One of these snowmen will be made out of paper. The second will be made out of Styrofoam balls and other objects. The students will then compare the two snowmen to notice the differences. The goal is to notice that the snowman made out of paper is flat and the one made out of Styrofoam has “depth”. (Indiana standards and resources) Partition circles into two equal shares- splitting a circle into two equal pieces(NGA Center & CCSSO, 2010)

Partition circles into four equal shares- splitting a circle into four equal pieces(NGA Center & CCSSO, 2010)

Describe shares using halves- describe that a rectangle or circle is divided into halves when it is split in half.(NGA Center & CCSSO, 2010)

Describe shares using fourths- describe that a rectangle or circle is divided into fourths when it is split into four equal pieces.(NGA Center & CCSSO, 2010)

Describe shares using quarters- Describe that a rectangle or circle is divided into quarters when it is split into four equal pieces(NGA Center & CCSSO, 2010)

Describe the whole as two of- describe the cookie, for example, as being two of the one half pieces put together to make one whole cookie.(NGA Center & CCSSO, 2010)

Describe the whole as four of- describe the cake, for example, as being four of the one-fourth pieces put together to make one whole cake.(NGA Center & CCSSO, 2010) In order to teach halves fourths and quarters, we are going to use examples like cakes, brownies and cookies. Start by asking the students a question like “ I have one cake but I want to share it with three of my friends, how could we each get an equal size piece?” We would ask similar questions for getting halves and with using a circle shape like a cookie or pie. Then follow up with “One of my friends decided she didn’t want a piece so I ate her piece, how much of the cake did I eat? How many fourths or quarters?” The cake was divided into fourths. If we each just ate one piece and there were only three of us how much of the cake would be left? Big Idea Indiana standards and resources. Indiana department of education. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.(NGA Center & CCSSO, 2010) Big Idea Big Idea Big Idea Big Idea Big Idea Big Idea Recognize and draw triangles, quadrilaterals, pentagons, hexagons and cubes (NGA Center & CCSSO, 2010). Here students will be able to look at a triangle, quadrilateral, pentagon, hexagon or cube and be able to tell his/her teacher what that shape is by name. They will also be able to draw each shape when instructed by just hearing the name of the shape. Gumdrop Geometry: In this activity “students will construct various geometric shapes and describe the shapes according to the number of faces, edges, and vertices (corners)” (NGA Center & CCSSO, 2010). The students would first construct these shapes out of construction paper and then out of gum drops and toothpicks. Recognize the number of angles or equal faces for triangles, quadrilaterals, pentagons, hexagons and cubes (NGA Center & CCSSO, 2010). Students can do this by “finding measures of the interior and exterior angles of polygons, justifying the method used” (website). They can also do this for the other shapes. Activity: An activity I could do here is give each student a piece of paper and ask them to draw a triangle, quadrilateral, pentagon and hexagon then count the number of angles each one has. I could then ask the students to see if they can find a relationship between the shapes using the angles as a clue. I would ask them in they see a pattern and to redraw the shapes in a pattern they see and then justify why they thought that pattern worked. “Partition circles and rectangles into two, three, or four equal shares” (NGA Center & CCSSO, 2010). Student are going to need to “recognize...shapes that have symmetry” (National Standards). They can then use circles and rectangles to create these equal shares.

Example:

Activity: In this activity I could bring in some small pies and give each group of four students a pie. I could ask them how they think the pie needs to be cut so that everyone has an equal share. I could then go around to each table and cut the pies for them in the way they tell me. We could do the same thing with a rectangular candy bar. After doing this the students would then be asked to draw a picture on a piece of paper of how the pie/candy was cut up. I would then ask them to prove to me why each piece was equal. “Describe shares of circles and rectangles using the words halves, thirds, half of, a third of, etc, and describe the whole as two halves, three thirds, four fourths” (NGA Center & CCSSO, 2010). To accomplish this students could “use a variety of problem-solving strategies, such as drawing a diagram, making a chart” (website) and draw out real life situations to learn about halves, thirds and so on. These real life situations could be things like drawing a present and then drawing a ribbon around the present cutting it into four equal parts. Activity: I could divide the class in groups of four. I could then write half, third, half of the group, a third of the group, two halves, three thirds, four fourths up on the wipe off board. I would then ask the groups to figure out how to cut their group in half and one third and so on to make these words come to life. I would assess their work by having them write how they cut their groups up by drawing stick figures on a piece of paper. “Recognize that equal shares of identical wholes need not have the same shape” (NGA Center & CCSSO, 2010).

This best way to understand this big idea is by looking at an example. These identical squares are divided into equal shares and yet the divided shapes on the right square are different than the divided shapes on left square. The square on the left is broken up into four equal squares while the square on the right is broken up into four equal triangles.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.(NGA Center & CCSSO, 2010) I will use a math work page from the book Indiana Mathematics grade K(2005, p. 209). The children will be given simple shapes and an outline of a biger shape. The children will uses the manupulative shapes to see how many of one particular shape it will take to reconstruct the bigger shape. They will then write how many smaller shapes it took to do it.

Now if there are some students who need a little harder problem. I will give them the next page from the book Indiana Mathematics grade K(2005, p. 210). In this page the students will be given different shapes to make a bigger shape. Foresman, S., Addison, W. (2005) Indiana Mathematics Grade K. Glenview, Il: Person Education, Inc. I got this idea from the book Indiana Mathematics Grade 1 Teachers Edition (2005, P. 165B). This page shows mulitple intelegence, and how to teach for each one. One lesson would be to take simple shapes and make different animals out of them, another would be to cut out the simple shapes with construction paper and then give the students clay and some toothpicks. The students would then take the small balls of clay and make the shape of the construction paper with the toothpicks. I would have a couple of different options for kids to play around with this core standard. I would have a regualar page with differnt squars to have kids count, then i would have a station where there is blocks put together and the kids could then try to count with the blocks together, or if they needed to they could take the blocks apart to count or even to check their answers. I would also have a more advanced station, I got this from the book Indiana Mathematics Grade 2 Teachers Edition (2005, P. 251). This would be about netting, where instead of just counting the blocks they could see that the blocks put in specific locations could be brought together and make a 3D shape. Foresman, S., Addison, W. (2005) Indiana Mathematics Grade 2 Teachers Edition. Glenview, Il: Person Education, Inc. Foresman, S., Addison, W. (2005) Indiana Mathematics Grade 1 Teachers Edition. Glenview, Il: Person Education, Inc. Task Task Task Task Task Activity: I could give each student two or three pieces of square paper, two or three pieces of circular paper, two or three pieces of pentagonal paper, two or three pieces of hexagonal paper and two or three pieces of rectangular paper. I would then ask each student to try to divide these papers into three or four equal shares. They could do this by drawing on the paper or folding it. I would tell them to try to figure out different ways to divide them. Task Task Task Task Task By:

Kayla Cole

Brittany Brugh

Melissa Dean

Amanda James