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- 1. Just Say “NO” to Worksheets Presented by Michelle Flaming [email_address] ESSDACK
- 2. Agenda <ul><li>Opening Activities </li></ul><ul><li>Cheap Manipulatives </li></ul><ul><ul><li>Foam Creatures, Toothpicks, Cards, Dice etc. </li></ul></ul><ul><li>Color Tiles </li></ul><ul><li>Geoboards </li></ul><ul><li>Base-Ten Blocks </li></ul><ul><li>Pattern Blocks </li></ul>
- 3. Opening Activity <ul><li>Probability lies between 0 and 1 </li></ul><ul><ul><li>Please line up according to where your event falls on the continuum. </li></ul></ul><ul><ul><li>No Way Jose </li></ul></ul><ul><ul><ul><li>Please line up according to where your event would fall in the story setting - The Titanic </li></ul></ul></ul>
- 4. Another Opening Activity <ul><li>Draw a line segment on a 1/2 piece of paper. </li></ul><ul><li>Measure your line to the nearest inch. </li></ul><ul><li>Write your measurement next to the line. </li></ul><ul><li>Line up in order from minimum to maximum. </li></ul>
- 5. Why Use Manipulatives? <ul><li>Helps children to understand mathematics. </li></ul><ul><li>Provide concrete ways for students to bring meaning to abstract mathematical ideas. </li></ul><ul><li>Helps children to learn new concepts and relate new concepts to what they have already learned. </li></ul><ul><li>Assist children with solving problems </li></ul>
- 6. Important to focus on … <ul><li>Thinking and reasoning must be the TOP priorities when students are engaged in learning manipulative materials. </li></ul><ul><li>Scaffolding from concrete to pictorial to abstract. </li></ul><ul><li>Connections of multiple representations. </li></ul><ul><ul><li>Pictures - Written Examples </li></ul></ul><ul><ul><li>Spoken Language - Written symbols </li></ul></ul>
- 7. Cheap Manipulatives <ul><li>My Function Book </li></ul><ul><ul><li>Proportion Functions </li></ul></ul><ul><ul><ul><li>Foam Creatures from Oriental Trading </li></ul></ul></ul><ul><ul><li>Linear Functions </li></ul></ul><ul><ul><ul><li>Toothpicks -- Bridges </li></ul></ul></ul>
- 8. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used during this activity? </li></ul>
- 9. The Toothpick Problem <ul><li>Using four toothpicks, determine all the different ways you can put the four together. </li></ul><ul><ul><li>All right angles </li></ul></ul><ul><ul><li>Touch end to end </li></ul></ul>
- 10. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used during this activity? </li></ul>
- 11. Close to 100 <ul><li>Shuffle a deck of cards (using Aces - 9 and Jokers) </li></ul><ul><li>Deal 6 cards to each player. </li></ul><ul><li>Each player chooses 4 of the 6 cards to make a 2 two digit numbers. The goal is to be make the sum as close to 100 as possible. </li></ul><ul><li>The score for this round, for this player, would be: 1 </li></ul><ul><li>Play 5 rounds, add the total from each round. </li></ul>
- 12. Where’s The Mathematics? <ul><li>Make a list of mathematical concepts and processes used during this activity? </li></ul>
- 13. Exploring Probability <ul><li>What are all the possible sums when rolling two dice? </li></ul><ul><li>Play the dice game. </li></ul><ul><li>Exploring experimental probability. </li></ul><ul><li>Determining theoretical probability. </li></ul>
- 14. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used during this activity? </li></ul>
- 15. Making Useable Clocks X Marks The Spot
- 21. Color Tiles <ul><li>Presenting Color Tiles </li></ul><ul><li>Viewing the Videotape </li></ul><ul><ul><li>Focus on the teachers’ thinking and reasoning. </li></ul></ul><ul><li>More Explorations </li></ul><ul><li>Discussion of the Videotape </li></ul>
- 22. Color Tile Uses: <ul><li>Counting </li></ul><ul><li>Estimating </li></ul><ul><li>Measuring </li></ul><ul><li>Investigating multiplication patterns. </li></ul><ul><li>Solving problems with fractions </li></ul><ul><li>Exploring geometric shapes </li></ul><ul><li>Carrying out probability experiments </li></ul>
- 23. Four-Color Square <ul><li>Make a square using 16 tiles, four of each color. </li></ul><ul><li>Arrange the tiles so there is exactly one of each color in each row and column. </li></ul><ul><li>See if you can do this so that the diagonals also have one color each. </li></ul>
- 24. The Changing Perimeter <ul><li>If you count the length of a side as one unit, the perimeter of the 16-tile square is 16 units. </li></ul><ul><li>Rearrange the same tiles into another shape. (A shape must have at least one whole side of each tile touching at least one whole side of another) Figure it’s perimeter. Record on grid paper your new shape. </li></ul><ul><li>How would you arrange the tiles to make the shape with the longest perimeter? Record on grid paper. The shortest perimeter? Record. </li></ul>
- 25. Viewing the Videotape <ul><li>What did you observe about the teaching? Refer to specific examples from the videotape. </li></ul><ul><li>What did you observe in the children’s responses? Refer to specific examples from the videotape. </li></ul><ul><li>What questions do you have as a result of watching the videotape. </li></ul>
- 26. Videotape discussion <ul><li>In the first activity, in which the 4th graders build rectangles for different number of tiles, the students were given a chance to explore with materials, then they were given a worksheet so that they could reflect on what they had done, and finally the class was brought together at the end for a class discussion. Discuss the benefits of each aspects of the lesson. </li></ul>
- 27. Videotape discussion <ul><li>You saw Hal and Myisha and then Russ and Shane working together on the Fraction Rectangles activity. Explain why it is beneficial for students to work together. </li></ul>
- 28. Videotape discussion <ul><li>Explain how finding areas with Color Tiles can help develop the second graders’ understanding of place value. </li></ul>
- 29. Videotape discussion <ul><li>The 4th graders on the videotape were playing Four in a Row for the first time. What would you have the children do now? </li></ul>
- 30. Color Tile Explorations <ul><li>Probability with Color Tiles (Grades 1-6) </li></ul><ul><li>5 Squares (Grades 2-6) </li></ul><ul><li>Rectangle Constructions (Grades 3-6) </li></ul><ul><li>Rectangles and Fractions (Grades 4-6) </li></ul>
- 31. Probability with Color Tiles (1-6) <ul><li>On a recording sheet, record which paper bag you have A, B, or C. </li></ul><ul><li>Without looking inside, take out a color tile. Record it’s color. Put it back in the bag, and shake the bag to mix the tiles. </li></ul><ul><li>Take 10 samples this way. </li></ul><ul><li>Predict how many color tiles of each. </li></ul><ul><li>Take another 10 samples. </li></ul><ul><li>Make a new prediction of how many color tiles of each. </li></ul><ul><li>Write about how you made your prediction. </li></ul>
- 32. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used in this exploration? </li></ul>
- 33. 5 Squares (2-6) <ul><li>With a partner, choose 5 square tiles of the same color. </li></ul><ul><li>Explore all the ways to put 5 squares together to make a larger shape. (Each side must match the same length as another side.) </li></ul><ul><li>Transfer all possibilities to grid paper. </li></ul>
- 34. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used in this exploration? </li></ul>
- 35. Rectangle Constructions (3-6) <ul><li>Work with a partner to determine which number of tiles will produce rectangles. </li></ul><ul><li>Select 12 tiles to use. </li></ul><ul><li>Construct all the possible rectangles using a 12 tiles. You may want to draw the rectangles on grid paper for better clarity. </li></ul><ul><li>How many, and what dimension of rectangles did you create? </li></ul><ul><li>The different dimensions represent the factors of a number. </li></ul><ul><li>Factors of 12 are: </li></ul><ul><ul><li>1, 2, 3, 4,6, and 12 </li></ul></ul>
- 36. Rectangle Constructions <ul><li>Work with a partner to determine which number of tiles will produce rectangles. </li></ul><ul><li>Select tiles between the numbers 1- 25. </li></ul><ul><li>Construct all the possible rectangles using a variety of number of tiles and record results the results on grid paper. </li></ul><ul><li>How many, and what dimension of rectangles did you create for each number? </li></ul><ul><li>Note: The different dimensions represent the factors of a number. </li></ul>
- 37. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used in this exploration? </li></ul>
- 38. Rectangles and Fractions (4-6) <ul><li>With a partner, find possible rectangles which meet certain criteria and then record. </li></ul><ul><li>For example, make a rectangle that is 1/2 red and 1/2 blue. </li></ul><ul><li>Possible solutions are: </li></ul>
- 39. Possible Rectangles: <ul><li>#1 Make a rectangle that is 1/2 blue, 1/4 red, and 1/4 yellow. Record on grid paper. </li></ul><ul><li>#2 Make a rectangle that is 1/8 red, 4/8, yellow, 2/8 green, and 1/8 blue. Record on grid paper. </li></ul><ul><li>#3 Make a rectangle that is 3/4 red, 1/8 yellow, and 1/8 green. Record on grid paper. </li></ul><ul><li>#4 Make a square that is 1/3 yellow and 2/3 blue. </li></ul>
- 40. Where’s the Mathematics? <ul><li>Make a list of mathematical concepts and processes used in this exploration? </li></ul>
- 41. Geoboards <ul><li>Presenting Geoboards </li></ul><ul><li>Viewing the Videotape </li></ul><ul><ul><li>Focus on the teachers’ thinking and reasoning. </li></ul></ul><ul><li>More Explorations </li></ul><ul><li>Discussion of the Videotape </li></ul>
- 42. Geoboards <ul><li>Geometric Figures </li></ul><ul><li>Fractions </li></ul><ul><li>Area/Perimeter </li></ul>
- 43. Rotational Symmetry <ul><li>Make a shape that is not a square and looks the same no matter on which side you rest the Geoboard. </li></ul><ul><li>How many different shapes were made in the class? </li></ul><ul><li>Similar Shapes: Make two figures that have the same shape, are different sizes, and are not squares. </li></ul>
- 44. Viewing the Videotape <ul><li>What did you observe about the teaching? Refer to specific examples from the videotape. </li></ul><ul><li>What did you observe in the children’s responses? Refer to specific examples from the videotape. </li></ul><ul><li>What questions do you have as a result of watching the videotape. </li></ul>
- 45. Videotape Discussions: <ul><li>1. What are the benefits of having children hypothesize how the shapes were sorted, as did the 4th graders in the first activity on the videotape? </li></ul>
- 46. Videotape Discussions: <ul><li>2. What could students learn from the Things That Fly activity, where they constructed shapes and organized them into a graph? </li></ul>
- 47. Videotape Discussions: <ul><li>3. Why is it important for students to know the definition of a polygon? </li></ul><ul><li>What is the value in having students create and read their definitions of a polygon? </li></ul>
- 48. Videotape Discussions: <ul><li>4. How does the approach to teaching about polygons shown on the videotape differ from a textbook approach? </li></ul>
- 49. Geoboard Explorations <ul><li>Free Explorations (all grades) </li></ul><ul><li>Polygons on the Geoboard (1-6) </li></ul><ul><li>Sorting Shapes (3-6) </li></ul><ul><li>Area on the Geoboard (4-6) </li></ul>
- 50. Free Explorations <ul><li>Make numerals on your geoboard. </li></ul><ul><li>Make your initials. Can you make other letters as well? </li></ul><ul><li>Make a car. How many rubber bands did you use? How is your car like others? How does it differ? </li></ul><ul><li>Using just one rubber band, make a shape that touches four pegs and has one peg inside. </li></ul>
- 51. Where is the Mathematics? <ul><li>Make a list of mathematical concepts and processes used in this exploration? </li></ul>
- 52. Polygons on the Geoboard <ul><li>Make a 3-sided polygon with 1 square corner and no 2 sides the same length. </li></ul><ul><li>Make a 4-sided polygon with no parallel sides. </li></ul><ul><li>Make a 4-sided polygons with all sides different lengths. </li></ul><ul><li>Make a 5-sided polygon that has exactly one pair of parallel sides. </li></ul><ul><li>Make a 6-sided polygon with one pair of sides perpendicular. </li></ul><ul><li>Make a polygon that is NOT a square and looks the same no matter on which side your rest the geoboard. </li></ul>
- 53. Base Ten Blocks <ul><li>Presenting Base Ten Blocks </li></ul><ul><li>Viewing the Videotape </li></ul><ul><ul><li>Focus on the teachers’ thinking and reasoning. </li></ul></ul><ul><li>More Explorations </li></ul><ul><li>Discussion of the Videotape </li></ul>
- 54. Race to 100 <ul><li>Play with a partner. </li></ul><ul><li>Take turns rolling the dice. </li></ul><ul><li>When it is your turn, roll the dice, calculate the sum, and put that number of units on the place value mat. </li></ul><ul><li>Do any necessary exchanges as you go. </li></ul><ul><li>The first to get 100 wins. </li></ul>
- 55. Base-Ten Blocks Uses: <ul><li>Physical model of our place value system. </li></ul><ul><li>Solving missing addend problems. </li></ul><ul><li>Operations: addition, subtraction, multiplication, and division. </li></ul><ul><li>Comparing and Ordering: whole numbers, decimals, fractions, etc. </li></ul>
- 56. Viewing the Videotape <ul><li>What did you observe about the teaching? Refer to specific examples from the videotape. </li></ul><ul><li>What did you observe in the children’s responses? Refer to specific examples from the videotape. </li></ul><ul><li>What questions do you have as a result of watching the videotape. </li></ul>
- 57. Videotape discussion <ul><li>When playing Race for a Flat, the children are involved only with the Base Ten Blocks. No numerical symbols are used. This is done purposely, with the belief that the numerical symbolization can interfere with concept development. Describe what you think is the basis of this belief. Discuss whether you agree or disagree, and why. </li></ul>
- 58. Videotape discussion <ul><li>Why do you think the children are asked to work together, as when Jean and Shannon solve 164 + 257? </li></ul>
- 59. Videotape discussion <ul><li>The fourth graders working on How Many More to Make Five Flats? Are capable of solving problems such as these by using the paper-pencil algorithm. What is the benefit of having them work with the blocks? What can the teacher learn about their understanding from this small-group interaction? </li></ul>
- 60. Videotape discussion <ul><li>There are three levels to the lesson on long division - the concrete, the connecting, and the symbolic. The children first experience the concept of the division algorithm concretely. The teacher then connects their experience to the standard algorithm. Finally, the boys work independently and use the symbolic algorithm. Discuss the reasons for and benefits of each of these levels of instruction. </li></ul><ul><li>Shown on the videotape are the boys doing only one problem at each of the levels. In actuality, they did many more examples. How can you tell when it is appropriate to move on to the next level of instruction? </li></ul>
- 61. Videotape discussion <ul><li>When ordering decimals on their worksheets, the children are grouped heterogeneously. Nguyen takes the role of explaining to Zal and Suzanne as they work. What do you think the effect of this heterogeneous grouping has on Nguyen, Zal, and Suzanne’s learning? </li></ul>
- 62. Base Ten Block Explorations <ul><li>Addition with Base Ten blocks (2-4) </li></ul><ul><ul><li>CGI Problems </li></ul></ul><ul><ul><li>Connecting Concrete to an Algorithm </li></ul></ul><ul><li>How Many More? (3-6) </li></ul><ul><li>Long Division with Base Ten (4-6) </li></ul><ul><li>Decimals (4-6) </li></ul><ul><ul><li>Compare and Order </li></ul></ul>
- 63. Addition with Base Ten Blocks (2-4) <ul><li>Show me 125 </li></ul><ul><li>Show me 304 </li></ul><ul><li>Bentley collects baseball cards. He has 227 baseball cards. For his birthday he received 178 more. How many birthday cards does Bentley have? </li></ul><ul><li>Terri has 389 seashells. While walking at the beach she found 99 more. How many seashells does Terri have? </li></ul>
- 64. How Many More? (3-6) Show with Base-Ten Blocks <ul><li>You have 228 pennies. How many more do you need to have 400 pennies? </li></ul><ul><li>Sammy has read 147 pages in his book. He would like to finish the book tonight. The book has 500 pages in it. How many more pages does Sammy need to read to finish the book? </li></ul><ul><li>Lori wants to work out 300 minutes a week, so far she has worked out 128 minutes. How many more minutes does she need to work out. </li></ul>
- 65. Long Division with Base Ten (4-6) <ul><li>Provide enough time for students to work at the concrete level. Time used at this level is well spent. </li></ul><ul><li>Then the teacher helps connect the concrete to the symbolism of the algorithm. </li></ul><ul><li>Play in both worlds </li></ul>
- 66. Long Division with Base Ten (4-6) <ul><li>Two different types of division: </li></ul><ul><ul><li>24 cherries are shared among 3 children. </li></ul></ul><ul><ul><li>24 cherries are placed on plates with three on each plate. How many plates are needed? </li></ul></ul>
- 67. Long Division with Base Ten (4-6) <ul><li>Show using base-ten blocks. </li></ul><ul><ul><li>372 pennies are to be shared amongst three students. How many pennies will each get? </li></ul></ul><ul><ul><li>530 students are going to the zoo. Each car holds 4 students. How many cars need to be taken to hold all the students? </li></ul></ul><ul><ul><li>421 baseball cards are being shared between three friends. How many baseball cards will each friend get. </li></ul></ul><ul><li>Practice writing down the numbers. </li></ul><ul><ul><li>1. How many did you put in each group? </li></ul></ul><ul><ul><li>2. How much did you use altogether? </li></ul></ul><ul><ul><li>3. How much is left on your board? </li></ul></ul>
- 68. Decimals (4-6) <ul><li>A prerequisite understanding is fractions. </li></ul><ul><li>First work with tenths, after students have a firm understanding, move on to hundredths. </li></ul><ul><li>Relating decimals to money </li></ul><ul><li>Write in two or more ways, plus represent the value by making a drawing. </li></ul><ul><ul><li>13/100, .13, or .1 + .03 </li></ul></ul>
- 69. Decimals (4-6) <ul><li>Show me 7/10 or .7 </li></ul><ul><li>Show me 5/10 or .5 </li></ul><ul><li>Show me 6/10 or .6 </li></ul><ul><li>Using your three pictures, put the decimals in order. </li></ul>
- 70. Decimals (4-6) <ul><li>Show me 0.21 </li></ul><ul><li>Show me 0.07 </li></ul><ul><li>Show me 0.15 </li></ul><ul><li>Using your three pictures, put the decimals in order. </li></ul>
- 71. Decimals (4-6) <ul><li>Show me 0.8 </li></ul><ul><li>Show me 0.81 </li></ul><ul><li>Show me 0.09 </li></ul><ul><li>Using your three pictures, put the decimals in order. </li></ul>
- 72. Decimals (4-6) <ul><li>Show me 0.30 </li></ul><ul><li>Show me 0.3 </li></ul><ul><li>Write >, <, or = for each. </li></ul><ul><li>0.30 0.3 </li></ul>
- 73. Pattern Blocks <ul><li>Presenting Pattern Blocks </li></ul><ul><li>Viewing the Videotape </li></ul><ul><ul><li>Focus on the teachers’ thinking and reasoning. </li></ul></ul><ul><li>More Explorations </li></ul><ul><li>Discussion of the Videotape </li></ul>
- 74. Pattern Block Exploratory Activity: <ul><li>Designing Puzzle Shapes: </li></ul><ul><ul><li>Make a design with the blocks and trace around the outside of it. </li></ul></ul><ul><ul><li>On your paper, record how many blocks you used for your design. </li></ul></ul><ul><ul><li>Trade papers and see if you can fit the indicated number of pieces into each other’s designs. </li></ul></ul>
- 75. Viewing the Videotape <ul><li>What did you observe about the teaching? Refer to specific examples from the videotape. </li></ul><ul><li>What did you observe in the children’s responses? Refer to specific examples from the videotape. </li></ul><ul><li>What questions do you have as a result of watching the videotape. </li></ul>
- 76. Videotape Discussion: <ul><li>The fourth graders are introduced to the mathematical names for the Pattern Blocks after their exploration time. What do you think are the benefits of introducing vocabulary at this time rather than when the students are first given the blocks? What other names for the blocks could be used? </li></ul>
- 77. Videotape Discussion: <ul><li>At the end of the fifth graders’ exploration of building larger shapes, Hal explained why it wasn’t possible to build larger hexagons using only the yellow blocks. What does this explanation mean to you? What other explanation can you offer? </li></ul>
- 78. Videotape Discussion <ul><li>The exploration for fifth graders in which they build shapes with specific fractional proportions is an example of teaching fractions through a problem-solving approach. What do you think this statement means? What follow-up do you think makes sense after the students solve fraction puzzles? </li></ul>
- 79. Videotape Discussion: <ul><li>How did the teaching done on the tape support children’s learning? What would you like to have seen the teachers do differently or in addition to what was shown? </li></ul>
- 80. Scoop and Sort (K-2) <ul><li>Take a handful of pattern blocks from the tub. </li></ul><ul><li>Make a concrete graph of the blocks. All hexagons go together and lined up. </li></ul><ul><li>Make a record of the concrete graph by drawing on smaller grid paper. </li></ul><ul><li>Answer the following questions: </li></ul><ul><ul><li>1. Which block is there the most of? (mode) </li></ul></ul><ul><ul><li>2. Which block is there the least of? </li></ul></ul><ul><ul><li>3. How many blocks did you scoop in all? </li></ul></ul><ul><ul><li>4. How many blocks with four sides did you scoop? </li></ul></ul><ul><ul><li>5. Did you scoop more triangles or squares? How many more? </li></ul></ul>
- 81. Advanced Hexagon Shape Covering (3-6) <ul><li>Cooperative Group Activity: </li></ul><ul><ul><li>Work together to discover how to cover the hexagon using different number of blocks. </li></ul></ul><ul><ul><li>Record on Hexagon Puzzle Sheet </li></ul></ul>
- 82. Things I want to Remember … <ul><li>Share one idea you wrote down. </li></ul><ul><li>What is the purpose of manipulatives? </li></ul><ul><li>What is the teacher’s role as students work with manipulatives? </li></ul><ul><li>Share one idea you will take back and share with a colleague. </li></ul>

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