1. DEEP with Pattern Blocks in Fractions Jeremy Winters jwinters@mtsu.edu Funded by MTSU Public Service Grant

2. Rationale 1.Building Conceptual Understanding to increase Procedural Fluency 2.CRA Model 3.Connecting the concrete and abstract

3. 8 Research Based Teaching Strategies 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking.

4. Various meanings of a Fraction 1.Part-Whole Meaning 2.Division 3.Ratio (later focus, not this workshop)

5. Three elements of the meaning 1)The unit (or whole) is clearly in mind What is equal to 1 ? 2)The denominator tells how many pieces of equal size the unit is cut into. (size of the pieces) 3)The numerator tells how many such pieces are being considered. (how many pieces)

6. Two types of Wholes 1.Discrete 2.Continuous

7. Equipartitioning Examples

8. Developing the Whole Using pattern blocks, take the yellow hexagon, the red trapezoid, the blue rhombus, and the green triangle. – A. let the hexagon =1. give the value for each of the other three pieces. – B. Let the trapezoid =1. Give the value for each of the other three pieces. – C. Let a pile of two hexagons = 1. Give the values for the hexagon and each of the other three pieces.

9. Developing the Whole In how many different ways can you cover the hexagon? Write an addition equation for each way.

10. Pattern Block Riddles 1.The area of all the blocks together is the same as the area of 24 green triangles. Three of the blocks together make up 75% of the total area. The green blocks cover one-half as much area as the blue blocks. 2.There are 9 blocks. The area covered by the yellow blocks is equal to the area covered by the blue blocks. The area covered by the red block is one-eighth the area covered by the yellow and blue blocks combined. 3.There are 8 blocks. 50% are blocks that would each cover one-third of the largest block. 25% are blocks that would each cover one-half of the largest block. The bag contains red, blue, green and yellow blocks.

11. Pattern Block Riddles 1.The blocks can be arranged to cover a yellow hexagon. They can also be arranged to make a parallelogram. There are only 2 colors of blocks. There are no red blocks. 2.There are 2 blocks. The blocks can be arranged to make a hexagon. This hexagon has 2 right angles. The perimeter of this hexagon is 7 units. (1 unit = the length of a side of a green triangle.)

12. Can you think of some really good clues to use in your own Pattern Block riddle? With a partner, choose up to 6 pattern blocks to write clues about. Examine your blocks. Notice things about them Decide on 3 to 5 clues for your riddle and write them down. For example, if you choose 2 green blocks and 2 blue blocks, your riddle might say: – Together the blocks form a hexagon the same size and shape as the yellow pattern block. There are 2 different kinds of blocks in the bag. There is the same number of each type of block. Talk about each clue. Is it too hard? Does it give away the riddle too soon? When you have all your clues, test your riddle and make sure it works. Then put your pattern blocks in the paper bag, close it, and clip the riddle to the bag. Exchange riddle bags with another pair and try to solve their riddle. Then look in the bag to check your solution.

13. 1.Sharing Equally division (partitive division) a)a whole would be partitioned into b equal parts 2.Repeated Subtraction division (measurement or quotitive division) a)How many (or much of) b fits in a.

15. Equivalence Using the Pattern Blocks, discuss with a partner how equivalent fractions can be visualized. Show at least 3 examples.

16. Simplify the following

17. Comparing and ordering

18. Convert the following to an improper fraction.

19. Convert the following to a Mixed Number

20. Operations with Fractions Using the CRAW Model Use Table 1 and Table 2

21. Operations with Fractions

22. Solve the following

24. Operations with Fractions

25. Solve the following

26. Operations with Fractions

27. Generalizations

28. Questions and Thoughts