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Rationale
1. Building Conceptual Understanding to increase Procedural Fluency
2. CRA Model
3. Connecting the concrete and abstract
8 Research Based Teaching Strategies1. 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.
Various meanings of a Fraction
1. Part-Whole Meaning
2. Division
3. Ratio (later focus, not this workshop)
Part-Whole Meaning of
• Three elements of the meaning1) 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)
Two types of Wholes
1. Discrete
2. Continuous
Equipartitioning
• Examples
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.
Developing the Whole
• In how many different ways can you cover the hexagon? Write an addition equation for each way.
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.
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.)
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.
Division Meaning of
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.
Describe the various meanings of
Equivalence
• Using the Pattern Blocks, discuss with a partner how equivalent fractions can be visualized. Show at least 3 examples.
Simplify the following
Explain using pattern blocks.
Comparing and ordering
• How can pattern blocks be used to answer the following?
Determine which fraction is larger.1. or 2. or 3. or
Convert the following to an improper fraction.
Explain using the pattern blocks.1. 2
2. 4
3. 1
Convert the following to a Mixed Number
Explain using the pattern blocks
1.
2.
Operations with Fractions
Using the CRAW Model
Use Table 1 and Table 2
Operations with FractionsChoose a unit and do each of the following. Write numerical expressions for your work with pattern blocks. Do in multiple ways if possible.
1. Combine and 2. Combine 2 and 3. How much more is than ?4. Take 1 from 2 5. Take 2 from 46. What needs to be combined with to get 2 ?
Solve the following
Jim ate of a whole pizza at lunch and of the whole pizza later for a snack. How much of the pizza did Jim eat? Explain using pattern blocks.
You have 1 2/3 cups of sugar. The recipe calls for 4 cups of sugar. How much more sugar do you need? Explain using pattern blocks.
Solve the following
Juanita mowed of a lawn. Jose raked of the mowed part. How much of the lawn has been raked?
Operations with Fractions
• Show each of these with pattern blocks or drawings. Make sketches of your pattern block work for later reference. 1. x 2. x 63. x 64. x 1 5. x (x 4)
Solve the following
A company is promoting its pizza by giving away slices. In the first hour they gave away 5 pizzas cut into, how many slices did they give away?
Operations with Fractions
• Show each of these with pattern blocks or drawings. Make sketches of your pattern block work for later reference. 1. 2. 4 3.
Generalizations
• Multiplication – a x b = c– Case 1 : a = 1, then b = c– Case 2 : a > 1, then c > b– Case 3 :0 < a < 1, then c < b
• Division – a b = c (a and b cannot equal 0)– Case 1: a = b, then c = 1– Case 2 : a > b, then c > 1– Case 3 : a < b, then c < 1
Questions and Thoughts
