You Mean Three Can Be One? Fractional Representations Presented by: Sherilyn Stratton, Carnegie Learning 60 minutes You Mean Three Can Be One? Participants work with pattern blocks to create different representations for the whole. They will create different representations for a hexagon using the denominators of the fractions to determine all possible combinations. Next, three hexagons are used to represent the whole and students determine what fractional part each pattern block shape represents. Then, combinations of different pattern blocks represent the whole and in each case students will identify the fractional values of various other pattern blocks. Participants build and sketch shapes such as a triangle that is two-thirds red, one-ninth green, and two-ninths blue. Finally, participants are given models of the whole in the forms of shapes, dot patterns, line segments, and adjoined rectangles and will sketch a model that represents some fractional part of the whole.

Goals for the Day To deepen your own understanding of fraction and their operations. To develop your mathematical reasoning and problem solving capabilities. To provide you with opportunity to reflect on and develop your own teaching practice. It may seem tedious, but many participants need a reminder about the goals of this type of professional development. Remind them each day with this slide that this professional development is about them deepening their current understandings. We will take some time for Classroom Connections, but we need to focus on content deepening first.

The Whole: Yellow Hexagon Start with the yellow hexagon. Cover the hexagon with other pattern block pieces. Record your design. Repeat the process to create as many representations as possible. Facilitator notes: This is a partner activity. May need to prompt participants to create combinations of different shapes. People typically cover the hexagon using only the same kind of shapes first. Give them plenty of time before prompting them (you don’t want to lower the cognitive demand if participants can figure it out themselves). As participants are working, ensure that they are recording their designs. If there are enough pattern blocks, they can keep their designs intact. If not, they will need to record them on pattern block paper. Some teachers have a tendency to “stack” their designs. Discourage that because it will make it difficult to compare the designs later. To go over answers, have them go through the number sentences that they wrote. If you have a document camera, have a participant build the designs under the document camera. Make sure that participants don’t repeat designs – if it has the same pieces, it doesn’t matter how they put them together.

The Whole: Yellow Hexagon How many different designs can you create? How did you know you determined all of the combinations? Write fraction number sentences to describe each of your designs. Sample Answers: Number of different designs: 7 (if you must use other pieces); 8 (if you allow a hexagon to cover a hexagon) 2 trapezoids: 1= 1 2 + 1 2 3 blue rhombus: 1= 1 3 + 1 3 + 1 3 6 triangles: 1= 1 6 + 1 6 + 1 6 + 1 6 + 1 6 + 1 6 1 trapezoid and 3 triangles: 1= 1 2 + 1 6 + 1 6 + 1 6 1 trapezoid, 1 rhombus, and 1 triangle: 1= 1 2 + 1 3 + 1 6 2 blue rhombi and 2 triangles: 1= 1 3 + 1 3 + 1 6 + 1 6 1 blue rhombus and 4 triangles: 1= 1 3 + 1 6 + 1 6 + 1 6 + 1 6 1 hexagon: 1=1 May need to prompt participants to create combinations of different shapes. People typically cover the hexagon using only the same kind of shapes first. Be sure to take time to discuss how participants knew they found all the ways.

The Whole: Triple Hexagon Create the whole: On a blank sheet of pattern block paper, put 3 hexagons together to form a “triple hexagon”. Trace around your triple hexagon shape(s). Determine what fractional part each pattern block shape represents: Hexagon Trapezoid Rhombus Triangle Participants should continue to work with their partners. Use the document camera to share answers after they complete #3 - #6. Hopefully participants will make different shapes with their three hexagons. Ask what shapes were used as wholes, and point out the different shapes used as wholes. If there are no differences, ask whether it is possible to make a different shape with three hexagons? Do the hexagons have to be touching to form a single region? Were the fraction names of the different pattern block pieces the same or different? Sample Answers: 1 hexagon: 1 3 , 2 hexagons: 2 3 , 3 hexagons: 3 3 =1 1 trapezoid: 1 6 , 2 trapezoids: 2 6 = 1 3 , 4 trapezoids: 4 6 = 2 3 , 5 trapezoids: 5 6 1 rhombus: 1 9 , 3 rhombi: 3 9 = 1 3 , 5 rhombi: 5 9 , 6 rhombi: 6 9 = 2 3 1 triangle: 1 18 , 2 triangles: 2 18 = 1 9 , 8 triangles: 8 18 = 4 9 , 15 triangles: 15 18 = 5 6

The Whole: Large Hexagon Cover the large hexagon using one or more trapezoids, rhombi, triangles, and hexagons. Use each shape at least once. Draw the result on the hexagon. Label each part with a fraction. Participants should work with their partners. Use the document camera to have participants share their solution strategies. How did you find the size of the yellow hexagon? One of the challenges of this larger shape is that it cannot be covered completely by hexagons. Participants will have to determine the value of the hexagon block based on the values of the other pattern blocks. Let participants realize this added constraint and grapple with the mathematics. Only 3 yellow hexagons will fit entirely in the space, but there are 3 remaining pieces that total 1 yellow hexagon. Therefore, the yellow hexagon is 1 4 the size of the Large Hexagon. Answers: Hexagon = 1 4 Trapezoid = 1 8 Rhombus = 1 12 Triangle = 1 24

How is this possible? From her work with pattern blocks in third grade, Lynn always thought that the trapezoid was called 1 2 . But when she made her triple hexagon, the trapezoid wasn’t called 1 2 anymore! What happened? How is this possible? Give participants individual think time to write their answers to this question. This should serve as a quick formative assessment for you – are the participants understanding the changes that happen to fractional pieces when the whole changes? The key idea is that the wholes are different. This is an example of where the same shape represents different fractional parts because the wholes are different. Names of fractional parts depend on the relationship between the parts and the whole. It is important to emphasize that fractions can be meaningless unless one thinks of them in reference to the whole. For example, half of a minute is different from half of an hour. It is extremely important that students have experiences in which the whole varies, and that we don’t give them the misunderstanding that “the trapezoid is 1 2 ”, since it is only 1 2 when the whole is 1 yellow hexagon. Explicitly ask about the relationship between the fraction and the size of the whole. For example, in the first activity, the trapezoid was 1 2 . What was it in triple hexagons? What was it in the large hexagon? In the activity that follows the next question, the participants will have an opportunity to articulate the relationship between the scale factor relating the wholes and the fractional part.

How is this possible? Lynn was trying to figure out which was larger, 1 3 or 1 2 . “My third grade teacher said that in fractions, larger is smaller and smaller is larger, so 1 2 is larger than 1 3 .” This is the first of 2 slides that contain question 8. Have a participant read through the question (in it’s entirety) and then ask participants to individually respond in writing. After 2 minutes of writing time, have them share their responses with their partner/group.

How is this possible? But then she looked at the three pattern block problems she just did. “The hexagon is 1 3 and the trapezoid is 1 2 . The hexagon is bigger than the trapezoid. So, 1 3 IS larger than 1 2 . I knew larger couldn’t be smaller!” What happened? How is this possible? This is the second of 2 slides that contain question 8. Have a participant read through the question (in it’s entirety) and then ask participants to individually respond in writing. After 2 minutes of writing time, have them share their responses with their partner/group. What were the big ideas that were common in the groups’ responses? Again, the key idea is that the wholes are different and names of fractional parts depend on the relationship between the parts and the whole. This is an example of how students can develop misconceptions if they do not pay attention to the whole in naming and comparing fractions, or if the teacher doesn’t give them opportunities to experience varying wholes.

Fractional Names of Pattern Block Pieces The Whole Pattern Block Piece Hexagon Triple Hexagon Large Hexagon Trapezoid Rhombus Triangle This activity is an opportunity to synthesize their learning from the pattern block activities and to extend their understanding to a generalization. First, they should fill in the 3 labeled columns with the fraction names for each pattern block piece. Participants shouldn’t worry about the blank columns at this point. They should fill out the 3 labeled columns first based on the work that they just did. The next slide provides the answers.

The Whole Pattern Block Piece Hexagon Triple Hexagon Large Hexagon 1 1 3 1 4 Trapezoid 1 2 1 6 1 8 Rhombus 1 9 1 12 Triangle 1 18 1 24 Now they will look for any and all patterns they notice in the table and try to explain mathematically why they believe those patterns exist. If participants focus on comparing between columns, ask them to describe any patterns they notice between rows (or vice versa). We are looking to see if anyone notices that as we increase the area of the whole the value of the pattern block decreases. You might hear people talk about scaling the whole. So if the Hexagon was the original, we scaled up the whole by a factor of 3 to get the triple hexagon. What happens to the fraction? You might hear some mistakenly say you scale the fractions up by a factor of 3 also. But actually, they are scaling down by a factor of 1 3 – the physical size of the piece doesn’t change; it’s size compared to the whole scales down by a factor of 1 3 . See if they can explain why this happens. Then have them look at original hexagon compared to the large hexagon which had an area equivalent to 4 hexagons. With a partner describe any patterns you notice in the table and explain why you think the patterns exist.

The Whole Pattern Block Piece Hexagon Triple Hexagon Large Hexagon Mega-Hexa gon 1 1 3 1 4 Trapezoid 1 2 1 6 1 8 Rhombus 1 9 1 12 Triangle 1 18 1 24 We are now adding another figure to the table – the Mega-Hexagon. Our new whole, the Mega-Hexagon, is equivalent to 7 hexagons. Without building a physical model, can the participants determine the fractional part of each piece? Have participants fill in the column in the table labeled, “Mega-Hexagon”. If they cannot determine parts without building the Mega-Hexagon, allow them to build a physical model. However, we want them to move towards seeing the relationship without building the model. The answers are on the next slide. Determine the fractional part of each piece if the whole is now a Mega-Hexagon (equivalent to 7 hexagons).

The Whole Pattern Block Piece Trape-zoid Hexagon Triple Hexagon Large Hexagon Mega-Hexagon 1 1 3 1 4 1 7 Trapezoid 1 2 1 6 1 8 1 14 Rhombus 1 9 1 12 1 21 Triangle 1 18 1 24 1 42 We will now scale down the whole. What if the trapezoid was now the whole or 1? Have them work with their partners/groups to fill in the column in the table labeled “Trapezoid”. The answers are on the next slide. Determine the fractional part of each pattern block piece if the whole is now a trapezoid.

The Whole Pattern Block Piece Rhom-bus Trape-zoid Hexagon Triple Hexagon Large Hexagon Mega-Hexagon 2 1 1 3 1 4 1 7 Trapezoid 1 2 1 6 1 8 1 14 Rhombus 2 3 1 9 1 12 1 21 Triangle 1 18 1 24 1 42 We are going to scale down again – the blue rhombus is now 1. The answers are on the next slide. Determine the fractional part of each piece if the whole is now a rhombus.

The Whole Pattern Block Piece Tri-angle Rhom-bus Trape-zoid Hexagon Triple Hexagon Large Hexagon Mega-Hexagon 3 2 1 1 3 1 4 1 7 Trapezoid 3 2 1 2 1 6 1 8 1 14 Rhombus 2 3 1 9 1 12 1 21 Triangle 1 18 1 24 1 42 The final scaling down – the triangle is 1. The answers are on the next slide. Determine the fractional part of each piece if the whole is now a triangle.

Describe the patterns that you see in the table. The Whole Pattern Block Piece Tri-angle Rhom-bus Trape-zoid Hexagon Triple Hexagon Large Hexagon Mega-Hexagon 6 3 2 1 1 3 1 4 1 7 Trapezoid 3 2 1 2 1 6 1 8 1 14 Rhombus 2 3 1 9 1 12 1 21 Triangle 1 18 1 24 1 42 Participants should take some time to write down and describe to their groups the patterns that they see in the table. Sample Answers: The hexagon is always 6 times the size of the triangle. The trapezoid is always 3 times the size of the triangle. The rhombus is always 2 times the size of the triangle. The hexagon is always 3 times the size of the rhombus. The trapezoid is always 1 1 2 times the size of the rhombus. When the size of the whole increases by a factor of 𝑛, the size of the pieces decreases by a factor of 1 𝑛 . When the size of the whole decreases by a factor of 1 𝑛 , the size of the pieces increases by a factor of 𝑛. Describe the patterns that you see in the table.

Mathematical Practices Describe ways in which you can connect the mathematical practices to the essential ideas of these tasks. (5 minutes) What mathematical practices were addressed in this problem? As teachers mention mathematical practices that were addressed in this problem, require them to defend their choices with evidence from the problem. Make sense of problems and persevere in solving them. Model with mathematics. Look for and make use of structure. Look for and express regularity in repeated reasoning.

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