Student Probe Figure A Fraction Values and Changing Wholes At a Glance Figure B Name the fractional part of Figure A for each of the following colored pattern blocks: blue rhombus, green triangle, red trapezoid. Name the fractional part of Figure B for each of the following colored pattern blocks: blue rhombus, green trangle, red trapezoid. Figure C List and explain how you found the fractional amount for each of the colored pattern blocks using Figure C as your whole: blue rhombus, green triangle, red trapezoid. Answers: Figure A: • Blue Rhombus: 1/3 of the yellow hexagon • Green Triangle: 1/6 of the yellow hexagon • Red Trapezoid: 1/2 of the yellow hexagon Figure B: • Blue Rhombus: 2/3 of the red trapezoid • Green Triangle: 1/3 of the red trapezoid • Red Trapezoid: 3/3 of the red trapezoid What: Understand that as the defined whole change so does the name for that fractional piece. Common Core Standards: CC.4.NF.1 Extend understanding of fraction equivalence and ordering. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.) Matched Arkansas Standard: AR.5.NO.1.1 (NO.1.5.1) Rational Numbers: Use models and visual representations to develop the concepts of the following: -­‐-­‐-­‐Fractions: parts of unit wholes, parts of a collection, locations on number lines, locations on ruler (benchmark fractions), divisions of whole numbers; -­‐-­‐-­‐Ratios: part-­‐to-­‐part (2 boys to 3 girls), part-­‐to-­‐whole (2 boys to 5 people); -­‐-­‐-­‐Percents: part-­‐to-­‐100 Mathematical Practices: Attend to precision. Look for and make use of structure. Who: Students who think that individual fractional pieces have a specific name regardless of the whole. (Such as a green triangle is always 1/6.) Grade Level: 4 Prerequisite Vocabulary: None Prerequisite Skills: None Delivery Format: individual, small group Lesson Length: 15-­‐30 minutes Materials, Resources, Technology: Pattern blocks Student Worksheets: Pattern Block Charts 1, 2, 3 and Summarization Chart Figure C: • Blue Rhombus: 2/9 of the yellow figure • Green Triangle: 1/9 of the yellow figure • Red Trapezoid: 1/3 of the yellow figure Most students should be able to correctly name the parts for Figure A. It is with Figure B and Figure C that special attention needs to be placed when looking at students’ responses. Explanations for Figure C need to be closely examined when determining level of understanding. Lesson Description The lesson is intended to help students develop an understanding of the relationship that exists between different patterns blocks when the defined whole is changed. Students will use a particular set of patterns blocks to find fractional amounts for a specific pattern block designated as the whole. There are multiple opportunities for students to record the information in a table for reference throughout the lesson. It is the intent of the lesson to provide enough repeated exposure to the patterns and relationships that are formed between the part and the whole that students will be able to make some generalizations about what happens when the whole is changed but the pattern block stays the same. Rationale Fraction manipulatives can be useful tools when introducing and helping students to establish a conceptual understanding of fractions. With all the benefits that these tools offer they can also lead students to a superficial understanding of fractions if not addressed properly. Students may think that a particular pattern block or fractional piece is always 1/3 or always 1/4 because of its use within that set of manipulatives. It is the focus of this particular lesson to put an emphasis on the relationship between the part and the whole that exists. Not only does this lesson focus on part to whole relationships but also part to a changing whole. Giving students opportunities to explore and disprove these misconceptions will help to place the emphasis on the concept instead of the tool. Preparation Provide students with pattern blocks (hexagons, trapezoids, triangles, and rhombi) and copies of the handouts “Pattern Block Charts 1, 2, and 3” and the “Summarization Chart. Lesson The teacher says or does… Expect students to say or do… 1. Let’s review the names of Students should be able to the pattern blocks we are state the names of the going to use today in our following pattern blocks: lesson. yellow-­‐hexagon blue-­‐rhombus red-­‐trapezoid green-­‐triangles 2. Take one of the yellow Students should be able to hexagons from your pattern do this in three different blocks. We are going to ways (using the blocks designate this block as our provided). whole for this activity. 2-­‐ red trapezoids 3-­‐blue rhombi Using the yellow hexagon as 6-­‐green triangles your whole, find all the ways that you can cover the hexagon using only one type of block. If students do not, then the teacher says or does… If students do not know the names, write the names and a description on the board or a chart to revisit throughout the course of the lesson. 3. Now that we have covered the hexagon all the possible different ways, identify the fraction name of each of the blocks used (trapezoid, blue rhombus and triangle) in relation to the whole we determined. Record the information for each of the pieces in Pattern Block Chart 1. If students are having problems finding the fraction name for any of the smaller blocks, use the chart title Pattern Block Chart 1 to help students see the connection between the pieces, the whole, and the fractional value. If students continue to struggle with the idea of naming a fractional amount, this could mean that students need additional work before attempting to master the concept of “changing wholes”. Students should be focusing on how many pieces of the same size does it take to cover the yellow hexagon. Size of the pieces (denominator) is an important concept to make sure students have a firm grasp. If students are not able to find all three ways, the teacher will need to directly model how to find/cover the hexagon with the pieces. The teacher says or does… Expect students to say or do… 4. NOTE: After students have Students should see that if identified the fraction they cover the blue rhombus names for each of the with the green triangles it pieces used to cover the will only take 2 blocks yellow hexagon, students instead of 6. will repeat the process with the blue rhombus as the The green triangle is now whole. 1/2 of the designated whole, the blue rhombus; however If our new whole is changed earlier it was 1/6 when using from the yellow hexagon to the hexagon as the whole. a blue rhombus, how does that affect the fractional Students will record that it names of the other pattern takes 2 triangles to cover blocks? the blue rhombus and the fractional value is 1/2. Record the information for the green triangle on the table titled Pattern Block Chart 2. 5. Why has the fraction name Student should state that for the triangle changed? the number of pieces needed to cover/fill the region has changed so therefore the label for that piece must change. It must be called something different and this difference is found in the denominator. If students do not, then the teacher says or does… If students are not making the connection between the new whole and the new fractional piece represented by the green triangle, the teacher may want to record information for the Blue Rhombus in the table labeled Pattern Block Chart 2. (This is used when the blue rhombus is the whole. If students do not state that the size of the designated whole has changed, ask the students what was different about the second problem (what did we change?) Did we change the size of the triangle or the size of the whole? It is important to make sure students see that when the size of the whole changed, the fractional value of the triangle changed as well. The teacher says or does… Expect students to say or do… 6. What is the current The students will record the relationship between the fact that if the rhombus is rhombus and the hexagon the defined whole then the in our new problem? hexagon is more than one whole. Follow-­‐Up question (for (This will go in the space further prompting if under the label “number needed) needed to cover the What is considered the whole”.) whole amount? In the appropriate space Record the information for under “Fraction Name for the red trapezoid on the each Block” students will table titled Pattern Block record that the value of the Chart 2. hexagon is 3 because it takes three of the blue rhombi to take make one yellow hexagon. 7. What is the relationship Students must use their between the blue rhombus knowledge about the (the defined whole) and the relationship between red trapezoid? triangles and rhombi to determine how many How many rhombi are the rhombi are the same as a same sizes as a red trapezoid. trapezoid? Students should explain that Record the information for 1 rhombus and 1 green the red trapezoid on the triangle will cover a red table titled Pattern Block trapezoid. Chart 2. The students will record the fact that if the rhombus is the defined whole then the red trapezoid is more than one whole. (This will go in the space under the label “number needed to cover the whole”.) If students do not, then the teacher says or does… Finding the fractional amount for the trapezoid is more complex than the triangle and the hexagon. The teacher may need to revisit the relationship between the triangles and rhombus with the physical model using pattern blocks. The teacher needs to focus students’ attention on the fact that there is not a direct relationship between the rhombi and trapezoid. The teacher says or does… 8. If 1 rhombus and 1 triangle cover the red trapezoid, what fraction name should be given to the red trapezoid? 9. We will repeat the process one more time. This time we will use the red trapezoid as our designated whole. What is the value for the yellow hexagon? Use the pattern blocks provided to determine the fractional amounts of the other pattern blocks (triangle, rhombus, and hexagon). 10. What is the value of the green triangle if the red trapezoid is our whole? Record the information for the red trapezoid on the table titled “Pattern Block Chart 3”. Expect students to say or do… Students should be focusing their arguments on the following logic: A triangle is ½ of a blue rhombus. So therefore 1 whole rhombus and ½ of another rhombus should yield 1 ½ rhombi. In the appropriate space under “Fraction name for each Block” students will record that the value of the red trapezoid is 1 ½ because it takes one blue rhombus and half of another (1 green triangle) to take make one red trapezoid. The value of the yellow hexagon is two because the red trapezoid is worth one whole and it takes two to make up a yellow hexagon. If students do not, then the teacher says or does… The teacher needs to model the thought process out loud for the student when describing how to use the knowledge about triangles to find the value for the trapezoid. Students should be focusing on the green triangle. It takes 3 triangles to cover the red trapezoid, so therefore each triangle is worth 1/3. The teacher says or does… 11. How can you use this information (about the green triangle) to help us find the value for the blue rhombus if the red trapezoid is the newly defined whole? 12. Now that we have explored various different pattern blocks as our designated whole, we will construct a “Summarization Chart” to help solidify our learning and provide us with a clear picture of how the values for each pattern block changed. Teacher Notes None Variations None Expect students to say or do… It takes one blue rhombus and a green triangle to cover a red trapezoid. 1 whole blue rhombus is equivalent to 2 green triangles. Each green triangle is worth 1/3 of a red trapezoid. So, 1 blue rhombus is 2/3. Students will complete the chart and discuss the relationship that exists between the parts and the changing whole in the chart. Conversation should focus on the size of the piece in relation to the size of the whole. (How many pieces do you see versus how many pieces does it take to cover/fill the area or region.) If students do not, then the teacher says or does… If each green triangle is 1/3 of the figure, then how many of the green triangles does it take to cover a blue rhombus? When putting all this information together, what fractional value does this produce for the red trapezoid? Why do the fractional values of the blocks keep changing each time? Formative Assessment
Figure A List and explain how you found the fractional amount for each of the colored pattern blocks using Figure A as your whole: blue rhombus, green triangle, red trapezoid, yellow hexagon. References
Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide -­‐ Response to Intervention in Mathematics. Retrieved 2 25, 2011, from rti4sucess: http://www.rti4success.org/images/stories/webinar/rti_and_mathematics_webinar_presentati
on.pdf An Emerging Model: Three-­‐Tier Mathematics Intervention Model. (2005). Retrieved 1 13, 2011, from rti4success: http://www.rti4success.org/images/stories/pdfs/serp-­‐math.dcairppt.pdf Marjorie Montague, Ph.D. (2004, 12 7). Math Problem Solving for Middle School Students With Disabilities. Retrieved 4 25, 2011, from The Iris Center: http://iris.peabody.vanderbilt.edu/resource_infoBrief/k8accesscenter_org_training_resources_
documents_Math_Problem_Solving_pdf.html Pattern Block Chart 1 Block Used as One Whole Blocks Number Needed to Cover Whole Fraction Name of Each Block Pattern Block Chart 2 Block Used as One Whole Blocks Number Needed to Cover Whole Fraction Name of Each Block Pattern Block Chart 3 Block Used as One Whole Blocks Number Needed to Cover Whole Fraction Name of Each Block Summarization Chart Whole Whole Whole Whole