PA MDC GRADE 3 MATCH MY FRACTION Concept Development Formative Assessment Lesson This lesson is intended to help you assess how well students are able to use pattern blocks to demonstrate an understanding of s with an emphasis on equivalent s This lesson was modeled from the Utah Education Network: Match Fractions and revised by the PA-MDC Writing Committee. Additional resource Region Relationships from the National Council of Teachers of Mathematics: http://illuminations.nctm.org
With Sincere thanks and appreciation for the effort and work of the Members of the PA- MDC Writing Committee and for the unwavering support from their home districts: Camp Hill School District, Cumberland Valley School District, Lower Dauphin School District and Shippensburg School District. Joan Gillis, State Lead MDC Dan Richards, Co- Lead MDC PA MDC Writing Committee Jason Baker Heather Borrell Susan Davis Carrie Tafoya Richard Biggs Carrie Budman Miranda Shipp Review Committee Katherine Remillard, Saint Francis University Renee Yates, MDC Specialist, Kentucky Carol Buckley, Messiah University Josh Hoyt, Berks IU Dr. Karla Carlucci, NEIU Kate Lange, CCIU Lori Rogers, CAIU If you have any questions please contact Joan Gillis at jgillis@caiu.org
MATCH MY FRACTIONS This Formative Assessment Lesson is designed to be part of an instructional unit. This concept development task should be implemented approximately two-thirds of the way through the instructional unit. The results of this task should then be used to inform the instruction that will take place for the remainder of your unit. Mathematical Goals: This lesson unit is intended to help you assess how well students are able to use pattern blocks to demonstrate an understanding of s with an emphasis on creating equivalent s. Common Core Standards: This lesson involves mathematical content in the standards from across the grade, with emphasis on: 3. NF.1 Understand a 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a a/b as the quantity formed by a parts of size 1/b. 3. NF.3 a. Explain equivalence of s in special cases, and compare s by reasoning about their size. a. Understand two s as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent s, (e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the s are equivalent, (e.g., by using a visual model.) This lesson involves a range of Standards for Mathematical Practice, with emphasis on: MP.2 Reason abstractly and quantitatively MP.4 Model with mathematics MP.7 Look for and make use of structure PA Core State Standards: CC2.2.1.3.C.1 Explore and develop an understanding of s as numbers Introduction: This lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task, Fractions, which is designed to reveal their current understanding and difficulties. You then review their solutions and create questions for students to consider in order to improve their work. In a whole-class introduction to the lesson, students review their understanding and al parts using pattern blocks. They then share some sample student work and evaluate the different approaches taken to express/ show equivalent s. Students then take part in a collaborative task, in pairs as they do this, they learn more about how s are related to each other and can express the same quantity. 1
Then in a whole-class discussion, students consider their approaches and share their conclusions. In a follow-up lesson, students receive your comments on the assessment task and use these to attempt a similar task, Fractions (revisited), approaching it with insights that they have gained from the previous lesson. Time Needed Estimated 80 to 95 minutes Timings are approximate: exact timings will depend on the needs of your students. Pre-assessment: 10 minutes Whole Class Introduction: 20 minutes Collaborative Activity: 35 40 minutes Whole Class Discussion: 10 15 minutes Post-assessment: 10 minutes: Materials Required Teacher may want to have a document camera or overhead projector for teacher/student demonstrations. Teacher will need Fraction Pattern Block Manipulatives. Each pair of students will need: Fraction Pattern Block pieces Recording sheet for Fraction Game: Match Fraction Set of laminated cards Set of laminated equivalent cards Set of Fraction Pattern Blocks (template at the end of guide.) Dry erase markers and eraser (if cards are laminated) Before the Lesson: Pre-Assessment Task: Fractions (10 minutes) Have students individually complete the pre-assessment in class a day or more before the Formative Assessment Lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson. Note: Teachers should take the time to complete the pre-assessment prior to students attempting the task. Using the Formative Assessment Anticipation Guide to anticipate where students will show their strengths and their weaknesses. It is also recommended that teachers create their own questions to address the misconceptions uncovered prior to using the questions provided in the guide. 2
Framing the Task: Give each student a copy of Fractions. Introduce the task briefly and help the class to understand the problem and its context. It is important that students answer the question without assistance, as far as possible. If students are struggling to get started,ask them quesitons that help them understand what is required but do not do the task for them. Assessing student s responses: Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. The purpose of this is to forewarn you of the issues that will arise during the lesson, so that you may prepare carefully. We suggest that you do not score students work. The research shows that this is counterproductive, as it encourages students to compare scores,and distracts their attention from how they may improve their mathematics. Instead, help students to make further progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given on below. These have been drawn from common difficulties anticipated. We suggest that you write your own lists of questions, based on your own students work, using the ideas below. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the beginning of the lesson. 3
Common Issues No clear understanding of what a represents Confuses numerator and denominator The understanding that the larger the denominator there are more pieces to equal the same with a smaller denominator (1/2 = 3/6 but you need smaller pieces to be equal) Cannot accurately place s on a number line Suggested questions and prompts Can you explain what means to you is? What does the numerator represent? What does the Denominator represent? Draw this : 3/4. How many more to equal 1? If you have 6 pie slices how many represent 1/2 of those slices? How do you know? If you have 6 slices how many represent 2/3 of the pizza? Draw a picture to model your answer. If 1/2 = 3/6 how would you explain this to your classmates? Does your number line include a starting point of 0? Draw a number line and show me where 3/4 would be? How many segments are there on a number line expressing fourths? Why? Student work lacks precision Do the al parts of your drawing represent equal parts of your whole? Student treats s with whole number rules. For example, thinks that 3/8 is larger than 3/4 based on denominator. Place 3/8 and 3/4 on a number line. Which is larger? Draw a model of 3/4 and 3/8. Which is larger? Please use the blanks to add your class misconceptions and directed questions. 4
Pre-Assessment Task: Fractions Name: _ Write a that represents the shaded portion of the square. The square represents the unit whole. Then, draw a line to match the equivalent s. Below or on the back of your paper Draw and label two models that show equivalent s. Draw a number line that proves your thinking about your equivalent models. 5
Whole Class Introduction (20 minutes) Making Fractions Group students in homogeneous pairs based on their pre-assessments. Pass out a set of Fraction Pattern Block pieces to each pair. Show students the Fraction Pattern Block pieces. Identify the yellow hexagon as the whole. Have them discuss with their partner what other pieces are equivalent, too. This should be just a quick review, since students should have already modeled the benchmark s using these pattern blocks. Walk them through the pattern blocks and the they represent in relationship with the hexagon. Use the chart to the right to assure they can determine the al value that the triangle (1/6), trapezoid (1/2), and rhombus (1/3) have with the hexagon (whole). The green triangle is what of the yellow hexagon? The blue rhombus is what of the yellow hexagon? The red trapezoid is what of the yellow hexagon? Model the 1/2. Next model how to create a that is equivalent to 1/2 using different pattern block pieces. For example, you could choose to model 3/6. Be sure to name the new, and talk about how you know that 1/2 and 3/6 are equivalent s. Instruct the students to: Take a few minutes to model two s you believe are equivalent to each other. Choose someone to come up from each team and represent the s his/her team made. Be sure to ask whether another pair came up with a different solution. Always be sure to ask the students how they know the s are equivalent and ask them to prove their certainty on a number line. Once you have observed that students are feeling comfortable modeling equivalent s, students are now ready to participate in an Card Game Activity. 6
Collaborative Activity (35 40 minutes) Card Game Activity: Match Fraction Run 2 sets of the Card Set A: Fraction Cards on cardstock and laminate for each pair. Run 6 sheets of the Card Set B: Fraction Cards. It is recommended that you use different color of cardstock than the cards above and laminate for multiple use. Give each set of partners the following supplies: Card set A & B Fraction Pattern Block Set: triangles, rhombus, trapezoid or cut-outs on page 15 Dry erase marker and eraser Recording sheet/ rulers Student 1 draws a card from the Fraction Cards stack. He/she then builds a with the Fraction Pattern Block Set, covering the WHITE portion of the hexagon only, and labels the created in the box below with a dry erase marker. Student 2 draws a card from the Fraction Cards stack. He/she then builds a that is equivalent to the built by Student 1. Student 2 cannot use the same Fraction Pattern Block pieces as Student 1. Both students record the equivalent s they made on their recording sheet Match Fraction. Students then trade jobs. Student 2 will now be the one to draw first from the Fraction Cards and Student 1 will draw from Fraction Cards, and match the that Student 2 made. Recording sheets can be turned in as a formative assessment. Whole Class Discussion (10-15 minutes) Teacher s role is one of Facilitator this is a student led conversation. Use questioning to drive the discussion. Choose students to share how they figured out which method was used most efficiently. If time is limited, only choose 2-3 groups to present. Can you describe how you and your partner created equivalents? How did you know you were right? Take a few minutes to look for a pattern with creating equivalent s 1 4 X 2 = 2 8 2 4 X 3 = 6 12 Your goal is to have students discover a faster method for creating equivalent s and gain conceptual understanding of s. 7
Match Fraction Player 1_ Player 2 Player 1 Player 2 Player 2 Player 1 8
Match Fraction Player 1_ Player 2 Player 1 Player 2 Player 2 Player 1 9
10 SET A Fraction Cards
11 SET A Fraction Cards
12 SET A Fraction Cards
SET B Fraction Cards Fractions Fractions Fractions Fractions Fractions Fractions 13
SET B Fraction Cards Fractions Fractions Fractions Fractions Fractions Fractions 14
If you do not have enough pattern blocks, Students could use the following template for a hands on experience: 15
Post-Assessment Task: Fractions Name: _ Each image below is 1 whole. Write the that is shaded. Draw a number line and place on it the s from least to greatest. Andre thinks it is strange that 1 8 of the cake would be less than 1 4 of the cake since eight is bigger than five. Explain this to Andre, using the 2 identical rectangles to represent the cakes. Show 1 eighth shaded on one and 1 fourth shaded on the other. Label the unit s and explain to him which slice is bigger. 16
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Tear-Off Sheet with Suggestions for Helping Students Access Information Barrier to Learning Suggested Strategy Student lacks understanding of math language Review domain specific vocabulary; create a picture dictionary with student. Student lacks basic perimeter/area knowledge Review prior lessons in perimeter and area; use manipulatives to explore perimeter/area concepts. Use key words and "picture stories" to help students identify the appropriate operation. Students have problems in understanding math word problems (reading comprehension) Build vocabulary through repeated classroom use and picture dictionary. Work on reading and understanding problems through modeling in small groups and peer-to- peer situations. Student struggles with multi-step problems Student struggles with writing explanations and math reasoning Student struggles with creating a rectangle with a given perimeter. Break the problem into smaller tasks, an understandable sequence. Continued use of Math Journaling and Share Time in which classmates critique each other can help strengthen this. Explain to the student that written explanation takes the place of verbal communication and the reader needs to understand how you solve the problem. Identify the perimeter of objects by measuring with a ruler. Develop skill through use of geoboard. Have student count each line drawn creating a rectangle. Student struggles with identifying the area of a rectangle. Provide multiple methods of find area: counting squares, repeated addition, composite units, multiplying the length and width. 18