Lesson Questions: How can I analyze a pattern to determine its rule and to determine future or missing elements in the pattern?

Transcription

1 Name of Lesson: What s Missing? Topic: Primary Algebraic Thinking Lesson #11 Approximate time: 3 sessions Gifted Standard and element(s): G2CG1. Students will reason logically using induction, deduction, and abduction. a. Explore and develop critical thinking skills through the process of convergent thinking. d. Solve problems using logical reasoning. f. Develop verbal and nonverbal communication skills to convey logical reasoning. Supports CCGPS: 5.OA.3 Analyze patterns and relationships fits but the specific element does not it involves two patterns and comparing corresponding elements of each, etc. Writing Gr. 2: 2. Write informative/explanatory texts in which they introduce a topic, use facts and definitions to develop points, and provide a concluding statement or section. Language Gr. 2: 1. Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. Language Gr. 2: 2. Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7. Look for and make use of structure. Essential Question(s): How can I use the mathematical meaning of equivalence to solve problems? How can I use variables to solve problems? *How can I use patterns, rules, and relationships to solve problems? Lesson Questions: How can I analyze a pattern to determine its rule and to determine future or missing elements in the pattern? Lesson Summary: Students will analyze various patterns, determine the next element or missing elements, and then determine future elements by using its rule rather than expressing all intermediate elements. Key idea: There are many strategies for analyzing patterns. These strategies need to be logically and systematically applied until a solution is reached. Assessment Description/Performance Task: Constructed response Informal assessment Performance task Selected response Brief Description of Assessment: Students complete several patterns on the student sheet, Fill in the Blank. During the discussion, the teacher assesses (informally) whether students can find the next element(s) and explain their reasoning. The student sheet, Analyzing Patterns has students complete patterns and explain their strategies and reasoning independently in writing.

2 Instructional Methods: Hook/Activator: Use the students themselves to create patterns. Place colored sheets of construction paper on the floor at the front of the classroom in this order: black, blue, brown, green, orange, purple, red, yellow. Give 7 students the direction cards for pattern 1. Have each follow the directions on his/her card. The students will then be arranged to show a pattern with a missing element. Have another child go stand in the empty location to complete the pattern correctly. Ask if others agree. Discuss reasons for agreement/disagreement. Make sure it gets corrected if the child was wrong. (The child on yellow should have his/her eyes open and be waving.) Ask students to name the pattern. (Eyes pattern is AB; motion pattern is ABC; together ABCDEF) Distribute cards for pattern 2. Have each follow the directions on his/her card. The students will then be arranged to show a pattern with a missing element. Have another child go stand in the empty location to complete the pattern correctly. Ask if others agree. Discuss reasons for agreement/disagreement. Make sure it gets corrected if the child was wrong. (The student on green should have his/her back to the audience and be on one foot.) Ask students to name the pattern. (Which way to face is AB; position is ABC; together ABCDEF) Discuss how finding the element in the middle is different from extending a repeating pattern was it easier? harder? Why? Distribute cards for pattern 3. Have each follow the directions on his/her card. The students will then be arranged to show a pattern with a missing element. Have another child go stand in the empty location to complete the pattern correctly. Ask if others agree. Discuss reasons for agreement/disagreement. Make sure it gets corrected if the child was wrong. (The child on yellow should be with 7 snap.) If no one points out that this is a growing rather than a repeating pattern, ask how it is different from the first two. NOTE: An interesting discussion may occur if you ask students to name this pattern. The position pattern is simple- ABC; the snap cube pattern is growing, so it makes it much harder to name together. You may wish to have students try to create a notation for this. Distribute cards for pattern 4. Have each follow the directions on his/her card. The students will then be arranged to show a pattern with a missing element. Have another child go stand in the empty location to complete the pattern correctly. Ask if others agree. Discuss reasons for agreement/disagreement. Make sure it gets corrected if the child was wrong. (The child on orange should be holding 7 yellow snap.) Discuss that this is also a growing pattern. Discuss how finding the element in the middle is different from extending a growing pattern was it easier? harder? Why? NOTE: Another interesting discussion may occur if you ask students to name this pattern. The color pattern is simple- AB; the number of snap is growing by one if you look at those holding the same color, but each color started at a different number. Some students may look at the number of together and see a rule like subtract 2, add 3. You may wish to have students try to create a notation for this pattern. Teaching Strategy: Define (or remind students of the definition of) the words analyze (examine carefully and in detail), metacognition (thinking about your thinking), and explanation (communicate the meaning). In this case, we are analyzing patterns to determine missing or next elements, and then thinking about our strategies for doing this and explaining them to others. Have students brainstorm (in groups, or individually and then in groups) things they look for when analyzing different kinds of patterns. (Note: One graphic organizer is provided. A key with ideas that should be brought up if students do not name them is provided. Only have students brainstorm the ones for Shape and Word. Number will be added later. This sheet should be saved for future reference. As students generate or recognize more strategies in the future, they may continue to add to the list.) Create a Pattern Analysis Tools poster for

3 reference somewhere in the classroom, and allow students to add to their personal Pattern Analysis Tools chart as each student or group shares ideas. Lead students to discover any strategies from the key that they did not mention. As students generate or recognize more strategies in the future, they may continue to add to the class list and their own lists. Remind students that sometimes a single pattern might have more than one predictable change in it like some of the student patterns from the beginning of class. Have students complete Fill in the Blank independently (or in pairs or small groups as needed). Summary by the Learner: Have each student choose one pattern from Fill in the Blank and explain his/her strategies (even the ones that did not work) and the reasons for his/her answer to a friend. The friend should verify the answer and that the reasoning is valid. Ask students to add ideas to the class Pattern Analysis Tools poster and their personal Pattern Analysis Tools charts. Next, the teacher should model writing an analysis, possibly using one of the patterns from the beginning of class or one from Fill in the Blank. Focus on the critical thinking skills of metacognition and explanation. This modeling should focus on explaining one s thinking in writing, including some strategies that did not work and their results. This will be important when students complete the assessment. The most effective way to do this is to show the pattern to the students and then think out loud as you write or type something similar to the example below. Example for modeling: This is number 4 from Fill in the Blank. I first looked at the pattern of shapes. It was confusing because the shapes change, the directions change and some shapes are shaded. I decided to look at one thing at a time. I started with just the shapes. I see that there is an arrow followed by two stars, another arrow followed by a blank and a star, another arrow followed by two stars, a blank followed by two stars, and then another arrow. It seems that the shape pattern is an arrow followed by two stars, so the first blank would have a star and the second would have an arrow. Then I looked at the arrows. I looked for a color or shading pattern, but there did not seem to be one. I also looked for whether the arrow changed in size or shape. It did not, but it did change direction, alternating between up and down. Therefore the arrow I put in the second blank should point up. I looked at the stars for shape changes or size changes or direction changes. There were not any. Then I looked at the shading of the stars. In the first pair of stars, the one on the right was shaded. There was a blank where I put a star in the next pair, but the star to the right of the blank was NOT shaded, so I knew that the star on the right wasn t always shaded. In the next pair, the star on the right was shaded, but in the next pair the star on the left was shaded. It appears that the shading alternates between the left and right stars in each pair, so the star I put in the first blank should be shaded. Have students complete Analyzing Patterns for assessment. (Note: For a student whose written explanations are minimal or unclear, the teacher may wish to have him or her explain orally in order to for the teacher to more accurately assess the child s mathematical under.) Differentiation: More capable: Have students generate patterns with more complicated combinations of rules. (This could be used to teach the critical thinking skill of synthesis because they will be putting different rules/patterns together as one.) Have students explain their patterns, either in writing, or in sharing with the class. They could also be challenged to determine a pattern for the combination patterns they create. (Ex. AB + ABC = ABCDEF; ABC + ABCD = ABCDEFGHIJKL; it will be the least common multiple of the number of elements in each basic pattern unit, though students are not likely to use that term.) Less capable: Suggest specific strategies for students to use for a particular pattern.

4 Materials for this Lesson: class set of Fill in the Blank class set of Analyzing Patterns class set of Pattern Analysis Tools one sheet of paper in each of these colors: black, blue, brown, green, orange, purple, red, yellow pattern cards (provided, but should be printed and cut apart) various colored snap ; need many yellow and blue large paper or poster board (for class version of Pattern Analysis Tools ) Vocabulary for this Lesson: strategy pattern synthesis analyze/analysis metacognition explanation

5 Name Pattern Analysis Tools Number Shape Word

6 Name _KEY_ Pattern Analysis Tools- Suggested answers Number Shape Word (to be completed in another lesson) number of sides- increasing or decreasing position- facing up/down/left/right, etc. number of shapes in each element arrangement of shapes in each element (if more than one shape per element) color or shading of each shape/element size or shape changes number of letters spelling pattern singular/plural meaning of words (synonym, antonym, etc) beginning letter alphabetical order double letters number or position of vowels ending letter rhyming

7 Name Fill in the Blank Use the strategies discussed to analyze each pattern. Then draw or write the missing element. Be ready to explain how you know what to write or draw. 1) 2) 3) 1,2, Z, 3, 4, Y, 5,, X, 7, 8,, 9, 10,... 4) 5) sullen, least, robbed, black, recess, snake,, model... 6) 7) abba effe ijji dccd lkkl 8) 1,2,3,, 2, 3, 4,, 3, 4, 5,, 4,,,, 5, 6, 7,

8 Name KEY Fill in the Blank Use the strategies discussed to analyze each pattern. Then draw or write the missing element. Be ready to explain how you know what to write or draw. 1) 2) 3) 1,2, Z, 3, 4, Y, 5, 6, X, 7, 8, W, 9, 10,... 4) 5) sullen, least, robbed, black, recess, snake, any word with double m, model 6) any pentagon any shaded hexagon 7) abba effe ijji dccd hggh lkkl 8) 1,2,3,, 2, 3, 4,, 3, 4, 5,, 4, 5_, 6_,, 5, 6, 7,

9 Name Analyzing Patterns Analyze the chart. Draw or write what belongs in the empty cells. ant bird dog elephant Explain your thinking. Tell what strategies you used, even if they did not work at first. Then tell how you decided what to draw in each cell.

10 Name Key Analyzing Patterns Analyze the chart. Draw or write what belongs in the empty cells. ant bird any animal that starts with c dog elephant Explain your thinking. Tell what strategies you used, even if they did not work at first. Then tell how you decided what to draw in each cell. Answers will vary, but students should mention strategies from the chartor others they used. There should be some mention of what did/did not work. Their explanation of how they knew what to draw should have some mention of position and shading and the alphabetical pattern to the words, and the fact that all the words named animals.

11 Pattern 1 Pattern 1 Go to the black paper. Close your eyes. Clap slowly. Go to the blue paper. Wave with one hand. Keep your eyes open. Pattern 1 Pattern 1 Go to the brown paper. Close your eyes. Jump up and down in place. Go to the green paper. Clap slowly. Keep your eyes open. Pattern 1 Pattern 1 Go to the orange paper. Close your eyes. Wave with one hand. Go to the purple paper. Jump up and down in place. Keep your eyes open. Pattern 1 Go to the red paper. Close your eyes. Clap Slowly.

12 Pattern 2 Pattern 2 Go to the black paper. Stand on one foot. Go to the blue paper. Turn your back to the audience. Kneel. Pattern 2 Pattern 2 Go to the brown paper. Sit on the floor. Go to the orange paper. Kneel. Pattern 2 Pattern 2 Go to the purple paper. Turn your back to the audience. Sit. Go to the red paper. Stand on one foot. Pattern 2 Go to the yellow paper. Turn your back to the audience. Kneel.

13 Pattern 3 Pattern 3 Go to the black paper. Sit on the floor. Go to the blue paper. Stand holding one snap cube. Pattern 3 Pattern 3 Go to the brown paper. Kneel holding 2 snap. Go to the green paper. Sit holding 3 snap. Pattern 3 Pattern 3 Go to the orange paper. Stand holding 4 snap. Go to the purple paper. Kneel holding 5 snap. Pattern 3 Go to the red paper. Sit holding 6 snap.

14 Pattern 4 Pattern 4 Go to the black paper. Stand holding 5 yellow snap. Go to the blue paper. Stand holding 3 blue snap. Pattern 4 Pattern 4 Go to the brown paper. Stand holding 6 yellow snap. Go to the green paper. Stand holding 4 blue snap. Pattern 4 Pattern 4 Go to the purple paper. Stand holding 5 blue snap. Go to the red paper. Stand holding 8 yellow snap. Pattern 4 Go to the yellow paper. Stand holding 6 blue snap.

15 Quick reference key for student patterns in Hook activity Pattern Color black blue brown green orange purple red yellow 1 (all facing audience) eyes closed clapping eyes open waving eyes closed jumping eyes open clapping eyes closed waving eyes open jumping eyes closed clapping eyes open waving 2 facing on one foot back to kneeling facing sitting back to on one foot facing kneeling back to sitting facing on one foot back to kneeling 3 (all facing audience) sitting with one snap cube kneeling with 2 snap sitting with 3 snap with 4 snap kneeling with 5 snap sitting with 6 snap with 7 snap 4 (all facing audience) with 5 yellow snap with 3 blue snap with 6 yellow snap with 4 blue snap with 7 yellow snap with 5 blue snap with 8 yellow snap with 6 blue snap Shading indicates the empty location that is the blank in the pattern.