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Big Ideas in Mathematics for Grades 4-6 with Dr. Marian Small Session 3 of 4 Focus on Patterns & Relations and Statistics & Probability Overview of Learning Opportunity: This recorded webinar will focus on such questions as: What are the big ideas you should bring to students’ attention when teaching Math in Division 2? How do they link to curriculum? How do teachers ask those critical questions that help students see those ideas? How can thinking about big ideas help shape lessons? This conversation guide is intended for Professional Learning Communities, instructional leaders or as a self-paced study to help guide instruction, conversations and reflections on using the big ideas in math as a means to deepen student mathematical understanding. *Please visit the following link to access the handouts for this session http://erlc.wikispaces.com/Big+Ideas+in+Math+4+-+6 Webinar Outline/Table of Contents Time Clip Information & Key Ideas 0:00 Recap from previous session What are the Big Ideas in Patterns & Relations and Statistics & Probability? 2:08 Would you… 5:47 22 x 13 10:34 What about division? 11:15 Which representation? 17:07 And what procedure? 24:06 What about estimating? 27:32 I added... 31:03 I subtracted... Page 1 of 8 31:07 33:39 34:17 34:49 39:33 45:56 48:51 53:07 1:02:33 1:04:39 1:10:41 The product is... I wonder... Patterns Multiplication table Look at this Picture a 100 chart How else could you... Variables and equations Data Which graph would you use? How do these graphs differ in the impression they give? Closing Remarks End of the webinar This “Conversation Guide” provides an overview of the webinar with ideas for continuing the conversation as well as questions for extended learning. This guide may also assist with self-paced study when viewing the archived webinar. Time Code Clip Introductions Clip Information, Key Points & Suggested Activities to use this Webinar for your own PD Sessions. Introductions Introduce yourself Ask participants to introduce themselves *Session handouts are available at Questions for Extended Learning Opportunities http://erlc.wikispaces.com/Big+Ideas+i n+Math+4+-+6 2:08 Agenda Outline of Marian’s 4 Math Webinar sessions • Session 1– A focus on number • Session 2 – A focus on operations • Session 3 – A focus on patterns and relations and statistics and probability • Session 4 – A focus on shape and space Would you… As a teacher, would you calculate 532 – 99 the same way you would calculate 532 – 111? Marian left a thought for us: should our students calculate these two questions the same way? What do you think? Page 2 of 8 5:47 22 x 13 Discussion question: would you prefer the method on the left or the visual method on the right? When the teachers in the clips discuss their preference and why they prefer a specific method, do any of their comments speak to you? Are there pros and cons to both methods? 10:34 What about division? Division: • what it means and • how to calculate Marian commented that division is tricky because it builds on the other three operations. What does she mean by this? As a group, can we come up with examples as to how most case in addition and subtraction all fit in to division? Is it important/helpful for students to see how the other three operations fit into division? 11:15 Which representation? Which picture best shows what 72 ÷ 3 means? Marian continues to show us multiple representations. As you can see from the teachers’ responses different people preferred a different representation. Are there additional ways we can represent 72 divided by three? or Discussion question: how can we increase our inclusion of multiple representations in our daily classroom work? Do we think this is necessary? Note: rectangles represent 10, small squares represent 1 If we think it is important, are there ways we can collect and share student samples so that we all have a larger pool of resources to draw from? Teachers’ responses: Page 3 of 8 17:07 And what procedure? What do each of the procedures below help a student see better than the other procedures? Discussion question: which method do you commonly use in class? Why do you think you use your preferred method? Do you think we should try all three methods in our classes? Or do you think multiple strategies might lead to confusion? Discussion question: when you look at the three methods, do any of them emphasize process over understanding? Do any of them emphasize understanding over process? Does it matter what we emphasize in our classrooms? Note: larger images are available in the presentation. 24:06 27:32 What about estimating? I added... It’s no longer just about rounding rules I added 3 numbers. One is little. One is close to double the other. The sum is 5000. What could the numbers be? 31:03 I subtracted... I subtracted a number from 3000. The result had the digits 3 and 4 in it. What could the subtraction have been? 31:07 The product is... The product of two numbers is almost 400. What might the numbers be? This example is considered an open question. The possibilities are infinite. This question combines the ideas of adding and multiplication and estimation. Is there a way to bring in the vague descriptions or how could be open this up so it didn't necessarily have to be exactly 5000? Is that a good idea or might we have too many possibilities? This is an example of a double question. First we have to find which two numbers could give us the exact product of 400. Then we have to adjust those numbers so that the product is almost 400. Discussion question: what are some answers you might get from your Page 4 of 8 students? Where can you see some of your classroom discussions going based on your students answers? 33:39 I wonder... 34:17 Patterns 34:49 Multiplication table I divided _3_ by 4. The answer was a 3 digit number. Tell me anything about _3_ that you’re sure of. Big idea: there are a variety of appropriate ways to estimate sums, differences, products and quotients depending on the numbers involved and the context. WNCP Pattern Outcomes Grade 4- focus on identifying patterns and relationships in tables and charts Grade 5- focus on pattern rules to predict Grade 6- focus on table of value relationships Big idea: many geometric attributes, measurements and calculations involving numbers are simplified by using patterns. What is a pattern in the table that might help you figure out 12x5? Big idea: patterns represent identified regularities. There is always an element of repetition, whether the same item repeats, or whether a “transformation”, for example adding one, repeats. In this section of the clip, a teacher volunteered a strategy she saw and Marian shared her strategy. Did you come up with a strategy different from either of these? Did seeing these strategies help you come up with one of your own? How can sharing and discussing pros and cons of strategies help students develop strategies that work for them? 39:33 Look at this To show the final digits of the multiples of 8: Marian noted that many of these multiples result in visual patterns, and the six star is also the four star, In fact, the following multiples all have the same shape: • • • • 6 and 4 9 and 1 8 and 2 Any two numbers that add up to 10 Do we believe this? Why does this Page 5 of 8 happen? Big idea: the mathematical structure of a pattern can be represented in a variety of ways. To show the final digits of the multiples of 6: Teacher sample in the clip: 45:56 Picture a 100 chart Marian noted that patterns in which students continue to add the same amount each time, are the most important patterns they will work on in grades 7 -11. Why might this be the case? Note: this visual is larger in the presentation. 48:51 How else could you... How else could you represent the pattern 2, 5, 8, 11,…. Discussion idea: Can we think of examples in which pattern work leads to algebra? Big idea: some ways of displaying data highlight patterns. These visual representations are a strong foundational piece for algebra. Page 6 of 8 A visual representation really helps students to create a formula in division three. Side note, Formula: y = 3n-1 53:07 Variables and equations • • • Grade 4- problems as equations with unknowns; solve one-step equations Grade 5- single-variable, one-step equations Grade 6- generalizations written using variables; model equality preservation Big idea: algebra is a way to represent and explain mathematical relationships and to describe and analyze change. Big idea: using variables is a way to efficiently and generally describe the relationships that can also be described. Discussion idea: what do these two big ideas mean to us? What are some possible strategies we could use to help students understand these big ideas? 1:02:33 Data • • • 1:04:39 Grade 4- many-to-one correspondence for pictographs and bar graphs Grade 5- double bar graphs Grade 6- line graphs Which graph would you use? In this clip but there is a discussion about bar graphs versus pictographs, and about the scale used. A. B. C. D. 1:10:41 How do these graphs differ in the impression they give? Big idea: graphs are powerful data displays since visual displays quickly reveal information about data. bar graph with scale of 2 bar graph with scale of 5 pictograph with scale of 2 pictograph with scale of 5 How do these differ in the impression they give? Discussion idea: what types of graphs do we most commonly used in our classrooms? Do we ask students why we prefer these types of graphs? Is the “why” a useful discussion? The way we display data, really affects the impression we get from that data. Discussion idea: does anyone have an example of an extremely biased graph we could show our students? It might be a great learning experience to try to get students to buy into the misleading graph and then dissect the graph together. Alternately, if you are working with a group of teachers, perhaps you could bring in an extremely biased (or subtly Page 7 of 8 biased) graph and dissect the graph with the teachers. 1:13:24 Closing Remarks Note: Larger graphs are available in the presentation Closing Remarks Write a word or two that stood out for you from today’s session. Reflect on your learning – What is one thing you will try in your classroom to engage students in math’s big ideas? Contact Info. [email protected] End of the webinar Resources Additional Resources Share additional handouts with participants http://erlc.wikispaces.com/Big%20Ideas% 20in%20Math%204%20-%206 If participants are interested in Marian’s PowerPoint slides or the archives from the 4 sessions, they can be found on the ERLC wiki Page 8 of 8