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Using Models to Make Math Visual

Using Models to Make Math Visual. GRADES K-2 January 14, 2015. Session goal and agenda. Session Goal: Explore visual representation models as tools to build number sense, solve word problems, and support student learning. Session Agenda: Number towers Number paths Number bonds

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Using Models to Make Math Visual

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  1. Using Models to Make Math Visual GRADES K-2 January 14, 2015 Coweta Committed to Student Success

  2. Session goal and agenda • Session Goal: Explore visual representation models as tools to build number sense, solve word problems, and support student learning. • Session Agenda: • Number towers • Number paths • Number bonds • Bar models Coweta Committed to Student Success

  3. A picture is worth a thousand words … Coweta Committed to Student Success

  4. Visual mathematical models • Mathematical models are a set of concrete and pictorial models that students use repeatedly across grade levels. • Over time, students become familiar with these models and use them in more complex ways to solve problems. • Mathematical models become part of their tool box, which will help them have a quicker understanding of concepts as they are introduced. Coweta Committed to Student Success

  5. Making connections 9 + 6 = 15 Coweta Committed to Student Success

  6. Number towers Coweta Committed to Student Success

  7. Number towers • Representations of quantity made by joining together interlocking cubes • Explores “one more” and “one less” • Color change at 5 • Reinforces 5 as a benchmark • Helps students see relationships between quantities • Encourages counting on from 5 Coweta Committed to Student Success

  8. Number towers • Used to compare length and quantities • Used to see the parts that make up a number • Foundational to understanding part-whole models, addition and subtraction, fact fluency, and the commutative property Coweta Committed to Student Success

  9. Number towers • Helps with preparing for multiplication “Each cube is worth 3. What is the value of a tower with 5 cubes?” • Develops understanding of the distributive property Coweta Committed to Student Success

  10. Number paths Coweta Committed to Student Success

  11. Number paths • Visual or pictorial representation of number towers • Foundational to understanding the number line Coweta Committed to Student Success

  12. Number paths - K • Visual representation of one-to-one correspondence and the concept of whole numbers • Reinforces number sequence • Used to work with number pairs of 10 • Used to count forward beginning with a number other than 1 Coweta Committed to Student Success

  13. Number paths – 1st • Color change at 10 to emphasize 10 as a benchmark • More advanced strategies • Counting on – begin the count at 8 • Converting to an easier problem – make 10 Coweta Committed to Student Success

  14. Number paths – 1st • Builds foundation for understanding the number line • Whole number units • Fraction • Measurement • Decimals • Negative numbers Coweta Committed to Student Success

  15. Number bonds Coweta Committed to Student Success

  16. Number bonds Number bonds are a pictorial representation of part-part-whole relationships. Smaller numbers (the parts) make up larger numbers (the whole) part + part = whole whole – part = part Coweta Committed to Student Success

  17. Developing conceptual understanding Concrete → Pictorial → Abstract Coweta Committed to Student Success

  18. Important to remember Orientation of the number bond diagram does not change its meaning. Number bonds of 10 are the most important in the lower grade levels. Coweta Committed to Student Success

  19. Number bonds – K 6 Coweta Committed to Student Success

  20. Number bonds – K Coweta Committed to Student Success

  21. Number bonds – 1st • Fluency with bonds of 10 • Composition and decomposition of teen numbers into 10 and some ones http://www.topmarks.co.uk/maths-games/hit-the-button Coweta Committed to Student Success

  22. Number bonds – 1st Coweta Committed to Student Success

  23. Number bonds – 2nd Coweta Committed to Student Success

  24. Number bonds – 2nd Coweta Committed to Student Success

  25. Grade 3 and beyond • Fractions • Time • Conversion of units • Adding/subtracting larger numbers Coweta Committed to Student Success

  26. Number bonds – 3rd Coweta Committed to Student Success

  27. Number bonds – 4th Coweta Committed to Student Success

  28. Number bonds – 5th Coweta Committed to Student Success

  29. Bar models Coweta Committed to Student Success

  30. Bar models • Students begin by drawing pictorial models • Evolves into using bars to represent quantities Enables students to become more comfortable using letter symbols to represent quantities and transition to algebra 7 ? 15 Coweta Committed to Student Success

  31. Steps to bar models: K-1 Sara has 2 apples. Jon has 5 apples. How many apples do they have altogether? How many more apples does Jon have than Sara? Coweta Committed to Student Success

  32. Foundation for bar models:Comparison model – K-1 Students are asked to match the dogs and cats one to one and compare their numbers. Example: There are 6 dogs. There are as many dogs as cats. Show how many cats there would be. Coweta Committed to Student Success

  33. Comparison model – 1st There are 2 more dogs than cats. If there are 6 dogs, how many cats are there? There are 6 dogs. There are 2 more dogs than cats. The difference between the two numbers is 2. There are 4 cats. Coweta Committed to Student Success

  34. First basic problem type Part + Part = Whole Whole – Part = Part Part – Part – Whole 8 = 3 + 5 8 = 5 + 3 3 + 5 = 8 5 + 3 = 8 8 – 3 = 5 8 – 5 = 3 5 = 8 – 3 3 = 8 – 5 number bond Coweta Committed to Student Success

  35. Comparison model – 2nd Students may draw a pictorial model to represent the problem situation. Example: Coweta Committed to Student Success

  36. Part-whole model – 2nd Ben has 6 toy cars. Stacey has 8 toy cars. How many toy cars do they have altogether? ? toy cars 6 toy cars 8 toy cars 6 + 8 = 14 They have 14 toy cars altogether. Coweta Committed to Student Success

  37. Forms of bar models • Part-Whole Model • Also known as the ‘part-part-whole’ model, shows the various parts which make up a whole • Comparison Model • Shows the relationship between two quantities when they are compared Coweta Committed to Student Success

  38. Part-whole model Part + Part = Whole Whole – Part = Part Coweta Committed to Student Success

  39. Given 2 parts, find the whole. Part-whole model: Ben has 6 toy cars. Stacey has 8 toy cars. How many toy cars do they have altogether? ? toy cars 6 toy cars 8 toy cars 6 + 8 = 14 They have 14 toy cars altogether. Coweta Committed to Student Success

  40. Given the whole and one part, find the other part. Part-whole model: At summer camp, there were 174 children. If there were 93 boys, how many girls were there? 174 93 ? 174 – 93 = 81 There were 81 girls. Coweta Committed to Student Success

  41. Practice 1 Shannon has 5 candy bars. Her friend, Meghan, brings her 4 more candy bars. How many candy bars does Shannon have now? ? 5 4 5 + 4 = 9 Shannon has 9 candy bars now. Coweta Committed to Student Success

  42. Practice 2 Chris has 16 matchbox cars. Mark brings him 4 more matchbox cars. How many matchbox cars does Chris have now? 16 4 ? 16 + 4 = 20 Chris has 20 cars now. Coweta Committed to Student Success

  43. Practice 3 Caleb brought 4 pieces of watermelon to a picnic. After Justin brings him some more pieces of watermelon, he has 9 pieces. How many pieces of watermelon did Justin bring Caleb? 4 ? 9 9 – 4 = 5 Justin brought Caleb 5 pieces of watermelon. Coweta Committed to Student Success

  44. Comparison model larger quantity – smaller quantity = difference smaller quantity + difference = larger quantity Coweta Committed to Student Success

  45. Comparison model There are 6 dogs. There are 2 more dogs than cats. How many cats are there? 6 dogs ? cats 2 6 – 2 = 4 There are 4 cats. Coweta Committed to Student Success

  46. Practice 4 Anthony has 5 baseball cards. Jeff has 2 more cards than Anthony. How many baseball cards do Anthony and Jeff have altogether? 5 Anthony ? Jeff 2 5 – 2 = 3 5 + 3 = 8 They have 8 cards altogether. Coweta Committed to Student Success

  47. Practice 5 A pencil costs 59 cents, and a sticker costs 20 cents less. How much do a pencil and a sticker cost together? 59 pencil ? sticker 20 59 – 20 = 39 59 + 39 = 98 They cost 98 cents together. Coweta Committed to Student Success www.illustrativemathematics.org

  48. Practice 6 Tracy had 30 Jolly Ranchers. She gave 12 Jolly Ranchers to her friend. How many Jolly Ranchers does Tracy have now? 30 Tracy 12 ? friend 30 – 12 = 18 Tracy has 18 Jolly Ranchers now. Coweta Committed to Student Success

  49. Next steps … Coweta Committed to Student Success

  50. Coweta Committed to Student Success

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