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Algebra for All: What, How & Why

Algebra for All: What, How & Why. Grades 3-5 Dr. Janet H. Caldwell Rowan University. Themes in Algebra. Patterns Functions and Relationships Modeling Procedures. Recognize, describe, extend & create patterns. Using concrete materials, pictures, rhythms, and whole numbers

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Algebra for All: What, How & Why

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  1. Algebra for All: What, How & Why Grades 3-5 Dr. Janet H. Caldwell Rowan University

  2. Themes in Algebra • Patterns • Functions and Relationships • Modeling • Procedures

  3. Recognize, describe, extend & create patterns. • Using concrete materials, pictures, rhythms, and whole numbers • Descriptions using words and number sentences or expressions, graphs, tables, variables • Repeating patterns • Sequences that stop or that continue infinitely • Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying, or dividing by a fixed number • Sequences can often be extended in more than one way

  4. Patterns 3 7 11 15 19 . . .

  5. How Does It Grow? • Draw the next figure. • When there are 6 triangles, how many squares are there? • When there are 5 triangles, how many shapes are there in all? • Can there be 7 squares in a figure? Why?

  6. Representations The number of squares is twice the number of the figure. S = 2 x N

  7. Functions & Relationships • Use concrete and pictorial models of function machines to explore the basic concept of a function. • Input/output tables, T-charts • Combining two function machines • Reversing a function machine

  8. Input/Output Tables

  9. 4 Add 3 ? ? X 6 Function Machines

  10. ? ? Add 3 X 6 Reversing Function Machines 72

  11. Cutting String • Fold a piece of string in half. While it is folded, make 1 cut. How many pieces of string do you have? Continue with another piece of string folded in half, making 2, 3, 4, and 5 cuts.

  12. Linear function

  13. What’s the pattern?

  14. Quadratic function

  15. Modeling • Recognize and describe changes over time • Graphs representing change over time • How change in one physical quantity can produce a corresponding change in another

  16. You are mowing the lawn. As you mow, the amount of grass to be cut decreases. You mow at the same rate until about half the grass has been cut. Then you take a break for a while. Then, mowing at the same rate as before, you finish cutting the grass. Sketch a graph that shows how much uncut grass is left as you mow, take your break, and finish mowing.

  17. The graph represents the relationship between the profit and the amount of lemonade sold at a lemonade stand. Write a story about how the lemonade stand’s profit is determined. Include an explanation of what is indicated when the line is below zero and when the line crosses the horizontal axis. (This graph assumes that the seller is not paid and that there is no overhead.)

  18. Modeling • Construct and solve simple open sentences 4 + 3 = 7 4 + 3 = 7

  19. Word Problems • Bobby read some books, then said, “If I read 5 more, I will have read 17.” How many books has he read? X + 5 = 17

  20. Jack & Jill • If Jack fetched 3 more pails of water than Jill did, and together they carried 21 buckets, how many pails did each fetch? X + (x + 3) = 21

  21. = x 3 + 6 8 Card Game = + 7 9 8

  22. Connect Algebra to Arithmetic • Juan, Maria, and Clarissa collect hats. Juan has 5 fewer hats than Maria and 10 fewer hats than Clarissa. If Maria has 20 hats, how many do Juan and Clarissa have? Explain your thinking in both words and symbols. M has 20 hats. Juan has 20 – 5 = 15 hats. Clarissa has 15 + 10 = 25 hats.

  23. Extension A • Juan, Maria, and Clarissa collect hats. Juan has 5 fewer hats than Maria and 10 fewer hats than Clarissa. If Maria has H hats, how many do Juan and Clarissa have? Explain your thinking in both words and symbols. M has H hats. Juan has H – 5 hats. Clarissa has H - 5 + 10 = H + 5 hats.

  24. Juan, Maria, and Clarissa have been adding to their collections. Juan and Clarrisa now have 15 more hats than they had before, and Maria has 2 times as many hats as she had before. How many hats does each have now? How many do they have all together?

  25. Procedures • Understand and apply the properties of operations and numbers • Solve simple linear equations with manipulatives, informally, graphically, and using formal symbolic methods.

  26. 3037 – 258 = 2779 Now find the answers to these problems, without working them out. Explain how you know the answers are correct. 3037 3037 3037 3037 3037 - 259-257- 358- 158- 288

  27. 3 x 5 + 9 ÷ 3 = ? Carla said, “I know that 3 x 5 is 15, and I know that 9 ÷ 3 is 3, so it’s 15 plus 3. The answer is 18.” Dave said, “No, that’s wrong. There’s 3 at the front and then ÷ 3 at the end. The 3s cancel out, so you’ve just got 5 plus 9, that’s 14.” Evan said, “I think the answer is 8 because you have to do it in order. Three times 5 is 15, then add 9 and you get 24, and then divide by 3 and you get 8.” Who is right?

  28. Understanding equivalence relationships Recognizing the operations Using a wide range of numbers Understanding important properties of numbers Describing patterns and functions Aspects of Number Essential for Algebra

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