Integer Division. Student Probe

Transcription

1 Student Probe What is 24 3? Answer: 8 Integer Division Lesson Description This lesson is intended to help students develop an understanding of division of integers. The lesson focuses on using the array model on the coordinate plane as a tool for students to develop the conceptual foundation. Rationale Integers are arguably the most important subset of the number system. The understanding of integers is essential for entry into higher level mathematics. Students are frequently confused by which sign (positive or negative) gets placed on the quotient. Preparation Prepare coordinate grid paper for student use. Each student will need a set of two-color counters, and red and blue markers. Lesson At a Glance What: Division of integers Common Core State Standards: CC.7.NS.2b. Apply and extend previous understanding of multiplication and division and of fractions to multiply and divide rational numbers. (b) Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = (-p)/q = p/(- q). Interpret quotients of rational numbers by describing real-world context. Mathematical Practices: Model with mathematics. Who: Students who have difficulty dividing integers Grade Level: 7 Prerequisite Vocabulary: positive, negative, quotient, set, expression, equation, divisor, dividend, value Prerequisite Skills: division of whole numbers Delivery Format: Individual, small group Lesson Length: 15 to 20 minutes Materials, Resources, Technology: Visual display, two-color counters, coordinate graph paper, use of interactive white board with interactive integer multiplication coordinate grid (optional) The teacher says or does 1. What are some operations you do with numbers? How have you modeled division of numbers? Expect students to say or do Addition, subtraction, multiplication and division arrays, skip counting, repeated addition If students do not, then the teacher says or does Offer hints to remind students of previous experience with these models.

2 The teacher says or does Expect students to say or do 2. Today we are going to extend our understanding of division of numbers to integers. (Distribute two-color counters to students and explain that yellow is the positive side and red is the negative side.) 3. Ask students to find 40 5 using an array model. ******** 5 ******** ******** 40 ******** If students do not, then the teacher says or does Refer to Fluently Divide within 100 Using Strategies. 4. Look at your grid paper. On the x- axis the positive region would be represented where? 5. What does the part of the x-axis to the left of the origin represent? 6. Show me the region that represents negative numbers on the y-axis. 7. So the section containing the y-axis, above the x-axis, represents what types of numbers? 8. Where would you find the ordered pair if the coordinates were (+, +)? 9. Where would you find the ordered pair if the coordinates were (-, -)? Students should then show 8 columns or rows depending on the orientation of the array. To the right of zero Negative numbers Students should indicate a downward movement on the y-axis, which is beneath the x-axis. Positive numbers Quadrant I Quadrant III Refer to Graph Ordered Pairs on a Coordinate Plane. Refer to Graph Ordered Pairs on a Coordinate Plane.

3 The teacher says or does 10. When we multiply using the coordinate grid, the product of the positive coordinates is a positive array in Quadrant I. The product of two negative coordinates is a positive array in Quadrant III. 11. Locate Quadrants I and III. Place large positive signs in each of those quadrants. 12. If the array (dividend) is positive then place it in Quadrant I or Quadrant III. Positive arrays are always placed in positive quadrants (Quadrants I or III). 13. Let s divide (Demonstrate to students 12 3 using 3 counters along the positive x-axis, filling in a rectangular array of 12.) (See Teacher Notes.) Expect students to say or do Students should place positive signs in Quadrants I and III. If students do not, then the teacher says or does Model. Model. We have a positive dividend and a positive divisor. Which quadrant will we work in? (As you are discussing the dividend divided by the divisor, point out that if the dividend is positive then the focus would be Quadrants I and III. If the divisor is positive as well, then Quadrant I will be used.) 14. How many rows is the array along the positive y- axis? 15. What is 12 3? If we have the expression 12 3, we will place three counters along the negative x-axis since the divisor is 3. How many rows is the array along the negative y-axis? Is the 4 positive or negative? Negative 18. What is 12 3? 4 4 Model. We have a positive dividend and a negative divisor. Which quadrant will we work in?

4 The teacher says or does 19. When we multiplied using the coordinate grid, the product of the positive and negative coordinates is a negative array in Quadrant II or in Quadrant IV. 20. Locate Quadrants II and IV. Place a large negative sign in those 2 quadrants. 21. If the array (dividend) is negative then place it in Quadrant II or Quadrant IV. Negative arrays will always be placed in negative quadrants (Quadrant II and Quadrant IV). 22. Demonstrate (-12) 3 using 3 counters along the positive x-axis and filling in a rectangular array of 12 in Quadrant IV. (Use the red side of the counters.) (See Teacher Notes.) Expect students to say or do Students should place negative signs in Quadrants II and IV. If students do not, then the teacher says or does Model. Model. We have a negative dividend and a positive divisor. Which quadrant will we work in? As you are discussing the dividend divided by the divisor, point out that if the dividend is negative the focus is on Quadrants II or IV. If the divisor is positive we are on the positive x-axis and looking at Quadrant IV. 23. How many rows is the array along 4 the negative y-axis? 24. Is the 4 positive or negative? negative 25. What is (-12) 3? 4

5 The teacher says or does 26. Demonstrate 12 3 using 3 counters along the negative x-axis. Fill in a rectangular array of 12 in Quadrant II. (Use the red side of the two-color counters.) (See Teacher Notes.) Expect students to say or do If students do not, then the teacher says or does Model. We have a negative dividend and a negative divisor. Which quadrant will we work in? As you are discussing the dividend divided by the divisor, point out that if the dividend is negative the focus is on Quadrants II or IV. If the divisor is negative we are on the negative x- axis and we are looking at Quadrant II. 27. How many rows is the array along 4 Model. the positive y-axis? 28. Is the 4 positive or negative? positive 29. What is the quotient of (-12) (-3)? 4

6 Teacher Notes

7 12 (-3) -3-4

8 (-12) 3 3-4

9 (-12) (-3) 4-3

10 Variations None Formative Assessment Determine the following quotients (-4) (-24) 8 (-24) (-6) Answers: 5,-5,-3, 4 Resources: Mathematics Preparation for Algebra. (n.d.). Retrieved 12 9, 2010, from Doing What Works: