Understanding Decimal and Fraction Addition and Subtraction

1. Content Session 10 July 14, 2009

2. Addition & Subtraction M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations. e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators. f. Model addition and subtraction of decimal fractions and common fractions with like denominators. g. Use mental math and estimation strategies to add and subtract decimal fractions and common fractions with like denominators.

3. Addition & Subtraction M4N5. Students will further develop their understanding of the meaning of decimals and use them in computations. • Add and subtract both one and two digit decimals. M4N6. Students will further develop their understanding of the meaning of decimal fractions and common fractions and use them in computations. b. Add and subtract fractions and mixed numbers with like denominators. (Denominators should not exceed twelve.)

4. Addition & Subtraction M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them. g. Add and subtract common fractions and mixed numbers with unlike denominators.

5. Goals (Grades 3 & 4) • Grade 3 • The meaning of addition and subtraction remains the same even when numbers become decimal numbers or fractions • Decimal numbers and fractions are numbers, just like whole numbers • Grade 4 • Fluency with decimal addition and subtraction • The sum of two fractions may exceed 1 (improper fractions or mixed numbers), and the minuend may also exceed 1.

6. Goals (Grades 5) • Fluency with fraction addition and subtraction

7. Key Ideas • Unitary perspective of numbers • 0.3 is 3 0.1-units; 3/5 is 3 1/5-units; etc. • Relative size of (decimal) numbers • 0.32 is 32 0.01-units • Addition/Subtraction can be performed only when the two numbers are referring to the same unit

8. How do these relate to 3 + 4? • 30 + 40 • 300 + 400 • 3000 + 4000 • etc. How about these? • 300 + 40 • 30 + 4000

9. If you put a tape that is 0.3 meters long and another tape that is 0.4 meters long together, end to end, how long will it be? • What math sentence will represent this problem? • What does 0.3 meters mean? • What does 0.4 meters mean? • How many 0.1 meter will there be altogether? • What is the answer?

10. If you put a tape that is 0.6 meters long and another tape that is 0.8 meters long together, end to end, how long will it be? What math sentence will represent this problem? What does 0.6 meters mean? What does 0.8 meters mean? How many 0.1 meter will there be altogether? What is the answer?

11. If you put a tape that is 0.07 meters long and another tape that is 0.05 meters long together, end to end, how long will it be? What math sentence will represent this problem? What does 0.07 meters mean? What does 0.05 meters mean? How many 0.01 meter will there be altogether? What is the answer?

12. If you put a tape that is 3.6 meters long and another tape that is 2.2 meters long together, end to end, how long will it be? What math sentence will represent this problem? What does 3.6 meters mean? What does 2.2 meters mean?

13. If you put a tape that is 3.6 meters long and another tape that is 2.2 meters long together, end to end, how long will it be? What math sentence will represent this problem? What does 3.6 meters mean? What does 2.2 meters mean? 3 meters + 2 meters, and 6 0.1-meters + 2 0.1-meters 5 meters and 8 0.1-meters, or 5.8 meters

14. If you put a tape that is 3.6 meters long and another tape that is 2.2 meters long together, end to end, how long will it be? What math sentence will represent this problem? What does 3.6 meters mean? What does 2.2 meters mean? 36 0.1-meters + 22 0.1-meters 58 0.1-meters, or 5.8 meters

15. If you put a tape that is 3.73 meters long and another tape that is 2.2 meters long together, end to end, how long will it be? What math sentence will represent this problem? What does 3.73 meters mean? What does 2.2 meters mean?

16. Can your students answer these: • 4 and eight 0.1’s will make _______. • Twenty-six 0.1’s will make _______. • 5 and thirty-four 0.01’s will make _______. • How many 0.01’s are in 1.57? • What number is 0.01 less than 5? • What number is 0.1 more than 4.83?

17. 2.14 is: • 2 and one 0.1 and four 0.01’s. • 2 and fourteen 0.01’s • Two-hundred fourteen 0.01’s • Twenty-one 0.1’s and four 0.01’s • Two-thousand one-hundred forty 0.001’s • etc.

18. If you put a tape that is 3/8 meters long and another tape that is 3/8 meters long together, end to end, how long will it be? • What math sentence will represent this problem? • What does 3/8 meters mean? • What does 4/8 meters mean? • How many 1/8 meter will there be altogether? • What is the answer?

19. If you put a tape that is 5/8 meters long and another tape that is 7/8 meters long together, end to end, how long will it be? • What math sentence will represent this problem? • What does 5/8 meters mean? • What does 7/8 meters mean? • How many 1/8 meter will there be altogether? • What is the answer?

20. If you put a tape that is 5/8 meters long and another tape that is 3/4 meters long together, end to end, how long will it be? • What math sentence will represent this problem? • What is different about this problem? • What does 5/8 meters mean? • What does 3/4 meters mean? • What can we do?

21. If you put a tape that is 5/8 meters long and another tape that is 3/4 meters long together, end to end, how long will it be? • Let’s make these fractions refer to the same unit. • We can change 3/4 into 6/8 [or we can change both to 10/16 and 12/16, or some other equivalent fraction pairs] • Now, we know how to add those fractions.

22. Do we need the least common denominator? • No – we just need a common denominator (common unit) in order to add.

23. Multiplying & Dividing Decimals M4N5. • Multiply and divide both one and two digit decimals by whole numbers. M5N3. • Explain the process of multiplication and division, including situations in which the multiplier and divisor are both whole numbers and decimals.

24. Multiplying & Dividing Decimals M4N5. • Multiply and divide both one and two digit decimals by whole numbers. M5N3. • Explain the process of multiplication and division, including situations in which the multiplier and divisor are both whole numbers and decimals.

25. Which problem can we use our whole number multiplication knowledge to solve? • 1m of wire weighs 1.4 lb. How much will 6m of the same wire weigh? • 1m of wire weighs 6 grams. How much will 1.4m of the same wire weigh?

26. What does 6 x 1.4 mean?

27. Decimal Unit Approach • We have 6 groups of 14 0.1 grams. • 14 x 6 = 84; Altogether, we have 84 0.1 grams. • 84 0.1 grams  8.4 grams • 1.4 x 6 = 8.4

28. Which problem can we use our whole number division knowledge to solve? • 9 m of iron pipe weighs 3.6 kg. How much will 1m of the same pipe weigh? • 3.6 m of iron pipe weighs 9 kg. How much will 1m of the same pipe weigh?

29. What does 3.6 ÷ 4 mean?

30. Decimal Unit Approach • Divide 36 0.1-kg to make 9 equal groups. • 36 ÷ 9 = 4; Each group will have 9 0.1-kg. • 9 0.1-kg = 0.9 kg. • 3.6 ÷ 4 = 0.9

31. How about 2.6 ÷ 5? • 26 0.1-kg: make 5 equal groups • BUT, 26 ÷ 5 = 5 rem. 1 • Each group will get 5 0.1-kg and there will be 1 0.1-kg left over. • 2.6 ÷ 5 = 0.5 rem. 0.1

32. Dividing on: 2.6 ÷ 5 • Model 2.6 ÷ 5 using base-10 blocks – use a flat as 1. • What will be left over? • Can we trade it? With what?

33. What is 8 ÷ 5?