1. 5 th Grade Critical Focus Areas in the Common Core State Standards 1: Operations with Fractions 2: Fluency with whole number and decimal operations 3: Understanding and solving problems with volume

2. 6 th Grade Critical Focus Areas in the Common Core State Standards 1: Ratios, rates and proportions 2: Division of fractions 3: Negative numbers 4: Using expressions and equations 5: Statistical thinking

3. 5 th Grade Critical Focus Areas in the Common Core State Standards Critical Area 1: Fractions Add and subtract fractions with unlike denominators Multiply whole numbers and fractions by fractions Divide a unit fraction by a whole number and a whole number by a unit fraction.

4. Add and subtract fractions with unlike denominators

6. Multiply whole numbers and fractions by fractions – Case 1 (4 th grade) What do students need to know, coming into 5 th grade, to do this? This is multiplying a fraction by a whole number.

7. Multiply whole numbers and fractions by fractions – Case 1 (4 th grade)

8. Multiply whole numbers and fractions by fractions – Case 2 What do students need to know, coming into 5 th grade, to master this? This is multiplying a whole number by a fraction, or rather, taking a fraction of a whole number.

9. Multiply whole numbers and fractions by fractions – Case 2

10. Multiply whole numbers and fractions by fractions – Case 3 What do students need to know, coming into 5 th grade, to master this?

11. Multiply whole numbers and fractions by fractions – Case 3 Can you draw a picture that shows ⅔ of ¾? What do these pictures show? What’s the generalization?

12. Multiply whole numbers and fractions by fractions – Case 3 What does the product of the denominators represent in this picture? What does the product of the numerators represent in this picture?

13. Divide a unit fraction by a whole number 1 2 3 4

14. and a whole number by a unit fraction

15. Division with whole numbers and unit fractions Students need to learn that these types of problems are fraction division problems: How much chocolate will each person get if 3 people share ½ lb of chocolate equally? How many ⅓-cup servings are in 2 cups of raisins?

16. 5 th Grade Critical Focus Areas in the Common Core State Standards Critical Area 2: Fluency with whole number and decimal operations Divide with 2-digit divisors Finalize fluency with whole number arithmetic Extend fluency with whole numbers to operations with decimals

17. Acquisition – Fluency – Generalization Practice – Drill – Application Practice involves learning strategies. Drill involves repetitive use of the strategies until the pattern is habituated. Application generally takes the student to the next level in the learning progression. Knowing that multiplication is repeated addition allows students to multiply a fraction times a whole number (6 x ¾). Knowing how to divide a multi-digit number by a 1 digit-divisor allows students to generalize the procedure with 2-digit divisors.

18. Divide with 2-digit divisors 475 ÷ 25 What do students need to know, coming into 5 th grade, to master this?

19. Divide with 2-digit divisors 475 ÷ 25 Since 475 = 400 + 70 + 5 400 ÷ 25 = 16 and 75 ÷ 25 = 3 so 475 ÷ 25 = 16 + 3 = 19

20. Finalize fluency with whole number arithmetic Practice in all kinds of real-world contexts, every day Two schools are getting together for a carnival. One of the schools has 378 students. The other school has 453 students. How many students would there be at the carnival if every student in both schools came? The key to this problem is for students to see it as an adding problem, since two groups are being brought together.

21. Extend fluency with whole numbers to operations with decimals 56.75 - 4.824.7 x 9.3 45.25 ÷ 2.50 What do students need to know from 4 th grade to accomplish this in 5 th grade?

22. Extend fluency with whole numbers to operations with decimals Addition and subtraction are the same: add or subtract hundreds and hundreds, tens and tens, ones and ones, tenths and tenths, hundredths and hundredths. Ah ha! This is the reason we line up the decimal points! 56.75 - 4.8 51.95

23. Extend fluency with whole numbers to operations with decimals 24.7 x 9.3 Students need a reason for the algorithm – why it works. 1)Estimating gives ~ 240. Calculator says 22971 2)Area model

24. Extend fluency with whole numbers to operations with decimals 2) Area model 1.4 x 1.3 = 1 + 0.4 + 0.3 + 0.12 = 1.82 tenths x tenths yields hundredths

25. Extend fluency with whole numbers to operations with decimals 45.25 ÷ 2.50 How would you extend your understanding of division with 2-digit divisors to solve this problem?

26. 5 th Grade Critical Focus Areas in the Common Core State Standards Critical Area 3: Volume Understand the concept of volume Relate the measurement of volume to l x w x h or b x h Solve problems using the formulas V = l x w x h or V = b x h

27. 6 th Grade Critical Focus Areas in the Common Core State Standards 1: Fluency with whole number and decimal operations (continuation from 5 th grade) 2: Introduction to negative numbers 3: Division of fractions (con’t from 5 th grade) 4: Ratios, rates and proportions 5: Using expressions and equations 6: Statistical thinking

28. 6 th Grade Critical Focus Areas in the Common Core State Standards Critical Area 1: Fluency with whole numbers and decimals All students should be fluent with multiplication and division “facts” when they enter 5 th grade. All students should be fluent with multi-digit operations with whole numbers and decimals when they leave 6 th grade.

29. 6 th Grade Critical Focus Areas in the Common Core State Standards Critical Area 2: The meaning of negative numbers –(3) = –3–(–3) = 3 |–30| = 30 the distance from zero

30. 6 th Grade Critical Focus Areas in the Common Core State Standards

31. Critical Area 4: Ratios and Rates Understand ratio and rate concepts Use ratio reasoning to solve proportion problems If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Hours714212835 Lawns48121620