Cool Cash Confusion

1. Cool Cash Confusion • In your color groups • Read the article – yellow sheet • Think of a way to teach and model this confusing concept to the people of Camelot • Use any material on the middle table, or something else that you think of to teach this concept

2. AMSTI Preservice Training September 25 Skyland Elementary School  - 408 Skyland Blvd. East – Tuscaloosa, AL   Hours:  8:30 – 3:30           Lunch 11:30 – 12:30 Park in the grassy/gravel area of the school grounds closer to the Locklear Automobile dealership   Go through the gate in the chain link fence and enter the building from the side door.  Drinks will be available for a nominal charge.   Proceeds will benefit the Skyland 4th and 5th grade Ambassadors.  Dress appropriately.  No shorts or revealing tops.

3. Preparing for Observations& the Midterm Using the Knowledge Quartet

4. Observation Practice • Take notes on what you seeand hear • Teacher • Students

5. During Observation • Take notes on what you seeand hear • Teacher • Students • Identify evidence • Foundation • Transformation After Observation

6. Concept or Procedure? • Three Column • Addition • Subtraction • Multiplication • Area • Polygon • Place Value • Addition • Fractions • Measurement • Skip Counting • Long Division

7. Concept, Concept, Concept! • The Why • Number Relationships • More than the correct answer • Making Connections

8. Concept Procedure • Place Value • Addition • Subtraction • Multiplication • Division • Fractions • Whole numbers • Equivalent fractions • Decimal • Polygon • Least Common Multiple • Multiple • Factor • Divisor • Long Division • Addition of 2 numbers (subtraction, multiplication, division) • Simplify fractions • Finding equivalent fractions • Finding the area or perimeter • Measuring • Skip counting • Making a graph or display

9. Addition & Subtraction Class 5

10. Addition & Subtraction • Models of addition & subtraction • Stages in early addition & subtraction • Strategies for addition &subtraction

11. How do you defineaddition?

12. Models of Addition • Aggregation– two or more quantities are combined into a single quantity and the operation of addition is used to determine the total. Examples? Think about the domino lesson. Ask an aggregation question. Haylock, D ( 2001) Mathematics Explained for Primary Teachers, London: Paul Chapman

13. Models of Addition • Augmentation – a quantity is increased by some amount and the operation of addition is required in order to find the augmented or increased value. Examples? Think about the domino lesson. Ask an augmentation question. Haylock, D ( 2001) Mathematics Explained for Primary Teachers, London: Paul Chapman

14. Models of Addition Make a story problem for both using 3 + 5 = 8 • Aggregation • Augmentation Haylock, D ( 2001) Mathematics Explained for Primary Teachers, London: Paul Chapman

15. Models of Addition Make a story problem for both using 3 + 5 = 8 • Aggregation: There are 3 brown dogs and 5 black dogs. How many dogs? 3 + 5 = • Augmentation: There are 3 brown dogs. Some black dogs come. Now there are 8 dogs. How many are black? 3 + = 8 Haylock, D ( 2001) Mathematics Explained for Primary Teachers, London: Paul Chapman

16. Early Stages in Addition • Counting all – a child doing 5 + 8 counts out five bricks and then eight bricks and then finds the total by counting all the bricks. • Counting on from the first number - a child finding 5 + 8 counts on from the first number: ‘five; six, seven …thirteen’, possibly keeping a tally of the eight (which is being added) with their fingers or more likely using a number line. • Counting on from the larger number– a child chooses a larger number, even when it is not the first number, and counts on from there.

17. Using a known addition fact – where a child gives an immediate response to facts known by heart, such as 6 + 4 or 3 + 3 or 10 + 8 • Using a known fact to derive a new fact – where a child uses a number bond that s/he knows by heart to calculate one that s/he does not know, e.g. using knowledge that 5 + 5 = 10 to work out 5 + 6 = 11 and 5 + 7 = 12. • Using knowledge of place value – where a child uses knowledge that 4 + 3 = 7 to work out 40 + 30 = 70, or knowledge that 46 + 10 is 56 to work out 46 + 11 = 57

18. Addition Strategies • Addition Properties: The zero property states that zero added to any number is the same as the original number. The commutative (or order) property states that the order of addends does not matter: 3 + 4 = 4 + 3. • Doubles and near doubles: If you have two groups of 8 objects, you have double 8, or 16, objects. Doubles facts are usually easy to remember, and can be used to learn other facts. Since 8 + 8 = 16, and 9 is one more than 8, 8 + 9 will be one more than 16, or 17. • Using 10 to add 9: The place-value system makes adding 10 to a number easy – just increase the digit in the tens place by 1. You can use this to help add 9 to a number. Just add 10 to the number, then subtract 1. • Fact families: A fact family is a group of related facts using the same numbers. One example would be 4 + 3 = 7, 3 + 4 = 7, 7 – 3 = 4, and 7 – 4 = 3. Fact families are a very powerful tool for mastering facts; once you know one fact in a family, you can work out the other facts in the same family. Fact families are also useful for solving problems with missing addends, such as 4 + __ = 7.

19. How do you define subtraction?

20. Subtraction Subtraction is used to model four related processes: • From a given collection, take away (subtract) a given number of objects. For example, 5 apples minus 2 apples leaves 3 apples. • From a given measurement, take away a quantity measured in the same units. If I weigh 200 pounds, and lose 10 pounds, then I weigh 200 − 10 = 190 pounds. • Compare two like quantities to find the difference between them. For example, the difference between $800 and $600 is $800 − $600 = $200. Also known as comparative subtraction. • To find the distance between two locations at a fixed distance from starting point. For example if, on a given highway, you see a mileage marker that says 150 miles and later see a mileage marker that says 160 miles, you have traveled 160 − 150 = 10 miles.

21. How might children do: 11 – 3 = 8 11 – 7 = 4

22. Early stages in subtraction • Counting out – a child finding 9 – 3 holds up cubes or fingers and folds down three. Then counts the remaining fingers 1, 2, … 6. (‘taking away’)

23. Counting back from – a child finding 9 – 3 counts back three numbers from 9: ‘eight, seven, six’ (‘taking away’). • Counting back to – a child doing 11 – 7 counts back from the first number to the second, keeping a tally using fingers of the number of numbers that have been said, ‘ten, nine, eight, seven, holding up four fingers. (Finding the difference)

24. Counting up – a child doing 11 – 7 counts up from 7 to 11, ‘eight, nine, ten, eleven’, sometimes keeping a count of the spoken numbers using fingers (not a ‘natural’ strategy for many children because of the widely held perception of subtraction as ‘taking away’) (Finding the difference)

25. Using a known fact – a child gives a rapid response based on facts known by heart, such as 8 – 3 or 20 – 9. • Using knowledge of place value – a child knows that 25 – 10 is 15 • Using a derived fact – a child uses a known fact to work out a new one, e.g. 20 – 5 = 15 so 20 – 6 must be 14 (more unusual in subtraction than in addition). Or 25 – 9 = 16

26. Subtraction Strategies • Subtraction Rules: There are two rules for using zero in subtraction. Zero subtracted from any number is the original number (this is the counterpart of the zero property of addition), and any number subtracted from itself equals zero. • Counting back: For facts such as 6 – 1, you can count back. (Counting back is often harder to master than counting on.) • Using 10 to subtract 9: The place-value system makes subtracting 10 to a number easy – just decrease the digit in the tens place by 1. You can use this to help subtract 9 from a number. Just subtract 10 from the number, then add 1. • Fact families: A fact family is a group of related facts using the same numbers. One example would be 4 + 3 = 7, 3 + 4 = 7, 7 – 3 = 4, and 7 – 4 = 3. Fact families are a very powerful tool for mastering facts; once you know one fact in a family, you can work out the other facts in the same family. Fact families are also useful for solving problems with missing subtrahends, such as 7 - __ = 3.

27. Children see problems differently than adults. • Young children have a rich informal knowledge of mathematics that can serve as a basis for developing understanding of mathematics. • Children can figure out how to solve problems without instruction. Solving many problems enables children to develop their own understanding of mathematics.

28. Young children can solve many mathematical problems by directly modeling the action and relationship in the problem • The structure of a problem and the placement of the unknown determines how difficult it is for children to solve and determines their initial solutions strategies • Addition and subtraction problems vary by: • Action/no action • Type of action • Location of the unknown (x)

29. Common Core Standards

30. Solve this problem…without the algorithm! • Mary had 114 spaces in her photo album. So far she has 89 photos in the album. How many more photos can she put in before the album is full?

31. Three types of computational strategies Progression Page 217

32. Everyday Math Strategies • Addition • Subtraction Let’s Try It!

33. Handout: Sample Diagnostic Interview • Next week… come with topic ideas

34. Three truths & a lie • Adorable Alabama Athletes • Link sheet • Closing reflection Class 3 • 4 quadrants • Round Robin Centers • Exiting Thoughts (Balloons) Getting our notebooks in order

35. For next week…