Location: Room H111, Nipissing University North Bay, Ontario, Canada

1. Education in Primary/Junior Division Mathematics (EDUC 4605/4615) Workshop 2: Number Sense and Numeration, Patterning and Algebra, Geometry, and Measurement Location: Room H111, Nipissing University North Bay, Ontario, Canada Instructor: Daniel H. Jarvis, Ph.D. Email: danj@nipissingu.ca Website: http://www.nipissingu.ca/faculty/danj Office Number: H331 Office Phone: (705) 474 3461 x 4445 Office Hours: by appointment (see embedded schedule)

2. Activity 1:“Are you in the Zone?” Mental Estimation [ESSO Family Math Program (London, Ontario)] Traditional and Invented Algorithms Strategies for Addition & Subtraction Using Base 10 Blocks/Place Value Mats Activity 2: Mathematics and Music (e.g., “Toe-Knee Chest-nut”onESSO Family Math Website) Multiplication & Division Facts and Strategies Mathematics and Literature (Math Curse) Exploring Smart Notebook & GeoGebra 3-D Solids: Prisms & Pyramids Activity 3: Geometry (Isometric Drawing & Modeling, MIRA) Teaching Measurement: Key Strategies, Developmental Stages, Unit Comparisons, Direct vs Indirect Measures, Standardized Measure, SI Units Measuring 1-, 2-, 3-Dimensional Space, Time (Analogue/Digital) Activity 4: Geoboards (Measuring Perimeter and Area) Workshop 2 Agenda 2

3. ESSO Family Math and the “Zone” Family Math (2001-05): Are You in the Zone? (NCTM Article) Estimation of Value(e.g., estimate a “zone” on sticky notes; have students post on relevant number line; locate the most focused area of sticky notes with frame; count out half of the jar of cookies/markers—this becomes the important “benchmark” or “anchor” for further estimation; have students re-evaluate estimate and agree on best “zone” estimate for class (i.e., move frame up or down); finish counting and locate the actual answer; discuss the estimation activity and the various strategies) 100s Chart Extension: Students cross out the small/large numbers on a 100s chart which they are fairly sure do not represent the number being estimates. Family Math Website (UWO): http://www.edu.uwo.ca/essofamilymath/ 3

4. Using the methods that you were taught in elementary school, compute the following two mathematics questions on paper. 212 - 45 145 +66 Make 10, 20, and 100 in your head: Addition or Subtraction? Add 23 and 16 in your head? How did you do it? Try 45 and 29. 4

5. The methods that you just used are called “algorithms”, or mathematical “recipes” • As teachers we must … • Work from our own personal knowledge • Search for alternative processes, strategies, or methods for doing the skill • Recognize the individual differences • observed in our students; ask good questions to see if this is a special case or if the method is generalizable 5

6. Main goal of Elementary/Primary mathematics was to master basic algorithms for whole numbers, decimals and fractions Paper-and-pencil computations were viewed as an important (and often the only valued) numeracy and everyday life skill Calculators and computers have changed the focus Understanding the process is vital Processes still important but a greater emphasis is now on estimation, alternate solution strategies, checking answers for reasonableness, and approximation skills Reform Pre-Reform 6

7. Rote (Memorized) Learning 2 x 5 = 10 6 x 5 = 30 8 x 5 = 40 e.g., students memorize lists without understanding 1 x 5 = 5 2 x 5 = 10 3 x 5 = 15 4 x 5 = 20 More Learning with Understanding e.g., using fact families arranged in arrays with manipulatives or drawn symbols (2-dimensional models) 7

8. Key Concept 1 Addition (Assembly) Subtraction (Disassembly) Inverse Repeated Repeated Multiplication (Assembly) Division (Disassembly) Inverse 8

9. Key Concept 2 Input Operation Output 9

10. Key Concept 3 “Delay! Delay! Delay!” “Once having begun with traditional methods, it is extremely difficult to suggest to students that they learn other methods. . . . If you plan to teach the traditional algorithms, you are well advised to first spend a significant time with invented strategies. Do not feel that you must rush to the traditional methods. Delay! Spend most of your effort on invented methods. The understanding children gain from working with invented strategies will make it much easier for you to teach the traditional methods.” Van de Walle, J. (2004). EMSM: Teaching Developmentally, pp. 202-205 • Working with invented/flexible/personal strategies helps children: • Develop a strong understanding of the operations • Develop efficient strategies for fact retrieval • Practice their selection and use of these strategies when needed 10

11. Comparing Learning Strategies Traditional Algorithms: Step-by-step procedures for teaching computational skills (e.g., 36 x 7, line up the numbers on left, multiply 6 by 7, “carry the 4”, add the 4 and 3, arrive at 252). Direct Modeling: The use of manipulatives and/or drawings along with counting to represent directly the meaning of an operation or story problem (e.g., 7 + 5 using the base-ten blocks; modeling of 36 x 7 with base-ten blocks). Invented (Flexible/Personal) Strategies: Any strategy other than the traditional algorithm (e.g., 75 + 19 can be done mentally as 75 + 20 – 1 = 94). Note: In standards-based mathematics there is still often one correct “answer” to a given problem, but many possible “solutions” or ways of thinking about or solving the problem. Other times, there are different answers possible with more “open-ended” problems. 11

12. 145 +66 11 100 100 211 145 = 100 + 40 + 5 +66 = 60 + 6 100 + 100 + 11 Using Expanded Notation Use expanded notation to advantage. It takes the student’s knowledge of place value and builds new structure. 12

13. Flash Card Practice 6 4 + 2 +/- 4 2 Much better for student understanding! 13

14. Strategies for Addition Think Turnarounds (Commutative Property: Changing the order of the numbers) 14 + 21 = 21 + 14 Think Any Combination Order (Associative Property: Changing the order of the additions for 3+ numbers) 7 + 50 + 123 = 50 + 7 +123 = 123 + 7 + 50 Think Compensation (i.e., give and take) 16 + 29 = 15 + 30 = 45 Think Compatible (i.e., pairs that make 10) 3 + 32 + 17 = 32 + 20 Think Place Value(i.e. add 10s then 1s) 45 + 78 = 110 + 13 14

15. Assembly Addition Model this on a place value mat: 145 +66 • Model both numbers on place value mat. • Model totals with new blocks. • Do trades from right to left. • Compare to traditional algorithm (if already taught). • (N.B. This method keeps a visual record of original question; it is not the only way in which students can use the blocks, e.g., they often “squish & sort” out trades for new total) 15

16. Disassembly Subtraction Model this question on a place value mat: 212 - 45 1. Model both numbers on place value mat. 2. Make necessary “trade” for ones column. 3. Model difference in bottom row for ones column with new blocks. 4. Repeat steps 2 and 3 for 10s, 100s, 1000s, etc columns as necessary. (N.B. This method maintains a visual record of original question, albeit in rearranged [after trades] form.) Alternate Methods with the Place Value Mats: • You can choose to model both numbers (in same/different colours) & then do trades and then “match & move” (like crokinole game scoring) • You can model the smaller number first and then introduce blocks until the larger number target is reached (i.e., change to addition) 16

17. Strategies for Subtraction Think Related Addition Facts and Equations 17 – 9 = ? relates to 9 + ? = 17 Think Decomposition 34 – 23 = 34 – 20 – 3 = 11 Think Commutative Property Problem (not like addition) 12 – 9  9 – 12 Think Equal Offset 68 – 14 = 70 – 16 (N.B. Both numbers are either raised or lowered, as we are concerned about the difference between the two numbers. But be careful here, as it’s easy for students to make mistakes) Make 20 and 100 using mental mathematics. Is this an addition or a subtraction practice exercise? Does it involve both? When? 17

18. Extra Practice First do each question with the traditional algorithm and then use the Base 10 blocks on the Place Value Mats. Think carefully about the connections between the two models as you work. 358 +64 527 - 39 18

19. Assembly Multiplication 216 x 12 846÷35 • Use a calculator to model repeated addition: • Do 3 x 4 So, 3 + 3 = , then hit = twice more • Now do, from above, 216 + 216 = • Then press the “equal” sign 10 more times to • add 216 a total of 12 times. • N.B. This method is inefficient and prone to errors: we need a more sophisticated method to calculate large groups (hence, students see the logical need for multiplication operation). • Show traditional, expanded, and array methods from simple to more complex questions: practice these in class. 19

20. Strategies for Multiplication: • Think Patterns: Multiplication Tables • Think Turnarounds (Commutative Property) 5 x 7 = 7 x 5 • Think Separations (Distributive Property) • 9 x 14 = 9 x (10 + 4) • 9 x 14 = 9 x 10 + 9 x 4 • Think Expanded Form (i.e., show all steps separately, without “carrying” in the algorithm) 20

21. 20 7 10 200 70 6 120 42 320 112 432 Multiplication with Arrays

22. Multiplication Table Analysis Notice Carefully: • Symmetry • 1’s and 2’s are learned quickly • 5’s are usually easily mastered • 9’s with fingers • 6-8’s with fingers • 2 perfect squares • Only one left: 12 = 3 x 4 22

23. Multiplication Table Patterns • Every multiplication fact (product) creates the same kind of ‘quadrilateral’. What is this multiplication shape? • Every number multiplied by itself makes a special shape too. What is this shape? Why does 52 and 72 make sense? • We’ll choose one of the products from the table: “24”.Find the different rectangles that make this product. Place a centicube on each square that displays the product 24. This forms a multiplication curve. • Can you find other curves? 23

24. Disassembly Division 846÷35 Do you use the“Gizinta” Method? • Model simple example of sharing objects/blocks equally among groups/individuals. Stress that we are still trying to equally share 846 among 35 groups. • Ask children to describe what is happening at the various stages of the algorithm (e.g., Why do we place the 2 above the 4 and not the 6? It really is 20 sets of 35.) • Allow for alternative methods of tracking [e.g., (i) if not enough 35s were taken out (e.g., 10 instead of 20), have this step repeated above the original 10 in a second iteration; (ii) have students write “35 x 20” out in the right-hand margin beside “700” after first step so that they understand what is actually happening] 24

25. Strategies for Division: Think Expanded Notation 96  4 = (80 + 16)  4 Think Related (Multiplication) Facts 24  6 = 4 or rearranged as 4 x 6 = 24 Think “leftovers” 32 = ( 7 x ______ ) + _______ 55 = ( 9 x ______ ) + _______ Division Mats / “fair sharing” Have students write division sign horizontally on the page rather than on a slant. This avoids problems later on in algebra. [Note: To use “Equation Editor” in Word 2007, simply click on Insert > Equation, then choose from many built-in Design features.] 25

26. Division by Zero 0 0/6 = 0 can be explained using an analogy: no cookies shared among six people is zero cookies each However, 6/0 is a special case and is more easily explained using patterns once again for clarification: 6/2 = 3 or we can write as 6 = 2 x 3 6/1 = 6 or we can write as 6 = 1 x 6 6/0 = ? or we can write as 6 = 0 x ? [no number exists] 3/0 = ? 3 = 0 x ? [no number exists] 0/0 = ? 0 = 0 x ? [any number] * Therefore, 0/0 is undefined because it is indeterminate (infinite number of solutions), whereas a/0, where a is not 0, is undefined because it is impossible(no solution). 26

27. Other Practice Ideas Navigating the 100s Chart with Operations What’s My Rule? Magic Squares + Bingo x 27

28. Multiple Meanings of Fractions 3/5 shaded • However, one can ask for challenging and creative ways to show “all the ways to shade ½ or ¼ of a square” [I] Fraction as a Part of the Whole • Using “equal parts” is most the common way of representing fractions

29. Multiple Meanings of Fractions [II] Fractions as Parts of a Set (the set is the whole)

30. Multiple Meanings of Fractions [III] Fractions as a Quotient • Begin with 3 cookies; share among 5 • Divide each into 5 parts • Each person gets 1/5 of each cookie • Thus, each person gets 1/5 + 1/5 + 1/5 or 3/5 • Or, 3 divided by 5 = 3/5

31. Multiple Meanings of Fractions [IV] Fractions as a Ratio • Conceptually very difficult because the same symbolic representation takes on a new meaning • What is the ratio of men to women? (3/5) Women to men? (5/3) • Students will want to say 3/8 and 5/8 (relating to the whole). Note: Both relating to the other group and to the whole group are valid. • Ratio can be written as 3/8 or 3:8 (“three is to eight”) • Also, 7/1 cannot be reduced to 7 within ratio notation (we lose track of the 1); and 7/3 is not “improper form” within ratio notation

32. Equivalent Fractions Note the many forms of equivalent fractions located on the Multiplication Table N.B. Students must learn to multiply using the “Identity Element” (i.e., “x 1”), choosing whichever identity fraction (e.g., 2/2) to use based on the goals of any given problem.

33. Finding the Whole Given a Part: A Giant Leap for Students 3/5 of bar whole bar • This picture represents three-fifths of a chocolate bar: draw the whole chocolate as it would appear.

34. Models, Words, and Symbols • Important for children to match models, words, and symbols One third of the pie is green 1/3 • Also important for children to explain their thinking and to link their learning to the real world (“authentic context”) • The same is true when discussing percent or ratio; It is important for students to understand that percent (Fr. “per cent”) is just another fraction with 100 equal parts (note: % is just 100 rearrange with the 1 shown between the two zeroes) • You need to represent percent concretely and pictorially – e.g. base 10 flat with coloured counters as parts; colouring in parts of a 100 square grid

35. Many Ways To Represent Fractions Fraction Graffiti: Small group rotation of sheet; brainstorm multiple representations Rope: Volunteers position equal segments; position/create numbers between two given endpoints (e.g., “1/5 and 2 ¾”) Number Line: Whole group orders themselves with cards from smallest to largest (i.e., multiple representations on the cards) Note duplicates and numbers very close together (1 and 0.99) “Fraction Machine”: Comparing fractions with colour strips “Elastic Meter Manipulative”: Emphasizes proportional reasoning in a very visual, tactile way Ontario Leading Math Success and Learn Alberta Websites 0.5 50% 1 2

36. Fractions in Different Formats Common, Improper (i.e., “larger number on top”), and Mixed Fractions “Simplify” / “Reduce” / “Lowest Terms” (i.e., the vocabulary of fractions) Show 5/2 = 2/2 + 2/2 + ½ = 2 ½ (understanding algorithm) Show 3 4/5 = 5/5 + 5/5 + 5/5 + 4/5 = 19/5 (again, why “steps” work)

37. Convert between Fractions/Decimals/Percent

38. An Introduction to GeoGebra Open program in iTeach > Education > Math > GeoGebra. Download Introductory file and try first activity, as time permits. GeoGebra Institute of Canada www.geogebracanada.org

39. Some Past Ideas “We do geometry in June—it’s a fun way to end the course.” “We don’t usually even get to geometry at the end of the year—too much to cover.” “Boys are better than girls at geometry and spatial sense.” “There is little connection to the curriculum documents.” “We don’t have resources and software.”

40. Geometry Systems Topological Coordinate Euclidean Transformational

41. 2-D Space: Polygons (‘many sides’) Integrate with measurement curriculum “Edges are straight” / “Faces are flat” Regular and irregular polygons Classification of polygons (hierarchy) Triangles (“trigons”): 3 sides Quadrilaterals (“tetragons”): 4 sides Pentagons: 5 sides Hectagon (“centagon”): 100 sides Chiliagon: 1000 sides Myriagon: 10 000 sides Polygons: many (i.e., more than two sides) (Many books like Burn’s Greedy Triangle available) Art Idea Build Mobiles Triangles  Trigons  Trigonometry

42. Transformations • Teaching Aids: • Computer Tools (Sketchpad, Cabri, GeoGebra, TABS+) • Grid paper transformations • Math/Learning Carpet

43. What Can Be Measured? Dimensional Space 1-D (Linear) 2-D (Area) 3-D (Volume / Capacity) Mass (Amount of Matter in an Object) and Weight (Pull of Gravity on an Object) Time Heat Combinations Thereof: Speed or Velocity (e.g., kph, mph, kps) Density (mass per unit of volume)

44. System International (SI) • Prefixes: • Sub-Units: milli- / centi- / deci- (smaller than unit) • Combined Units: deca- / hecto- / kilo- (larger) • Mnemonic: “King Henry’s daughter usually drinks chocolate milk” • Base Units: • Metre (Linear) • Gram (Mass) • Litre (Capacity/Volume)

45. Measuring 1-Dimensional Space Participation (concrete materials) is essential Non-standard > Standardization (video clip) Need for sub-units for precision (e.g. m > mm) Frequent and meaningful estimation activities

46. Exploring with Geoboards • Set a perimeter, then find • as many different areas that • can be enclosed. • Try It: Perimeter = 14 units • What different polygons (area) • can be made? • Set an area, then find as many different shapes (perimeters) that can be generated. • Try It: Area = 12 units 2 • What different polygons (area) • can be made?

47. Understanding Formula Derivation 5 cm Generalizations to discover formula most often extend out of one another 3 cm P=2 (l+w) A= l x w

48. Developing Formulae using Paper Area of a Rectangle Area of a Parallelogram Area of a Triangle Area of Trapezoid A= b x h A= 1/2 (b x h) A= b x h A= 1/2[h(a + b)] All four are related.

49. Measuring 3-Dimensional Space Model building a key element (e.g., linking cubes, blocks) Need for a new unit (3-D) What “packs” together in space and leaves no gaps? (cubes) Volume = Area of the base shape multiplied by the height/depth as a general pattern. Mysterious Volume Demonstration: Two equal-sized sheets of paper both folded into cylinders and taped, but along the two different edges Estimate the two volumes within each shape Place longer cylinder (A) within shorter cylinder (B) and fill A with rice. Now slowly remove the inner cylinder and note the results Question: What is the key factor in the difference between these two volume formulas? (radius squared—greater effect) Note the relationship between the volume of a cone and related cylinder, and between a rectangular pyramid and related rectangular prism (hence the formulas involving a factor of 3)

50. Measurement Formula Development For circle circumference formula, have students cut out circles and a string length equal to the diameter; measure edge of circle with the string length and notice how many it takes. For circle area formula, divide a circle into equal pie sections; rearrange pie pieces in an alternating pattern; imagine smaller and smaller pie pieces being used for this model; eventually we have radius (width) x half of the circumference (length) Be aware that prisms (straight up from base) and pyramids (edges from base meet at single point) are the two main classification categories for 3-D figures in K-6