**Elementary Math Focus Day** December 1, 2014 Bing Elementary

Workshop: **Problem Solving in Visibly Random Groups on Temporary Vertical Surfaces**

Dr. Peter Liljedahl’s (SFU) research on visibly random groups working on temporary vertical surfaces has shown dramatic improvement in student’s problem solving ability, math comprehension and appreciation. Engagement is high and open-ended for differentiation. We will discuss the techniques while modeling these lessons with Gr. 4 – 7+ examples but teachers of all levels can gain from this workshop. (Fred Harwood, Retired Richmond Teacher, SFU, Teacher Consultant)

**Research: **http://www.peterliljedahl.com/wp-content/uploads/Visibly-Random-Groups.pdf

**Key Points: **Results indicate that the use of visibly random grouping strategies, along with ubiquitous (predominant learning activity) group work, can lead to:

(1) students becoming agreeable to work in any group they are placed in,

(2) the elimination of social barriers within the classroom,

(3) an increase in the mobility of knowledge between students,

(4) a decrease in reliance on the teacher for answers,

(5) an increase in the reliance on co-constructed intra- and inter-group answers, and

(6) an increase in both enthusiasm for mathematics class and engagement in mathematics tasks.

**Today’s first workshop **focused on the Andorran kissing problem with 7 groups of 3. The groups were randomly arranged by lining up in their birthdate order (Month & Day NOT year). The question was set up just as is written below. I didn’t answer any clarity questions that stopped their thinking. I wanted them to become aware of the assumptions that often have to be made to solve a problem. Sometimes those assumptions split the solution into several tracks. Initially, every group arrived at a different answer. We mobilized the data by having one person stay with their board and the other two visited other boards to discuss what they were doing. Afterwards, they compared notes and looked at their own approach. Answers were converging after this with some happy to accept some group’s differing assumptions. We then did a gallery walk together where we looked at each group’s approach. I could have orchestrated it. Or I could have asked a member of a different group to explain what one group did (as this makes a group more accountable for communicating their approach clearly.) I had not pre-loaded this possibility since our classroom community wasn’t fully formed so I went with a representative of each group sharing what they had done. It was revealing to see one catching a mistake because he was verbalizing the solution out loud. This correction lead to a final answer that was repeated by multiple groups. The ‘class’ then shared how they appreciated the mathematics that had arisen through the process. This is what happens when this is a pedagogical approach as differentiation needs are met with students listening to others’ ideas. Different people will make connections to other ideas in different ways as they are ready to receive them. No one found the whole problem easy but all entered in and grew through the process. Connections to prior problems like the handshake problem were suggested. We were focusing on the core competencies of communication, thinking and personal/social skills. Problem solving on temporary vertical in random groups meet all these competencies and help to develop them further.

We finished the session with a look at the loading of the last question and then generated questions the ‘class’ might want to pursue. Teachers could see why this would be motivational when they were pursuing solutions to their own questions. Are there any numbers less than 100 that can’t be written as the sum of consecutive natural numbers? What numbers can be written as the sum of three consecutive numbers? Four consecutive? Are there patterns in which numbers work multiple ways or only one way? Which number less than 100 is the sum of the maximum number of consecutive numbers?

Following is how the handout and list of problems looked:

**Elementary Math Focus Day** December 1, 2014 Bing Elementary

Workshop: **Problem Solving in Visibly Random Groups on Temporary Vertical Surfaces**

Dr. Peter Liljedahl’s (SFU) research on visibly random groups working on temporary vertical surfaces has shown dramatic improvement in student’s problem solving ability, math comprehension and appreciation. Engagement is high and open-ended for differentiation. We will discuss the techniques while modeling these lessons with Gr. 4 – 7+ examples but teachers of all levels can gain from this workshop. (Fred Harwood, Retired Richmond Teacher, SFU, Teacher Consultant)

**Research: **http://www.peterliljedahl.com/wp-content/uploads/Visibly-Random-Groups.pdf

**Key Points: **Results indicate that the use of visibly random grouping strategies, along with ubiquitous (predominant learning activity) group work, can lead to:

(1) students becoming agreeable to work in any group they are placed in,

(2) the elimination of social barriers within the classroom,

(3) an increase in the mobility of knowledge between students,

(4) a decrease in reliance on the teacher for answers,

(5) an increase in the reliance on co-constructed intra- and inter-group answers, and

(6) an increase in both enthusiasm for mathematics class and engagement in mathematics tasks.

**Questions:**

1.) In Andorra, men greet men with a kiss on one cheek. Men greet women with a kiss on the hand. Women greet women with two kisses. At a party, 10 couples, 3 single men and 5 single women gather and greet each other. How many kisses were there?

2.) (Math language): Find all the natural (counting) numbers n that have the property: n + sum of the digits of n = 2014. (Kid language) 1813 has a digit sum of 1+8+1+3 = 13. Adding 1813 + 13 we get 1826. What counting numbers add up with their digit sums to equal 2014?

3.) How many natural numbers can be added to 2014 to make a four-digit palindromic number? [For different grade levels this question could be “what numbers add to 12 to make a 2-digit palindrome like 33?” Or “how many numbers can be added to 110 to make a 3-digit palindrome, a number the same frontwards and backwards?”

4.) Fibonacci-like number patterns are created when the two prior numbers add to be the current number. For example, the sixth number in a sequence is the sum of the fourth and fifth numbers. Your question: What starting numbers have 100 as the fifth number in a Fibonacci-like pattern?

5.) Take any three-digit number and repeat the three digits to make a six-digit number. Why is it divisible by 7, 11 and 13?

6.) **Seven** can be written as the sum of two consecutive integers 3 and 4. 7 = 3+4

** Six** can be written as the sum of three consecutive integers. 6 = 1+2+3

** Fifteen** can be written as the sum of consecutive integers 3 different ways!

15 = 7+8 or 15 = 4+5+6 or 15 = 1+2+3+4+5

Generate some questions that you might be interested in knowing the answers to:

Now explore the problems you have raised to find some answers.

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Workshop II: **Number Sense Problem Solving with Pattern Blocks (3-7)**

This session is focused on building number sense with pattern blocks in a problem solving setting with a group gathering around a single desk. This is an excellent format for utilizing manipulatives as thinker toys and for having students communicate mathematical reasoning. The growth climate is fed with feedback and provides amazing structure for building upon in other units.

(Fred Harwood, Retired Richmond Teacher, SFU, Teacher Consultant)

The last workshop was on using temporary vertical surfaces to solve problems. Another powerful problem solving format is visibly random student groups standing around central small tables. Here they can work with manipulatives in solving the problem. The formats are similar with sharing of resources, ideas and turn-taking with pens and manipulatives to ensure engagement. The ‘bar table’ setting is conducive to conversation, reasoning and question posing.

Problem 1: Background: A yellow hexagon is worth 6 red trapezoids, a red trapezoid is worth 6 squares, a square is worth 6 rhombi, and a rhombus is worth 6 triangles. If needed, we’ll make a light brown ‘diamond kite’ worth 6 hexagons.

Each group has a random amount of the main pieces on their table (no diamond kites).

Question 1: Which group do you think has the most values if the pieces were money? Which group has the least value? This is not a calculation but a gut-check intuition. It creates a student-generated need to verify their predictions. [Participants pointed to the table they thought contained the most. Fingers pointed four different ways.] “Hmmm, still hard to see? Let’s put our data into a table under the headings: Hexagons (H), Trapezoids (T), Squares (S), Rhombi (R), Triangles (t)

Question 2: We’ll create a chart. Each team will count and record their group’s values in our chart. Afterwards we’ll re-ask question 1. [I expected people to be more sure of the answer but fingers still pointed different ways.] “How will we determine it? Okay, we’ll make reduced into the biggest units by trading up. Go to the bank of pattern blocks and do any changing into bigger units. Remember 6 triangles equals one Rhombi and 6 rhombi equals one Square etc.”

Question 3: How might we know for sure how to compare the groups? We’ll then rewrite our chart with the new information. [They traded their smaller pieces for bigger pieces with the bank.] [Groups started making comparisons of this problem with place value and our base 10 system. “This is a Base 6 system!”] They converted their original levels to show the reduced levels. Still different people thought different groups had the most so we then did paper calculations to convert their set of tiles into the smallest units of triangles. It is like taking the information for problem 3 below and making them into total number of cents. Most of the groups had answers around 12 000 triangles but one was at 69 000. The class expected a mistake had been made (they had did one extra conversion so their correct conversion was 11 500). Now everyone could clearly see group 1, the group with the most hexagons to begin with, did have the most. 1 Hexagon (H) was more than 5 T, 5S, 5R and 5t combined just as 1 x 10^4 or 10 000 is more than 9×10^3+9×10^4+9×10^2+9×10+9. Participants remarked on how clear this process of problem solving made the concept of place value well anchored in the concrete.

We then used pattern blocks to represent fractions noting that we define the value of “ONE” and not the pattern blocks determining “ONE”. 1 yellow hexagon could be “1” so trapezoids were ½, rhombi were 1/3 and triangles were 1/6. However, 3 yellow hexagons could be ‘1’ so red trapezoids are now 1/6, a blue rhombus would be 1/9 and the triangles were each 1/18. Etc. This makes the concepts of fractions very powerful with pattern blocks as we can use them to determine and demonstrate any fraction. Students SEE why there are 5/6 and that ½ is not a number but a concept since ½ of a yellow hexagon is a different size than ½ of 4 hexagons but both are called ½. Participants then solved the two magic squares included at the end with the blocks and much trading and talking and reasoning happened in the process.

The foundations of fractions were well established through these problems.

When students add a) 3 ¾ + 2 ½ + 1 ¾ , they see that they can regroup the whole numbers and then combine the fraction parts while trading them to common fractions (shapes) to add the bits. We finished the workshop with a straight fraction addition problem of c) 5/24 + 7/12 + 1/3

PROBLEM 2: Magic Squares have the same sums for every row, column and the two diagonals. Use pattern blocks to complete the following magic squares:

PROBLEM 3: Making Sense: How much money does each person have?

Person 1. 6 loonies, 17 quarters, 16 dimes, 30 nickels and 52 pennies

Person 2. 19 loonies, 22 quarters, 22 dimes, 22 nickels, 22 pennies

Person 3. 40 loonies, 36 quarters, 20 dimes, 15 nickels, 75 pennies

PROBLEM 4: Who in Problem 3 has more money and by how much?

PROBLEM 5: Which is bigger?

- a) 1 triangle, 1 rhombus, 2 squares, 0 trapezoids and 4 hexagons

or b) 7 triangles, 12 rhombi, 8 squares, 14 trapezoids and 1 hexagon

PROBLEM 6: Show me adding

- a) 3 ¾ + 2 ½ + 1 ¾ b) 1 5/6 + 5/12 + 1/3 c) 5/24 + 7/12 + 1/3

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**For problem 2: MAGIC SQUARE PROBLEMS TO BE DONE WITH PATTERN BLOCKS:**

Picture a 3×3 magic square on a full 8.5×11 sheet with the bottom row containing: 2/3 1 1/2 1/3 and the middle of the square being 5/6. Students then use pattern blocks to figure out the missing 5 squares so that every row, every column and the two diagonals all add to the same total. (2 1/2 for this first one)

The back side of the sheet contains another 3×3 magic square with the middle square being 1 1/4 and the bottom left being 1 and the bottom right being 1/2. Students need to find the six missing squares’ values. Some students will need to notice that the central square on their previous square was one third of the total for all the rows, columns and diagonals so they need to triple 1 1/4 to get a magic square total of 3 3/4 for every row, etc.