Visualizing as a Tool to Extend Numeracy by Fred Harwood @HarMathWhistler Pacific Northwest Math Conference: Oct 24, 2015 fharwood@sfu.cahttps://harmath.wordpress.com/

37ish years ago, one-third of each of my classes could only see 3 sides to this representation of a cube. Elementary schools began focusing on visualization and as a result, today no student seems to only see the visible three sides. They can visualize the back, left and bottom sides. Naming these sides: back and front, left and right, top and bottom helps.

Draw a 3-D rectangular prism on a piece of paper where the height is smaller than the width and the depth is more than the width. Compare it with your neighbour’s box. Does it show front, top and right sides? This perspective is the usual.

Now draw it from a different perspective (the cube above is drawn from an above right perspective). Change to a bottom left perspective so the front, bottom and left sides are in view. Discuss how difficult you found this activity with your table group. Practice is needed to switch perspective views.

Draw a triangle. About 70% of you will have the base parallel to the bottom of the page. Many will have it as isosceles or equilateral. Many others will have a right angle in their drawing. Hopefully none will have two or more right angles. The minority position is scalene.

Can you draw an upside down triangle? Most can thus exposing a weakness in our understanding.

Visualizing allows us to address some key concepts and vocabulary. “Mathematics is the art of giving the same name to different things.” (Poincaire) Naming gives us access to student thinking.

Stand up and then turn left. I expect most people will be able to do this even though I did not give a direction on how far left to turn. Turn left 15° or turn left 120° gives the real information. This form of coding can be very powerful in training students’ visualizing skills. In elementary, a checkerboard carpet with symbols and letters could lead to students coding the instructions to get from L to W. Or, “starting on W facing up, go 3 up, turn left go forward two, turn right and go forward 1. Where are we?” In intermediate or secondary, I would recommend using LOGO language with Turtle Academy or Turtle Playground to do the same. Have students direct one student to navigate from the back of the room to a position in the front is another visualizing skill where precision of language emerges if they need to create the entire code before motion begins. You will also notice body language emerging with hand gestures driving the steps of their visualization.

“I was walking down a lane one day and, when passing by a backyard with a three foot high fence, I heard a bark and growl. Looking to my right, I saw a large German Shepherd racing towards me. My anxiety spiked until I noticed it was chained to the clothesline. I relaxed until I saw the chain sliding along the clothesline and the dog was clearing the fence. I sprinted left to get away but the dog’s front paws struck my chest driving me onto my back into the blackberry bushes. The snarling dog lunged at my throat . . . and the chain finally caught one foot from my face! While crawling backwards and sidling away from the slathering beast, I thought unkind

thoughts about the lack of numeracy of the owners. But perhaps it was planned so the dog could pee outside of their yard and reducing the messes they needed to clean up. They should have cared though if their pet dog, Toodles, could decide to snack on the neighbour’s toddler.”

Have you heard me tell this story before? Visualizing with a story makes for memorable learning. It can give a powerful context for problem solving so that students will care.

List what mathematics the following diagrams ‘show’:

I have had table groups talking non-stop for minutes with the multitude of mathematical ideas that flow from these. To create a situation for everyone with a chance to contribute, use a numbered heads approach so the ideas flow around the circle. A student may pass but must add an idea after the pass. Big ideas, operations, measurement, problem solving, fractions, and much more can flow. The richness leads to lots of eye-opening exchanges and opportunities for building vocabulary and conceptual understanding all in one visual. “A picture is worth a hundred times a thousand words!” (Ron Lancaster?) You can ask if anyone said something in your group that surprised you. For example, in the second picture, someone might have said “3” which indicates a different perspective most would not see.

Now visualize these on a square table before three students on three different sides. Does it change their perspective of the mathematics? If you have not explored 10-frames before, do some research and look at activities that can be done with them. Think about how they could have changed your own number sense development!

What can this diagram represent?

Does visualizing numbers affect how students strategize mathematical operations?

The big idea of decomposing and recomposing flows from seeing the ability to slide 3 tiles from the four to complete the right hand ten making it 10 + 10 + 1 = 21. It moves us from numbers being just sticks on paper. Remember the weaknesses of 28+35 8+5 = 13 so we write down the 3 and carry the ‘one’. Two + three is five plus the one is 6 so the answer is 63. The weak conceptual understanding of this would be better as carry the ten, twenty plus thirty plus the ten is sixty so the answer is 63.

English numbering defeats knowing the structure of 12 as 10 + 2 because it is called “twelve”. Visualizations like this can reinforce its true nature.

Again the richness of the image leads to incredible numerical insights. When students are working in pairs or small groups on this, you will hear mathematical talk exchanged. How do you see 5 plus 5? Can you show me your 4+4+4–2? Can you see the powerful big idea of inclusion and exclusion emerge in this insight?

What mathematical expressions could be used to represent this set of dots?

Find the area of the side of this stairway as many ways as you can: Assume all the risers and treads are 10 inches long.

What happens when ‘students’ are given 20 blocks to do this stairway problem with? It frees them to manipulate the blocks, which frees their minds to new avenues.

Is it similar to when students can play with manipulatives when doing addition or subtraction problems? An awareness of ‘number’ being a collection of things allows for them to be manipulated. 12 + 19 can easily be 10 + 1 + 1 + 19 or 10 + 20 + 1 = 31.

The associative and commutative principles become naturalized in their intuitive understandings of moving the blocks around.

In the measurement problem, the moving of blocks provide key thoughts around the big ideas of breaking the shape apart in simpler shapes, moving some blocks to create an easier shape or adding blocks to complete an easier shape then subtracting the extras.

https://nrich.maths.org/6447 is an nRich article on ‘visualizing being more than what meets the eye.’ Here is one of their problems. Imagine we have an unlimited supply of interlocking blocks of the same size but in three different colours. We begin with one yellow cube and completely surround it by red cubes.

1. How many red cubes are needed for this surrounding?2. How many red cubes meet the yellow cube face to face?3. How many red cubes share an edge with the yellow cube?4. How many red cubes share only a single point with the yellow cube?5. If we then surround the red tiles completely in blue tiles, how many blue cubes are needed?6. What is the pattern of successive layers completely covering the previous one?

If this problem is worked on with just paper, you can again see students visualizing by their fingers visually filling in the gaps on the middle diagram. Recording their work shows their thinking and organizing. The structures of data can give us insight, allow us to ask probing/clarifying questions and provide avenues of ‘nudging’ should a group need it. Often just seeing another group’s data/organization is enough of a nudge to get them reoriented to pursue the problem deeper.

Show your reasoning on paper

Blue tiles

Yellow tile

Red tiles B

R

R

Y

Y

Draw the next two figures in this growing pattern. Determine how many shaded squares would be in the eighth figure in the pattern (several different ways). How many in the nth figure? Figure 1 Fig. 2 Fig. 3

from Fawn Nguyen’s Visual Patterning: http://www.visualpatterns.org/101-120.html

If you haven’t been exposed to Visual Patterning, read the introductions and recommendations on her website for how to utilize these visualizations in your classrooms to build student problem solving, reasoning and communication skills. http://fawnnguyen.com/patterns-poster-algebra-1/

Ron Lancaster said “This un-reduced equation speaks to me. The simplified equation, not so much. It is easier to use but doesn’t speak of the how.” How did students arrive at each of these equations for the number of squares in the nth figure? How do these equations speak to the reader?

1) S = nxn + n + n–1

2) S = (n+1)2 – 2

3) S = (n–1)2 +2n + 2(n–1)

Show their thinking on the grid and colour to identify the story.

I derived some of these with my students by creating lists of multiples of that divisor and then looking for patterns. This is my consolidation of these that had similar patterns. What do you think of them? I was very proud of the crystallization simplifying the multiplicity of tests. But they did not answer the question of ‘why’. They were the ‘hows’. Visualizing created the whys.

Divisibility Tests

Test for divisibility by 10 (and 2 & 5 since they are factors of 10)Last digit is divisible by 10 ( or 2 or 5)

Test for any power of 10 (or power of 2 or 5)Check the same number of last digits as the power for divisibility by the numberEg. Test for 8: check last 3 digits for ÷ by 8 because 8 = 23.

Test for 9 note: it is almost our base value of 10 (and 3 since it is a factor of 9) Add up the digits to see if divisible by 9 (or by 3)

Test for 11 note: it is almost our base value of 10 (one of many tests but this one is similar to 9’s)

Alternate the digits + - + - + - + etc until you are out of digits and then add up the digits to see if ÷ by 11. Remember that 0 can be ÷ by 11.

Test for 7 (a student discovery that works for 3 digit and larger numbers)Starting on the right, break the number up into groups of 2 digits, the first digit is

normal (x1), the second group is doubled (x2), the third group is x4, 4th is x8 etc until you are out of groups. Add up the products of the groups to see if divisible by 7. If the total is still too large to tell, repeat the process until it is a two digit total. (Note: 91 & 98 are ÷ by 7)

Ex: 1572354: breaks up into 1 57 23 54: The total = 8x1 + 57x4 + 23x2 + 54x1 = 8+228+46+54 = 336 and then 336 breaks into 3 36 so 3x2+36x1 = 42 which is ÷ by 7 so 1572354 is also ÷ by 7. [Note: this is especially useful for 3 & 4 digit numbers.]

Test for composites: break into mutually exclusive factors (they must not have a factor in common other than 1)6 = 2 x 3 so the tests for 2 and 3 must work.

Eg. 2 must ÷ last digit & digits add up to ÷ by 312 = 3 x 4 so tests for 3 and 4 must work.

Eg. 4 must ÷ last two digit s & digits add up to ÷ by 315 = 3 x 5 so the tests for 5 and 3 must work.

Eg. 5 must ÷ last digit & digits add up to ÷ by 318 = 2 x 9 (not 3 x 6) etc.

Find other tests to tell that 11 is a factor of the following numbers. “Think to find multiples of 11 within these numbers.110 121 143 209 231 319 715

1001 1331 7843 8855 2442 3245 5016

13574 11033 55176 26950 42361 61105 62436

121363 686796 597652 6574172 1235421

Marc Garneau provided me with a beautiful perspective that was very empowering. In a 2014 NCTM conference presentation he showed how the divisibility test for nine could be visualized. Picture the number 3231. For decades I taught the test of adding the digits to see if the total is divisible by 3 or 9 because “this is what works.” Here 3+2+3+1 = 9 so both 3 and 9 would divide into 3213. I never thought to visualize it. Marc said to look at the place values of 1000’s, 100’s, 10’s and 1’s. To do this we use Base-10 materials to aid the visualization.

[Note dividing can also be visualized more easily. Using the thousand-block-less-one as an example: Peel of the top layer that has the one missing. There are now 100 stacks of 9 underneath. From the top layer, peel off the 9 from the 99. There are 10 sets of 9 plus the one set of 9 you pulled off. There are 100+10+1 = 111 sets of 9 in 999.]

If we pull off one from a hundred, we leave 99 or 11 x 9 so a multiple of 9. If we pull off one from a ten, we leave 9. For 3231 we can pull three ones off of the thousands, two ones off the hundreds, three ones of the tens and combine with the single one. All the remainders are divisible by 9 and the pullouts add to nine as well so 3231 must be divisible by 9.

Now consider the previous page’s divisibility tests. How are these to be visualized?

Start with the 2, 5 and 10. Every thousand cube is already divisible by 2, 5 and 10. Every hundred square is divisible by 2, 5 and 10. Every ten is divisible by 2, 5 and 10. So the only part that has not been tested are the leftover ones and this is why we only need to look at the last digit of any number to tell if it is divisible by 2, 5 or 10.

For 11, 1001 is divisible by 11. 99 is divisible by 11. 11 is divisible by 11. So for every 1000 cube, we need to add 1. For every 100, we will subtract 1. For every 10, we add 1. This explains the +, –, +, – . . . pattern of our divisibility test. There are many other tests for 11. Can you envision them on the base ten blocks? What other tests can you invent by beginning with the visual blocks and logic? What can your students discover?

3231= 3x1000+2x100+3x10+1

Taking the one green cube away from the 1000 cubes leaves 999 cubes. If students don’t see this is divisible by 9 then look at the fact there are 99 stacks of ten and one stack of nine. Taking one from each of the 99 tens give us 99 nines and the 99 ones can be made into 11 sets of 9. This gives us 111 sets of 9.