We must address the imbalance in our expectations that place a priority on written work at the expense of the oral.. Students need to build, explain, represent and then compare all three in order to really “learn”. Say it, record or match it, read the record.

Here’s my card

Here’s my Approach: I thought about how to make the first shape I decided to Focus

I thought I saw a 5 and a 3 I had a flash of intuition

Soon as I got the pieces out I felt the uh-oh! COMPARE/EVALUATE Soon as I started to Build I had to Evaluate

It’s really this: 5 + 2 Read 5 plus 2

I needed just the 4 inside the 5. It is five plus 2

I transformed COMPARE/EVALUATE

What do they all have in common? How are they different?

Mathematically: SHAPE Same shape when traced: turns do not change properties. I can name the shape by number of sides and measure of angles. It has how many sides? Therefore it is a?

Sides are not all equal length. 1 2 Irregular octagon Sides are not all equal length. The perimeter is based on the side length of a one unit. So this perimeter is 12 units. If I change the shape I make with the 5 and 2 to show 7 I will get a different perimeter.

Mathematically: SPACE I can find the 5 + 2 different ways by turning or by chunking in different places

Mathematically: SPACE They all make the same space that covers 7 square units BUT there are other possible shapes for 7 square units.... Grade 5 could go there. Can you see any in your head? The perimeter is not 7 units. What is it? Which 7 unit shape has the greatest perimeter? The least?

Mathematically: Number I can do all kinds of matches: 7 2 + 5 = 7 seven 5 + 2 = 7 5 + 2 7 = 5 + 2 7 = 2 + 5 2 + 5 I can study combinations I am practicing my facts: Match See and Say the Symbolic

I first thought 5 + 2 but then I saw 4 + 3 and then I saw 3 + 3 + 1. Can you see them? Build them.... Represent them.... COMPARE/EVALUATE I can study equalities

I can study equalities. These all equal each other. 5 + 2 4 + 3 3 + 3 + 1 6 + 1 2 + 3 + 2 (still 4 + 3) If the change in colours or materials bother you you need to work on your ability to abstract, visualize and think symbolically. The colour is not the relationship. I can study equalities. These all equal each other. I am practicing my facts: Match See and Say the Symbolic

Mathematically: Number I can “play” with equalities: 2 + 5 = 1 + 6 2 + 5 = 3 + 4 I am practicing my facts: Match See and Say the Symbolic

I can play with : Commutative Property 2 + 5 = 1 + 6 2 + 5 = 7 2 + 5 = 1 + 6 2 + 5 = 7 5 + 2 = 7 1 + 6 = 2 + 5 7 = 5 + 2 6 + 1 = 2 + 5 7 = 2 + 5 6 + 1 = 5 + 2 I am practicing my facts: See, say, build, match

I can predict, generalize + practice Do I have all the ways ? Have I discovered anything? Can I generalize? 2 + 5 = 7 5 + 2 = 7 7 = 5 + 2 7 = 2 + 5 Yes I do if I stick with 7,2,5 and addition. But what if I include subtraction which is the inverse of addition.

Do I have all the ways for these numbers? Have I discovered anything? Can I generalize? 2 + 5 = 1 + 6 No I do not think I do. Can you find more? Can you organize to know you have them all? 1 + 6 = 2 + 5 6 + 1 = 2 + 5 6 + 1 = 5 + 2 I am practicing my facts: See, say, build, match

You have to decide your goal as the teacher. In grade one and two I really care about knowing how the number combinations to 10 are related. A focus on relationships lets all students participate. As we repeat repeat repeat investigations, recall builds for all. Build Explain Represent Compare: keep students engaged in learning.

Can you see all the “facts” in a 7 puzzle? In Grade 3, I might care more about making equalities... Understanding the embeddedness and actions that maintain the equality no matter how I move things. Can you see all the “facts” in a 7 puzzle? Can you explain how they are related by looking in just one puzzle? Notice If you are going to stay with the actual question which is important if you want to learn to be logical, then 10 – 3 has no place here. This is not about showing off, it is about answering the question. Evaluating your answer. Being flexible, self reflective and logical in your evaluation. I DID NOT SAY 10-3 does not equal 7. But that has nothing to do with the problem at hand.

Mental Transformations and Gesture both impact learning Mental Transformations and Gesture both impact learning. They help link the physical world and body to the mental representations and symbols. So I can use ChunkitZ to build my thinking skills as I practice facts.

Here’s my card Do these both represent the same quantity or area? If I have already identified one I might ask if the other is also 7 or I might ask if the other one can be transformed to be congruent

Ooo! I see it. Yes, they are congruent I can move the parts in my head without touching them, can you?

Here’s my series of moves 1 3 2

Here’s my series of moves

I can represent like this I am mapping. slide the 2 Then rotate Rotate the whole shape

I could map other versions of 7 and see if they can be made with the Chunks, so keep moving the 2 Chunk into different positions and recording what I did. It will still be 2 + 5 = 7