1. Numeration Quantity/Magnitude Base Ten Equality Form of a Number ProportionalReasoning Algebraic and Geometric Thinking The Components of Number Sense © 2007 Cain/Doggett/Faulkner/Hale/NCDPI Language

2. Students don’t see the connection between X, ÷ and geometry when we inundate them with formulas If they understand these relationships they will become fluent in area and perimeter problems

3. Note the connection between the linear measurements and the area measurements.

4. What we say now: Okay, what formula are you going to use? A = l*w But it’s a square right? So what formula do we use for a square? A = s*s Good! Now plug in what you know and…

5. Our new habit: Remember, you all already understand the relationships between one dimension and two dimensions. Let’s look at these again: We know from our work With multiplication that if we know these two lengths we know how many little one x one squares are in here, right? These two factors give us this product. 14 6

6. Arrays: Build the fundamental understanding of these relationships and watch them connect and grow! 112 little one by one squares! 7 linea r units long 16 linear units long

7. Link Mult/Div to Geometry! This is deepening knowledge But look, this is Geometry! Our two factors tell us how long the linear sides are, right? And what about our product? That is our AREA. We measure AREA in 1x1 squares. If we say 1 square unit, that’s exactly what we mean!

8. What we do now: Area is the thing on the inside; rug; carpet, etc. Perimeter is the measurement around; fence;string, etc. This is good but it is not precise enough to have clear impact on student thinking about measurement.

9. Thinking in dimensions: the measurement/geometry link Teachers should always have a piece of string one unit long, and a square that is 1 x 1 unit long available! Hold these up whenever talking about area and perimeter or area versus perimeter Do we mean 3 of these? Or 3 of these? How do you know?

10. Fluency - Practice lots of these! So if I know this and this What else can I figure out? Great, now what if I knew this And this? I have the Area and one side length. Can I figure out the other side length? Yes! Multiplication and Division help us go back and forth depending on what we need! 7 3 21 7

11. Symbolic Level - Make connection So what we have been doing the past week is figuring out area and perimeter because we understand how they are related. Mathematicians write this out with symbols, but they just mean what we already know how to do! Area = base * height

12. Symbolic Level - BIG IDEA Area = base * height You will see lots of different ways to write this. You need to always go back to what you know: it’s just what we do with our multiplication arrays! A = l*w A = s*s A = b*h A = 1/2 b*h Don’t be fooled! It is all the same big idea! Practice: pairing symbolic with representational (arrays).

13. Arrays Distributive property Base-ten 6*7 7 10 6 10*7 16 x 7

14. Arrays and distributive property (X + 6) 7 6*7 7 (X + 6) X*7

15. Several Warm-ups where students move from problems in the form of 7(x + 6) to 7x + 42. Simplifying. And also from the form of 7x + 42 back to the area model and 7 (x +6). Factoring.

16. Arrays and distributive property (X + 6) 7 6*7 7 (X + 6) X*7 We know that we have 7x + 42. Still don’t know How much the 7x is because we are only given One dimension (the 7) and not the other (x).

17. Finally, Problems that have students solve for x given what they know. For instance, give them the area of a rectangle, one dimension and part of the other dimension and solve for x. 7(x + 6) = 77. Draw out the rectangle and fill in everything you know. What does the x have to be in order for the area to be 77? Figure this through guess and check. Students will build on this algebraically by the end of the course. Finally, more Warm-ups that have students figure this out using the algebra they learn from Hands-on- Equations AND modeling it using geometry (these rectangles) should be put into Units 5 and 6.

18. Arrays and distributive property (X + 6) 7 6*7 7 (X + 6) X*7 But what if we knew the whole area was 77? Could we Work backwards to figure out what x is?