The Path to Math

A few reminders before the examples from the workshop:

• Students learn math concepts through the concrete-representational-abstract instructional approach. This means that you must guide them in creating their own mental models from concrete materials before you simply present the representational and abstract steps (such as an addition expression or an equation to be solved). Concrete materials include Algebra tiles, counting bears, etc.
   
• When we spiral instruction, students are “scaffolded” to higher and higher levels by revisiting a concept on multiple occasions, each time with an increased understanding of the task. This is a great way to reach students who struggle in math!
   
• Some students use alternate forms of communication; we must prepare the right words in advance so those students can jump into the conversation and build mathematical understanding!
   

• Number and Number Sense: counting; classifying; number relationships (more than/less than; simple fractions; whole numbers; quantity; decimals
• Measurement: weight; volume; money; fractions
• Patterns, functions & algebra: compare, sort and classify
• Probability and Statistics: 
• Computation & estimation: addition & subtraction; use appropriate computation tools; multiplication & division intro. Use equal grouping etc.

1. Sort grocery list items into categories based on locations in store---Meat, canned food, produce, dairy, bakery
2. Enter store and locate cart
3. Collect items from produce department: 1lb. red grapes; 4 red apples; 3 lbs bananas; 3 green peppers
4. From bakery: 1 loaf of bread for $2 or less, 4 bagels/$1
5. From meat department: 1.5 lbs chicken; 2 lbs. steak for less than $7
6. Canned food: 1 pound rice; 2 15oz cans of black beans; 4oz green chilies; 1 6pk applesauce (best price)
7. Dairy: 6 yogurts; 1 lb cream cheese; 1 quart half/half; 2 cups shredded cheese
8. Purchase items: Using cash pay for items---

• Chart with measurements for mass, weight and volume
• Calculator
• Strategy for “counting up”
• List adapted with pictures; include map of store with location of items

• Do you have any…
• Can you help me find…
• I need…
• Is this the same as...
• That’s not enough
• Put it here
• Turn it around
• I like that one
• That's not the same
• I think...
• I know...

1. Count out [x] pieces.
2. What does the rectangle look like?
3. What do the sides look like?
4. Are there other ways to arrange the pieces?
5. How does your rectangle compare to other students’ rectangles?
6. Break this rectangle into groups or sets of the same size.
7. How did you break the rectangle into groups?
8. How many pieces were in each group?
9. How many groups did you create?
10. Were there any groups that did not work?
11. Why didn’t they work?
12. What chart do we use that looks like a rectangle?
13. How could we describe these rectangles in terms of multiplication?
14. Why can we describe these rectangles in terms of multiplication?
15. Try making a different rectangle with your pieces.
16. What does the new rectangle look like?
17. What do the new rectangle’s sides look like?
18. How could we describe this new rectangle in terms of multiplication?
19. Why can we describe this new rectangle in terms of multiplication?

 Demonstration Lesson: Grade 4

1. We have a bathroom that has a really yucky floor. [Hold up a piece of old flooring or show a photo.]
2. What should we do?
3. How should we do it?
4. [Optional: Take a quick vote on floor color and make graph with summary using fractions and decimals]
5. What do we need to know about the bathroom before we begin?
6. Pick up a piece and hold it in the air so we can all see it.
7. What can you tell me about the sides of this piece?
8. What is the name of this shape?
9. Is there another name for this shape?
10. This piece represents one square foot. It can be written like this: 1 square foot or 1 sq. ft. or 1 ft²
11. [Optional: Take a quick vote on which notation we want to use today and make graph with summary using fractions and decimals]
12. Hold up pieces to represent two square feet!
13. Hold up pieces to represent one square foot!
14. Let’s say that we measured our bathroom and determined that it is x feet long and y feet wide. Do we think that sounds like a big bathroom or small bathroom? [Optional: Take a quick vote on perception of bathroom size and make graph with summary using fractions and decimals]
15. Use your pieces and grid to make a plan of what that bathroom floor would look like if you looked at it from the air.
16. What shape did you make?
17. What do the sides look like?
18. How does your rectangle compare to other students’ rectangles?
19. Take one piece and hold it in the air where everyone can see it. What does each of the pieces represent?
20. Let’s say that each of our [selected color] tiles are about 1 square foot in size and these tiles are sold in packs of 4. How many packs would we need?
21. How did you determine how many packs we would need?
22. Did anyone find another strategy that worked?
23. We also need to know what won’t work. Were there any strategies that would not work?
24. Why won’t they work?
25. When we determine area, we work with what shapes?
26. What chart do we use that looks like a rectangle?
27. How could we describe these rectangles in terms of multiplication?
28. Why can we describe these rectangles in terms of multiplication?

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