ð

ð

DE K-12 Mathematics Partnership Project

LEARNING SEQUENCES

A Context for Mathematical Arguments:

The Properties & the Behavior of Operations

A Context for Mathematical Arguments:

The Properties and the Behavior of Operations

1. Noticing Regularities: Show a set of problems that embody the relationship. (SMP8- Look for and Express Regularities)

2. Articulating Conjectures: Have students state explicitly what their claim includes. (SMP6- Attend to precision)

3. Investigating Through Representations: Use objects, drawings, diagrams, number lines, arrays and story contexts. (SMP6- Look for and make use of Structure)

4. Constructing Arguments Based on Representations: Students start with specific examples that embody the relationship and then move to generalizations for all whole numbers, and finally all numbers. (SMP3 Construct Viable Arguments)

5. Comparing and Contrasting Operations: Students need to try a conjecture with more than one operation to finalize the conjecture. Working between operations leaves a lasting impression about operations, not number alone.

The Teacher’s Role:

• Help students develop a curiosity as well as the ability to see the structure. Improve the habit of looking for a pattern and regularity as well as making sense of that pattern and regularity. (Expand student thinking beyond finding a pattern to seeing how a pattern works)

• Lead students deeper into familiar content, not assuming that operations are “understood” just because the students can use them to compute. (Help students focus on the meaning of the operations)

• Don’t take for granted that ideas are settled because they have been stated. (Use representations to make ideas visible)

Topics for Instructional Sequences 1: Changing the order of the addends in an addition expression does not change the sum.

a + b = b + a Given two addends and their sum, the difference between the sum and one of the addends is the other addend.

a + b = c c – b = a c – a = b

2: If 1 (or some amount) is added to an addend, the sum increases by 1 (or that amount).

If a + b = c, then (a +1) + b = c + 1

If a + b = c, then (a + m) + b = c + m Given a subtraction expression, if 1 is added to the starting amount, the difference increases by 1.

a – b = c, then (a + 1) – b = (c + 1)

Given a subtraction expression, if 1 is added to the amount taken away, the difference decreases by 1.

a – b = c, then a – (b + 1) = (c - 1)

Topics for Instructional Sequences 3: If you add an amount to one addend and subtract the same amount from another addend, the sum remains the same.

(a + m) + (b – m) = a + b In a subtraction expression, if you increase both numbers by the same amount, the difference remains the same.

(a + m) - (b + m) = a - b

4: Changing the order of the factors in a multiplication expression does not change the product.

a * b = b * a Given two factors and a product, the quotient of the product and one factor (not equal to 0) is the other factor.

If a * b = c, then c/b = a and c/a = b

5: If you add some amount to an addend, the sum increases by that amount.

If a + b = c, then (a + m) + b = c + m

5. If you add 1 to a factor, the product increases by the other factor.

If a * b = c, then (a + 1) * b = c + b If you add some amount to a factor, the product increases by the other factor multiplied by that amount. These claims are investigated with whole numbers.

If a * b = c, then (a + m) * b = c + m* b

6: Students investigate the claims from Sequence 5 with fractions.

7: In a multiplication expression, if you multiply one factor by an amount (not equal to 0) and divide the other factor by the same amount, the product remains the same.

If a * b = c, then (a * m) * (b/m) = c In a division expression, if you multiply (or divide) the dividend and divisor by the same amount (not equal to 0), the quotient remains the same.

If a/b = c, then (a * m)/ (b * m)= c

8: When a number is multiplied by 10, every digit moves one place to the left. When a number is divided by 10, every digit moves one place to the right. This claim is investigated first with whole numbers, then with decimals.