reasoning Math processes

Teacher Wait Time

the silence that often comes after a question has been asked but before students have finished considering their answer and/or find the courage to speak up

Metacognition

the awareness and understanding of one's own thought processes, enabling students to monitor, control, and reflect on their learning strategies and problem-solving techniques to enhance their cognitive abilities

Decomposing

Breaking up numbers by place value

Example.

125→100+20+5

Deductive Reasoning

using two or more known premises to draw a conclusion

Example.

All cats say meow. (premise #1)

Jackie is a cat. (premise #2)

Therefore we can deduce that Jackie says meow. (conclusion)

Partitioning

A mental math strategy that involves breaking up numbers by place value before adding, subtracting, multiplying or dividing

Think-Pair-Share

learning activity in which the teacher provides a prompt, the students consider it individually (Think), then pair up and brainstorm responses or solutions (Pair), and then share their results with the class (Share)

Benchmarking

Using benchmark numbers to estimate a mathematical expression

Inductive Reasoning

generalizing knowledge from one area to another

Example.

If a random sample of a population shows a correlation in improved health with a new drug, it can be induced that the drugs will be helpful for others in the population.

Informal Reasoning

used to answer questions and solve problems that are complex and open-ended (without a definitive solution)

Example.

Compare pros and cons

Subtraction Property of Equality

If the quantities on each side of an equal sign have the same amount subtracted from them, the resulting statement will still be equal.

Example.

If a=b, then a–c=b–c

Conjecture

Term definition.

an inference created without proof

Example.

Conjecture: all quadrilaterals are rhombuses

Counterexample: rectangle

Compensation

A mental math strategy in which one number is changed to a value that simplifies the computation, such as a multiple of 10

Addition Property of Equality

If the quantities on each side of an equal sign have the same amount added to them, the resulting statement will still be equal.

Example.

If a=b, then a+c=b+c

Division Property of Equality

If the quantities on each side of an equal sign are both divided by the same amount, the resulting statement will still be equal.

Example.

If a=b and c≠0, then a÷c=b÷c

Algorithms

systematic methods that always produce a correct solution to a problem

Example.

Area = Base x Height

Composing

Recombining numbers broken up by place value

Example.

100+20+5→125

Estimating

rounding numbers or approximating them to quickly perform math operations

Example.

23+39≈20+40 so the answer is about 60

Multiplication Property of Equality

If the quantities on each side of an equal sign are both multiplied by the same amount, the resulting statement will still be equal.

Example.

If a=b, then ac=bc

Formal Reasoning

the use of logic and algorithms to reach conclusions

Example.

answer questions and solve problems that have a single solution

Benchmark Numbers

Numbers that are easily added, subtracted, multiplied, or divided.

Example.

10, 25, 100