Find all different size squares on your geoboard

Number Side length Area

Irrationality

· Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and so it was irrational.·· However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!

Division rule - explain

· “When you divide two fractions you flip and multiply. “

From your reading

Rational numbers

· Can I write every fraction as a decimal?

· Can I write 0.238238238238238238…. as a fraction?

· Can I write 0.1010010001000010000010000001……… as a fraction?

Why?

· What are negative numbers?

· Why do we need them?

· What is your relationship with negative numbers? · One thing you like about negative numbers.· One thing that you wonder about negative numbers.

Accounting as model

· If I have money in my bank account then the amount is written in black numerals: $34571· If I am in debt, then the amount I owe is written in red numerals: $3221· My accountant has recorded the following: · $721· $250· $600How much money do I have?

Using chips

· Again use different colors

Making a zero

· Credit of $1 and debt of $1 dollar means you have nothing:

Multiple representations of a number

Opposites

· Notice that these two sets have something in common

· Opposite of a is the integer number that is represented by the same number of chips, but of different color.

4-4negative four

Integer number line

The opposite of integer a is the integer that is equally distant from 0 as a and is different from a.

Motivation for addition

Using measurement model

Walking the number line

· Moving forwards models the addition operation.· Moving backwards models the subtraction operation.· Facing the positive direction models a positive integer. · Facing the negative direction models a negative integer.

Couple of questions

· What is

· Is always negative?

· What is

Subtraction via patterns

Subtraction via take-away

Few more

Adding the opposite

Missing addend approach

Multiplication

· We can again think of multiplication as repeated addition: · 3 × 5 is three groups of 5, which is 15 (black chips)· 3 × (-5) is three groups of -5 which is -15 (red chips)· (-3) × 5 makes less sense, but it should be the same as 5 × (-3) which we know is -15· (-3) × (-5)=?

Patterns

Few word problems

· After spending $30 on a dress, Mary had of her money left. How much money did she have at first?

· of a group of children are boys. If there are 18 more boys than girls, how many children are there altogether?

Few word problems

· A tank is full of water. If 40 gal more are needed to fill the tank completely, find the capacity of the tank.

· Mr. Ramirez had $600. He gave of it to his wife and spent of the remainder. How much money did he spend?

· John and Matthew had equal amounts of money. After John spent $25 and Matthew spent $18, the ratio of John’s money to Matthew’s was 2:3. How much money did each boy have at first?

· The ratio of John’s money to Mili’s was 5:2 at first. After John spent half of his money, he has $20 more than Mili. How much money did they both have at first?

· In a test, there were 50 problems. Tim answered 80% correctly, and Carlos 90%. · How many more questions did Carlos answer correctly than Tim? · How many percent more questions did Carlos answer correctly than Tim?

· John spent 20% of his money on food. He spent 2/5 of the remainder on a toy. The toy cost $12. · What percentage of his money did he spend on the toy?· How much money did he have at first?

Equation or not?

x2 - 3x + 4

3x + 6 = 8

3 = 1 + 2

3 = 4

What is happening here?

x2-3x−4 = 0

x2 − 3x = 4

x(x − 3) = 4

x=2 or x − 3 = 2

x=2 or x = 5