Presentation on theme: "Without using number values for x, explain how you know that the expression 2x-1 will always equal an odd number. Journal Writing Warm-Up Bell Quiz 1-5."— Presentation transcript:
Without using number values for x, explain how you know that the expression 2x-1 will always equal an odd number. Journal Writing Warm-Up Bell Quiz 1-5
Quantity – Modeling - Ratio - Rate - Unit Rate – Unit conversion – Proportion - Words to Know
Kayla and Torrey are both in Mrs. Henderson’s Language arts class. On the first day of school, Mrs. Henderson posed a reading challenge to the class. The challenge is to see how many books they can each read throughout the entire year. Kayla and Torrey have been participating in the challenge since day one. By October, eight weeks into school, Kayla had already finished 6 books. “I think I am reading at the same pace,” Torrey said. “After four weeks, I had already finished 3 books.” “Are you sure,” Kayla asked. “Well, I need to count, but I am pretty sure that we have both finished reading the same number of books.” Is Torrey correct? Have the girls each read the same number of books now that 8 weeks have passed?
In math and in real-life, we compare things all the time. We look at what we have and what someone else has or we look at the difference between values and we compare them. Comparing comes very naturally to us as people. Using ratios comes naturally too, because ratios are a way that we can compare numbers and values.
A ratio compares two numbers or quantities called terms. For example, suppose there are 3 green (G) apples and 4 red (R) apples in this basket. We can express the ratio of green apples to red apples in the basket as a fraction. We can also express this ratio in words, 3 to 4, or using a colon, 3:4. The ratio above compares one part of the apples in the basket to another part. For example, the ratio above compares the apples that are green to the apples that are red. What is a ratio?
A ratio may also express a part to a whole. For example, we can express the ratio of green apples to total apples in the basket as a fraction, too. That’s just like probability
Being able to rewrite ratios in different forms can help us solve real-world problems. Think for a minute about all of the times that we compare things in life. This will give you a minute to imagine all of the places where ratios are very useful. Let’s look at using ratios to solve some real-world problems.
Ratios are all around us. Throughout history, the ratio for length to width of rectangles of 1:1.618 has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. The Golden Ratio The math of beauty
Even the human face abounds with examples of the Golden Ratio Scientists have found that the faces people rate as the most beautiful are those that fit the golden ratio
Elena and Jake have a box with only two colors of marbles in it. There are 28 blue marbles and 16 gray marbles in the box. Elena says that the ratio of gray marbles to blue marbles is. Jake says that the ratio of gray marbles to blue marbles is. Who is correct?
What is a rate? A rate is a special kind of ratio. It compares two different types of units, such as dollars and pounds. Suppose you are buying turkey at a supermarket, and you pay $12 for 2 pounds of turkey. That is an example of a rate. The turkey you bought cost $6 per pound for the turkey. That is another example of a rate.
These words signal us that we are talking about a rate. We use rates all the time. We use them when shopping for example, per pound. We use them with gasoline, think per mile. We use them with speed, per gallon or with pricing, $4.00 per yard of material. Per, Each and Every
Determine if these two rates are equivalent: 40 miles in 2 hours, and 80 miles in 4 hours.
There is a special kind of rate which is called a unit rate. A unit rate has a denominator of 1, meaning that it is the measure for one of whatever you are talking about: 1 mile, 1 pound, 1 foot, etc. When we talk about unit rates, we are talking about singular measurement and single rates. Unit Rates
To express this as a unit rate, we need to figure out Rochelle’s speed for one mile. Begin by setting up the rate as a fraction. Rochelle ran 15 miles in 2 hours. Express her speed as a unit rate. Use division to divide 15 by 2 The unit rate was 7.5 miles in 1 hour, or 7.5 miles per hour.
Sometimes, it is helpful to compare unit rates. For example, a unit price is a type of unit rate. In real life, you may want to compare two unit prices to determine which one is the better buy. Felicia needs to buy sugar. She could buy a 16-ounce box of sugar for $1.12, or she could buy a 24-ounce box of sugar for $1.44. Which is the better buy? How many cents per ounce cheaper is that buy? Comparing Unit Rates
Often it is necessary to change the units we are using for more effective communication. For example: 96 inches is more difficult to understand than 8 feet. Rates often need to be converted as well. 2 miles per minute is equal to 120 miles per hour. Conversions
So how can we convert 17.6 inches per second into miles per hour? We know that there are 12 inches per foot and 5280 feet per miles. We also know there are 60 seconds per minute and 60 minutes per hour. Let’s use these rates to make the needed changes. Converting Rates
First lets change lengthsNow lets change timeAnd we get Just like climbing a ladder, you take it one step at a time. That’s why this method is called a Ladder Conversion.