Chapter 7: Proportional Relationships
Section / Objective / Estimated Dates7-1: / Ratios and Rates / Tuesday, Feb 16th
7-2: / Using Tables to Explore Equivalent Ratios & Rates / Wednesday, Feb 17th
7-3: / Proportions / Thursday, Feb 18th
7-4: / Similar Figures / Friday, Feb 19th
7-5: / Indirect Measurement / Monday, Feb 22nd
7-6: / Scale Drawings and Maps / Tuesday, Feb 23rd
Quiz A / Wednesday, Feb 24th
7-7: / Percents / Thursday, Feb 25th
7-8: / Percents, Decimals, and Fractions / Friday, Feb 26th
7-9: / Percent Problems / Monday, Feb 29th
7-10: / Applying Percents / Tuesday, March 1st
Quiz B / Wednesday, March 2nd
Review / Chapter 7 Pages 338-397 / Thursday, March 3rd
Test / Friday, March 4th
Why Learn This?
"Scale models are used frequently to make very large of very small objects easier to visualize. For example, architects often use scale models of buildings they are designing in presentations to clients. The scale models allow the clients to see 'finished' buildings before they begin the costly task of constructing the real thing. Students will learn about scale and scale models in Lesson 7-6." (Holt authors)
Percents, proportions, ratios, and rates are used in a variety of ways throughout math. Many times using a proportion can help you understand which operation to use in a word problem. Percents are used in shopping and are an important concept to know how to use! This chapter is a big chunk of the Minnesota Math Standards for 6th grade! Enjoy :)
Mrs. Helmberger
Parent Signature ______
Math humor ----
Question: Why did the math book look so sad?
Answer: Because it had so many problems.
Section 7-1: Ratios and rates
You'll learn: to write ratios and rates and to find unit rates
ratio:
equivalent ratios:
rate:
unit rate:
Ratios can be written in 3 ways:
A basket of fruit contains 6 apples, 4 bananas, and 3 oranges. Write each ratio in all three forms.
A. bananas to apples B. bananas and apples to oranges C. oranges to total pieces of fruit
Equivalent Ratios
You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number. Write three equivalent ratios to compare the number of stars with the number of moons in the pattern.
Rates
Suppose a 2-liter bottle of soda costs $1.98. How would you write this as a rate?
When the comparison is to one unit this is a unit rate. Divide both terms by the second term to find the unit rate.
When the prices of two or more items are compares, the item with the lowest unit rate is the best deal.
Find the better buy: A 2-liter bottle of soda costs $2.02. A 3-liter bottle of the same soda costs $2.97. Which is the better deal?
Section 7-2: Using Tables to Explore Equivalent Ratios and Rates
You'll learn: to use a table to find equivalent ratios and rates.
Mrs. Kennedy's students are painting a mural in their classroom. They mixed yellow and blue paints for a green background and found that the ratio of the amount of yellow to the amount of blue is 3 to 2. Now they need to make more green paint, using the same ratio as before.
Pints of yellow / 3 / 6 / 9 / 12Pints of blue / 2 / 4 / 6 / 8
You can increase amounts but keep them in the same ratio by multiplying both the numerator and denominator of the ratio by the same number. You can also decrease amounts in the same ratio by dividing the numerator and denominator by the same number.
Making a Table to find Equivalent Ratios
Use a table to find three equivalent ratios. A. 8/3 Use a table to find three equivalent ratios. B. 4/7
4 / 8 / 12 / 167 / 14 / 21 / 28
Ratios in tables can be used to make estimates or predictions.
A group of 10 friends is in line to see a movie. The table shows how much different groups will pay in all. Predict how much the group of 10 will pay.
Number in Group / 3 / 5 / 6 / 12Amount Paid ($) / 15 / 25 / 30 / 60
Section 7-3: Proportions
You'll learn: to write and solve proportions.
Proportion:
Modeling Proportions: Write a proportions for the model.
Now separate
· First write the ratio of triangles to circles:
· Next separate the triangles and the circles into two equal groups. Write the ratio of triangles to circles in each group:
· Now combine them with an equal sign to show the proportion:
Using Cross Products to complete proportions: Cross products in proportions are equal.
Find the missing value in the proportions
3 = n 5 = 30 x = 43
4 16 25 x 90 35
The label from a bottle of pet vitamins shows recommended dosages. What dosage would you give an adult dog that weighs 15 lb?
· Adult dogs: 1 tsp per 20 lb body weight
· Puppies, pregnant dogs, or nursing dogs: 1 tsp per 10 lb body weight
· Cats: 1 tsp per 12 lb body weight
What dosage would you give an adult cat that weights 9 lbs?
Section 7-4: Similar Figures
You'll learn: to use proportions to find missing measures in similar figures.
Corresponding sides:
Corresponding angles:
9 cm
3 cm
2 cm 6 cm
Similar Figures: Two figures are similar if:
· the measures of the corresponding angles are equal
· the ratios of the lengths of the corresponding sides are proportional
Find Missing Measures in Similar Figures:
Section 7-5: Indirect Measurement
You'll learn: to use proportions and similar figures to find unknown measures
indirect measurement:
The Statue of Liberty has been standing in New York Harbor since 1876. How could you measure the height of the statue?
Suppose that on a sunny day, the statue case a shadow that was 610 feet long. A 6-foot -tall person standing by the statue case a 12-foot long shadow. Both the person and the statue form 90° angles with the ground, and their shadows are cast at the same angle. This means we can form two similar triangles and use proportions to find the missing height!
A lighthouse casts a shadow that is 36 m long when a meter stick casts a shadow that is 3 m long. The height of a meter stick is obviously 1 meter. What is the height of the lighthouse?
Section 7-6: Scale Drawings and Maps
You'll learn: to read and use map scales and scale drawings
scale drawing:
scale:
Finding Actual Distances
On a map of Chicago, the scale is 1 in: 2 mi. This ratio means that 1 inch on the map represents 2 miles in Chicago.
A. On a map, the distance between Wrigley Field and the Shedd Aquarium is approximately 3.3 inches. What is the actual distance?
B. On a diagram of space the scale is 1 in: 30 million km. If Mercury is 3 inches away from Earth on the diagram, how far is the actual distance?
C. The actual distance from Mercury to Venus is 50 million km. How far apart should Mercury and Venus be on the diagram?
D. The scale on a map is 2 cm: 0.5 km. If the distance from the library to the school on a map is 5 cm, how far is the actual distance?
Section 7-7: Percents
You'll learn: to write percents as decimals and as fractions.
percent:
You can remember that percent means "per hundred".
For example, 8% means 8 per hundred, or 8 out of 100.
If a sales tax rate is 8%, the following statements are true:
· For every $1.00 you spend, you pay $0.08 in sales tax
· For every $10.00 you spend, you pay $0.80 in sales tax
· For every $100.00 you spend, you pay $8 in sales tax
Because percent means "per hundred", 100% means 100 out of 100. This is why 100% is often meant "all" or "the whole thing!"
Modeling Percents:
Use a 10-by-10 square grid to model 8%
Writing percents as fractions:
Write 40% as a fraction in simplest form. Write 5% as a fraction in simplest form.
Up to 55% of the heat lost by your body can be lost through your head. Write 55% as a fraction in simplest form.
Writing Percents as Decimals:
Write 24% as a decimal. Write 5% as a decimal. Write 50% as a decimal.
Section 7-8: Percents, Decimals, and Fractions
You'll learn: to write decimals and fractions as percents
Percents, decimals, and fractions appear in newspapers, on television, and on the Internet. To fully understand the data you see in your everyday life, you should be able to change from one number form to another.
1 = 0.5 = 50%
2
Writing Decimals as Percents:
Method 1: Use place value.
A. 0.3 B. 0.43 C. 0.98 D. 0.04
Method 2: Multiply by 100
A. 0.7431 B. 0.023 C. 0.04 D. 0.3
Writing Fractions as Percents:
Method 1: Write an equivalent fraction with a denominator of 100.
A. 4 B. 2 C. 10 D. 6
5 4 20 15
Method 2: Use division to write the fraction as a decimal.
A. 4 B. 2 C. 10 D. 6
5 4 20 15
Section 7-9: Percent Problems
You'll learn: to find the missing value in a percent problem.
The frozen-yogurt stand in the mass sells 420 frozen-yogurt cups per day, on average. Forty-five percent of the frozen-yogurt cups are sold to teenagers. On average, how many frozen-yogurt cups are sold to teenagers each day?
To answer this question, you will need to find 45% of 420.
To find the percent one number is of another, use this proportion:
% = is
100 of
Heather is downloading a file from the Internet. So far, she has downloaded 75% of the file. If 30 minutes have passed since she started, how long will it take her to download?
Think: 75% of the file has downloaded, so 30 minutes is 75% of the total time needed!
Multiplying to find a Percent of a number
Before you can do these problems you have to change the percents to decimals.
Find 20% of 150. Find 6% of 400. Find 44% of 35.
Section 7-10: Applying Percents
You'll learn: to solve percent problems that involve discounts, tips, and sales tax.
Common Uses of PercentsDiscounts / A discount is an amount that is subtracted from the regular price of an item.
Discount = price · discount rate
Total cost = price - discount
Tips / A tip is an amount added to a bill for service.
Tip = bill · tip rate
Total cost = bill + tip
Sales Tax / Sales tax is an amount added to the price of an item.
Sales tax = price · sales tax rate
Total cost = price + sales tax
Finding Discounts:
A music store sign reads, "10% off the regular price." If Nichole wants to buy a CD whose regular price is $14.99, about how much will she pay for her CD after the discount?
· First round $14.99 to $15.00 because the problem says "about how much..."
· Next, find 10% of $15 by either of the two methods you used yesterday.
· Now Subtract the answer from $15.00 to see what she will pay.
Finding Tips:
Leslie's lunch bill is $13.95. She wants to leave a tip that is 15% of the bill. About how much should her tip be?
· First round $13.95 to $14.00 because the problem says "about how much..."
· Next find 15% of $14 by either of the two methods you used yesterday.
· How much should she leave for a tip?
Finding Sales Tax:
Marc is buying a scooter for $79.65. The sales tax rate is 6%. About how much will the total cost of the scooter be?
· First round $79.65 to $80.00 because the problem says, "about how much..."
· Next find 6% of $80 using either of the 2 methods we used yesterday.
· Finally add the amount to $80.00 to see what Marc will pay.