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# GRADE MATH TUTOR- NYC TOP TUTORS ### WE travel to YOUR home:

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So what exactly do students learn in 6th-grade math? We broke down 6th grade math into the following problems and topics, like breaking down a language into parts.

If you need checklists for a breakdown of 6th grade math, we highly recommend Khan Academy and IXL for Common Core Sixth-Grade Math Standards & K5 learning for their 6th Grade Worksheets. We also thank Embarc for breaking down 6th grade math into the following standards:

### Module 1: Ratios and Unit Rates

#### Topic A: Representing and Reasoning About Ratios (6.RP.A.1, 6.RP.A.3a)

##### The Term Ratio

A 6th grade math tutor can help students develop an understanding and definition of the term ratio. A ratio is always a pair of numbers like 2:3 and never a pair of quantities like 2 cm:3 sec. They write the ratio in two forms: 2:3 or 2 to 3. They understand that order matters. If I am comparing 2 cm to 3 sec. Then I read and write the ratio 2:3 or 2 to 3. If I compare 3 sec to 3 cm, I write the ratio 3:2 or 3 to 2.

Example Problem and Solution
A recipe for fruit salad requires 2 oranges for every 3 apples. Write this situation as a ratio of oranges to apples.

Answer: The ratio of oranges to apples is 2:3. Students are encouraged to complete a table to show the relationship. Students expand the table to show that there are other options to demonstrate 2 oranges for every 3 apples in the recipe. Students are guided to articulate that there are 32 as many apples as there are oranges. Another way to articulate the same situation is that there are 23 as many apples as oranges. Students develop an intuitive understanding of equivalent ratios by using tape diagrams to explore possible quantities when one of the quantities is given. If the student is having a problem articulating ratio, a 6th grade math tutor can help.

Example Problem and Solution
Patrick and Dwight are keeping track of the number of grams of sugar they consume daily as part of their diet regimen. The ratio of the number of grams of sugar Patrick consumes to the number of grams of sugar Dwight consumes in a day is 5 to 2. If they consume 105 grams of sugar per day, how many grams do each consume? The tape diagram models the situation.

To determine how many grams of sugar each of them consumes, students need to know how many grams of sugar each of the sections of the tape diagram represents. Students divide the number of total grams of sugar they consume together (105 grams) by 7 total sections of the tape diagram.

105 divided by 7=15; therefore, each section of the tape diagram represents 15 grams of sugar.

Patrick consumes 75 grams of sugar per day (15 x 5), and Dwight consumes 30 grams of sugar per day (15 x 2).

##### Value of a Ratio

The ratios of 5:2 and 75:30 are not the same ratios, but we say they have the same value. The value of 75:30 is 5:2.

In our first example, 2:3 and 6:9 are not the same ratio, but we say the ratios have the same value.

A 6th grade math tutor can help students begin to apply a basic understanding of the relationship between ratios and fractions. We say if there are 2 boys to every 5 girls, then the fractional relationship is there are 25 as many boys as girls.

###### Words to Know:

Ratio – A pair of nonnegative numbers, A: B, where both are not zero, and that are used to indicate that there is a relationship between two quantities such that when there are A units of one quantity, there are B units of the second quantity.

Equivalent Ratios – Ratios that have the same value. Associated Ratios (e.g., if a popular shade of purple is made by mixing 2 cups of blue paint for every 3 cups of red paint, not only can we say that the ratio of blue paint to red paint in the mixture is 2:3, but we can discuss associated ratios such as the ratio of cups of red paint to cups of blue paint, the ratio of cups of blue paint to total cups of purple paint, the ratio of cups of red paint to total cups of purple paint, etc.)

Ratio Table – A table listing pairs of numbers that form equivalent ratios.

#### Topic B: Collections of Equivalent Ratios (6.RP.A.3a)

Lesson 15: No video provided*
We can use repeated addition or multiplication to create a ratio table. Students are to demonstrate an understanding of the additive and multiplicative properties of ratio tables. If adding a certain number to each entry in one column, you may need to add that same number to the entries in the other column and keep the same ratio. For example, in the ratio table above, 2 is added to the column labeled “Orange.” If we added 2 down the column labeled “Blue,” we would not get the same entries as the table.

Instead, the numbers you add to the entries must be related to the ratio used to make the table. However, if multiplying the entries in one column by a certain number, you can multiply the entries in the other column by the same number, and the ratio remains. Ratio tables can be used to compare two ratios. You can do this by extending the table or comparing the values of the ratios. Students will build on their experience with number lines by representing collections of equivalent ratios on a double number line.

Example Problem and Solution
Julie has been biking at a constant speed for 30 minutes, during which time she traveled 10 miles. Julie would like to know how long it will take her to run 25 miles, assuming she maintains the same speed. Help Julie determine how long the trip will take. Include a table or diagram to support your answer. Create a double number line using 10 miles and 30 minutes increments. 25 miles falls halfway between 20 and 30 miles, so its equivalent minutes would be between 60 and 90 miles, which is 75 miles.  It will take Julie 75 minutes to travel 25 miles.

Example Problem and Solution
Paul wants to buy a new bike. The bike costs 52 more dollars than he has. If Paul charges \$8 to cut grass, how many lawns will he have to cut to have enough money to buy his bike? Use a double-number line to support your answer. Create a double number line using the increments 1 lawn and \$8/lawn. \$52 falls halfway in between \$48 and \$56, so its equivalent lawns would be halfway between 6 and 7 lawns, which is 6 1⁄2. Paul would need to cut 6 1⁄2 lawns.

Students will relate their knowledge of ratio tables to equations using the value of the ratio.

Example Problem and Solution
The ratio of boys to girls in Mrs. Smith’s math class is 1:3. Use the ratio table to create two equations that would show the relationship between the number of boys and the number of girls in the class. There are three times as many girls in the class as boys. The equation to determine the number of girls in Mrs. Smith’s would be G = 3B.

There are 1/3 the number of boys in the class than girls. The equation to determine the number of boys in Mrs. Smith’s class would be B = 1⁄3G.

If there were 8 boys in the class, how many girls are there be? Use the equation to find your answer. Since you know the number of boys, you would use the formula G = 3B to find the answer. 3 x 8 boys = 24 girls. There are 24 girls in the class.

**A ratio table, equation, or double-number line diagram can be used to create ordered pairs. These ordered pairs can then be graphed on a coordinate plane to represent the ratio.

Example Problem and Solution
Using the ratio table from the previous example, graph the ordered pairs on the coordinate plane. Each entry on the ratio table creates an ordered pair that can be graphed on a coordinate plane.

Ex: 1 boy to 3 girls creates the ordered pair
(1, 3) 2:6 creates (2, 6)…

What would be the number of boys if there were 24 girls in the class? To find the number of boys, find the y coordinate that matches up with 24 girls on the line. There are 8 boys in the class.

###### Words to Know:

Ratio – A pair of nonnegative numbers, A: B, where both are not zero, and that are used to indicate that there is a relationship between two quantities such that when there are A units of one quantity, there are B units of the second quantity.

Equivalent Ratios – Ratios that have the same value. Associated Ratios (e.g., if a popular shade of purple is made by mixing 2 cups of blue paint for every 3 cups of red paint, not only can we say that the ratio of blue paint to red paint in the mixture is 2:3, but we can discuss associated ratios such as the ratio of cups of red paint to cups of blue paint, the ratio of cups of blue paint to total cups of purple paint, the ratio of cups of red paint to total cups of purple paint, etc.)

Value of a Ratio– For the ratio A: B, the value of the ratio is the quotient A/B.

Ratio Table – A table listing pairs of numbers that form equivalent ratios.

#### Topic C: Unit Rates (6.RP.A.2, 6.RP.A.3b, 6.RP.A.3d)

##### Computing Unit Rate

The students recognize that they can associate a ratio of two quantities, such as 5 miles per 2 hours, to another quantity called the rate. Given a ratio, students precisely identify the associated rate. They identify the unit rate and the rate unit.

A publishing company is looking for new employees to type novels that will soon be published. The publishing company wants to find someone who can type at least 45 words per minute. Trent discovered he could type at a constant rate of 704 words in 16 minutes. Does Trent type at a fast enough rate to qualify for the job? Explain why or why not? After completing the table, students conclude that Trent can only type 44 words per minute which is slower than the 45 words per minute required.

Sample Problems:
How many words would Trent have to type in 20 minutes to qualify for the job? What is the rate unit for this problem?

20 minutes x 45 words per minute = 900 words. Trent would have to type 900 words in 20 minutes to qualify for the job with the publishing company. The rate unit is words per minute.
Example Problem and Solution

Question:
At the Crystal Clean and Shine carwash, the average employee can wash 4 cars per hour. If the carwash wants to accommodate the large volume of customers that go through their business on a typical day, they must continue this rate for 8 hours. At this rate, how many cars can a typical employee wash in 8 hours? Students should be answering these questions:

At what rate does an average employee wash cars?
4 cars an hour
How long does the employee need to wash cars in order to keep up with the volume of customers on a typical Day?
8 hours.

Then, students multiply the rate (4 cars an hour) times the time (8 hours). 4 x 8 = 32. An average employee can wash 32 cars in 8 hours.

Students transfer this skill to convert measurements:

How many cups are in 5 quarts?

##### Percent as a Rate per 100

Students model percentages as a rate per hundred using 10 x 10 grids and write them as fractions over 100 or as a decimal. Students use the model to represent a percent; more importantly, they begin to connect and recognize the other representations of the same number in the form of a fraction and decimal.

Students solve problems by analyzing different unit rates given in tables, equations, and graphs. Question:
Emilio wants to buy a new motorcycle. He wants to compare the gas efficiency of each motorcycle before he makes a purchase. The dealerships presented the data below.

##### Leisure Bikes Gallons vs Miles Which motorcycle dealership has the most gas-efficient motorcycle? Use the graph and the table to make your decision.

At this point in the module, students are proficient in finding the unit rate from a graph and a table. They are encouraged to use a calculator to divide the quantities. In the graph of Leisure Bikes, a bike travels 110 miles on 2 gallons of gas; therefore, the unit rate is 55 miles per gallon. According to the values recorded in the table for Sports Bikes, their bike can travel 287.5 miles on 5 gallons of gas; therefore, the bike from Sports Bikes can travel 57.5 miles on one gallon of gas. Sports Bikes get 2.5 miles more per gallon of gas. Therefore, it is the most gas efficient.

###### Words to Know:

Ratio – A pair of nonnegative numbers, A: B, where both are not zero, and that are used to indicate there is a relationship between two quantities such that when there are A units of one quantity, there are B units of the second quantity.

Rate – A rate indicates, for a proportional relationship between two quantities, how many units of one quantity there are for every 1 unit of the second quantity. For a ratio of A:B between two quantities, the rate is A/B units of the first quantity per unit of the second quantity.

Unit Rate – The numeric value of the rate, e.g., in the rate of 2.5 mph, the unit rate is 2.5.
Value of a Ratio – The value of a ratio is a ratio whose denominator is one. The value of the ratio 9:3 is 3:1.

Ratio table – A table of values listing pairs of numbers that form equivalent ratios. Rate Unit– Miles per hour; dollars per pound; words per minute

#### Topic D: Percent (6.RP.A.3c)Lessons 24-29

##### Part-to-Whole Ratio

Students understand that percentages are related to part-to-whole ratios and rates where the whole is 100. Students model percentages and write a percent as a fraction over 100 or a decimal to the hundredth place.

Imagine you are shopping. You want to purchase an item for \$100. Today it is 20% off of the original price. What does this mean?

What does the grid show?
100 blocks.

20 blocks.

80 blocks.

How can we use this model to help us think about 20% off of \$100.00?

From the grid, we can see that when I save 20%, I am paying 80% of the original value. We can see I would be saving \$20 and paying \$80 when a \$100 item is 20% off. Students write a fraction and a decimal as a percent of a whole quantity and write a percent of a whole quantity as a fraction or decimal.

Example Problem and Solution
Star Middle school was having a Spring Fair. Terry was responsible for the workers in the dunking booth. Here is a list of the workers and the percentage of time they would have to work. Complete the table with the missing values. Answer: Serrie – Percentage is 30%
Paul – Fraction is 45100
Juan – Decimal is .25

###### Words to Know:

Rate – A rate indicates, for a proportional relationship between two quantities, how many units of one quantity there are for every 1 unit of the second quantity. For a ratio of A:B between two quantities, the rate is A/B units of the first quantity per unit of the second quantity.

Unit Rate – The numeric value of the rate, e.g., in the rate of 2.5 mph, the unit rate is 2.5.

Percent– Rate per hundred

*The following information was obtained from Embarc.Online (a website that was originally created to support local teachers using Eureka Math, but we now serve 30,000 users each day across the nation). This information applies to all math users.

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#### Craig Selinger

Owner at Themba Tutors
Craig Selinger, CEO of Themba Tutors (serving Manhattan, Brooklyn, Queens, Staten Island, Bronx, Westchester, Long Island, New Jersey, and Connecticut), is a NY State licensed speech-language pathologist, executive functioning coach, and learning specialist with over 18 years of experience working professionally with over a thousand families. His expertise includes language-based learning issues, e.g. reading, writing, speaking, and listening, executive functioning, ADHD/ADD, and learning disabilities. Check out his interviews with top-notch professionals in the field on Spotify.

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