01 6 .rp teacher's guide & lesson 8 1v5.pdf engageny

6.RP

EduTron Corporation Draft for NYSED NTI Use Only

66..RRPP RRAATTIIOOSS AANNDD PPRROOPPOORRTTIIOONNAALL RREELLAATTIIOONNSSHHIIPPSS Understand ratio concepts and use ratio reasoning to solve problems

DRAFT 2012.11.29

TTEEAACCHHEERR’’SS GGUUIIDDEE

Teacher’s Guide: Common Core Mathematics 6th

Grade Ratios and Proportions of 30

© 2012 EduTron Corp. All Rights Reserved. (781)729-8696 [email protected]

TTaabbllee ooff CCoonntteennttss

Page

I. Overview 3

II. Advanced Content Knowledge for Teachers 5

1. In short, what is the gist of this topic? 5

2. Show me the key issues and ideas here. 5

3. Are ratios and rates really different? 6

4. When do I use a tape diagram? 7

5. How do I use a tape diagram to solve ratio problems? 7

6. When and how do I use a double number line diagram? 9

7. How do I demonstrate proportionality using a graph in the x-y plane and establish the correspondence among a table, graph and equation?

11

8. Does it always make sense to connect the points and draw the line?

12

9. How can I express the relationship in a collection of equivalent ratios as a linear equation?

12

10. Are all linear relationships proportional? 13

11. Are all relationships (between two related sets of data) linear? 13

III. Lesson 15

A. Assumptions about what students know and are able to do coming into this lesson

15

B. Objectives 15

C. Anticipated Student Preconceptions/ Misconceptions 16

D. Assessments 17

E. Lesson Sequence and Description 18

F. Closure 20

G. Teacher Reflection 20

IV. Worksheets 21

A. Class Practice 21

B. Homework 23

V. Worksheets with Answers 26

A. Class Practice 26

B. Homework 28




LLeessssoonn 88:: UUssee RRaattiioo aanndd RRaattee RReeaassoonniinngg ttoo

SSoollvvee RReeaall--wwoorrlldd aanndd MMaatthheemmaattiiccaall PPrroobblleemmss

II.. OOvveerrvviieeww Essential Questions to be addressed in the lesson

How are tables of equivalent ratios, graphs and equations related to one another? How do they allow us to make predictions and solve problems?

Standards to be addressed in this lesson

According to the Common Core Standards Map, 6.RP.3 (a and b) and 6.EE.9 are inherently connected; this connection should be reflected in the teaching sequence (See highlight in the Common Core Standards Map, Figure 1).

Figure 1.




6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then, at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

Two additional standards in 6th grade that are strongly related to this lesson are:

6.RP.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times a quantity); solve problems involving finding the whole, given the part and the percent. This standard will be addressed in the following lesson. 6.RP.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. This standard will be addressed in the following unit.

Standards of Mathematical Practice Although all eight Standards of Mathematical Practice should be instilled in students in these topics, three of them were chosen to be highlighted. They are denoted with these symbols: MPX.

MP4: Model with mathematics. MP7: Look for and make use of structure. MP8: Look for an express regularity in repeated reasoning.

The three components of rigor in the Common Core Standards (Computation Fluency, Conceptual Understanding and Problem Solving) will be denoted by Fluency Concept Application . Additional materials should be used to make the Rigor and Mathematical Practice Standards come alive. (For example, see a separate document Challenging Problems and Tasks.)

II. Advanced Content Knowledge Teacher’s Guide: 6th



IIII.. AAddvvaanncceedd CCoonntteenntt KKnnoowwlleeddggee ffoorr TTeeaacchheerrss 1. In short, what is the gist of this topic? If A is (directly) proportional to B, then the ratio A/B is a constant.1 We express this as A/B = r, where r is known as a (constant) ratio, or rate. It can also be written as A = rB or B = A/r. 2. Show me the key issues and ideas here. This lesson is about representing and reasoning about ratios and collections of equivalent ratios. Because the multiplication table is familiar to sixth graders, situations that give rise to columns or rows of a multiplication table can provide good initial contexts when ratios and proportional relationships are introduced. Pairs of quantities in equivalent ratios arising from whole number measurements such as “3 lemons for every $1” or “for every 5 cups grape juice, mix in 2 cups peach juice” lend themselves to being recorded in a table.6.RP.3a Initially, when students make tables of quantities in equivalent ratios, they may focus only on iterating the related quantities by repeated addition to generate equivalent ratios (See Figure 2. Additive Structure). To understand this topic, they must become equally comfortable with multiplicative relationships (See Figure 2. Multiplicative Structure). As students work with tables of quantities

1 Example: A is the distance between two locations on the map; B is the actual distance on the ground. The ratio A/B is the same for any map-distance A and its actual distance B. Typical town map scale (ratio) is 1:5000, or 1/5000. The rate is also called a proportionality constant.

Figure 2.




in equivalent ratios, they should practice using and understanding ratio and rate language. 6.RP.1

and 6.RP.2 It is important for students to focus on the meaning of the terms “for every,” “for each,” “for each 1,” and “per” because these equivalent ways of stating ratios and rates are at the heart of understanding the structure in these tables, providing a foundation for learning about proportional relationships in Grade 7. This meaning can be reinforced by gradually encouraging students to consider entries in ratio tables beyond those they find by skip counting, including larger entries and fraction or decimal entries. These other entries will require the explicit use of multiplication and division by a constant factor, not just repeated addition or skip counting (See Figure 2. Multiplicative Structure)2MP7. For example, if Seth runs 9 meters every 2 seconds, then at this rate, Seth will run 45 meters every 10 seconds, because in 5 times the amount of time he will run 5 times the distance. This can also be scaled down: he will run 4.5 meters in 1 second because in half the time he will go half as far. In other words, when the elapsed time is divided by 2, the distance traveled should also be divided by 2. More generally, if the elapsed time is multiplied (or divided) by N, the distance traveled should also be multiplied (or divided) by N. Double number lines can be useful in representing ratios that involve fractions and decimals entries (See Introduction to Double Number Line3 MP4 Document). This understanding of Multiplicative Structure allows students to not only better apply ratios to real world problems but also determine unit rates. It is vital to this lesson: building tables, graphing and writing the algebraic expression. Students graph the pairs of values displayed in tables of equivalent ratios on coordinate axes. The graph of such a collection of equivalent ratios lies on a line through the origin, and the pattern of increases in the table can be seen in the graph as coordinated horizontal and vertical increases. (6.EE.9)4 MP4 Real-world scenarios involving constant rates are modeled with tables and graphs. Then the regular patterns in tables and graphs are extracted and expressed in the form of equations.5 MP8 3. Are ratios and rates really different? Ratios, rates, and proportional relationships arise in situations in which two (or more) quantities are related. Sometimes the quantities have the same units and can be represented as points on the same number line (e.g., 3 cups of apple juice and 2 cups of grape juice); other times they do not (e.g., 3 meters and 2 seconds) and have to be represented as points on different number lines. The division of quantity A by quantity B results in a quotient. The

2 MP7: By looking for the repeated multiplicative structure students are able to make sense of the table.

3 MP4: Double number lines offer students the ability to model the rate problems visually and thereby gain

understanding of the problem. 4 MP4: A 2D graph is a visual model of a mathematical relationship between 2 sets of quantities.

5 MP8: Once the students have found the patterns in the tables and graphs, they are ready to express this relationship

in the form of an equation after repeated reasoning.




quotient can be called either ratio, or a rate, depending on context. Some authors distinguish ratios from rates, using the term “ratio” when units are the same and “rate” when units are different; others use ratio to encompass both kinds of situations. The Standards chose to use ratio in the second sense, applying it to situations in which units are the same as well as to situations in which units are different. Relationships of two quantities in such situations may be described in terms of ratios, rates, percents, or proportional relationships. A ratio associates two or more quantities. Ratios can be indicated in words as “3 to 2” and “3 for every 2” and “3 out of every 5” and “3 parts to 2 parts.” This use might include units, e.g., “3 cups of flour for every 2 eggs” or “3 meters in 2 seconds.” Notation for ratios can include the use of a colon, as in 3:2. The quotient of the two quantities being compared is sometimes called the value of the ratio 3:2. Ratios have associated rates. For example, the ratio 3 feet for every 2 seconds has the associated rate 3/2 feet for every 1 second; the ratio 3 cups apple juice for every 2 cups grape juice has the associated rate 3/2 cups apple juice for every 1 cup grape juice. In Grades 6 and 7, students describe rates in terms such as “for each 1,” “for each,” and “per.” The unit rate is the numerical part of the rate; the “unit” in “unit rate” is often used to highlight the 1 in “for each 1” or “for every 1.” Proportionality can be modeled using equivalent fractions, tables, tape diagrams6 MP4 and double number line diagrams. Both tape diagrams and double number line diagrams visually depict the relative sizes of the quantities and are a key to truly grasping the underlying structure of ratios. Read on (and see additional files) for introductions to Tape Diagrams and Double Number Lines. 4. When do I use a tape diagram? Tape diagrams are best used to model ratios when the two quantities have the same units. They can be used to solve problems and also to highlight the multiplicative relationship between the quantities. Although it is not the focus of this lesson, it is good to be aware of this visual modeling technique. See a separate document Introduction to Tape Diagrams. 5. How do I use a tape diagram to solve ratio problems? Using Figure 3, it is also possible to see how tape diagrams can be used to solve simple ratio problems. Example: The grapple fruit juice is made by mixing apple and grape juice in a ratio of 3:2. How much apple juice and grape juice is needed

6 MP4: The tape diagram is a visual model of ratios.

Figure 3.




to make 15 gallons of grapple? In this problem there are 3 units (rectangles) apple and 2 units (rectangles) grape making a total of 5 units. Therefore:

5 units = 15 gallons 1 unit = 3 gallons 3 units (apple juice) x 3 gallons = 9 gallons 2 units (grape juice) x 3 gallons = 6 gallons

9 gallons of apple juice and 6 gallons of grape juice are needed to make 15 gallons of grapple. It is also possible to represent multi-step problems using the same method. See next problem. Figure 4.

For this problem I start with the yellow and blue in the 5 to 3 ratio (Figure 4, tape diagram on the left). I can then see that I need two more blue units to make them equal as the problem states (Figure 4, tape diagram on the right). According to the problem, the amount added is equal to 14 liters. Therefore:

2 units = 14 liters 1 unit = 7 liters 8 units (total at start) x 7 liters = 56 liters

There was 56 liters of green paint at first.




6. When and how do I use a double number line diagram? Double number line diagrams are best used when the quantities have different units (otherwise the two diagrams will use different length units to represent the same amount). Double number line diagrams can help make visibleMP4 that there are many, even infinitely many, pairs in the same ratio, including those with rational number entries. As in tables, unit rates (R) appear in the pair (R, 1). Figure 5

Directions on how to make a double number line:

Set up two number lines. Both number lines start with 0.

Mark the first pair of data on the two number lines. In the example above it is 5 meters in 2 seconds. The convention is to use the upper line for the dependent variable and the lower line for the independent variable. Label the number lines with the corresponding measurement units, meters and seconds in this case.

As in building up a table, mark additional pairs of data on the number lines, e.g., by multiplying (or dividing) both by the same quantity. In this case 5 x 2 = 10 meters and 2 x 2 = 4 meters.

Continue placing equivalent ratios on the number lines in this manner, similar to populating an equivalent ratio table.




Directions on how to find the unit rate using a double number line:

Unit rates are based on “per” something or “for 1” something. Divide the bottom number on the number line by itself to get it to 1. In the above example we would divide it by 2.

Do the same to the meters. In this case it is 5 meters divided by 2. This means the unit rate is 2.5 meters per second.

Another example: If I use 3 cups of flour for 9 batches of cookies. I can make a double number line to represent many equivalent ratios and to find the unit rate. I have started this one by beginning at 0 and placing my first pair of data (3, 9). What would be the missing number?

The missing number is 18. That means if I have 6 cups of flour, I can make 18 batches of cookies. To find the unit rate I will take 3 and divide it by 3 getting it to the desired “1” in the lower line.

I now divided the 9 also by 3 which equals 3. The unit rate is 3 batches per cup of flour.

0 ? 9 batches

3 6 9 ? homework

0 1 2 3 6 9 12 cups

0 9 ? batches

3 6 9 ? homework

0 3 6 9 12 cups




7. How do I demonstrate proportionality using a graph in the x-y plane and establish the correspondence among a table, graph and equation? Reminder: if A is (directly) proportional to B, then A/B is a constant. A collection of equivalent ratios can be graphed in the x-y plane. The graph represents a proportional relationship. The unit rate appears in the equation and graph as the slope of the line, and in the table’s coordinate pair with first coordinate 1 (See Figure 6). Notice that if a student makes a graph using the data in the table, he or she is able to connect the data points to make a straight line. This allows students to make predictions visually based on the graph. After establishing the correspondence between the unit rate and the slope of the line, students are ready to express the “rule” of the table and graph, and to write down the equation of the line.7 MP4,7,8 Figure 6.

7 MP4,7, and 8 Graphs in the x-y plane are visual models of mathematical relationships - By using these visual

representations of the models students develop a sense of the correspondence between the rate and the slope. The

students then express this understanding in an equation.




8. Does it always make sense to connect the points and draw the line? Concept MP4

It depends. If a table lists the number of tricycles versus the total number of wheels (W = 3T), then the points in between the whole-number data points do not make sense in real life. For

example, between 1 tricycle and 2 tricycles, one cannot have 1.29 tricycles or 2 tricycles. In general, if the actual quantities represented by the values in the table are discrete (consisting of distinct or unconnected elements, i.e., non-continuous), then it does not make sense to connect the dots to form the line. Furthermore, in the tricycle example, the line y = 3x goes through first and third quadrants. But the line is only useful in the first quadrant since we do not speak of negative numbers of tricycles or wheels. However, sometimes we still choose to connect the dots to form a (dotted) line as a visual aid to identify the linear relationship and to make predications—knowing full well that most of the points on the line cannot represent real scenarios. 9. How can I express the relationship in a collection of equivalent ratios as a linear equation? An equation can be written to describe the relationship in the table of equivalent ratios. Such equations are special in that they follow the specific form of direct variation or y = mx. The graphs of these equations are lines passing through the origin. In other words, the y-intercepts, where the lines cross the y-axis, are always (0,0). The slope of the equation is the unit rate. For example, if the unit rate (speed) of a traveling car is 60 mph and the rate remains constant, then the miles traveled (distance) is equal to the unit rate (60) multiplied by the number of hours travelled, or in algebraic notation: y = 60 x. As another example, in Figure 5, the algebraic equation for the double number line relating meters to seconds would be y = 2.5 x, since 2.5 is the unit rate, x is the number of seconds, and y is the number of meters. For the batches of cookies the equation would be y = 3x with x being the cups of flour and y being the batches of cookies. Look again at Figure 6. If I look at the first row in the table I see that 5 cups of grape juice (input) is mixed with 2 cups of peach juice (output). To find the unit rate one needs to get the independent variable (input) to 1. In order to do that, I will divide 5 cups of grape juice by 5, the same reasoning as in the double number line example. I have 1 cup of grape juice. I need to do the same thing to the output. I take 2 cups of peach juice and divide that by 5 also. I now have my unit rate of 2/5. I can write my equation y = 2/5 x.




10. Are all linear relationships proportional? Not all linear relationships are proportional. Take for instance conversion between Celsius and Fahrenheit: F = 9/5 C + 32, create a table for 0, 10, 20, 30, and 40 degrees Celsius. Plot the points on an x-y plane. Does it make a linear relationship? Does the line cross the y axis at the origin? Is the relationship proportional?

Concept 11. Are all relationships (between two sets of data) linear? Concept No. In general, two sets of data may or may not be related to each other. Sometimes there is very little or no correlation between them. For example, the population of a country has little to do with the alphabetical order (1 to 26) of the first letter of the name of the country. Imagine a plot of population versus first letter of the names of all countries. Will you see a line? Will you even see any recognizable pattern? When two sets of data are related to each other, the relationship can be represented in an input–output table, a graph, and sometimes an equation. The two sets of related data could have the following simple recognizable relationships: (1) Proportional and Linear, which when graphed shows a straight line passing through the origin, such as y = 2x, y = -3x, or y = 1.6x.




(2) Linear but not proportional, which when graphed shows a straight line NOT passing through the origin, such as y = 2x + 1, y = 1.7x + 8, or y = -2x + 3. See question 10 (last page) for another example about Celsius to Fahrenheit.

(3) Nonlinear, which when graphed shows a curve, not a straight line, such as y = x2, y = 1/x, y = 2x, or other, more complicated, relationships.

III. Lesson Teacher’s Guide: 6th



IIIIII.. LLEESSSSOONN

A. Assumptions about what students know and are able to do coming into this lesson

Fluency Concept

Students understand the ratios concept and can create and complete tables of equivalent ratios given an initial ratio.

Example: The ratio of boys to girls in the class is 2:3

Boys 2 4 6 1

Girls 3 6 8 1.5

Fluency Students can calculate a unit rate given two related quantities of different measures

Example: If I drive 240 miles in 4 hours what is my average speed? 240miles / 4hours = 60 miles per hour

B. Objectives

By the end of this lesson students will know:

MP4, 7 Fluency Application

How to apply a constant rate to construct a table, given a real-world problem. Concept How to graph an input/output table and make predictions—based on a constant unit rate.




C. Anticipated Student Preconceptions/ Misconceptions

1. A common error when working with ratios is to make additive comparisons. For example: 2:3 ≠ 4:5 by adding 2. 2:3 = 4:6 by multiplying by 2.

Remedy: The whole notion of ratio is a multiplicative comparison of two quantities. It should be made clear and contrasted with the difference of two quantities. Students need to focus on the rows (or columns) of a table as multiples of each other (multiplicative comparison). 2. A common language gap is that some students don’t understand phrases using the

word “per”. Remedy: The following needs to be explicitly taught. “Per something” means “for each something”. Example: 240miles/4hours = 60 miles per hour or 60 miles for each hour. 3. Students can be confused that the order of the input/ output or equivalent ratio

table and the order we read unit rates appear to be reversed. This may seem backwards to them. Example: We read mph (unit rate) as “miles per hour” with “miles” pronounced first, but in the table the input is hours (first) and the output is miles (last).

Remedy: In some countries, including the USA, “mph” is a convention. It is unfortunate that the order is not consistent with the x-y (independent-dependent) table in mathematics. Some other countries describe “60 mph” as “every-hour-60 miles.” The convention needs to be taught; the perceived “order flip” also needs to be dealt with explicitly in context. 4. A student can mistakenly reverse the x (independent variable) and y (dependent

variable) on his or her graph, creating an inverse function. Remedy: Although the students will be able to read the graph and answer questions off the graph, the convention should be corrected. Guide the student to change the two axes so they comply with the convention of the independent variable as the horizontal axis and the dependent variable as the vertical axis.




D. Assessments

Pre- assessment/ Formative Post – assessment/ Formative

Discussion: Review Give examples of a rate: Examples: Miles to hours

Dollars to pounds What is a unit rate? A unit rate is when one measure of a quantity is compared to a single unit of another. What are some examples of unit rates? Examples: Miles per gallon Dollars per pound Can you compute the unit rate? If I drive 125 miles and use 5 gallons of gas, what is my average rate of consumption?

Review homework




E. Lesson Sequence and Description

The following lesson sequence provides a baseline for building foundation skills and concepts. Additional materials should be used to make the practice standards come alive (See a separate document Challenging Problems and Tasks and many other documents with additional problems for enrichment). Students learn that when a relationship with a constant rate is graphed, it makes a line on the x-y plane which allows them to make further predictions.

The initial discussion will bring up any misconceptions students have had up until this point with rates. In groups have the students come up with two other unit rates they have heard in real life.

Take one they are familiar with from all student input, e.g., miles per hour, and make up a problem based on the chosen context. For example: I have driven for 240 miles and it has taken me 4 hours. Assuming the speed to be constant, at what rate am I driving? Use a double line diagram to determine the unit rate (this is a review from lesson 5).

Once students have found the unit rate, e.g., 60 mph, use this rate to complete a table of equivalent ratios.MP7

Hours Miles

1 60

2

3

4

5

Now we need to find the equation for the table. What operations do we use to relate hours to the number of miles8?MP8 Concept

Since we multiply the hours by 60 to get the number of miles, let’s write this in a verbal sentence.

The number of miles is equal to the number of hours times 60.

Next write it in a math statement. The number of miles = the number of hours x 60

8 Conceptual understanding: The students need to understand the operations and the "math relationship" that is

embedded in the table in order to derive the equation.

MP 8: This entire set of directions is to aid students in expressing regularity in the repeated reasoning in

mathematical terms,




Since the number of miles is the output, or y, and the number of hours is the input, or x, we can write it this way. y= 60·x

Take this same information and have the students work in pairs to graph the data. Remind students that hours will be the horizontal axis and miles the vertical axis. Circulate to check how they are setting up the graphs. The proper choices of relative lengths on the two axes may need to be discussed.9Application

Show one of the examples completed by students. Using a document camera allows all students to see the graph. Have each pair of students check their results for accuracy and consistency. What do we notice about the points graphed on the grid? When connected, they make a line. Encourage students to connect the points.

Using a sample: Have the students predict how far they would travel in 8 hours. Graph that point.

Predict how long it would take them to travel 180, (210 and 480) miles- after giving time demonstrate how to take that information off the graph by going y- axis to where 180, (210 and 480) is travel over to the line and down to the x-axis. Push to give a number that is off the graph. Have students determine ways of finding that information (e.g., for 180 you can double the results from 90). Stress that this works because the relationship between variables is proportional.10 Concept Application Extension: How long does it take to drive from Albany NY to San Francisco? With or without stopping for rest. Give a couple more questions for the students to answer using their graphs with their partners. Circulate to check for understanding.

Once students are confident hand out class practice sheet 1.

Review answers to practice sheet to check for understanding. Pass out home work.

Exit slip: If I travel 280 miles in 7 hours, how far will I travel in 4 hours?

9 Problem solving: Students need to determine how to set up the graph and work though choosing the proper

intervals for both axes. 10

Conceptual and Problem solving: The problem requires an understanding of the graph and reading the graph-

when asked to use a number off the graph students need to problem solve in order to determine an answer.




F. Closure

Review outcomes of lesson:

Students can give a unit rate given two related quantities of different measure. Fluency

Students can create a table of equivalent ratios given two related quantities—of same or different measure. MP4,7

Students can graph the information from the table of equivalent ratios on an x-y plane. Concept

Students can use the graph to predict outcomes based on the rates assuming the rates are constant. Application

G. Teacher Reflection

Did the students accomplish the outcomes? What evidence do I have? What would I do differently next time?

IV. Worksheets Teacher’s Guide: 6th



IV. Worksheets A. Class Practice

1. Emanuel read 150 pages in 5 hours. a. Find his average reading rate: Fluency c. Graph the data in the x-y plane: MP4

b. Complete the table of equivalent ratios assuming the rate remains constant: MP7

Hours Pages

d. Write the algebraic expression that represents the relationship between the input and the output columns in the table, assuming the rate remains constant: MP8 Answer the following questions using the table and graph and assuming the rate remains constant: MP7 Fluency Application

a. How many hours would it take Emanuel to read 210 pages? b. How many pages can he read in 4 ½ hours? c. How long will it take him to read 230 pages?




2. A car drove 148 miles using 4 gallons of gasoline.

a. Find the unit rate in miles per gallon: c. Graph the data in the x-y plane: b. Complete the rate table, assuming the rate remains constant:

d. Write the algebraic expression that represents the relationship between the input and output columns for the table:

Answer the following questions, assuming the rate remains constant:

a. How many miles would you travel on 3 gallons? b. How many gallons will it take to drive 6 ½ miles?




B. Homework

1. Nichole types 105 words in 3 minutes.

a. Find her average typing rate:

c. Graph the data in the x-y plane: b. Complete the rate table, assuming the rate remains constant:

Answer the following questions assuming the rate remains constant.

d. Write the algebraic expression that represents the relationship between the input and output columns for the table: e. How many minutes would it take Nichole to type 140 words? f. How many words can she type in 2 ½ hours? Assuming each page contains about 500 words, how many pages is that approximately? g. How long will it take her to type 315 words? Explain how you got your answer.




2. Create a rate problem that fits the following graph.

3. Extension questions: 1. Gloria started riding her bike at 11:30 a.m. By 1:30 p.m. she had covered a distance of 20 mi. a. What is her average speed in mph? MP4,7

Graph her average speed on x-y plane (hint: you may need to make a table first)

b. If she traveled 50 miles at this speed, at what time would she be done biking? c. If she biked 15 miles farther at the same rate, what time would it be? Concept Application

http://www.google.com/imgres?q=graph+direct+variation&hl=en&sa=X&biw=1192&bih=587&tbm=isch&prmd=imvns&tbnid=y72cItAA_mV4-M:&imgrefurl=http://www.mathsteacher.com.au/year10/ch17_variation/05_directvariation/25direct.htm&docid=eUPr0tk89AS-qM&imgurl=http://www.mathsteacher.com.au/year10/ch17_variation/05_directvariation/Image5889.gif&w=343&h=298&ei=2VyYUNWtA-uo0AGLy4DwCA&zoom=1&iact=hc&vpx=910&vpy=148&dur=1786&hovh=209&hovw=241&tx=141&ty=109&sig=102537433676198453227&page=1&tbnh=150&tbnw=173&start=0&ndsp=18&ved=1t:429,r:5,s:0,i:86




2. A motorcyclist took 7 hours to travel from Albany to upper state New York at an average speed of 35 km/h. A car took only 5 hours for the same trip. Find the average speed of the car. Fluency Concept Application

3. A car traveled on a highway for 2 hours at 80 km/h. It then traveled another 3 hours at 70 km/h. Concept Application

a. Find the car’s average speed for the whole trip. b. Graph the distance traveled versus time on the x-y plane.

c. Describe the features of the graph in part (b). Explain why the graph looks the way it does. Concept

V. Worksheets with Answers Teacher’s Guide: 6th



V. Worksheets with Answers A. Class Practice

1. Emanuel read 150 pages in 5 hours.

a. Find his average reading rate: Double number line can be used here c. Graph the data in the x-y plane: 150 pages /5 hours = 30 pages per hour

b. Complete the table of equivalent ratios assuming the rate remains constant:

Hours Pages 1 30 2 60 3 90 4 120 5 150

** The skill of choosing an appropriate y – interval must be explicitly taught**

d. Write the algebraic expression that represents the relationship between the input and the output columns in the table, assuming the rate remains constant:

y = 30x Answer the following questions using the table and graph and assuming the rate remains constant:

a. How many hours would it take Emanuel to read 210 pages?

7 hours b. How many pages can he read in 4 ½ hours?

135 pages c. How long will it take him to read 230 pages?

7 hours 40 minutes or reading off the graph acceptable answers are between 7 ½ and 7 ¾

0 1 2 3 4 5 6 7

Hours

210

180

150

120

90

60

30

Pag

es




2. The car drove 148 miles using 4 gallons of gasoline

a. Find the unit rate (in mpg): c. Graph the data in the x-y plane: Double number line can be used here 37 miles per gallon b. Complete the rate table, assuming the rate remains constant:

Gallon Miles 1 37 2 74 3 111 4 148 5 185

*Students may also use intervals for the y–axis that are in the same interval as the unit rate. d. Write the algebraic expression that represents the relationship between the input and output columns for the table: y = 37x

Answer the following questions, assuming the rate remains constant:

a. How miles would you travel on 3 gallons? 111 miles b. How many gallons will it take to drive 6 ½ miles? About 1/6 of a gallon

0 1 2 3 4 5 6 7

Gallons

175

150

125

100

75

50

25

Mil

es




B. Homework 1. Nichole types 105 words in 3 minutes.

a. Find her average typing rate: Double number line can be used here c. Graph the data in the x-y plane: 35 words per minute

b. Complete the rate table, assuming the rate remains constant:

Minutes Words 1 35 2 70 3 105 4 140 5 175

Answer the following questions assuming the rate remains constant. d. Write the algebraic expression that represents the relationship between the input and output columns for the table:

y = 35x e. How many minutes would it take Nichole to type 140 words?

4 minutes

f. How many words can she type in 2 ½ hours? Assuming each page contains about 500 words, how many pages is that approximately?

Using the graph between 85 and 90. g. How long will it take her to type 315 words? Explain how you got your answer.

It will take Nichole 9 minutes. If it takes her 3 minutes to type 105 words and 3 x 105 = 315 then 3 minutes x 3 = 9 minutes.

Minutes

245

210

175

140

105

70

35

W

ord

s




2. Create a rate problem that fits the following graph.

One possible answer: Jeff rides his bike 60 miles in 3 hours. Graph his distance in a given time. 3. Extension questions: 1. Gloria started riding her bike at 11:30 a.m. By 1:30 p.m. she had covered a distance of 20 mi.

a. What is her average speed in mph? 10 miles per hour Graph her average speed on x-y plane (hint: you may need to make a table first) b. If she traveled 50 miles at this speed what time would she be done biking? 4:30 c. If she biked 15 miles farther at the same rate, what time would it be?

6:00

hours Miles 1 10 2 20 3 30 4 40 5 50

0 1 2 3 4 5 6 7

Hours

70

60

50

40

30

20 10

Mil

es

http://www.google.com/imgres?q=graph+direct+variation&hl=en&sa=X&biw=1192&bih=587&tbm=isch&prmd=imvns&tbnid=y72cItAA_mV4-M:&imgrefurl=http://www.mathsteacher.com.au/year10/ch17_variation/05_directvariation/25direct.htm&docid=eUPr0tk89AS-qM&imgurl=http://www.mathsteacher.com.au/year10/ch17_variation/05_directvariation/Image5889.gif&w=343&h=298&ei=2VyYUNWtA-uo0AGLy4DwCA&zoom=1&iact=hc&vpx=910&vpy=148&dur=1786&hovh=209&hovw=241&tx=141&ty=109&sig=102537433676198453227&page=1&tbnh=150&tbnw=173&start=0&ndsp=18&ved=1t:429,r:5,s:0,i:86




2. A motorcyclist took 7 hours to travel from Albany to upper state New York at an average speed of 35 km/h. A car took only 5 hours for the same trip. Find the average speed of the car.

Step 1: 7 hours x 35 km / hr = 245 km Step 2: 245 km / 5 hours = 49 km/h

3. A car traveled on a highway for 2 hours at 80 km/h. It then traveled another 3 hours at 70 km/h.

a. Find the car’s average speed for the whole trip.

Step 1: 80km/h x 2 hours = 160 km Step 2: 70 km/h x 3 hours = 210 km Step 3: 370km / 5 hours = 74 km/ hr

b. Graph the distance traveled versus time on the x-y plane.

c. Describe the features of the graph in part (b). Explain why the graph looks the way it does.

There is a constant rate from 0 – 2 hours on the graph. Then the rate changes. There is a new constant rate from 2 – 5 hours. This leads to 2 different slopes (steepness) of the two line segments.

525

450

375

300

225

150

75

01 6 .rp teacher's guide & lesson 8 1v5.pdf engageny

Documents