A Story of Ratios

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosGrade 7 – Module 4

Session 1


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios

AgendaIntroduction to the ModuleConcept DevelopmentModule Review



Curriculum Overview of A Story of Ratios



Grade 7 Module 4 OverviewQuestions to address:• How many topics are there and what

are those topics?

• How many days are required for lessons? Assessments? Remediation and return?

• What are some of the important topics and concepts discussed in the narrative?

Module Overview (p. 3)



Topic AFinding the WholeWhat concepts do you expect to see in

Topic A?

Topic A Opener (p. 11)



Lesson 1: Percent• Students understand that percent is the number and that the symbol

means percent.

• Students convert between fraction, decimal, and percent; including percents that are less than or greater than .

• Students write a non-whole number percent as a complex fraction.

Lesson 1 / Student Outcomes



PercentColor in a 10 x 10 grid to represent each fraction:

Lesson 1 / Opening Exercise



What are equivalent representations of ?

Percent




What are equivalent representations of

Percent




PercentUse the meaning of the word “percent” to write each percent as a fraction and

then a decimal.Percent Fraction

Decimal

Lesson 1 / Example 1



Consider this…The following is not from the module, but is to help make a point…

I will name a sequence of colors and for each one, do one of the following:

If I name a red color, raise your right hand high.

If I name a blue color, clap your hands together once.



Lesson 2: Part of a Whole as a Percent• Students understand that the whole is , and use the formula to problem-

solve when given two terms out of three from the part, whole, and percent.

• Students solve word problems involving percent using expressions, equations, numeric, and visual models.




Part of a Whole as a PercentExample 1: In Ty’s math class, 20% of the students earned an A on a test. If

there were 30 students in the class, how many got an A?

Lesson 2 / Example 1 and Exercise 1



Part of a Whole as a PercentIs the expression equivalent to from our steps in Example 1? What does represent? What does represent? What does their product represent?

Lesson 2 / Discussion



Lesson 3: Comparing Quantities w/ Percent• Students use the context of a word problem to determine which of two

quantities represents the whole.

• Students understand that the whole is 100% and think of one quantity as a percent of another using the formula , to problem solve when given two terms out of three from a quantity, whole, and percent.

• When comparing two quantities, students compute percent more or percent less using algebraic, numeric, and visual models.




Comparing Quantities with PercentSix club members decided to evenly split the total number of bracelets to be

produced [300 bracelets]. Of the 54 bracelets produced over the weekend, Anna produced 32 bracelets. Compare the number of bracelets that Emily produced [22] as a percent of those that Anna produced.

Compare the number of bracelets that Anna produced as a percent of the number that Emily produced.

Lesson 3 / Example 1(a) and (b)



Comparing Quantities with PercentWhat percent more did Anna produce in bracelets that Emily?

What percent fewer did Emily produce in bracelets than Anna?

Lesson 3 / Example 1(a) and (b)



Lesson 4: Percent Increase and Decrease• Students solve percent problems when one quantity is a certain percent

more or less than another.

• Students solve percent problems involving a percent increase or decrease.




Percent Increase and DecreaseA sales representative is taking 10% off of your bill as an apology for any

inconveniences.

Lesson 4 / Discussion



Lesson 5: Find One Hundred Percent Given Another Percent• Students find 100% of a quantity (the whole) when given an quantity that

is a percent of the whole by using a variety of methods including finding 1%, equations, mental math using factors of 100, and double number line models.

• Students solve word problems involving finding 100% of a given quantity with and without using equations.




Find One Hundred Percent Given Another PercentWhat are the whole number factors of 100? What are the multiples of those

factors (up to 100)? How many multiples are there of each factor (up to 100)?




Find One Hundred Percent Given Another PercentNick currently has 7,200 points in his fantasy baseball league which is 20%

more points than Adam has. How many points does Adam have?

Lesson 5 / Exercise 2



Mental Math using Factors of 100If is ___% of a number, what is that number? How did you find your answer?




Lesson 6: Fluency with Percents• Students solve various types of percent problems by identifying the type of

percent problem and applying appropriate strategies.

• Students extend mental math practices to mentally calculate the part, the percent, or the whole in percent word problems.




Lesson 6: Fluency with PercentsRichard works from 11:00 a.m. to 3:00 a.m. His dinner break is of the way

through his work shift. What time is Richard’s dinner break?




Topic BPercent Problems Including More than One WholeWhat concepts do you expect to see in

Topic B?

Topic B Opener (p. 101)



Lesson 7: Markup and Markdown Problems• Students understand the terms original price, selling price, markup,

markdown, markup rate, and markdown rate.

• Students identify the original price as the whole and use their knowledge of percent and proportional relationships to solve multi-step markup and markdown problems.

• Students understand equations for markup and markdown problems and use them to solve markup and markdown problems.




Markup and Markdown ProblemsBlack Friday: A mountain bike is discounted by 30% and then discounted an

additional 10% for shoppers who arrive before 5:00 a.m. a) Find the sales price of the bicycle.

b) In all, by how much has the price of the bicycle been discounted in dollars? Explain.

c) After both discounts were taken, what was the total percent discount?




Markup and Markdown ProblemsExercise 4:a) Write an equation to determine the selling price, , on an item that is

originally price dollars after a markup of 25%.

b) Create a table (and label it) showing five possible solutions to your equation.

c) Create a graph (and label it) of your equation.

d) Interpret the points and .

Lesson 7 / Exercise (4)



Lesson 8: Percent Error Problems• Given the exact value, , of a quantity and an approximate value, , of the

quantity, students use the absolute error, , to compute the percent error by using the formula .

• Students understand the meaning of percent error: the percent the absolute error is of the exact value.

• Students understand that when an exact value is not known, an estimate of the percent error can still be computed when given a range determined by two inclusive values; (e.g., if there are known to be between 6,000 and 7,000 black bears in New York, but the exact number is not known, the percent error can be estimated at most , which is .




Understanding Percent ErrorHow Far Off?• Using a 12-inch ruler, measure the diagonal of a sheet of paper and record

your value.




Understanding Percent ErrorHow Far Off?• Use the formula for absolute error to find the absolute errors of the given

measurements.




Percent Error ProblemsThe attendance at a musical event was counted several times. All counts were

between and . If the actual attendance number is between and , inclusive, what is the most and least the percent error could be? Explain your answer.




Lesson 9: Problem Solving when the Percent Changes• Students solve percent problems where quantities and percents change.

• Students use a variety of methods to solve problems where quantities and percents change, including double number lines, visual models, and equations.




Solving Problems when the Percent ChangesTom’s money is of Sally’s money. After Sally spent and Tom saved all of his

money, Tom’s money is more than Sally’s money. How much money did each have at the beginning?




Problem Solving when the Percent ChangesKimberly and Mike have an equal amount of money. After Kimberly spent $50

and Mike spent $25, Mike’s money is 50% more than Kimberly’s. How much money did Mike and Kimberly have at first?




Lesson 10: Simple Interest• Students solve simple interest problems using the formula: , where , , ,

and .

• When using the formula , students recognize that units for both interest and time must be compatible; students convert the units when necessary.




Understanding Simple InterestLarry invests $100 in a savings plan. The plan pays interest each year on his

$100 account balance. The following chart shows the balance on his account after each year for the next five years. He did not make any deposits or withdrawals during this time.


Time (in years) Balance (in dollars)

1 104.50

2 109.00

3 113.50

4 118.00

5 122.50



Simple InterestComplete Lesson 10, Problem Set #3

Lesson 10 / Problem Set #3



Lesson 11: Tax, Commissions, Fees, and other Real-World Percent Problems• Students solve real-world percent problems involving tax, gratuities,

commissions, and fees.

• Students solve word problems involving percent using equations , tables, and graphs.

• Students identify the constant of proportionality (the tax rate, commission rate, etc.) in graphs, equations, tables, and in the context of the situation.




Tax, Commissions, Fees, and other Real-World Percent Problems• Complete modeling Exercise 5 (parts a, b, and c) from lesson 11.

• Write up your solutions on poster paper to present to the group.




Mid-Module AssessmentComplete Problem #1 from the Mid-Module Assessment

Mid-Module Assessment / Problem #1




Session 1



Topic CScale DrawingsWhat concepts do you expect to see in

Topic C?

Topic C Opener (p. 170)



Lesson 12: The Scale Factor as a Percent for a Scale Drawing• Given a scale factor in percent, students make a scale drawing of a picture

or geometric figure using that scale, recognizing that the enlarged or reduced distances in a scale drawing are proportional to the corresponding distances in the original picture.

• Students understand scale factor to be the constant of proportionality.

• Student make scale drawings in which the horizontal and vertical scales are different.




The Scale Factor as a Percent for a Scale Drawing

Create a scale drawing of the following drawing using a horizontal scale factor of , and a vertical scale factor of .




The Scale Factor as a Percent for a Scale Drawing

Create a scale drawing of the original drawing given below with a horizontal scale factor of 80% and a vertical scale factor of 175%. Write numerical equations to find the horizontal and vertical distances.

Lesson 12 / Problem Set #2



Lesson 13: Changing Scales• Given Drawing 1, and Drawing 2 (a scale model of Drawing 1 with scale

factor), students understand that Drawing 1 is also a scale model of Drawing 2, and compute the scale factor.

• Given three drawings that are scale drawings of each other, and two scale factors, students compute the other related scale factor.




Changing ScalesA regular octagon is an eight-sided polygon with side lengths that are all equal.

All three octagons are scale drawings of each other. Use the chart and the side lengths to compute each scale factor as a percent. How can we check our answers?




Changing ScalesThe scale factor from Drawing 1 to Drawing 2 is and the scale factor from

Drawing 1 to Drawing 3 is . Drawing 2 is also a scale drawing of Drawing 3. Is Drawing 2 a reduction or an enlargement of Drawing 3? Justify your answer using the scale factor. The drawing is not necessarily drawn to scale.




Lesson 14: Computing Actual Lengths from a Scale Drawing• Given a scale drawing, students compute the lengths in the actual picture

using the scale factor.




Computing Actual Lengths from a Scale Drawing

The distance around the entire small boat is 28.4 units. The larger figure is a scale drawing of the smaller sketch of the boat. State the scale factor as a percent, and then use the scale factor to find the distance around the scale drawing.




Lesson 15: Solving Area Problems Using Scale Drawings• Students solve area problems related to scale drawings and percent by

using the fact that an area, , of a scale drawing is times the corresponding area, , in the original picture, where is the scale factor.




Solving Area Problems Using Scale DrawingsWhat percent of the area of the large disk lies outside the smaller disk?




Solving Area Problems Using Scale DrawingsUse Figure 1 below and the enlarged scale drawing to justify why the area of

the scale drawing is times the area of the original figure.




Topic DPopulation, Mixture, and Counting Problems Involving Percents

What concepts do you expect to see in Topic D?

Topic D Opener (p. 227)



Lesson 16: Population Problems• Students write and use algebraic expressions and equations to solve

percentage word problems related to populations of people and compilations.




Population ProblemsA school has girls and boys. If of the girls wear glasses and of the boys wear

glasses, what percent of all students wears glasses?




Population ProblemsThe weight of the first of three containers is more than the second, and the

third container is lighter than the second. The first container is heavier than the third container by what percent?




Population ProblemsIn one year’s time, of Ms. McElroy’s investments increased by , of her

investments decreased by , and of investments increased by . By what percent did the total of her investments increase?




Lesson 17: Mixture Problems• Students write and use algebraic expressions and equations to solve

percent word problems related to mixtures.




Mixture ProblemsA 6-pint oil mixture is added to a 3-pint oil mixture. What percentage of the

resulting mixture is oil?

Lesson 17 / Exercise 1a



Mixture Problems is used to model a mixture problem. a) How many units are in the total mixture?

b) What percentages relate to the two solutions that are combined to make the final mixture?

c) The two solutions combine to make six units of what percentage solution?

d) When the amount of a resulting solution is given (for instance gallons) but the amounts of the mixing solutions are unknown, how are the amounts of the mixing solutions represented?




Lesson 18: Counting Problems• Students solve counting problems related to computing percentages.




Counting Problems1. How many 4-letter passwords can be formed using the letters “A” and

“B”?

2. What percentage of the 4-letter passwords contain:a) No “A’s”?b) Exactly one “A”?c) Exactly 2 “A’s”?d) Exactly 3 “A’s”?e) 4 “A’s”?

Lesson 18 / Exercises 1-2



End-of-Module Assessment

End-of-Module Assessment



Your Biggest TakeawayTake a moment to reflect on today’s module focus session and prepare 1 or 2

key takeaways that you think are important to share with the group.



Key Points• Percent is a part to whole relationship.• Focus on identifying the whole quantity (or quantities) in percent

problems.• Greater fluency with percent improves problem solving abilities.• Percent problems can be represented in a variety of models including

equations, visual models, and numeric models.• Percent can compare a part of a quantity to the whole quantity, or can

compare two separate quantities.• A percent of a set of quantities represents a proportional relationship.• Percent error is not just a formula to memorize … it has meaning.



Key Points• The scale factor is the unit rate (in percent form)• Scale drawings may have more than one scale factor (horizontal and

vertical scales).• Given a drawing , and scale drawing of drawing with a scale factor ,

drawing is a scale drawing of drawing with scale factor .• Work with percents in module 4 ushers in the topics of probability and

statistics in module 5.




Session 1

Documents

A Story of Ratios