6th Grade Unit 1: Ratios and Proportions September 9th – October 9th
6th Grade Mathematics
Ratios and Proportions Unit 1 Curriculum Map: September 9th – October 9th
Table of Contents
I. / Unit Overview / p. 2II. / CMP Pacing Guide / p. 3
III. / Pacing Calendar / p. 4-5
IV. / Math Background / p. 6
V. / PARCC Assessment Evidence Statement / p. 7-9
VI. / Connections to Mathematical Practices / p. 10
VII. / Vocabulary / p. 11
VIII. / Potential Student Misconceptions / p. 12
IX. / Teaching to Multiple Representations / p. 13-15
X. / Unit Assessment Framework / p.16
XI. / Performance Tasks / p.17-21
XII. / Assessment Check / p. 22
XIII. / Summative Task / p. 23
XIV. / Extensions and Sources / p. 24
Unit Overview
In this unit students will …
-Strengthen sense of and understanding of proportional reasoning
-Develop and use multiplicative thinking
-Develop the understanding that a ratio is a comparison of two numbers or quantities
-Find percents using the same processes for solving rates and proportions
-Solve real-life problems involving measurement units that need to be converted
Enduring Understandings
-A ratio is a number that relates two quantities or measures within a given situation in a multiplicative relationship (in contrast to a difference or additive relationship).
-Ratios can express comparisons of a part to whole, (a/b with b ≠0)
-Fractions are part-whole ratios, meaning fractions are also ratios. Percentages are ratios and are sometimes used to express ratios.
-Both part-to-whole and part-to-part ratios compare two measures of the same type of thing. A ratio can also be a rate.
-A rate is a comparison of the measures of two different things or quantities; the measuring unit is different for each value.
-Ratios use division to represent relations between two quantities.
CMP Pacing Guide
Activity / Common Core Standards / Estimated TimeUnit Readiness Assessment (CMP3) / 5.NBT.A.1, 4.NBT.A.2, 5.NF.B.3, 5.NF.B.7, 5.NBT.A.3b / 1 Block
Comparing Bits and Pieces
(CMP3) Investigation 1 / 6.RP.A.1, 6.RP.A.3, 6.RP.A.3a, 6.NS.B.4 / 4 Blocks
Assessment: Partner Quiz (CMP3) / 6.RP.A.1, 6.RP.A.3, 6.RP.A.3a, 6.NS.B.4 / ½ Block
Comparing Bits and Pieces (CMP3) Investigation 2 / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3, 6.RP.A.3b, 6.NS.B.4 / 2 Blocks
Assessment: Check Up 1 (CMP3) / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3, 6.RP.A.3b, 6.NS.B.4 / ½ Block
Performance Task 1 / 6.RP.A.2 / ½ Block
Comparing Bits and Pieces
(CMP3) Investigation 3 / 6.NS.C.6a, 6.NS.C.6c, 6.NS.C.7b, 6.NS.C.7c / 4 Blocks
Assessment: Check Up 2 (CMP3) / 6.NS.C.6a, 6.NS.C.6c, 6.NS.C.7b, 6.NS.C.7c / ½ Block
Comparing Bits and Pieces
(CMP3) Investigation 4 / 6.RP.A.1, 6.RP.A.3, 6.RP.A.3b, 6.RP.A.3c, 6.NS.B.2 / 2 Blocks
Assessment: Check Up 3 (CMP3) / 6.RP.A.1, 6.RP.A.3, 6.RP.A.3b, 6.RP.A.3c, 6.NS.B.2 / ½ Block
Decimal Ops
(CMP3) Investigation 4 / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3c, 6.NS.B.2, 6.NS.B.3 / 2½ Blocks
Unit 1 Assessment / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c / 1 Block
Performance Task 2 / 6.RP.A.3c / ½ Block
Total Time / 19 ½ Blocks
Pacing Calendar
SEPTEMBERSunday / Monday / Tuesday / Wednesday / Thursday / Friday / Saturday
1
Labor Day / 2
OPENING DAY
SUP. FORUM
PD DAY / 3
PD DAY / 4
PD DAY / 5
PD DAY / 6
7 / 8
PD DAY? / 9
1st Day for students / 10
Unit 1:
Ratios & Proportions
Readiness Assessment / 11 / 12 / 13
14 / 15 / 16 / 17
Assessment:
Partner Quiz / 18 / 19
Performance Task 1 Due / 20
21 / 22
Assessment: Check Up 1 / 23 / 24
12:30 pm
Dismissal for students / 25 / 26
Assessment: Check Up 2 / 27
28 / 29 / 30
OCTOBER
Sunday / Monday / Tuesday / Wednesday / Thursday / Friday / Saturday
1
Assessment: Check Up 3 / 2 / 3 / 4
5 / 6
Assessment:
Unit 1 Assessment / 7 / 8 / 9
Unit 1 Complete
Performance Task 2 Due / 10 / 11
12 / 13 / 14 / 15 / 16 / 17 / 18
19 / 20 / 21 / 22 / 23
12:30 pm Dismissal for students / 24 / 25
26 / 27 / 28 / 29 / 30
12:30 pm Dismissal for students / 31
Unit 1 Math Background
Rational numbers are a focal point for middle school students. The goal of this unit is to help students deepen their understanding of equivalent fractions and develop this understanding as they explore ratios. Throughout the unit students will learn to compare with ratios for specific cases. This will assist them in improving their multiplicative thinking and prepare them for proportional reasoning.
During their elementary mathematics education, students were exposed to the area model for fractions. In this unit the students work with more linear models in order to extend the manner in which they reason about rational numbers, understand equivalence, as well as perform operations on rational numbers which is explored further in a later unit. These models include fraction strips, percent bars, and number lines.
Throughout the unit, students use rate tables as a way to express equivalent ratios and compute unit rates. For most of this unit, ratios are not written as fractions. The intent is to keep the notation for part–whole fractions and rational numbers apart from the notation for ratio comparisons to help develop understanding. When the word fraction appears, it is used to represent part of a whole. The learning here will help lay the foundation for the work on ratios and unit rates that will come later in the year as well as the following year. In grade 7, students will use fraction notation to express ratios and will explore ratios in more detail.
PARCC Assessment Evidence Statements
CCSS / Evidence Statement / Clarification / Math Practices / Calculator?6.RP.1 / Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” / i) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be whole numbers. / 2 / No
6.RP.2 / Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” / i i) Expectations for unit rates in this grade are limited to non-complex fractions. (See footnote, CCSS p 42.)
The initial numerator and denominator should be whole numbers. / 2 / No
6.RP.3a / 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. / The testing interface can provide students with a calculation aid of the specified kind for these tasks.
i) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be whole numbers. / 2, 4, 5, 7, 8 / Yes
6.RP.3b / 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.
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? / i) See ITN Appendix F, Table F.c, “Minimizing or avoiding common drawbacks of selected response,” specifically, Illustration 1 (in contrast to the problem
“A bird flew 20 miles in 100 minutes. At that speed, how long would it take the bird to fly 6 miles?”)
ii) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
iii) Expectations for unit rates in this grade are limited to non-complex fractions. (See footnote, CCSS p 42)
iii) The initial numerator and denominator should be whole numbers. / 2, 8, 5 / Yes
6.RP.3c 1 / 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.
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity). / i) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
ii) Pool should contain tasks with and without contexts
iii) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS
p 42.) The initial numerator and denominator should be whole numbers. / 2, 7, 5, 8 / Yes
6.RP.3c-2 / 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.
c. Solve problems involving finding the whole, given a part and the percent. / i) The testing interface can provide students with a calculation aid of the specified kind for these tasks.
ii) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS
p 42.) The initial numerator and denominator should be whole numbers. / 2, 7, 5, 8 / Yes
6.RP.3d / 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.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. / i) Pool should contain tasks with and without contexts
ii) Tasks require students to multiply and/or divide dimensioned quantities
iii) 50% of tasks require students to correctly express the units of the result.
The testing interface can provide students with a calculation aid of the specified kind for these tasks.
iv) Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be whole numbers. / 2, 6, 7, 5, 8 / Yes
Connections to the Mathematical Practices
1 / Make sense of problems and persevere in solving them-Make sense of real-world rate and proportion problem situations by representing the context in tactile and/or virtual manipulatives, visual, or algebraic models
-Understand the problem context in order to translate them into ratios/rates
2 / Reason abstractly and quantitatively
-Understand the relationship between two quantities in order to express them mathematically
-Use ratio and rate notation as well as visual models and contexts to demonstrate reasoning
3 / Construct viable arguments and critique the reasoning of others
-Construct and critique arguments regarding the proportion of a whole as represented in the context of real-world situations
-Construct and critique arguments regarding appropriateness of representations given ratio and rate contexts, EX: does a tape diagram adequately represent a given ratio scenario
4 / Model with mathematics
-Model a problem situation symbolically (tables, expressions, or equations), visually (graphs or diagrams) and contextually to form real-world connections
5 / Use appropriate tools strategically
-Choose appropriate models for a given situation, including tables, expressions or equations, tape diagrams, number line models, etc.
6 / Attend to precision
-Use and interpret mathematical language to make sense of ratios and rates
-Attend to the language of problems to determine appropriate representations and operations for solving real-world problems.
-Attend to the precision of correct decimal placement used in real-world problems
7 / Look for and make use of structure
-Use knowledge of problem solving structures to make sense of real world problems
-Recognize patterns that exist in ratio tables, including both the additive and multiplicative properties
-Use knowledge of the structures of word problems to make sense of real-world problems
8 / Look for and express regularity in repeated reasoning
-Utilize repeated reasoning by applying their knowledge of ratio, rate and problem solving structures to new contexts
-Generalize the relationship between representations, understanding that all formats represent the same ratio or rate
-Demonstrate repeated reasoning when dividing fractions by fractions and connect the inverse relationship to multiplication
-Use repeated reasoning when solving real-world problems using rational numbers
Vocabulary
Term / DefinitionAbsolute Value / The absolute value of a number is its distance from 0 on a number line. Numbers that are the same distance from 0 have the same absolute value. For example,−3and 3 both have an absolute value of 3.
Equivalent Fractions / Fractions that are equal in value, but may have different numerators and denominators. For example,andare equivalent fractions. The shaded part of this rectangle represents bothand.
Mixed Number / A number that is written with both a whole number and a fraction. A mixed number is the sum of the whole number and the fraction. The number2 represents 2 wholes and aand can be thought of as 2 +
Opposite / Two numbers whose sum is 0. For example,−3and 3 are opposites. On a number line, opposites are the same distance from 0 but in different directions from 0. The number 0 is its own opposite.
Percent / A fraction or ratio in which the denominator is 100; a number compared to 100
Proportion / An equation which states that two ratios are equal
Rate / A comparison of two quantities that have different units of measure
Rate Table / A table that shows the value of a single item in terms of another item. It is used to show equivalent ratios of the two items.
Ratio / Compares quantities that share a fixed, multiplicative relationship
Rational Number / A number that can be written as a/b where a and b are integers, but b is not equal to 0
Tape Diagram / A thinking tool used to visually represent a mathematical problem and transform the words into an appropriate numerical operation. Tape diagrams are drawings that look like a segment of tape, used to illustrate number relationships. Also known as Singapore Strips, strip diagrams, bar models or graphs, fraction strips, or length models.
Unit Rate / A unit rate is a rate in which the second number (usually written as the denominator) is 1, or 1 of a quantity. For example, 1.9 children per family, 32 miles per gallon, andare unit rates. Unit rates are often found by scaling other rates.
Unit Ratio / Ratios written as some number to 1
Potential Student Misconceptions
-Often there is a misunderstanding that a percent is always a natural number less than or equal to 100. Provide examples of percent amounts that are greater than 100%, and percent amounts that are less than 1%.
-Students maynot distinguish between proportional situations and additive situations. Students may not realize that although they may have added to find equivalent ratios, they did not add the same amount on both sides.
-Students may still not understand the need to keep the same rate when thinking proportionally.
Teaching Multiple Representations
Assessment Framework
Unit 1 Assessment FrameworkAssessment / CCSS / Estimated Time / Format / Graded
?
Unit Readiness Assessment
(Beginning of Unit)
CMP3 / 5.NBT.A.1, 4.NBT.A.2, 5.NF.B.3, 5.NF.B.7, 5.NBT.A.3b / 1 Block / Individual / No
Assessment: Partner Quiz
(After Investigation 1)
CMP3 / 6.RP.A.1, 6.RP.A.3, 6.RP.A.3a, 6.NS.B.4 / ½ Block / Group / Yes
Assessment: Check Up 1
(After Investigation 2)
CMP3 / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3, 6.RP.A.3b, 6.NS.B.4 / ½ Block / Individual / Yes
Assessment: Check Up 2 (After Investigation 3)
CMP3 / 6.NS.C.6a, 6.NS.C.6c, 6.NS.C.7b, 6.NS.C.7c / ½ Block / Individual / Yes
Assessment: Check Up 3
(After Investigation 4)
CMP3 / 6.RP.A.1, 6.RP.A.3, 6.RP.A.3b, 6.RP.A.3c, 6.NS.B.2 / ½ Block / Individual or Group / Yes
Unit 1 Assessment
(Conclusion of Unit)
Model Curriculum / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c / 1 Block / Individual / Yes
Unit 1 Performance Assessment Framework
Assessment / CCSS / Estimated Time / Format / Graded
?
Performance Task 1
(Mid-September)
Mangos for Sale / 6.RP.A.2 / ½ Block / Group / Yes; Rubric
Performance Task 2
(Early October)
Gianna’s Job / 6.RP.A.3, 6.RP.A.3a / ½ Block / Individual w/ Interview Opportunity / Yes: rubric
Assessment Check 1 (optional) / 6.RP.A.1, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c,
6.NS.C.6c / Teacher Discretion / Teacher Discretion / Yes, if administered
Summative Tasks
(optional) / 6.RP.A.1, 6.RP.A.2, 6.RP.A.3 / Teacher Discretion / Teacher Discretion / Yes, if administered
Performance Tasks
Performance Task 1:
Mangos for Sale (6.RP.A.2)
A store was selling 8 mangos for$10 at the farmers market.
Keisha said,
“That means we can write the ratio 10 : 8, or$1.25 per mango.”
Luis said,
“I thought we had to write the ratio the other way, 8 : 10, or 0.8 mangos per dollar."
Can we write different ratios for this situation? Explain why or why not.
Solution:
Yes, this context can be modeled by both of these ratios and their associated unit rates. The context itself doesn’t determine the order of the quantities in the ratio; we choose the order depending on what we want to know.
Performance Task Scoring Rubric:
3-Point Response / The response shows complete understanding of the problem’s essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.2-Point Response / The response shows nearly complete understanding of the problem’s essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.
1-Point Response / The response shows limited understanding of the problem’s essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.
0-Point Response / The response shows insufficient understanding of the problem’s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.
Performance Task 2:
Gianna’s Job (6.RP.A.3, 6.RP.A.3a)
Gianna is paid$90 for 5 hours of work.
- At this rate, how much would Gianna make for 8 hours of work?
- At this rate, how long would Gianna have to work to make$60?
Solutions:
Solution:Making a table- This method uses a ratio table:
5 / 90
10 / 180
20 / 360
40 / 720
8 / 144
- The first row is the given information and to get to the second row we multiply both entries of the first row by 2. To get from the second to the third row of the table we multiply by 2 again. From the third to the fourth for we multiply by 2 for a third time. Now 40 hours can be divided by 5 to give 8 hours so this is the last step. There are many other possible ways to arrive at the answer with a table. For example, since
we could move from the first row to the last in one step, multiplying the first row by .
- We again make a table and this time the goal is to get$60 in the earnings column and find out how many hours it takes for Gianna to earn this amount of money. We see that 60 is not a factor of 90 so we can’t get to 60 directly by dividing by a whole number. But 60 is a factor of 180 which is 2×90 so we use this:
5 / 90
10 / 180
/ 60
- It takes Giannahours or 3 hours and 20 minutes to make$60.
Solution: Making a double number line
- We are given that Gianna makes$90 in 5 hours. We can plot this information on a double number line, with money plotted on one line and time on the other:
The goal is to use the information given to work out what dollar amount will go along with 8 hours. One way to do this would be to work out the hourly wage and then multiply by 8. This is shown below with the first step drawn in purple and the second step in blue:
To find the hourly wage we have to divide the number of given hours by 5 and so we also divide the wages by 5. Next, to find the wages for 8 hours we multiply the hourly wage by 8. There are many other alternatives. The quickest method would be to multiply the given values of money and time by .
- To find how long Gianna has to work to make$60 notice that$60 isof$90. So we can first take one third of the given values (in purple below) and then double these new values (in blue):
It takes Gianna hours or 3 and a third hours to earn$60.
Solution: Using a unit rate
- In order to find out how much Gianna makes in 8 hours, we can first find her hourly rate and then multiply by 8. Since Gianna makes$90 in 5 hours she will make$90÷5 in 1 hour. This means that Gianna makes$18 per hour. So in 8 hours she will make
- To find out how long it takes Gianna to make$60 we can find out how long it take her to make$1 and then multiply by 60. Since Gianna makes$90 in 5 hours she will make$1 in 5÷90 hours. This isof an hour. Since Gianna makes$1 inof an hour she will make$60 inhours. This is three and a third hours.
Performance Task Scoring Rubric: