Core Content Connectors by Common Core State Standards: Mathematics 5th Grade

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='''Grade 6 Overview'''=
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='''Grade 5 Overview'''=
  
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=='''Ratios and Proportional Relationships'''==
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=='''Operations and Algebraic Thinking'''==
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*'''Understand ratio concepts and use ratio reasoning to solve problems.'''
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*'''Write and interpret numerical expressions.'''
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*'''Analyze patterns and relationships.'''
  
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=='''The Number System'''==
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=='''Number and Operations in Base Ten'''==
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*'''Apply and extend previous understandings of multiplication and division to divide fractions by fractions.'''
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*'''Understand the place value system.'''
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*'''Compute fluently with multi-digit numbers and find common factors and multiples.'''
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*'''Perform operations with multi-digit whole numbers and with decimals to hundredths.'''
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*'''Apply and extend previous understandings of numbers to the system of rational numbers.'''
+
  
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=='''Expressions and Equations'''==
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=='''Numbers and Operations—Fractions'''==
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*'''Apply and extend previous understandings of arithmetic to algebraic expressions.'''
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*'''Use equivalent fractions as a strategy to add and subtract fractions.'''
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*'''Reason about and solve one-variable equations and inequalities.'''
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*'''Apply and extend previous understandings of multiplication and division to multiply and divide fractions.'''
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*'''Represent and analyze quantitative relationships between dependent and independent variables.'''
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 +
=='''Measurement and Data'''==
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*'''Convert like measurement units within a given measurement system.'''
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*'''Represent and interpret data.'''
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*'''Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.'''
  
 
=='''Geometry'''==
 
=='''Geometry'''==
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*'''Solve real-world and mathematical problems involving area, surface area, and volume.'''
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*'''Graph points on the coordinate plane to solve real-world and mathematical problems.'''
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*'''Classify two-dimensional figures into categories based on their properties.'''
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=='''Statistics and Probability'''==
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*'''Develop understanding of statistical variability.'''
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*'''Summarize and describe distributions.'''
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{|border=1
 
{|border=1
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|width = "625" style="background-color:#D9D9D9;"|'''Ratios and Proportional Relationships'''
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|width = "625" style="background-color:#D9D9D9;"|'''Operations and Algebraic Thinking'''
  
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|width = "1875" style="background-color:#D9D9D9;"|'''6.RP'''
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|width = "1875" style="background-color:#D9D9D9;"|'''5.OA'''
  
 
|-
 
|-
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| colspan=2|'''Understand ratio concepts and use ratio reasoning to solve problems.'''
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| colspan=2|'''Write and interpret numerical expressions.'''
  
 
|-
 
|-
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| colspan=2|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."''
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| colspan=2|1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
  
 
|-
 
|-
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| rowspan=5|'''''CCCs linked to 6.RP.A.1'''''
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||'''''CCCs linked to 5.OA.A.1'''''
  
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||6.NO.1f2 Write or select a ratio to match a given statement and representation.
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||5.SE.1a1 Given a real world problem, write an equation using 1 set of parentheses.
  
 
|-
 
|-
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| colspan=2|2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. ''For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.''
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||6.NO.1f31 Select or make a statement to interpret a given ratio.
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|-
 
|-
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||6.PRF.1c1 Describe the ratio relationship between two quantities for a given situation.
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||'''''CCCs linked to 5.OA.A.2'''''
  
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|-
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||5.SE.1a1 Given a real world problem, write an expression using 1 set of parentheses.
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||6.PRF.2b3 Complete a statement that describes the ratio relationship between two quantities.  
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|-
 
|-
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||6.NO.1f2 Write or select a ratio to match a given statement and representation.
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| colspan=2|'''Analyze patterns and relationships.'''
  
 
|-
 
|-
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| colspan=2|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."''
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| colspan=2|3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. ''For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.''
  
 
|-
 
|-
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| rowspan=3|'''''CCCs linked to 6.RP.A.2'''''
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| rowspan=4|'''''CCCs linked to 5.0A.B.3'''''
  
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||6.PRF.1c2 Represent proportional relationships on a line graph.
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||5.PRF.1b1 Given 2 patterns involving the same context (e.g., collecting marbles) determine the 1<sup>st</sup> 5 terms and compare the values.
  
 
|-
 
|-
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||6.PRF.2b4 Determine the unit rate in a variety of contextual situations.
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||5.PRF.1b2 When given a line graph representing two arithmetic patterns, identify the relationship between the two
  
 
|-
 
|-
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||6.NO.1f4 Find a missing value (representations, whole numbers, common fractions, decimals to hundredths place, percent) for a given ratio.
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||5.PRF.1b2 Generate or select a comparison between two graphs from a similar situation.
  
 
|-
 
|-
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| colspan=2|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.
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||6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs.
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|-
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| colspan=2|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.
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|-
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| colspan=2|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?''
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|-
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| colspan=2|c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
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|-
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| colspan=2|d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
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|-
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| rowspan=8|'''''CCCs linked to 6.RP.A.3'''''
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| |6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
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|-
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| |6.NO.1f5 Solve unit rate problems involving unit pricing.
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|-
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||6.ME.2a2 Solve one step real world measurement problems involving unit rates with ratios of whole numbers when given the unit rate (3 inches of snow falls per hour, how much in 6 hours).
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|-
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||7.NO.f6 Solve word problems involving ratios.
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|-
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||6.NO.1f1 Calculate a percent of a quantity as rate per 100.
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|-
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||6.ME.1b4 Complete a conversion table for length, mass, time, volume.
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|-
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||6.ME.1b5 Analyze table to answer questions.
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|-
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||7.NO.1h1 Identify an equivalent fraction, decimal and percent when given one of the three numbers.
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|}
 
|}
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{|border=1
 
{|border=1
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|width = "625" style="background-color:#D9D9D9;"|'''The Number System'''
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|width = "625" style="background-color:#D9D9D9;"|'''Numbers and Operations in Base Ten'''
  
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|width = "1875" style="background-color:#D9D9D9;"|'''6.NS'''
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|width = "1875" style="background-color:#D9D9D9;"|'''5.NBT'''
  
 
|-
 
|-
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| colspan=2|'''Apply and extend previous understandings of multiplication and division to divide fractions by fractions.'''
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| colspan=2|'''Understand the place value system.'''
  
 
|-
 
|-
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| colspan=2|1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. ''For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?''
+
| colspan=2|1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
  
 
|-
 
|-
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||'''''CCCs linked to 6.NS.A.1'''''
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| rowspan=2|'''''CCCs linked to 5.NBT.A.1'''''
  
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||6.NO.2c3 Solve one step, addition, subtraction, multiplication, or division problems with fractions or decimals.
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||5.NO.1a1 Compare the value of a number when it is represented in different place values of two 3 digit numbers.
  
 
|-
 
|-
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| colspan=2|'''Compute fluently with multi-digit numbers and find common factors and multiples.'''
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||5.SE.1a1 Given a real world problem, write an expression using 1 set of parentheses.
  
 
|-
 
|-
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| colspan=2|2. Fluently divide multi-digit numbers using the standard algorithm.
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| colspan=2|2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  
 
|-
 
|-
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||'''''CCCs linked to 6.NS.B.2'''''
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||'''''CCCs linked to 5.NBT.A.2'''''
  
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||6.NO.2c5 Divide multi-digit whole numbers.
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||6.NO.1i1 Identify what an exponent represents (e.g., 8³ = 8 x 8 x 8).
  
 
|-
 
|-
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| colspan=2|3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
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| colspan=2|3. Read, write, and compare decimals to thousandths.
  
 
|-
 
|-
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||'''''CCCs linked to 6.NS.B.3'''''
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| colspan=2|a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
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+
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||6.NO.2c3 Solve one step, addition, subtraction, multiplication, or division problems with fractions or decimals.
+
  
 
|-
 
|-
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| colspan=2|4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. ''For example, express 36 + 8 as 4 (9 + 2).''
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| colspan=2|b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
  
 
|-
 
|-
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||'''''CCCs linked to 6.NS.B.4'''''
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| rowspan=3|'''''CCCs linked to 5.NBT.A.3'''''
  
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||None
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||5.NO.1b1 Read, write, or select a decimal to the hundredths place.
  
 
|-
 
|-
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| colspan=2|'''Apply and extend previous understandings of numbers to the system of rational numbers.'''
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||5.NO.1b2 Read, write or select a decimal to the thousandths place.  
  
 
|-
 
|-
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| colspan=2|5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
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||5.NO.1b3 Compare two decimals to the thousandths place with a value of less than 1.
  
 
|-
 
|-
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||'''''CCCs linked to 6.NS.C.5'''''
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| colspan=2|4. Use place value understanding to round decimals to any place. Perform operations with multi-digit whole numbers and with decimals to hundredths.
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+
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||6.NO.1d4 Select the appropriate meaning of a negative number in a real world situation.
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|-
 
|-
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| colspan=2|6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
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| rowspan=3|'''''CCCs linked to 5.NBT.A.4'''''
  
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|-
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||5.NO.1b4 Round decimals to the next whole number.
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| colspan=2|a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
+
  
 
|-
 
|-
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| colspan=2|b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
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||5.NO.1b5 Round decimals to the tenths place.
  
 
|-
 
|-
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| colspan=2|c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
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||5.NO.1b6 Round decimals to the hundredths place.
  
 
|-
 
|-
−
| rowspan=6|'''''CCCs linked to 6.NS.C.6'''''
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| colspan=2|'''Perform operations with multi-digit whole numbers and with decimals to hundredths.'''
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||6.NO.1d5 Find given points between -10 and 10 on both axes of a coordinate plane.
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|-
 
|-
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||6.NO.1d6 Label points between -10 and 10 on both axes of a coordinate plane.
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| colspan=2|5. Fluently multiply multi-digit whole numbers using the standard algorithm.
  
 
|-
 
|-
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||6.NO.1d1 Identify numbers as positive or negative.
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||'''''CCCs linked to 5.NBT.B.5'''''
  
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|-
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||5.NO.2a1 Solve problems or word problems using up to three digit numbers and addition or subtraction or multiplication.
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||6.NO.1d2 Locate positive and negative numbers on a number line.
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|-
 
|-
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||6.NO.1d3 Plot positive and negative numbers on a number line.
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| colspan=2|6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  
 
|-
 
|-
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||6.NO.2e1 Determine the difference between two integers using a number line.
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| rowspan=3|'''''CCCs linked to 5.NBT.B.6'''''
  
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|-
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||5.NO.2a3 Find whole number quotients up to two dividends and two divisors.
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| colspan=2|7. Understand ordering and absolute value of rational numbers.
+
  
 
|-
 
|-
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| colspan=2|a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. ''For example interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.''
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||5.NO.2a4 Find whole number quotients up to four dividends and two divisors.
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+
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|-
+
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| colspan=2|b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. ''For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.''
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|-
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| colspan=2|c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. ''For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.''
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|-
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| colspan=2|d. Distinguish comparisons of absolute value from statements about order. ''For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.''
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|-
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||'''''CCCs linked to 6.NS.C.7'''''
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||6.NO.2e2 Compare two numbers on a number line (e.g., -2 > -9).
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|-
 
|-
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||6.NO.1e1 Determine the meaning of absolute value.
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||5.NO.2a5 Solve word problems that require multiplication or division.
  
 
|-
 
|-
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| colspan=2|8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
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| colspan=2|7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
  
 
|-
 
|-
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||'''''CCCs linked to 6.NS.C.8'''''
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||'''''CCCs linked to 5.NBT.B.7'''''
  
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||7.NO.2f4 Use a rate of change or proportional relationship to determine the points on a coordinate plane
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||5.NO.2c1 Solve 1 step problems using decimals.
  
 
|}
 
|}
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{|border=1
 
{|border=1
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|width = "625" style="background-color:#D9D9D9;"|'''Expressions and Equations'''
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|width = "625" style="background-color:#D9D9D9;"|'''Numbers and Operations--Fractions'''
  
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|width = "1875" style="background-color:#D9D9D9;"|'''6.EE'''
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|width = "1875" style="background-color:#D9D9D9;"|'''5.NF'''
  
 
|-
 
|-
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| colspan=2|'''Apply and extend previous understandings of arithmetic to algebraic expressions.'''
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| colspan=2|'''Use equivalent fractions as a strategy to add and subtract fractions.'''
  
 
|-
 
|-
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| colspan=2|1. Write and evaluate numerical expressions involving whole-number exponents.  
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| colspan=2|1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
  
 
|-
 
|-
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| rowspan=2|'''''CCCs linked to 6.EE.A.1'''''
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| rowspan=2|'''''CCCs linked to 5.NF.A.1'''''
  
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||6.NO.1i1 Identify what an exponent represents (e.g., 8³= 8 x 8 x 8).
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||5.NO.2b1 Add and subtract fractions with unlike denominators by replacing fractions with equivalent fractions (identical denominators).
  
 
|-
 
|-
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||6.NO.1i2 Solve numerical expressions involving whole number exponents.
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||5.NO.2b2 Add or subtract fractions with unlike denominators.
  
 
|-
 
|-
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| colspan=2|2. Write, read, and evaluate expressions in which letters stand for numbers.
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| colspan=2|2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
  
 
|-
 
|-
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| colspan=2|a. Write expressions that record operations with numbers and with letters standing for numbers. ''For example, express the calculation "Subtract y from 5" as 5 – y.''
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||'''''CCCs linked to 5.NF.A.2'''''
 +
 
 +
||5.NO.2c2 Solve word problems involving the addition, subtraction, multiplication or division of fractions.
  
 
|-
 
|-
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| colspan=2|b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. ''For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.''
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| colspan=2|'''Apply and extend previous understandings of multiplication and division to multiply and divide fractions.'''
  
 
|-
 
|-
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| colspan=2|c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). ''For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.''
+
| colspan=2|3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. ''For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?''
  
 
|-
 
|-
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||'''''CCCs linked to 6.EE.A.2'''''
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||'''''CCCs linked to 5.NF.B.3'''''
  
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||6.SE.1a2 Given a real world problem, write an equation using 1 set of parentheses.
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||5.NO.2b4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
  
 
|-
 
|-
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| colspan=2|3. Apply the properties of operations to generate equivalent expressions. ''For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.''
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| colspan=2|4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  
 
|-
 
|-
−
||'''''CCCs linked to 6.EE.A.3'''''
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| colspan=2|a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. ''For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)''
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||6.SE.1b2 Use properties to produce equivalent expressions
+
  
 
|-
 
|-
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| colspan=2|4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). ''For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.''
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| colspan=2|b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
  
 
|-
 
|-
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||'''''CCCs linked to 6.EE.A.4'''''
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||'''''CCCs linked to 5.NF.B.4'''''
  
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||6.SE.1b1 Evaluate whether or not both sides of an equation are equal.
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||5.NO.2b3 Multiply a fraction by a whole or mixed number.
  
 
|-
 
|-
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| colspan=2|'''Reason about and solve one-variable equations and inequalities.'''
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| colspan=2|5. Interpret multiplication as scaling (resizing), by:
  
 
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|-
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| colspan=2|5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
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| colspan=2|a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
  
 
|-
 
|-
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||'''''CCCs linked to 6.EE.B.5'''''
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| colspan=2|b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
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+
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||None
+
  
 
|-
 
|-
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| colspan=2|6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
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| rowspan=2|'''''CCCs linked to 5.NF.B.5'''''
 +
 
 +
||5.PRF.1a1 Determine whether the product will increase or decrease based on the multiplier.
  
 
|-
 
|-
−
||'''''CCCs linked to 6.EE.B.6'''''
+
||6.PRF.1a2 Determine whether or not the quotient will increase or decrease based on the divisor.
−
 
+
−
||6.PRF.2a2 Use variable to represent numbers and write expressions when solving real world problems.
+
  
 
|-
 
|-
−
| colspan=2|7. Solve real-world and mathematical problems by writing and solving equations of the form ''x'' + ''p'' = ''q'' and ''px'' = ''q'' for cases in which ''p'', ''q'' and ''x'' are all nonnegative rational numbers.
+
| colspan=2|6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
  
 
|-
 
|-
−
| rowspan=2|'''''CCCs linked to 6.EE.B.7'''''
+
||'''''CCCs linked to 5.NF.B.6'''''
  
−
||6.NO.2a6 Solve problems or word problems using up to three digit numbers and any of the four operations.
+
||5.NO.2b3 Multiply a fraction by a whole or mixed number.
  
 
|-
 
|-
−
||6.PRF.1d1 Solve real world, single step linear equations.
+
| colspan=2|7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
  
 
|-
 
|-
−
| colspan=2|8. Write an inequality of the form ''x'' > ''c'' or ''x'' < ''c'' to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form ''x'' > ''c'' or ''x'' < ''c'' have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
+
| colspan=2|a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. ''For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.''
−
 
+
−
|-
+
−
||'''''CCCs linked to 6.EE.B.8'''''
+
−
 
+
−
||6.SE.1a4 Given a real world problem, write an inequality.
+
  
 
|-
 
|-
−
| colspan=2|'''Represent and analyze quantitative relationships between dependent and independent variables.'''
+
| colspan=2|b. Interpret division of a whole number by a unit fraction, and compute such quotients. ''For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.''
  
 
|-
 
|-
−
| colspan=2|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. ''For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.''
+
| colspan=2|c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. ''For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?''
  
 
|-
 
|-
−
| rowspan=3|'''''CCCs linked to 6.EE.C.9'''''
+
| rowspan=2|'''''CCCs linked to 5.NF.B.7'''''
  
−
||6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another.
+
||5.NO.2b4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
  
 
|-
 
|-
−
||6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another.
+
||6.NO.2c4 Solve word problems involving the addition, subtraction, multiplication or division of fractions.
−
 
+
−
|-
+
−
||6.PRF.2a4 Analyze the relationships between the dependent and independent variables using graphs and tables, and relate to the equation
+
  
 
|}
 
|}
Line 350: Line 279:
  
 
{|border=1
 
{|border=1
−
|width = "625" style="background-color:#D9D9D9;"|'''Geometry'''
+
|width = "625" style="background-color:#D9D9D9;"|'''Measurement and Data'''
  
−
|width = "1875" style="background-color:#D9D9D9;"|'''6.G'''
+
|width = "1875" style="background-color:#D9D9D9;"|'''5.MD'''
  
 
|-
 
|-
−
| colspan=2|'''Solve real-world and mathematical problems involving area, surface area, and volume.'''
+
| colspan=2|'''Convert like measurement units within a given measurement system.'''
  
 
|-
 
|-
−
| colspan=2|1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
+
| colspan=2|1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. Represent and interpret data.
  
 
|-
 
|-
−
| rowspan=5|'''''CCCs linked to 6.G.A.1'''''
+
| rowspan=4|'''''CCCs linked to 5.MD.A.1'''''
  
−
||6.ME.1a2 Identify the appropriate formula (i.e., perimeter, area, volume) to use when measuring for different purposes in a real life context.
+
||5.ME.1b1 Convert measurements of time.
  
 
|-
 
|-
−
||6.ME.2a3 Apply the formula to find the area of triangles.  
+
||5.ME.1b2 Convert standard measurements of length.
  
 
|-
 
|-
−
||6.ME.2b3 Decompose complex shapes (polygon, trapezoid, pentagon) into simple shapes (rectangles, squares, triangles) to measure area.
+
||5.ME.1b3 Convert standard measurements of mass.
  
 
|-
 
|-
−
||6.GM.1d1 Find area of quadrilaterals.
+
||5.ME.2a1 Solve problems involving conversions of standard measurement units when finding area, volume, time lapse, or mass.  
  
 
|-
 
|-
−
||6.GM.1d2 Find area of triangles
+
| colspan=2|'''Represent and interpret data.'''
  
 
|-
 
|-
−
| colspan=2|2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas ''V = l w h'' and ''V = b h'' to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
+
| colspan=2|2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. ''For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.''
  
 
|-
 
|-
−
| rowspan=2|'''''CCCs linked to 6.G.A.2'''''
+
||'''''CCCs linked to 5.MD.B.2'''''
  
−
||6.ME.1a2 Identify the appropriate formula (i.e., perimeter, area, volume) to use when measuring for different purposes in a real life context.
+
||5.DPS.1c1 Collect and graph data: bar graph, line plots, picture graph (e.g., average height among 3 classrooms, # of boys and girls).
  
 
|-
 
|-
−
||6.ME.1c1 Find the area of a 2-dimensional figure and the volume of a 3-dimensional figure.
+
| colspan=2|'''Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.'''
  
 
|-
 
|-
−
| colspan=2|3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
+
| colspan=2|3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
  
 
|-
 
|-
−
| rowspan=2|'''''CCCs linked to 6.G.A.3'''''
+
| colspan=2|a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
  
−
||6.GM.1c7 Use coordinate points to draw polygons.
+
|-
 +
| colspan=2|b. A solid figure which can be packed without gaps or overlaps using ''n'' unit cubes is said to have a volume of ''n'' cubic units.
  
 
|-
 
|-
−
||6.GM.1c8 Use coordinate points to find the side lengths of polygons that are horizontal or vertical.
+
||'''''CCCs linked to 5.MD.C.3'''''
 +
 
 +
||5.ME.2b1 Use filling and multiplication to determine volume.  
  
 
|-
 
|-
−
| colspan=2|4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
+
| colspan=2|'''Represent and interpret data.'''
  
 
|-
 
|-
−
||'''''CCCs linked to 6.G.A.4'''''
+
| colspan=2|4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
  
−
||7.GM.1h2 Find the surface area of three dimensional figures using nets of rectangles or triangles.  
+
|-
 +
||'''''CCCs linked to 5.MD.C.4'''''
  
−
|}
+
||5.ME.2b1 Use filling and multiplication to determine volume.
  
 +
|-
 +
| colspan=2|5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
  
−
 
+
|-
−
 
+
| colspan=2|a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
−
{|border=1
+
−
|width = "625" style="background-color:#D9D9D9;"|'''Statistics and Probability'''
+
−
 
+
−
|width = "1875" style="background-color:#D9D9D9;"|'''6.SP'''
+
  
 
|-
 
|-
−
| colspan=2|'''Develop understanding of statistical variability.'''
+
| colspan=2|b. Apply the formulas ''V = l × w × h'' and ''V = b × h'' for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
  
 
|-
 
|-
−
| colspan=2|1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. ''For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.''
+
| colspan=2|c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
  
 
|-
 
|-
−
||'''''CCCs linked to 6.SP.A.1'''''
+
| rowspan=2|'''''CCCs linked to 5.MD.C.5'''''
  
−
||6.DPS.1a2 Identify statistical questions and make a plan for data collection.
+
||6.ME.2b3 Decompose complex 3-D shapes into simple 3-D shapes to measure volume.
  
 
|-
 
|-
−
| colspan=2|2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
+
||5.ME.2b2 Apply formula to solve one step problems involving volume.
  
−
|-
+
|}
−
| rowspan=2|'''''CCCs linked to 6.SP.A.2'''''
+
  
−
||6.DPS.1d4 Find the range of a given data set.
 
  
−
|-
 
−
||6.DPS.1d6 Explain or identify what the mode represents in a set of data.
 
  
−
|-
 
−
| colspan=2|3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
 
  
−
|-
+
{|border=1
−
| rowspan=4|'''''CCCs linked to 6.SP.A.3'''''
+
|width = "625" style="background-color:#D9D9D9;"|'''Geometry'''
  
−
||5.DPS.1d1 Select an appropriate statement about the range of the data for a given graph (bar graph, line plot) (i.e., range of data) up to 10 points.
+
|width = "1875" style="background-color:#D9D9D9;"|'''5.G'''
  
 
|-
 
|-
−
||5.DPS.1e1 Use measures of central tendency to interpret data including overall patterns in the data.
+
| colspan=2|'''Graph points on the coordinate plane to solve real-world and mathematical problems.'''
  
 
|-
 
|-
−
||6.DPS.1d2 Solve for mean of a given data set.  
+
| colspan=2|1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
  
 
|-
 
|-
−
||6.DPS.1d5 Explain or identify what the mean represents in a set of data.
+
| rowspan=5|'''''CCCs linked to 5.G.A.1'''''
  
−
|-
+
||5.GM.1c1 Locate the x and y axis on a graph.
−
| colspan=2|'''Summarize and describe distributions.'''
+
  
 
|-
 
|-
−
| colspan=2|4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
+
||5.GM.1c2 Locate points on a graph.
  
 
|-
 
|-
−
| rowspan=2|'''''CCCs linked to 6.SP.B.4'''''
+
||5.GM.1c3 Use order pairs to graph given points.
−
 
+
−
||6.DPS.1c2 Collect and graph data: bar graph, line plots, dot plots, histograms.
+
  
 
|-
 
|-
−
||7.DPS.1g1 Graph continuous data using line graphs, histograms, or dot plots.
+
||6.GM.1c4 Locate points on a graph.
  
 
|-
 
|-
−
| colspan=2|5. Summarize numerical data sets in relation to their context, such as by:
+
||6.GM.1c5 Use order pairs to graph given points.
  
 
|-
 
|-
−
| colspan=2|a. Reporting the number of observations.
+
| colspan=2|2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Classify two-dimensional figures into categories based on their properties.
  
 
|-
 
|-
−
| colspan=2|b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
+
||'''''CCCs linked to 5.G.A.2'''''
 +
 
 +
||6.GM.1c6 Find coordinate values of points in the context of a situation.
  
 
|-
 
|-
−
| colspan=2|c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
+
| colspan=2|'''Classify two-dimensional figures into categories based on their properties.'''
  
 
|-
 
|-
−
| colspan=2|d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
+
| colspan=2|3. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. ''For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.''
  
 
|-
 
|-
−
| rowspan=6|'''''CCCs linked to 6.SP.B.5'''''
+
||'''''CCCs linked to 5.G.B.3'''''
  
−
| |6.DPS.1d3 Select statement that matches mean, mode, and spread of data for 1 measure of central tendency for a given data set.  
+
||5.GM.1a1 Recognize properties of simple plane figures.
  
 
|-
 
|-
−
||7.DPS.1i1 Solve for the median of a given data set.  
+
| colspan=2|4. Classify two-dimensional figures in a hierarchy based on properties.
  
 
|-
 
|-
−
||6.DPS.1d7 Explain or identify what the median represents in a set of data.
+
||'''''CCCs linked to 5.G.B.4'''''
  
−
|-
+
| rowspan=2|5.GM.1b1 Distinguish plane figures by their properties.
−
 
+
−
||6.DPS.1e2 Use measures of central tendency to interpret data including overall patterns in the data.
+
−
 
+
−
|-
+
−
| |8.DPS.1i4 Identify outliers, range, mean, median, and mode.
+
  
 
|}
 
|}
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[[Category: CCCs]]
 
[[Category: CCCs]]
 
[[Category: CCSS]]
 
[[Category: CCSS]]
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[[Category:Middle]]
 
 
[[Category:Math]]
 
[[Category:Math]]
 +
[[Category:Elementary]]

Latest revision as of 13:53, 23 July 2014

BACK TO Core Content Connectors


Contents

[edit] Grade 5 Overview

[edit] Operations and Algebraic Thinking

  • Write and interpret numerical expressions.
  • Analyze patterns and relationships.

[edit] Number and Operations in Base Ten

  • Understand the place value system.
  • Perform operations with multi-digit whole numbers and with decimals to hundredths.

[edit] Numbers and Operations—Fractions

  • Use equivalent fractions as a strategy to add and subtract fractions.
  • Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

[edit] Measurement and Data

  • Convert like measurement units within a given measurement system.
  • Represent and interpret data.
  • Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

[edit] Geometry

  • Graph points on the coordinate plane to solve real-world and mathematical problems.
  • Classify two-dimensional figures into categories based on their properties.



Operations and Algebraic Thinking 5.OA
Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
CCCs linked to 5.OA.A.1 5.SE.1a1 Given a real world problem, write an equation using 1 set of parentheses.
2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
CCCs linked to 5.OA.A.2 5.SE.1a1 Given a real world problem, write an expression using 1 set of parentheses.
Analyze patterns and relationships.
3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
CCCs linked to 5.0A.B.3 5.PRF.1b1 Given 2 patterns involving the same context (e.g., collecting marbles) determine the 1st 5 terms and compare the values.
5.PRF.1b2 When given a line graph representing two arithmetic patterns, identify the relationship between the two
5.PRF.1b2 Generate or select a comparison between two graphs from a similar situation.
6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs.



Numbers and Operations in Base Ten 5.NBT
Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
CCCs linked to 5.NBT.A.1 5.NO.1a1 Compare the value of a number when it is represented in different place values of two 3 digit numbers.
5.SE.1a1 Given a real world problem, write an expression using 1 set of parentheses.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
CCCs linked to 5.NBT.A.2 6.NO.1i1 Identify what an exponent represents (e.g., 8³ = 8 x 8 x 8).
3. Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
CCCs linked to 5.NBT.A.3 5.NO.1b1 Read, write, or select a decimal to the hundredths place.
5.NO.1b2 Read, write or select a decimal to the thousandths place.
5.NO.1b3 Compare two decimals to the thousandths place with a value of less than 1.
4. Use place value understanding to round decimals to any place. Perform operations with multi-digit whole numbers and with decimals to hundredths.
CCCs linked to 5.NBT.A.4 5.NO.1b4 Round decimals to the next whole number.
5.NO.1b5 Round decimals to the tenths place.
5.NO.1b6 Round decimals to the hundredths place.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard algorithm.
CCCs linked to 5.NBT.B.5 5.NO.2a1 Solve problems or word problems using up to three digit numbers and addition or subtraction or multiplication.
6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
CCCs linked to 5.NBT.B.6 5.NO.2a3 Find whole number quotients up to two dividends and two divisors.
5.NO.2a4 Find whole number quotients up to four dividends and two divisors.
5.NO.2a5 Solve word problems that require multiplication or division.
7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
CCCs linked to 5.NBT.B.7 5.NO.2c1 Solve 1 step problems using decimals.



Numbers and Operations--Fractions 5.NF
Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
CCCs linked to 5.NF.A.1 5.NO.2b1 Add and subtract fractions with unlike denominators by replacing fractions with equivalent fractions (identical denominators).
5.NO.2b2 Add or subtract fractions with unlike denominators.
2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
CCCs linked to 5.NF.A.2 5.NO.2c2 Solve word problems involving the addition, subtraction, multiplication or division of fractions.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
CCCs linked to 5.NF.B.3 5.NO.2b4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
CCCs linked to 5.NF.B.4 5.NO.2b3 Multiply a fraction by a whole or mixed number.
5. Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
CCCs linked to 5.NF.B.5 5.PRF.1a1 Determine whether the product will increase or decrease based on the multiplier.
6.PRF.1a2 Determine whether or not the quotient will increase or decrease based on the divisor.
6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
CCCs linked to 5.NF.B.6 5.NO.2b3 Multiply a fraction by a whole or mixed number.
7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
CCCs linked to 5.NF.B.7 5.NO.2b4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
6.NO.2c4 Solve word problems involving the addition, subtraction, multiplication or division of fractions.



Measurement and Data 5.MD
Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. Represent and interpret data.
CCCs linked to 5.MD.A.1 5.ME.1b1 Convert measurements of time.
5.ME.1b2 Convert standard measurements of length.
5.ME.1b3 Convert standard measurements of mass.
5.ME.2a1 Solve problems involving conversions of standard measurement units when finding area, volume, time lapse, or mass.
Represent and interpret data.
2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
CCCs linked to 5.MD.B.2 5.DPS.1c1 Collect and graph data: bar graph, line plots, picture graph (e.g., average height among 3 classrooms, # of boys and girls).
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
CCCs linked to 5.MD.C.3 5.ME.2b1 Use filling and multiplication to determine volume.
Represent and interpret data.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
CCCs linked to 5.MD.C.4 5.ME.2b1 Use filling and multiplication to determine volume.
5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
CCCs linked to 5.MD.C.5 6.ME.2b3 Decompose complex 3-D shapes into simple 3-D shapes to measure volume.
5.ME.2b2 Apply formula to solve one step problems involving volume.



Geometry 5.G
Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
CCCs linked to 5.G.A.1 5.GM.1c1 Locate the x and y axis on a graph.
5.GM.1c2 Locate points on a graph.
5.GM.1c3 Use order pairs to graph given points.
6.GM.1c4 Locate points on a graph.
6.GM.1c5 Use order pairs to graph given points.
2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Classify two-dimensional figures into categories based on their properties.
CCCs linked to 5.G.A.2 6.GM.1c6 Find coordinate values of points in the context of a situation.
Classify two-dimensional figures into categories based on their properties.
3. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
CCCs linked to 5.G.B.3 5.GM.1a1 Recognize properties of simple plane figures.
4. Classify two-dimensional figures in a hierarchy based on properties.
CCCs linked to 5.G.B.4 5.GM.1b1 Distinguish plane figures by their properties.
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