Concept: Fractions can mean different things and be modeled in different ways: part of a set, part of a region, and as a measure.
Skills: Identify that a unit fraction is one part of a whole; indicate that the more parts a whole is divided into, the smaller the parts will be; use partitioning and iterations to represent the unit fractions; compare two unit fractions.
Big Idea: A fractional part is equal to, less than, or greater than one whole.
Essential Questions: How can I represent these fractions? What is the relationship between the two fractions? Are they equivalent? Which fraction is larger/smaller? Initial Precursor: •Recognize wholeness •Recognize a unit •Recognize parts of a given whole or unit
Initial Precursor: In order to compare unit fractions, students need to gain experience with parts and wholes. This concept can be taught in every area of mathematics (i.e., sets, number sense, counting, operations, patterns, measurement, data analysis, geometry, and algebra). Educators can start by having students work with sets, taking whole sets and breaking them into parts based on attributes. When counting, label what has been counted (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a row of 5 dots and a row of 4 dots are parts or subsets of 9).
Distal Precursor: •Model equal part •Partition any shape into equal parts
Distal Precursor: As students begin to develop the understanding of sets and numbers, educators highlight the differences between sets on the basis of overall area or discrete number using the words "more," "less," and "equal." Provide students with multiple opportunities to count and compare a wide variety of sets with an increasing number of items, label the sets (e.g., 8 balls, 12 bears, 15 blocks), and move items in and out of the sets, labeling and counting them again (e.g., "You just said this set has 11 cubes; if I take two cubes, how many will you have?"). Being able to partition shapes requires students to recognize a unit and recognize when basic objects are in whole and part forms. Work on this understanding by giving students an opportunity to observe, feel, or otherwise interact with objects and shapes in their whole and part forms. The general goal is to explore the differences between whole units or objects and parts of units or objects. As students explore shapes, label them and describe them as whole or part. Have students build (construct) and take apart (deconstruct) shapes.
Target •Explain relationships between unit fractions.
Successor •Explain numerator •Explain denominator •Compare fractions using models •Decompose a fraction into a sum of unit fractions with the same denominator •Add fractions with common denominators
Concept: Problems can be solved using various operations.
Skills: Use the values in a division equation to find the number of groups that can be made or the number of items in each group using the strategy of fair or equal shares; solve multiplication problems using 2 values whose product is less than or equal to 50; use concrete objects to prove the answer; use a calculator to prove the answer.
Big Idea: Some problems involving joining equal groups, separating equal groups, comparison, or combinations can be solved using multiplication; others can be solved using division.
Essential Questions: How can I make equal groups from this one large group? How do I know this is a fair share? What is the product? How can I solve this multiplication/division problem using objects? How can I solve this multiplication/division problem using a calculator?
Initial Precursor
Recognize separateness
Recognize set
Recognize subset
Initial Precursor: In order to understand division, students must learn to organize items into groups or sets based on a common characteristic such as size, color, shape, or texture. Students working at the Initial Precursor linkage level learn how to sort items by separating a group of items into two groups (e.g., music I like and music I don't like; red fidgets and black fidgets). As students become more comfortable sorting items into sets, they are encouraged to communicate their thought process by identifying and naming the characteristic that determines the set (e.g., color, length). Activities that require students to engage actively with the items will foster understanding of set, subsets, and separateness.
Distal Precursor
Partition sets
Partition sets into equal subsets
Distal Precursor: As students' understanding of labeling and counting sets develops, they begin working on adding and taking away items from a set. Educators provide opportunities for students to work on developing an understanding of partitioning by actively participating in one-to-one distribution of objects to person, objects to objects, and objects to available space (e.g., giving each person in the group two pencils; given four counters they can line up, then four more counters in front of or on top of the first set; given three chairs at a table, the students would place a cup on the table for each available chair) and taking equal shares away (subtracting) from each person, object, or space. Educators provide opportunities for students to connect their understanding of subtraction (starting with the whole and taking away a part) to repeated subtraction. For example, if the educator has 12 balls, and each team gets 4 balls, how many teams will there be? By subtracting 4 from the whole repeatedly, they can make 3 equal sets, so there are 3 teams.
Concept: Problems can be solved using various operations.
Skills: Use the values in a division equation to find the number of groups that can be made or the number of items in each group using the strategy of fair or equal shares; solve multiplication problems using 2 values whose product is less than or equal to 50; use concrete objects to prove the answer; use a calculator to prove the answer.
Big Idea: Some problems involving joining equal groups, separating equal groups, comparison, or combinations can be solved using multiplication; others can be solved using division.
Essential Questions: How can I make equal groups from this one large group? How do I know this is a fair share? What is the product? How can I solve this multiplication/division problem using objects? How can I solve this multiplication/division problem using a calculator?
Initial Precursor: In order to solve multiplication problems, students must learn to organize items into groups or sets based on a common characteristic such assize, color, shape, or texture. Students learn how to sort items by separating a group of items into two groups (e.g., music I like and music I don't like; red fidgets and black fidgets). As students become more comfortable sorting items into sets, they are encouraged to communicate their thought process by identifying and naming the characteristic that determines the set (e.g., color, length). Activities that require students to engage actively with the items will foster understanding of set, subsets, and separateness.
Distal Precursor • Explain repeated addition • Represent repeated addition with an equation • Solve repeated addition problems
Distal Precursor: As students' understanding of labeling and counting sets develops, they begin working on adding items to a set and combining sets to create a new set. Additionally, students work on developing an understanding of equal shares by actively participating in one-to-one distribution of objects to person, objects to objects, and objects to available space (e.g., giving each person in the group two pencils; given four counters, they would line up four more counters in front of or on top of the first set; given three chairs at a table, the students would place a cup on the table for each available chair). As students learn to work with sets and connect their understanding of equal shares to sets, educators provide students experience with combining multiple sets (e.g., 3 sets with 4 counters each) and represent the problem (e.g., 4 + 4 + 4 = ?). Students also learn to represent the problem in writing (e.g., students are shown 4 equal sets each with 2 counters. The students then count the first set and write a 2 or indicate 2, then write or indicate the plus sign. The students repeat for all 4 sets and then indicate the equal sign and solve the problem).
Proximal Precursor •Demonstrate the concept of multiplication
Target •Multiply by 1, 2, 3, 4, and/or 5
Successor •Apply the relationship between multiplication and division •Divide by 1, 2, 3, 4, and/or 5
Concept: Both positive and negative numbers represent a distance from zero on the number line.
Skills: Identify positive and negative numbers on a number line; identify real-world examples for the use of positive and negative numbers (e.g., temperature, owing money, working with a budget, elevations below sea level, the basement floor of a building, diving under water); explain that zero is the value between positive and negative numbers; show the direction of movement on a number line when working with positive and negative numbers.
Big Idea: Positive numbers are greater than zero. Negative numbers are less than zero and have a negative sign (−) in front of them. A negative number is the opposite of a positive number of the same size.
Essential Questions: Where can I find this number on a number line? Does this number have a positve or negative value? What are some examples I can use to show negative and positive numbers? If I start with a positive number and then add a negative number, what direction on the number line will I move? How far is this number from zero?
Initial Precursor •Recognize separateness •Recognize set
Initial Precursor: In order to use positive and negative numbers, students need to gain experience with creating sets. Educators can help students learn this by providing students with opportunities to take a set of objects (e.g., tiles, linking cubes, buttons) and separate them based on a given characteristic (e.g., shape, color, size) into two distinct sets. Then encourage them to separate them again based on another characteristic.
Distal Precursor •Count all objects in a set or subset •Recognize different number of •Recognize same number of •Recognize fewer number of •Recognize more number of
Distal Precursor: As students begin to develop the understanding of sets and numbers, educators highlight the differences between sets on the basis of overall area or discrete number using the words "same," "different," "fewer," and "more." Provide students with multiple opportunities to count and compare a wide variety of sets with an increasing number of items, label the set (e.g., 8 balls, 12 bears, 15 blocks), then move items in and out of the sets, labeling and counting them again (e.g., "You just said this set has 11 cubes; if I take two cubes, how many will you have?").
Proximal Precursor •recognize opposite numbers
Target •Use positive and negative numbers in real-world contexts
Successor •Relate the meaning of 0 to positive and negative numbers in real-world contexts •Explain inequalities from real world contexts
Concept: Numbers can be represented, displayed, converted, and compared.
Skills: Add fractions with like denominators; solve multiplication problems; solve divisions problems; convert a fraction with denominator of 10 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the sum of two fractions? Which part of the fractions do I add? Why do I not add the denominators? What is the product of this multiplication problem? What model can I use to help me solve this multiplication problem? What are the parts of division problem? What model can I use to help me solve this division problem? How can I express a fraction as a decimal? Which tenth is larger/smaller(from a real world example)?
Initial Precursor: Adding fractions requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a row of 5 dots and a row of 4 dots are parts or subsets of 9).
Distal Precursor •Recognize parts of a given whole or a unit
Distal Precursor: As students begin to understand labeling, counting small sets, and recognizing wholes and parts of objects and sets, use a variety of tools (e.g., ten-frames, egg cartons, a collection of items in a category [clothes: shoes, socks, pants], your hands) to label and count the sets, and label and count the subsets.
Proximal Precursor •Explain the concept of addition and subtraction of fractions •Decompose a fraction into a sum of unit fractions with the same denominator
Target •Add fractions with common denominators
Successor •Add or subtract fractions with denominators of 10 and 100
Concept: Numbers can be represented, displayed, converted, and compared.
Skills: Add fractions with like denominators; solve multiplication problems; solve divisions problems; convert a fraction with denominator of 10 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the sum of two fractions? Which part of the fractions do I add? Why do I not add the denominators? What is the product of this multiplication problem? What model can I use to help me solve this multiplication problem? What are the parts of division problem? What model can I use to help me solve this division problem? How can I express a fraction as a decimal? Which tenth is larger/smaller(from a real world example)?
Initial Precursor
Recognize separateness
Recognize set
Initial Precursor: Solving multiplication problems requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, they label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a row of 5 dots and a row of 4 dots are parts or subsets of 9).
Distal Precursor
Solve repeated addition problems
Represent repeated addition with an equation
Explain repeated addition
Distal Precursor: As students' understanding of labeling and counting sets develops, they will begin working on adding items to a set and combining sets to create a new set. Additionally, students will work on developing an understanding of equal shares by actively participating in one-to-one distribution of objects to person, objects to objects, and objects to available space (e.g., giving each person in the group two pencils; given four counters, they would line up four more counters in front of or on top of the first set; given three chairs at a table, the student would place a cup on the table for each available chair). As students learn to work with sets and connect their understanding of equal shares to sets, educators will provide students experience with combining multiple sets (e.g., 3 sets with 4 counters each) and represent the problem (e.g., 4 + 4 + 4 = ?). Students will also learn to represent the problem using a pencil or their communication system (e.g., the student is shown 4 equal sets each with 2 counters. The student counts the first set and writes a 2 or indicates 2, then writes or indicates the plus sign. The student repeats for all 4 sets and then indicates the equal sign and solves the problem.).
Proximal Precursor
Demonstrate the concept of multiplication
Target
Multiply by 1, 2, 3, 4, 5, 6, 7, 8, 9, and/or 10
Successor
Divide by 1, 2, 3, 4, 5, 6, 7, 8, 9, and/or 10
Apply the relationship between multiplication and division
Concept: Numbers can be represented, displayed, converted, and compared.
Skills: Add fractions with like denominators; solve multiplication problems; solve divisions problems; convert a fraction with denominator of 10 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the sum of two fractions? Which part of the fractions do I add? Why do I not add the denominators? What is the product of this multiplication problem? What model can I use to help me solve this multiplication problem? What are the parts of division problem? What model can I use to help me solve this division problem? How can I express a fraction as a decimal? Which tenth is larger/smaller(from a real world example)?
Initial Precursor
Recognize subset
Recognize set
Recognize separateness
Initial Precursor: In order to understand division, students must learn to organize items into groups/sets based on a common characteristic such as size, color, shape, or texture. Students learn how to sort items by separating a group of items into two groups (e.g., music I like/music I don't like; red fidgets/black fidgets). As students gain comfort sorting items into sets, they are encouraged to use their language to convey their thought process by identifying and naming the characteristic that determines the set (e.g., color, length). Activities that require students to engage actively with the items will foster understanding of set, subsets, and separateness.
Distal Precursor
Solve repeated subtraction problems
Represent repeated subtraction with an equation
Explain repeated subtraction
Distal Precursor: As students' understanding of labeling and counting sets develops, they will begin working on adding and taking away items from a set. Educators provide opportunities for students to work on developing an understanding of equal shares by actively participating in one-to-one distribution of objects to person, objects to objects, and objects to available space (e.g., giving each person in the group two pencils; given four counters they would line up, then four more counters in front of or on top of the first set; given three chairs at a table, the student would place a cup on the table for each available chair) and taking equal shares away (subtracting) from each person, object, or space. Educators will provide opportunities for students to connect their understanding of subtraction (starting with the whole and taking away a part) to repeated subtraction. For example, if the educator has 12 balls, and each team gets 4 balls, how many teams will there be? By subtracting 4 from the whole, we made 3 equal sets so there are 3 teams.
Proximal Precursor
Demonstrate the concept of division
Target
Divide by 1, 2, 3, 4, 5, and/or 10
Successor
Explain the relationship between multiplication and division
Concept: Numbers can be represented, displayed, converted, and compared.
Skills: Add fractions with like denominators; solve multiplication problems; solve divisions problems; convert a fraction with denominator of 10 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the sum of two fractions? Which part of the fractions do I add? Why do I not add the denominators? What is the product of this multiplication problem? What model can I use to help me solve this multiplication problem? What are the parts of division problem? What model can I use to help me solve this division problem? How can I express a fraction as a decimal? Which tenth is larger/smaller(from a real world example)?
Initial Precursor Initial Precursor: Expressing a fraction as decimal requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, they label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a set of 9 is part of 10) Recognize separateness Recognize set
Distal Precursor Recognize whole on a set model
Distal Precursor: As students work toward greater understanding of sets, educators will provide students with many set models (see below) of fractions using the same unit fraction, either halves, thirds, fourths, or tenths. Students will work on identifying the whole.
Proximal Precursor Target Explain the decimal point Represent a fraction with a denominator of 10 as a decimal
Successor Recognize tenths in a set model Recognize one tenth in a set model Explain place value for tenths Compare two decimals to tenths using symbols
EE.7.NS.3. Compare quantities represented as decimals in real-world examples to tenths
Concept: Numbers can be represented, displayed, converted, and compared.
Skills: Add fractions with like denominators; solve multiplication problems; solve divisions problems; convert a fraction with denominator of 10 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the sum of two fractions? Which part of the fractions do I add? Why do I not add the denominators? What is the product of this multiplication problem? What model can I use to help me solve this multiplication problem? What are the parts of division problem? What model can I use to help me solve this division problem? How can I express a fraction as a decimal? Which tenth is larger/smaller(from a real world example)?
Initial Precursor •Recognize separateness •Recognize set •Recognize subset
Initial Precursor: Adding fractions requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. Educators present a set, label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a row of 5 dots and a row of 4 dots are parts or subsets of 9).
Distal Precursor •Recognize one tenth in a set model •Recognize tenths in a set model
Distal Precursor: As students begin to understand labeling, counting small sets, and recognizing wholes and parts of objects and sets, use set models to provide a wide variety of sets of 10 to model tenths (e.g., individual shapes to match the fraction: “I have 10 cubes in my bag, 1/10 of them are blue”).
Proximal Precursor •Represent a decimal to tenths as a fraction
Target •Compare two decimals to tenths using symbols
Successor •Compare two decimals to hundredths using symbols
Concept: Division of whole into parts can be represented by fractions and decimals.
Skills: Identify when two fractions are divided into an equal number of parts (like denominators); subtract fractions with like denominators; convert a fraction with denominator of 100 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the difference of two fractions? Which part of the fractions do I subtract? Why do I not subtract the denominators? How can I express a fraction as a decimal? Which hundredths is larger/smaller (from a real world example)?
Initial Precursor: Subtracting fractions requires students to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, they label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a set of 9 is part of 10).
Distal Precursor Recognize parts of a given whole or unit
Distal Precursor: As students work toward greater understanding of sets, educators provide students with many set models (see below)of fractions using the same unit fraction: either halves, thirds, fourths, or tenths. Students will work on identifying the whole.
Proximal Precursor Decompose a fraction into a sum of unit fractions with the same denominator Explain the concept of addition and subtraction of fractions
Target Subtract fractions with common denominators
Successor Add or subtract fractions with denominators of 10 and 100
Concept: Division of whole into parts can be represented by fractions and decimals.
Skills: Identify when two fractions are divided into an equal number of parts (like denominators); subtract fractions with like denominators; convert a fraction with denominator of 100 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the difference of two fractions? Which part of the fractions do I subtract? Why do I not subtract the denominators? How can I express a fraction as a decimal? Which hundredths is larger/smaller (from a real world example)?
Initial Precursor • Recognize separateness • Recognize set
Initial Precursor: Converting a fraction to a decimal requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, they label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a row of 5 dots and a row of 4 dots are parts or subsets of 9).
Distal Precursor • Partition sets into equal subsets • Explain unit fraction
Distal Precursor: As students become more adept at tracking discrete objects, they will begin working on one-to-one distribution of objects to person, objects to objects, and objects to available space (e.g., giving each person in the group a pencil; given four counters, they would line up four more counters in front of or on top of the first set; given three chairs at a table, the student would place a cup on the table for each available chair). As students understanding of one-to-one distribution develops, provide students many opportunities to recognize equivalence in sets with same items and then sets with differing items. As students work on all these skills and concepts, continue to draw their attention to parts and wholes.
Proximal Precursor • Explain the decimal point • Represent a fraction with a denominator of 10 as a decimal
Target • Represent a fraction with a denominator of 100 as a decimal
Successor • Compare two decimals to the tenths using symbols • Compare two decimals to hundredths using symbols
EE.8.NS.2.b. Compare quantities represented as decimals in real-world examples to hundredths.
Concept: Division of whole into parts can be represented by fractions and decimals.
Skills: Identify when two fractions are divided into an equal number of parts (like denominators); subtract fractions with like denominators; convert a fraction with denominator of 100 to a decimal; compare decimals in real-world examples.
Big Idea: The concepts and properties of addition, subtraction, multiplication, and division are the same whether using whole numbers, fractions, or decimals.
Essential Questions: What is the difference of two fractions? Which part of the fractions do I subtract? Why do I not subtract the denominators? How can I express a fraction as a decimal? Which hundredths is larger/smaller (from a real world example)?
Initial Precursor Recognize separateness
Initial Precursor: Representing fractions as decimals requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, they label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. Use tools like the ten-frame to point out whole and parts (e.g., a row of 5 dots and a row of 4 dots are parts or subsets of 9).
Distal Precursor Recognize one tenth in a set model Recognize tenths in a set model
Distal Precursor: As students begin to understand labeling, counting small sets, and recognizing wholes and parts of objects and sets, use set models to provide a wide variety of sets of 10 to model tenths (e.g., for individual shapes to match the fraction, say, "I have 10cubes in my bag, 1/10 of them are blue.").
Proximal Precursor Represent a decimal to tenths as a fraction Represent a decimal to hundredths as a fraction
Target Compare two decimals to hundredths using symbols
Successor Compare two decimals to thousandths and beyond using symbols