Author: Alissa.Ouderkirk

Rational Numbers

Unit description:  In this unit the students will learn to develop the concept of opposite numbers and absolute values, and that zero is its own opposite. They will use positive and negative numbers to represent real-world quantities and compare and order integers and rational numbers with and without number lines. The students will describe the relationship between rational numbers in real-world contexts through comparison and using understanding of absolute value. The students will learn to plot points in all four quadrants, find the distance between points, identify reflections across both axes, and create polygons. 

Download the complete Rational Numbers framework to customize for your own planning.

Essential Outcomes of the Unit

The Number System 

Apply and extend previous understandings of numbers to the system of rational numbers.

• 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative numbers to represent quantities in real world contexts, explaining the meaning of 0 in each situation. 
• 6.NS.6 Understand a rational number as a point on the number line. Use number lines and coordinate axes to represent points on a number line and in the coordinate plane with negative number coordinates.
• 6.NS.6a 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, and that 0 is its own opposite.
• 6.NS.6c Find and position integers and other rational numbers on a horizontal or vertical number line. Find and position pairs of integers and other rational numbers on a coordinate plane.
• 6.NS.7 Understand ordering and absolute value of rational numbers. 

Other Standards Addressed in the Unit

The Number System 

Apply and extend previous understandings of numbers to the system of rational numbers

• 6.NS.6b 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.
• 6.NS.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line.
• 6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts
• 6.NS.7c 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.
• 6.NS.7d Distinguish comparisons of absolute value from statements about order. 
• 6.NS.8 Solve real-world and mathematical problems by graphing points on a coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Geometry

Solve real-world and mathematical problems involving area, surface area, and volume

• 6.G.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.

Essential Questions and Big Ideas

How are positive and negative numbers used?

• Quantities having more or less than zero are described using positive and negative numbers.

How do rational numbers relate to integers?

• Number lines are visual models used to represent the density principle: between any two whole numbers are many rational numbers, including decimals and fractions.
• The rational numbers can extend to the left or to the right on the number line, with negative numbers going to the left of zero, and positive numbers going to the right of zero.

What is modeled on the coordinate plane?

• The coordinate plane is a tool for modeling real-world and mathematical situations and for solving problems.

Geometry and Fractions/Money and Time

Unit description: In this unit students will learn about two-dimensional shapes, time and money. They will learn to describe and classify shapes as polygons or non-polygons. They will decompose and combine shapes to make other shapes and they analyze two-dimensional shapes to develop a foundation for understanding area, congruence, similarity, and fractions in later grades. Students will work to solve problems involving money up to $1 including counting mixed coins and making change. They will identify all coins and be able to arrange them in order from greatest to least value. Students will also be able to tell time using both analog and digital clocks to the nearest 5 minute and identify a.m. and p.m.

Download the complete Geometry and Fractions/Money and Time framework to customize for your own planning.

Essential Outcomes of the Unit

Geometry- Reason with shapes and their attributes.

• 2.G.1 Classify two-dimensional figures as polygons or non-polygons.
• 2.G.3 Partition circles and rectangles into two, three, or four equal shares. Describe the shares using the words halves, thirds, half of, a third of, etc. Describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Measurement and Data- Work with time and money.

• 2.MD.7 Tell and write time from analog and digital clocks in five minute increments, using a.m. and p.m. Develop an understanding of common terms, such as, but not limited to, quarter past, half past, and quarter to.
• 2.MD.8a Count a mixed collection of coins whose sum is less than or equal to one dollar.

Other Standards Addressed in the Unit

Geometry- Reason with shapes and their attributes.

• 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them

Measurement and Data- Work with time and money.

• 2.MD.8b Solve real world and mathematical problems within one dollar involving quarters, dimes, nickels, and pennies, using the ¢ (cent) symbol appropriately.

Essential Questions and Big Ideas

How do I tell time? 

• Time is told in hours and minutes.  
• An hour is made of 60 minutes. 
• In a day there are 24 hours.  
• Hours are broken into AM (morning) and PM (afternoon).  

How do I count coins?  

• A penny is worth 1 cent. 
• A nickel is worth 5 cents
• A dime is worth 10 cents. 
• A quarter is worth 25 cents. 
• When counting coins, it is efficient to start with the largest value.  

How do we reason with shapes and their attributes?

• Polygons are closed two-dimensional figures made of line segments.
• Shapes can be combined to make new shapes called composite shapes.

What does dividing shapes into equal shares tell us?

• Understanding equal shares helps us understand area.
• Understanding equal shares helps us understand fractions.

Download the complete Geometry and Fractions/Money and Time framework to customize for your own planning.

Classifying 2-D Shapes

Unit description: Students will build on their work in third grade classifying triangles, quadrilaterals, pentagons, and hexagons. Students will differentiate between acute, obtuse, and right angles, as well as parallel, perpendicular, and intersecting lines. Students will be able to classify triangles as acute, obtuse, or right, and quadrilaterals as parallelograms, trapezoids, rectangles, rhombi, or squares. Students will also consider symmetry.

Essential Outcomes of the Unit

Geometry- Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

• 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
• 4.G.2a. Identify and name triangles based on angle size (right, obtuse, acute).
• 4.G.2b Identify and name all quadrilaterals with 2 pairs of parallel sides as parallelograms.
• 4.G.2c Identify and name all quadrilaterals with four right angles as rectangles.
• 4.G.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Essential Questions and Big Ideas

How can I describe lines and angles?

• Lines can be described using the terms parallel, perpendicular, and intersecting.
• Angles can be described using the terms acute, obtuse, and right.

How can I classify triangles?

• Triangles can be classified as acute, obtuse, or right.

How can I classify quadrilaterals?

• Quadrilaterals can be classified based on their lines (parallelograms, trapezoids).
• Quadrilaterals can be classified based on their angles (rectangle)
• Quadrilaterals can be classified based on the length of their sides (rhombi, squares)

What is symmetry?

• Symmetric objects can be folded in half and match up.

Download the complete Classifying 2-D Shapes framework to customize for your own planning.

Perimeter and Area

Unit description: In this unit students will deepen their understanding of 2-D shapes by considering area and perimeter.  They will focus on rectangles and how to find their area and perimeter. They will relate area to multiplication and addition. The students will develop arrays of unit squares to solve area and perimeter problems.

Download the complete Grade 3  Unit 4 framework to customize for your own planning.

Essential Outcomes of the Unit

Measurement and Data- Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

• NY-3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.
• NY-3.MD.6 Measure areas by counting unit squares. 
• NY-3.MD.7 Relate area to the operations of multiplication and addition. 
• NY-3.MD.7c Use tiling to show in a concrete case that the area of a rectangle with whole-number side length a and side length b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
• NY-3.MD.7d Recognize area as additive. Find areas of figures composed of non-overlapping rectangles, and apply this technique to solve real world problems.

Other Standards Addressed in the Unit

Measurement and Data- Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

• NY-3.MD.5a Recognize a square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
• NY-3.MD.5b Recognize a plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
• NY-3.MD.7a Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
• NY-3.MD.7b . Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

Measurement and Data- Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
NY-3.MD.8a Finding the perimeter given the side lengths or find one unknown side length given the perimeter and other side lengths. 

• NY-3.MD.8b Identify rectangles with the same perimeter and different areas or with the same area and different perimeters.

Essential Questions and Big Ideas

What is area?

Area is the amount of space a shape covers, or the measure of the inside of shape.

What is perimeter?

Perimeter is the distance around the outside of a shape.

How do we solve area and perimeter problems with rectangles? 

The area of a rectangle can be determined using L x W. 

The perimeter of a rectangle can be determined by using the formula 2L + 2W.

How do we find the area of a shape made of multiple rectangles?

The area of a shape made of multiple rectangles can be determined by finding the area of each rectangle using L x W and adding the total of all areas within the larger shape.

Download the complete Grade 3  Unit 4 framework to customize for your own planning.

Measurement and Data

Unit description: In this unit students will learn to tell time to the hour and half-hour using analog and digital clocks. The students will learn to identify coins and their values and to use the cent (¢) sign. The students will count combinations of dimes and pennies within 100 cents.The students also learn to measure and compare lengths using standards and nonstandard measuring tools. The students will organize, represent and interpret data, including asking and answering questions and comparing amounts across categories.

Essential Outcomes of the Unit

Measurement and Data- Tell and write time and money

1.MD.3a . Tell and write time in hours and half-hours using analog and digital clocks. Develop an understanding of common terms, such as, but not limited to, o’clock and half past.

1.MD.3b Recognize and identify coins (penny, nickel, dime, and quarter) and their value and use the cent symbol (¢) appropriately. 

1.MD.3c Count a mixed collection of dimes and pennies and determine the cent value (total not to exceed 100 cents)

Measurement and Data- Represent and interpret data.

1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Other Standards Addressed in the Unit

Measurement and Data- Measure lengths indirectly and by iterating length units

1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object.

1.MD.2 Measure the length of an object using same-size “length units” placed end to end with no gaps or overlaps. Express the length of an object as a whole number of “length units.” 

Essential Questions and Big Ideas

How do we know the time?

• Analog and digital clocks tell us the time.

What is money and how do we know its worth?

• Money is made up of bills and coins that have a certain value that can be exchanged for goods and services.
• American money is valued in dollars(bills) and cents(coins).
• The coins are:
  • Penny: 1 cent, 1¢
  • Nickel: 5 cents, 5¢
  • Dime: 10 cents, 10¢
  • Quarter: 25 cents, 25¢

How do we measure and compare lengths?

• Standard and non-standard tools can be used to measure lengths.
• Lengths of objects can be compared when measured.

How can data be displayed and analyzed?

• Data can be displayed using charts and graphs.
• Data can be analyzed by asking and answering questions and comparing amounts across categories.

Download the complete Grade 1 Math Unit 4 framework to customize for your own planning.

Convert Units of Measure

Unit description: Students will extend their knowledge of metric and customary conversions from fourth grade to work with metric units of more sizes, as well as to convert customary volume and mass units.  

Essential Outcomes of the Unit

Measurement and Data- Convert like measurement units within a given measurement system.

• 5.MD.1 Convert among different-sized standard measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

Essential Questions and Big Ideas

How do I convert metric units? 

• The metric system is based on powers of 10.  
• Conversions in the metric system can be completed by multiplying or dividing by powers of 10.  

How do I convert customary length units?

• Conversion factors can be used to convert length units.  
• When converting units, you must consider the relationship between units and their sizes, which is represented as a conversion factor.

How do I convert customary volume units? 

• Conversion factors can be used to convert length units.  
• When converting units, you must consider the relationship between units and their sizes, which is represented as a conversion factor.

How do I convert customary mass units?  

• Conversion factors can be used to convert length units.  
• When converting units, you must consider the relationship between units and their sizes, which is represented as a conversion factor.

Download the complete Grade 5 Math Unit 4 framework to customize for your own planning.

Three-Digit Addition and Subtraction

Unit description: In this unit the students will learn to represent and solve three digit addition and subtraction problems within 1,000. Students will recognize when it is necessary to compose or decompose tens or hundreds. They will apply their understanding of the relationship between addition and subtraction to solve two-step word problems.

Download the complete Three Digit Addition and Subtraction framework to customize for your own planning.

Essential Outcomes of the Unit

Number and Operations in Base Ten- Use place value understanding and properties of operations to add and subtract.

• NY-2.NBT.7a . Add and subtract within 1000, 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 representation. Note: A written representation is any way of showing a strategy using words, pictures, or numbers.

Other Standards Addressed in the Unit

Number and Operations in Base Ten- Use place value understanding and properties of operations to add and subtract.

• NY-2.NBT.7b Understand that in adding or subtracting up to three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones, and sometimes it is necessary to compose or decompose tens or hundreds.

Operations and Algebraic Thinking- Represent and solve problems involving addition and subtraction

• NY-2.OA.1b Use addition and subtraction within 100 to develop an understanding of solving two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

Essential Questions and Big Ideas

How do we solve three digit addition problems?

• Strategies such as modeling, using 10s blocks, extended form, mental math and standard algorithms can be used to solve addition within 1,000.
• Some multi-digit addition problems involve regrouping 10s to the next place value.

How do we solve three digit subtraction problems? 

Strategies such as modeling, using 10s blocks, extended form, mental math and standard algorithms can be used to solve subtraction within 1,000.
Some multi-digit subtraction problems involve regrouping 10s to the prior place value.

How do we check our answers when adding and subtracting within 1,000?

• The reverse relationship between addition and subtraction can be used to check answers.

How do we solve two step word problems involving addition and subtraction within 100?

• Identifying key words and understandings within word problems can help determine whether to add or subtract to solve problems in our daily life.
• Understanding of properties of operations and the relationship between addition and subtraction can help to solve word problems.
• Marking up the question and drawing pictures can help to visualize word problems.

Download the complete Three Digit Addition and Subtraction framework to customize for your own planning.

Fractions

Unit description: Students will extend their knowledge of fractions from third grade by looking at fractions with denominators of 10 and 12.  They will also extend their understanding of fractions equal to 1 whole to multiply and divide fractions by 1 whole to create equivalent fractions.  Students will begin to interpret and solve word problems that require combining or separating fractions within the same whole and with the same denominator.  Students will also interpret word problems that involve equal groups of a fraction.  Students will understand fractions larger than one and how to convert them into mixed numbers.  

Essential Outcomes of the Unit

• NY-4.NF.1 Explain why a fraction abis equivalent to a fraction a x nb x nby using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
  • Note: Grade 4 expectations are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• NY-4.NF.2 Compare two fractions with different numerators and different denominators. Recognize that comparisons are valid only when the two fractions refer to the same whole. e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 12. Record the results of comparisons with symbols >, =, or <, and justify the conclusions. e.g., using a visual fraction model.
  • Note: Grade 4 expectations are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• NY-4.NF.3 Understand a fraction abwith a > 1 as a sum of fractions 1b Note: 1brefers to the unit fraction for ab
• NY-4.NF.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
• NY-4.NF.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions. e.g., by using a visual fraction model such as, but not limited to:
  • 3/8 = 1/8 + 1/8 + ⅛
  • 3/8 = 1/8 + 2/8
  • 218= 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
• NY-4.NF.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
• NY-4.NF.4 Apply and extend previous understandings of multiplication to multiply a whole number by a fraction.
• NY-4.NF.4a Understand a fraction abas a multiple of 1b e.g., Use a visual fraction model to represent 54 as the product 5 × 14, recording the conclusion with the equation 54= 5 × 14.
• NY-4.NF.4b Understand a multiple of abas a multiple of 1b, and use this understanding to multiply a whole number by a fraction. e.g., Use a visual fraction model to express 3 x 25as 6 x 15, recognizing this product as 65, in general, n × ab= (n x a)b.
• NY-4.NF.4c . Solve word problems involving multiplication of a whole number by a fraction.

Essential Questions and Big Ideas

How can I find equivalent fractions and compare fractions? 

• Equivalent fractions can be found by drawing models or by multiplying or dividing by a whole.  
• Fractions can be compared by drawing models or by giving the fractions the same denominator and comparing their number of pieces.  

How can I use addition to represent non-unit fractions? 

• Non-unit fractions can be written as the sum of unit fractions.  

How can I use addition and subtraction to relate fractions?

• Fractions with the same denominator or referring to the same whole can be added or subtracted by focusing on the number of parts.   

How can I show equal groups relationships with fractions?

• Fractions can exist in equal groups.  
• If you have a whole number of equal groups of a fraction, you can multiply the whole number by the numerator to figure out how many total parts you will have.  The denominator would remain the same.    

Download the complete Fractions framework to customize for your own planning.