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Connections across the Grades
K-3rd
The recently developed Michigan Focal Points represent the focus of instruction that should surround the GLCE. Core GLCE are now those GLCE linked to a Michigan Focal Point and as such will be tested with two items on the MEAP. Extended core GLCE will be tested with 1 item and are those expectations either not linked to a Focal Point at that grade level or are a subset of a core expectation so 2 items are not needed.
None of the GLCE were changed or eliminated. The GLCE have always represented the measurable skills and concepts that result from a good curriculum. The Michigan Focal Points represent the organization of curriculum and instruction; they identify those big mathematical ideas that should be considered when planning instruction at each grade level.
Those extended expectations that are not linked directly to a Focal Point at that grade level are fall under the heading of Connections. This means they are connected foundational to a focal point at higher grade level or are an extension of work done at an earlier grade. Some dont necessarily link to any mathematics focal point but are essential tools needed for success in science and social studies.
This document attempts to highlight those connections. Each table represents a particular 3rd grade Michigan focal point and shows the expectations from earlier grades that lay the foundation for that focal point. The purpose is to lend perspective to the content expectations that are not considered core. They are still important; while not the focus of instruction for that grade some time must be spent on those topics in preparation for learning at a later grade.
Connections across the Grades
K-3rd
3rd grade Focal Point
Number and Operations and Algebra (NAX-03)
Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts (NCTM-3rd)
Benchmark for the Critical Foundations of Algebra
Multiply and divide whole numbers (National Mathematics Advisory Panel) K 1st Grade2nd Grade3rd Grade
Count, write, and order numbers*
N.ME.00.05 Count orally to 100 by ones. Count to 30 by 2s, 5s and10s using grouped objects as needed.
Compose and decompose numbers*
N.ME.00.06 Understand the numbers 1 to 30 as having one, or two, or three groups of ten and some ones. Also count by tens with objects in ten-groups to 100.
Count, write, and order numbers*
N.ME.01.01 Count to 110 by 1s, 2s, 5s, and 10s, starting from any number in the sequence; count to 500 by 100s and 10s; use ordinals to identify position in a sequence, e.g., 1st, 2nd, 3rd.
Explore place value*
N.ME.01.07 Compose and decompose numbers through 30, including using bundles of tens and units
Count, write, and order whole numbers*
N.ME.02.01 Count to 1000 by 1s, 10s and 100s starting from any number in the sequence.
N.ME.02.04 Count orally by 3s and 4s starting with 0, and by 2s, 5s, and 10s starting from any whole number.
Understand meaning of multiplication and division
N.MR.02.15 Understand division () as another way of expressing multiplication, using fact families within the 5 x 5 multiplication table; emphasize that division undoes multiplication, e.g., 2 x 3 = 6 can be rewritten as 6 2 = 3 or 6 3 = 2. [Ext - NC]
N.MR.02.13 Understand multiplication as the result of counting the total number of objects in a set of equal groups, e.g., 3 x 5 gives the number of objects in 3 groups of 5 objects, or 3 x 5 = 5 + 5 + 5 = 15. [Ext - NC]
N.MR.02.14 Represent multiplication using area and array models. [Ext - NC]
N.MR.02.16 Given a simple situation involving groups of equal size or of sharing equally, represent with objects, words, and symbols; solve. [Ext - NC]
N.MR.02.17 Develop strategies for fluently multiplying numbers up to 5 x 5. [NASL]
Count in steps, and understand even and odd numbers
N.ME.03.04 Count orally by 6s, 7s, 8s, and 9s starting with 0, making the connection between repeated addition and multiplication. [NASL]
N.ME.03.05 Know that even numbers end in 0, 2, 4, 6, or 8; name a whole number quantity that can be shared in two equal groups or grouped into pairs with no remainders; recognize even numbers as multiples of 2. Know that odd numbers end in 1, 3, 5, 7, or 9. Work with patterns involving even and odd numbers. [Ext - NC]
Multiply and divide whole numbers
N.MR.03.09 Use multiplication and division fact families to understand the inverse relationship of these two operations, e.g., because 3 x 8 = 24, we know that 24 8 = 3 or 24 3 = 8; express a multiplication statement as an equivalent division statement. [Core - NC]
N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding How many groups? and How many in a group? and write mathematical statements to represent those situations. [Core - NC]
N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication and division relationships. [Core - NC]
N.MR.03.12 Find solutions to open sentences, such as 7 x q = 42 or 12 q = 4, using the inverse relationship between multiplication and division. [Ext]
N.FL.03.13 Mentally calculate simple products and quotients: up to a three-digit number by a one-digit number involving multiples of 10, e.g., 500 x 6, or 400 EMBED Equation.3 8. [NASL]
N.MR.03.14 Solve division problems involving remainders, viewing the remainder as the number left over; interpret based on problem context, e.g., when we have 25 children with 4 children per group then there are 6 groups with 1 child left over. [Core]
Problem-solving with whole numbers
N.MR.03.15 Given problems that use any one of the four operations with appropriate numbers, represent with objects, words, (including product and quotient), and mathematical statements; solve. [Core]
3rd grade Focal Point
Measurement (MXX-03)
Developing an understanding of area and perimeter and determining the areas and perimeters of two-dimensional shapes ( from NCTM-4th)
Benchmark for the Critical Foundations of Algebra:
[Working toward] Solve problems with perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (National Mathematics Advisory Panel) K 1st Grade2nd Grade3rd Grade
Explore other measurement attributes*
M.UN.00.04 Compare two or more objects by length, weight and capacity, e.g., which is shorter, longer, taller?
M.PS.00.05 Compare length and weight of objects by comparing to reference objects, and use terms such as shorter, longer, taller, lighter, heavier.
Estimate and measure length*
M.UN.01.01 Measure the lengths of objects in non-standard units, e.g., pencil lengths, shoe lengths, to the nearest whole unit.
M.UN.01.02 Compare measured lengths using the words shorter, shortest, longer, longest, taller, tallest, etc.
Solve problems
M.PS.01.08 Solve one-step word problems using addition and subtraction of length, money and time, including how much more/less, without mixing units.
Understand the concept of area
M.UN.02.03 Measure area using non-standard units to the nearest whole unit. [Ext - NC]
M.TE.02.04 Find the area of a rectangle with whole number side lengths by covering with unit squares and counting, or by using a grid of unit squares; write the area as a product. [Ext - NC]
Measure, add, and subtract length*
M.UN.02.01 Measure lengths in meters, centimeters, inches, feet, and yards approximating to the nearest whole unit and using abbreviations: cm, m, in, ft, yd. [Ext - NC]
M.PS.02.02 Compare lengths; add and subtract lengths (no conversion of units). [Core -NC]
Solve measurement problems*
M.PS.02.10 Solve simple word problems involving length and money. [Core - NC]
M.TE.02.11 Determine perimeters of rectangles and triangles by adding lengths of sides, recognizing the meaning of perimeter. [Core - NC]
Understand meaning of area and perimeter and apply in problems
M.UN.03.05 Know the definition of area and perimeter and calculate the perimeter of a square and rectangle given whole number side lengths. [Core]
M.UN.03.06 Use square units in calculating area by covering the region and counting the number of square units. [Core NC]
M.UN.03.07 Distinguish between units of length and area, and choose a unit appropriate in the context. [Core - NC]
M.UN.03.08 Visualize and describe the relative sizes of one square inch and one square centimeter. [Ext - NC]
Estimate perimeter and area
M.TE.03.09 Estimate the perimeter of a square and rectangle in inches and centimeters; estimate the area of a square and rectangle in square inches and square centimeters. [Core - NC]
Solve measurement problems
M.PS.03.10 Add and subtract lengths, weights and times using mixed units, within the same measurement system. [Ext]
M.PS.03.11 Add and subtract money in dollars and cents. [Ext]
M.PS.03.12 Solve applied problems involving money, length, and time. [Core]
M.PS.03.13 Solve contextual problems about perimeters of rectangles and areas of rectangular regions. [Core]
3rd grade Focal Point
Geometry (GXX-03)
Describing properties of two-dimensional shapes (part of NCTM-3rd) and classifying 3D shapes (MI)
Benchmark for the Critical Foundations of Algebra:
[Working toward]Analyze the properties of two-dimensional shapes K 1st Grade2nd Grade3rd Grade
Create, explore, and describe shapes*
G.GS.00.02 Identify, sort, and classify objects by attribute and identify objects that do not belong in a particular group.
G.GS.00.01 Relate familiar three-dimensional objects inside and outside the classroom to their geometric name, e.g., ball/sphere, box/cube, soup can/cylinder, ice cream cone/cone, refrigerator/prism.
Create and describe shapes
G.GS.01.01 Create common two-dimensional and three-dimensional shapes, and describe their physical and geometric attributes, such as color and shape.
G.LO.01.02 Describe relative position of objects on a plane and in space, using words such as above, below, behind, in front of.Identify and describe shapes(
G.GS.02.04 Distinguish between curves and straight lines and between curved surfaces and flat surfaces. [Ext - NC]
G.GS.02.01 Identify, describe, and compare familiar two-dimensional and three-dimensional shapes, such as triangles, rectangles, squares, circles, semi-circles, spheres, rectangular prisms. [Core - NC]
G.GS.02.02 Explore and predict the results of putting together and taking apart two-dimensional and three-dimensional shapes. [Core - NC]
G.SR.02.05 Classify familiar plane and solid objects, e.g., square, rectangle, rhombus, cube, pyramid, prism, cone, cylinder, and sphere, by common attributes such as shape, size, color, roundness or number of corners and explain which attributes are being used for classification. [Core - NC]
G.TR.02.06 Recognize that shapes that have been slid, turned or flipped are the same shape, e.g., a square rotated 45 is still a square. [Ext - NC]
Recognize the basic elements of geometric objects
G.GS.03.01 Identify points, line segments, lines, and distance. [Core - NC]
G.GS.03.02 Identify perpendicular lines and parallel lines in familiar shapes and in the classroom. [Core - NC]
G.GS.03.03 Identify parallel faces of rectangular prisms, in familiar shapes and in the classroom. [Core - NC]
Name and explore properties of shapes
G.GS.03.04 Identify, describe, compare and classify two-dimensional shapes, e.g., parallelogram, trapezoid, circle, rectangle, square and rhombus, based on their component parts (angles, sides, vertices, line segment) and the number of sides and vertices. [Core - NC]
G.SR.03.05 Compose and decompose triangles and rectangles to form other familiar two-dimensional shapes; e.g., form a rectangle using two congruent right triangles, or decompose a parallelogram into a rectangle and two right triangles. [Core - NC]
Explore and name three-dimensional solids
G.GS.03.06 Identify, describe, build and classify familiar three-dimensional solids, e.g., cube, rectangular prism, sphere, pyramid, cone, based on their component parts (faces, surfaces, bases, edges, vertices). [Core - NC]
G.SR.03.07 Represent front, top, and side views of solids built with cubes. [Ext - NC]
3rd grade Focal Point
Number and Operations (NXX-03)
Developing an understanding of fractions and fraction equivalence (NCTM-3rd)
Benchmark for the Critical Foundations of Algebra:
[Working toward] Identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals K 1st Grade2nd Grade3rd Grade
Work with unit fractions*
N.ME.02.18 Recognize, name, and represent commonly used unit fractions with denominators 12 or less; model 1/2, 1/3, and 1/4 by folding strips. [Ext - NC]
N.ME.02.19 Recognize, name, and write commonly used fractions: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4.
N.ME.02.20 Place 0 and halves, e.g., 1/2, 1 1/2, 2 1/2, on the number line; relate to a ruler. [Ext - NC]
N.ME.02.21 For unit fractions from 1/12 to 1/2 understand the inverse relationship between the size of a unit fraction and the size of the denominator; compare unit fractions from 1/12 to 1/2. [Ext - NC]
N.ME.02.22 Recognize that fractions such as 2/2, 3/3, and 4/4 are equal to the whole (one). [Ext - NC]
Record, add and subtract money*
M.UN.02.07 Read and write amounts of money using decimal notations, e.g., $1.15. [Core - NC]
M.PS.02.08 Add and subtract money in mixed units, e.g., $2.50 + 60 cents and $5.75 - $3, but not $2.50 + $3.10. [Ext - NC]
Understand simple fractions, relation to the whole, and addition and subtraction of fractions
N.ME.03.16 Understand that fractions may represent a portion of a whole unit that has been partitioned into parts of equal area or length; use the terms numerator and denominator. [Core - NC]
N.ME.03.17 Recognize, name, and use equivalent fractions with denominators 2, 4, and 8, using strips as area models. Core - NC]
N.ME.03.18 Place fractions with denominators of 2, 4, and 8 on the number line; relate the number line to a ruler; compare and order up to three fractions with denominators 2, 4, and 8. Core - NC]
N.ME.03.19 Understand that any fraction can be written as a sum of unit fractions, e.g., 3/4 = 1/4 + 1/4 + 1/4. [Ext - NC]
N.MR.03.20 Recognize that addition and subtraction of fractions with equal denominators can be modeled by joining or taking away segments on the number line. [Ext - NC]
Understand simple decimal fractions in relation to money
N.ME.03.21 Understand and relate decimal fractions to fractional parts of a dollar, e.g., 1/2 dollar = $0.50; 1/4 dollar = $0.25. [Ext]
NCTM Focal Points Grade 3
Number and Operations and Algebra: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts
Students understand the meanings of multiplication and division of whole numbers through the use of representations (e.g., equal-sized groups, arrays, area models, and equal jumps on number lines for multiplication, and successive subtraction, partitioning, and sharing for division). They use properties of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to multiply whole numbers and apply increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving basic facts. By comparing a variety of solution strategies, students relate multiplication and division as inverse operations.
Number and Operations: Developing an understanding of fractions and fraction equivalence
Students develop an understanding of the meanings and uses of fractions to represent parts of a whole, parts of a set, or points or distances on a number line. They understand that the size of a fractional part is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less than, or greater than 1. They solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators. They understand and use models, including the number line, to identify equivalent fractions.
Geometry: Describing and analyzing properties of two-dimensional shapes
Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.
NCTM Focal Points Grade 4
Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes
Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle.
Based on the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved.
* This topic is a focus of instruction at this grade level
* This topic is a focus of instruction at this grade level
( This topic is a focus of instruction at this grade level
* This topic is a focus of instruction at this grade level
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