Math Grade 4

Standards for Mathematical Practice

Students will: (to be embedded throughout instruction as appropriate)

Make sense of Reason abstractly and Construct viable Model with Use appropriate tools Attend to precision. Look for and make use Look for and express
problems and quantitatively. arguments and critique mathematics. strategically. of structure. regularity in repeated
persevere in solving the reasoning of others.
reasoning.
them.

SMP.1 SMP.2 SMP.3 SMP.4 SMP.5 SMP.6 SMP.7 SMP.8

MAFS Domains: Measurement and Data Pacing: Weeks 27-39
Geometry February 21 – May 26
Operations and Algebraic Thinking
Number and Operations- Fractions

Learning Targets Standards Vocabulary

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. MAFS.4.G.1.1 acute angle

Students will: angle ()

 identify and draw the following two-dimensional attributes: point, line, line segment, ray, right angle (exactly 90), acute angle congruent sides (≅)
(less than 90), obtuse angle (greater than 90 and less than 180), straight angle, perpendicular lines, and parallel lines.
degree ()
 represent geometric objects with geometric notation. endpoint
geometric notation
E.g., line
line segment
point line line segment ray congruent sides obtuse angle
P X parallel lines
D A B Y perpendicular lines
C CD AB XY point
protractor
point P ray
right angle
right angle acute angle obtuse angle straight angle
A P two-dimensional
C
I attributes
G D G vertices/vertex
O
T

CAT, TAC, A PIG, GIP, I DOG, GOD, O

straight angle perpendicular lines parallel lines right angle
K
B M

ME T CD L
N

MET, TEM, E CD DB KL MN

….continued on the next page…

49 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

 classify angles of 2-dimensional figures as right, acute, obtuse or straight.
E.g.,

right angle acute angle straight angle obtuse angle

 identify the two-dimensional attributes shown above in two-dimensional shapes.
E.g., List the two-dimensional attributes within this trapezoid:

Student: two acute angles, four line angle
segments, a pair of parallel lines, etc. center
degree
HINT: Students can and should make geometric distinctions about angles without measuring or mentioning degrees. Angles endpoint
should be classified in comparison to right angles, such as larger than (obtuse) or smaller than (acute). interval
measure
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. MAFS.4.MD.3.5 one-degree angle
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc ray
between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can rotation
be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

Students will:

 define an angle (i.e., a geometric shape that is formed when two rays share a common endpoint; a measure of rotation).
 explain an angle as a series of “one-degree turns”.

E.g., A water sprinkler rotates one-degree at each interval. If the sprinkler rotates a total of 100º, how many one-degree turns
has the sprinkler made?

 explain that it takes 360 one-unit degrees to make a circle (i.e., 1/360 of a circle is a “one degree angle”).
E.g.,

45º 90º 180º 270º 360º

 explain the relationship between a circle and the number of degrees in an angle (i.e., an angle is measured in reference to a
circle—its center is the endpoint for each of the rays that make up the angle).

 use the geometric notation for degrees (°) to label the measure of an angle.

50 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. MAFS.4.MD.3.6 acute
additive
Students will: angle
angle measure
 use appropriate terminology (right, acute, obtuse and straight) to describe angles and rays. obtuse
protractor
HINT: Right angles measure exactly 90, acute angles measure less than 90, obtuse angles measure greater than 90 and right
less than 180, and straight angles measure exactly 180. straight angle
unknown angle
 compare a given angle to the benchmark right angle (90º) to determine which set of numbers to use on a protractor. variable
 measure an angle to the nearest whole number using a protractor.

E.g.,

 use a protractor to create an angle given a specific measurement.
 determine if the measure of an angle is reasonable based on the relationship of the angle to a right angle.

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the MAFS.4.MD.3.7
angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems,
e.g., by using an equation with a symbol for the unknown angle measure.

Students will:

 explain that the angle measurement of a larger angle is the sum of the angle measures of its decomposed parts.
 write an equation using a symbol to represent an unknown angle measurement.
 use addition and subtraction to solve for the missing angle measurements.

E.g.,

For this right angle, 60º
what is the value of x? x

 solve word problems involving unknown angles.

E.g., A lawn water sprinkler rotates a total of 90 degrees but pauses during its cycle according
to the diagram. How many degrees does it rotate between the 25 and 20 degree pause?
Write an equation to represent the situation. 25 + m + 20 = 90

51 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a MAFS.4.G.1.2 acute angle
specified size. Recognize right triangles as a category, and identify right triangles. attribute
category
Students will: classify
congruent
 classify two-dimensional shapes into the following categories: those with parallel lines, those with perpendicular lines, those with equilateral
both parallel and perpendicular lines, those with no parallel or perpendicular lines. isosceles
line symmetry
 classify two-dimensional shapes into categories based on the presence or absence of acute, obtuse, or right angles. obtuse angle
 identify different types of quadrilaterals based on defining attributes. parallel lines
 identify different types of right triangles: scalene (no congruent sides) and isosceles (two or more congruent sides). perpendicular lines
 classify right triangles. polygons
right angle
E.g., Do you agree with the label on each of the ovals in the Venn diagram? Why or why not? right triangle
Explain why some shapes fall in the overlapping sections of the ovals. scalene
two-dimensional

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 MAFS.4.G.1.3
parts. Identify line-symmetric figures and draw lines of symmetry.

Students will:

 identify and describe figures that have line symmetry (e.g., fold a figure or draw a line so it has two parts that match exactly).
 draw lines of symmetry in both regular and non-regular polygons.

E.g.,

HINT: All remaining standards in the
map are review standards.

HINT: This standard only includes line symmetry, NOT rotational symmetry.

52 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which MAFS.4.OA.1.3 equation
remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the estimation
reasonableness of answers using mental computation and estimation strategies including rounding. mental computation
multi-step
Students will: operation
reasonableness
 choose the correct operation to perform at each step of a multi-step word problem with whole-number answers. remainder
round
 solve multi-step word problems (i.e., up to 3 steps) that involve any of the four operations using various strategies. strategies
unknown quantity
 explain the strategies used to solve multi-step word problems. variable

 represent a multi-step word problem using equations involving a variable represented by a letter for the unknown number.

 interpret remainders that result from multi-step word problems. The context of a word problem must be
 assess the reasonableness of answers to multi-step word problems using considered when interpreting
remainders. Here are some ways
estimation strategies and mental computation.

E.g., Your class is collecting bottled water for a service project. The goal is to collect 300 remainders can be addressed:

bottles of water. On the first day, Max brings in 3 packs with 6 bottles in each  remain as a leftover

container. Sarah wheels in 6 packs with 6 bottles in each container.  discard leaving only the whole

About how many bottles of water still need to be collected? number answer

Student 1: Student 2:  increase the whole number answer
First, I multiplied 3 and 6 which equals 18. up one
First, I multiplied 3 and 6 which equals 18. Then I multiplied 6 and 6 which is 36. I
know 18 is about 20 and 36 is about 40  round to the nearest whole number
Then I multiplied 6 and 6 which is 36. I for an approximate result
know 18 plus 36 is about 50. I’m trying to

get to 300. 50 plus another 50 is 100. 40+20=60. 300-60=240, so we need

Then I need 2 more hundreds. So we about 240 more bottles.

need 250 bottles.

HINT: Refer to pages 5 and 6 in the Fourth Grade Mathematics Curriculum Map for clarification of Common Addition and
Subtraction Situations and Multiplication and Division Situations. It is expected that students will become proficient with all
situations.
The expectations for multi-step word problems include multiplication of 2-digit by 1-digit or a multiple of 10 by a 1-digit and
division of 2-digit by 1-digit.

53 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Understand a fraction a/b with a > 1 as a sum of fractions 1/b. MAFS.4.NF.2.3 decomposition
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. denominator
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an difference
equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 equivalent fraction
= 8/8 + 8/8 + 1/8. fraction greater than 1
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using mixed number
properties of operations and the relationship between addition and subtraction. model
d. Solve word problems involving the addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by numerator
using visual fraction models and equations to represent the problem. product
sum
Students will: unit fraction
whole number
 demonstrate with visual models (i.e., number lines, rectangles, squares, and circles) that adding fractions within the same
whole (i.e., fractions with the same denominator) is joining parts of that whole.

 demonstrate with visual models (i.e., number lines, rectangles, squares, and circles) that subtracting fractions within the same
whole (i.e., fractions with the same denominator) is separating parts of that whole.

HINT: Denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, 100.
 decompose a fraction in more than one way using visual models (i.e., number lines, rectangles, squares, and circles), including

decomposing a fraction into a combination of several unit fractions ( i.e., fractions with a numerator of one).
 record the decomposition of a fraction in an equation.

E.g.,

…continued...

54 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

 decompose a mixed number into fractions equal to one and unit fractions to find the fraction greater than one. decomposition
E.g., denominator
difference
2 1/8 equivalent fraction
fraction greater than 1
2 1/8 mixed number
/\ | model
1 + 1 + 1/8 numerator
||| product
8/8 + 8/8 + 1/8 sum
unit fraction
17/8 whole number

 convert fractions greater than one to mixed numbers by decomposing the fraction into a sum of fractions equal to one and
fractions less than one.

E.g.,

17/6

/\
6/6 + 11/6

| /\
6/6 + 6/6 + 5/6

|||
1 + 1 + 5/6

2 5/6

 add mixed numbers with like denominators using equivalent fractions and the properties of operations.
E.g., 3 ¾ + 2 ¼

Student 1: Student 2: Student 3:
3+2=5 and ¾ + ¼ = 1 3¾+2=5¾+¼= 3 ¾ = 15/4 and 2 ¼ = 9/4,
so 5 + 1 = 6 5 + 4/4 = 5 + 1 = 6 so 15/4 + 9/4 = 24/4 = 6

 subtract mixed numbers with like denominators using equivalent fractions and the properties of operations.
…continued…

55 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

 add or subtract mixed numbers with like denominators using the relationship between addition and subtraction. decomposition

E.g., Students use their knowledge of decomposition (17/6 = 12/6 + 5/6) to calculate 17/6 – 5/6 = 12/6. denominator
difference

 solve word problems involving addition of fractions with like denominators using models, drawings, pictures, and equations. equivalent fraction
fraction greater than 1

mixed number
E.g., Lindsey and Brooke need 3 3/8 feet of ribbon to design costumes for a performance. Lindsey has 11/8 and Brooke has 2 5/8 model
feet of ribbon. If they combine what they have, will that be enough for the project? Explain why or why not.
numerator

product

sum

unit fraction

whole number

Ribbon needed for the project 33/8 ft. = 8/8 ft. + 8/8 ft. + 8/8 ft.+ 3/8 ft.= 27/8 ft.
Lindsey’s 9/8 ft. + Brooke’s 21/8 ft. = 30/8 ft.
30/8 ft. is greater than 27/8 ft. So they have enough ribbon for the costumes.

 solve word problems involving subtraction of fractions with like denominators using models, drawings, pictures, and equations.

56 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or MAFS.4.MD.1.2 intervals of time
decimals2. Represent fractional quantities of distance and intervals of time using linear models. formerly known as
(1See Table 1 and Table 2) elapsed time
(2Computational fluency with fractions and decimals is not the goal for students at this grade level.)

Students will:

 represent fractional quantities of distance using linear models.

E.g., Billy has been training for a half-marathon. Each day he runs on the treadmill 2 2/4 miles and runs on the outdoor track for 3
1/4 miles. In all, how many miles does Billy run each day?

0 | | | 1 | | | 2 | | | 3 | | | 4 | | | 5 | | | 6 | | | 7 | | | 8 | | | 9 | | | 10

 use the four operations to solve word problems involving distances (i.e., mile, yard, foot, feet, inch; kilometer, meter, centimeter)
including problems that require expressing measurements given in a larger unit in terms of a smaller unit (converting units).

 represent intervals of time using linear models.

E.g.,

What time does Marla have to leave to be at her friend’s house by a quarter
after 3, if the trip takes 90 minutes?

 use the four operations to solve word problems involving intervals of time.
 use the four operations to solve word problems involving money including dollars and cents with decimal notation.

E.g., Helen bought popcorn at the movies. She bought 2 large bags, and each one cost $1.30.
How much change did she receive from $5?

 use the four operations to solve multistep word problems, including problems involving fractions or decimals and problems that
require expressing measurements given in a larger unit in terms of a smaller unit (converting units).

HINT: Decimal amounts should be limited to hundredths in multiples of 10 (e.g., $0.10, $0.20, $0.30, $ 0.40…) in order to relate
to fractions.
Computational fluency with fractions and decimals is not the goal for students at this grade level.
Refer to pages 5 and 6 in the Fourth Grade Mathematics Curriculum Map for clarification of Common Addition and
Subtraction Situations and Multiplication and Division Situations. It is expected that students will become proficient with
all situations.

57 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room MAFS.4.MD.1.3 area
given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. formula
length (l)
Students will: perimeter (P)
square units
HINT: It is a 3rd grade expectation that students are able to communicate their understanding of why the perimeter and area width (w)
formulas work.

 identify situations that require the use of the perimeter formula in real-world contexts. E.g., amount of fencing needed to make a
rectangular pig pen, length of ribbon needed to wrap around a box, distance around a city block.

 apply the perimeter formula (P = 2l + 2w) in real world and mathematical situations.
 identify situations that require the use of the area formula in real-world contexts (e.g., size of a carpet, area of a garden,

covering a wall with paint).
 apply the area formula (A = l × w) in real world and mathematical situations.
 write area measurements in square units (e.g., cm2, ft2, km2, etc.).
 solve for missing dimensions of rectangles when provided with the perimeter and/or area.

E.g., A rectangular garden has as an area of 80 square feet. It is 5 feet wide.
How long is the garden? 80 ft2 = 5ft x l

 apply methods of maximizing area using a given perimeter, and vice versa.
 solve problems involving area of a composite figure composed of rectangles.

58 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Unit 4 Suggested Instructional Resources

MAFS AIMS Lakeshore MFAS Internet enVision

Hall of Teacher Guide Locating Points, Lines, and www.k- www.IXL.com/signin/volusia 16-1A
Mirrors pp. 23-24 Rays 16-1B
5mathteachingresources.com P.13, P.30, P.31
Lines to Reproducible Parallel and Perpendicular
Design p. 3 Sides G.1 5th Grade: Z.12, Z.30, Z.31

G.1.1 Daily Math Journal All About Angles www.cpalms.org https://learnzillion.com/
pp. 66, 67, 68, 69, 70 Geometric Map Makers Unit 10
Lines, Rays, and Line Point, Lines, Angles Oh My! Lesson 2: Using plane figures
Pick A Problem Segments Geometry At The County Fair to understand geometric
#86, 87, 88, 89, 90, 91, Parallel & Perpendicular Lines components
92 Unit 13
hcpss.instructure.com/cours Lesson 2: Identify
How Did You Solve It? es/107 perpendicular and parallel
#75, 76, 77, 78, 79 G.1 Lessons lines
G.1 Formatives Video: Draw points, lines, line
2-D Geometric Shapes segments, rays, angles and
Tub parallel and perpendicular
lines
Geometry in the Real
World Photo Magnets http://achievethecore.org

Protractor Teacher Guide Lawn Sprinkler www.k- www.IXL.com/signin/volusia 16-1C
Ground p. 22 Circle the Angles 5mathteachingresources.com P.14
School Determining an Angle’s MD.5
Reproducible Measures https://learnzillion.com/
Flight Paths p. 3 www.cpalms.org Unit 10
This Angle Angle Your Way Around Lesson 4: Understand that a
MD.3.5 Daily Math Journal What’s My Degree? rotation of 1/360 of a circle is 1
p. 60 Runway Rotations degree angle
Lesson 5: Using circles to find
How Did You Solve It? hcpss.instructure.com/cours angle measurement
#69, 70 es/107
MD.5 Lessons http://achievethecore.org
Angles Measurement MD.5 Formatives
Center

enVisionMATH: SE = Student Edition; RMC= Ready-Made Centers; POD= Problem of the Day; A&R = Assessment and Reteaching Workbook

59 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Unit 4 Suggested Instructional Resources

MAFS AIMS Lakeshore MFAS Internet enVision

Protractor Teacher Guide Measuring Angles with a www.k- www.IXL.com/signin/volusia 16-2 SE,
Ground pp. 22-23 Protractor A&R
School 5mathteachingresources.com P.15
Reproducible Drawing and Measuring 16-3A
Flight Paths p. 3 Angles MD.6 5th Grade: Z.13

MD.3.6 Mirrors that Daily Math Journal Town of Happyville www.cpalms.org https://learnzillion.com/
Multiply pp. 59, 62 Angles for $500 Unit 10
Using a Protractor to Draw Protractor Power Lesson 8: Measuring angles
Hall of How Did You Solve It? Angles What’s Your Angle Lesson 9: Constructing
Mirrors #71, 72 angles using a protractor
hcpss.instructure.com/cours
Angles Measurement es/107 http://achievethecore.org
Center MD.6 Lessons
MD.6 Formatives

Protractors Turns on a Skateboard www.k- www.IXL.com/signin/volusia
Teacher Guide 5mathteachingresources.com P.16, P.17
pp. 22-23 Using Known Angles MD.7
https://learnzillion.com/
Reproducible Understanding Angles www.cpalms.org Video: Compose and
p. 3 Edible Angles decompose angles
What is the Measure of the Edible Angles: Decomposing Unit 13
Daily Math Journal Angle? Angles into Parts of a Whole Lesson 4: Compose and
pp. 56, 59, 62, 64 What’s Your Angle? decompose angle measures
MD.3.7
Pick A Problem
#82, 83, 84, 85 hcpss.instructure.com/cours http://achievethecore.org
es/107
How Did You Solve It? MD.7 Lessons
#73, 74 MD.7 Formatives

Angles Measurement
Center

Protractors
enVisionMATH: SE = Student Edition; RMC= Ready-Made Centers; POD= Problem of the Day; A&R = Assessment and Reteaching Workbook

60 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Unit 4 Suggested Instructional Resources

MAFS AIMS Lakeshore MFAS Internet enVision

Classifying Teacher Guide Grouping Triangles www.k- www.IXL.com/signin/volusia 16-3B
Quadrilateral p. 24 Sketching Quadrilaterals 16-3C
s Sketching Triangles 5mathteachingresources.com P.1, P.2, P.3, P.4, P.6, P.8, 16-3D
Reproducible
pp. 3, 16 G.2 P.9

G.1.2 Daily Math Journal www.cpalms.org https://learnzillion.com/
pp. 66, 68, 69, 70, 71 Polygon Express Unit 10
Geoboard Lesson 3: Use perpendicular
Pick A Problem A Closer Look at Quadrilaterals and parallel lines to describe
#93, 94, 95, 96, 98 2D figures
hcpss.instructure.com/cours
es/107 http://achievethecore.org
G.2 Lessons
How Did You Solve It? G.2 Formatives
#80, 81, 82, 83, 84, 85

Line Protractors Squares and Lines of www.k- www.IXL.com/signin/volusia 16-5 SE,
Symmetry, Teacher Guide Symmetry 5mathteachingresources.com P.29 A&R, RMC
Naturally p. 24 G.3 3rd Grade: V.5
Using Lines of Symmetry 5th Grade: Z.28
Mirror Twins Reproducible www.cpalms.org
p. 3 Line Symmetry Geometry in the World of Art https://learnzillion.com/
Paper Quilts Unit 13
G.1.3 Pick A Problem Identifying and Explaining Symmetrical Solutions Lesson 8: Create lines of
#97, 99, 100 Symmetry symmetry
hcpss.instructure.com/cours
Daily Math Journal es/107 Video: Recognize and draw
pp. 66, 67, 68, 70 G.3 Lessons lines of symmetry and line-
G.3 Formatives symmetric figures
How Did You Solve It?
#86, 87, 88, 89, 90 http://achievethecore.org

enVisionMATH: SE = Student Edition; RMC= Ready-Made Centers; POD= Problem of the Day; A&R = Assessment and Reteaching Workbook

61 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Unit 4 Suggested Instructional Resources

MAFS AIMS Lakeshore MFAS Internet enVision

Picturing Teacher Guide Picking Strawberries www.k- www.IXL.com/signin/volusia 1-6 SE
Clues p. 4 Estimating the Solution 1-7 SE,
Roller Coaster Rides 5mathteachingresources.com P.27 RMC
Party Reproducible Juice Boxes 4-6 SE
Planning p. 3 OA.3 5th Grade Z.22 5-4 SE, A&R

Hall Of Daily Math Journal www.cpalms.org https://learnzillion.com/ 13-8F
Mirrors pp. 3, 5, 7, 9, 11, 13, 15 Rockin’ Remainders Unit 14
OA.1.3 Carnival Tickets Lesson 1: Add and subtract
(review) Pick A Problem Karl’s Garden multi-digit numbers using
#1, 2, 32 various strategies
hcpss.instructure.com/cours Lesson 4: Multiply whole
How Did You Solve It? es/107 numbers using various
#7, 8, 9, 10 OA.3 Lessons strategies
OA.3 Formatives Lesson 5: Solve multi-step
multiplication problems
Lesson 8: Solve multi-step
division problems

Shady Teacher Guide Decomposing Three-Fifths www.k- http://achievethecore.org
Fractions pp. 13-16 5mathteachingresources.com www.IXL.com/signin/volusia
Anna Marie and the Pizza NF.3 R.4, R.5, R.6, R.7, R.11, R.1,
Cindy’s Reproducible R.2, R.3, R.12, R.9, R.10
Carpet pp. 3, 7, 11, 12 Fraction Word Problems
Emporium www.cpalms.org https://learnzillion.com/
Daily Math Journal Adding and Subtracting Learning to Love Like Unit 3
pp. 34, 35, 36, 39, 41, Mixed Numbers Denominators Lesson 7: Add fractions with
43, 45, 48, 50, 53, 55 Plastic Building Blocks like denominators
NF.2.3 Lesson 8: Subtract fractions
(review) Pick A Problem hcpss.instructure.com/cours with like denominators
#46, 47, 48, 49, 50, 55 es/107 Lesson 5: Understand
NF.3 Lessons fractions can be decomposed
How Did You Solve It? NF.3 Formatives in multiple ways
#38, 39, 40, 41, 42, 43,
44, 45 UNIT 9
Lesson 2: Adding and
Daily Dose of Fractions subtracting mixed numbers
& Decimals #1-120 using equivalent fractions
Lesson 8: Solving word
Giant Magnetic Fraction problems by adding and
Circles and Bars subtracting mixed numbers

http://achievethecore.org
enVisionMATH: SE = Student Edition; RMC= Ready-Made Centers; POD= Problem of the Day; A&R = Assessment and Reteaching Workboo

62 Volusia County Schools Grade 4 Math Curriculum Map
May 2016
Mathematics Department

Unit 4 Suggested Instructional Resources

MAFS AIMS Lakeshore MFAS Internet enVision

Step by Step Reproducible Shopping List www.k- www.IXL.com/signin/volusia 12-9C
p. 3 Kesha and Juan
Mix-Ups and Remote Control Motorcycle 5mathteachingresources.com O.1, O.3, O.5, O.6, O.7, O8
Mysteries Daily Math Journal Piano Lessons
pp. 56, 58, 61, 63, 65 MD.2 M.4, M.6, M.7

MD.1.2 Pick A Problem www.cpalms.org https://learnzillion.com/
(review) #56, 57, 58, 60, 61, 62, Money Managers Unit 7
64, 65, 66, 67, 68, 69, Lesson 7: Convert
70, 71, 72 hcpss.instructure.com/cours measurements in order to
es/107 compare amounts
How Did You Solve It? MD.2 Lessons Lesson 8: Practice converting
#59, 60, 61, 62, 63 MD.2 Formatives measurements in order to
compare amounts

http://achievethecore.org

Hall of Teacher Guide Using Area and Perimeter www.k- www.IXL.com/signin/volusia 15-3 SE,
Mirrors pp. 20-21 5mathteachingresources.com P.19, P.20, P.21, P.22, P.24, RMC
Fencing a Garden MD.3 P.25 15-6 SE,
Reproducible RMC
pp. 3, 15 What is the Perimeter of the www.cpalms.org https://learnzillion.com/ 15-7 SE
Lettuce Section? Its All Around But Covered Up Unit 2
MD.1.3 Daily Math Journal New Puppy’s Pen Lesson 13: Applying area and
(review) pp. 57, 58, 60, 64 Applying Area and perimeter
Perimeter hcpss.instructure.com/cours Lesson 15: Finding area,
Pick A Problem es/107 perimeter, and missing sides
#74, 75, 76, 77, 78 MD.3 Lessons
MD.3 Formatives http://achievethecore.org
How Did You Solve It?
#64, 65, 66

Area Tiles

Party Math Geometry
Problem Solving Kit
Activity Cards 5-6
enVisionMATH: SE = Student Edition; RMC= Ready-Made Centers; POD= Problem of the Day; A&R = Assessment and Reteaching Workbook

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Mathematics Department

Formative Assessment Strategies

Mathematics K-5

Name Description Additional Information
A & D Statements A & D Statements analyze a set of “fact or fiction” statements. First,
students may choose to agree or disagree with a statement or identify Statement How can I find out?
Agreement Circles whether they need more information. Students are asked to describe
their thinking about why they agree, disagree, or are unsure. In the 9/16 is larger than 5/8.
Annotated Student second part, students describe what they can do to investigate the
Drawings statement by testing their ideas, researching what is already known, or __agree __disagree
using other means of inquiry. __not sure __it depends on

Agreement Circles provide a kinesthetic way to activate thinking and My thoughts:
engage students in mathematical argumentation. Students stand in a
circle as the teacher reads a statement. They face their peers still http://www.mathsolutions.com/documents/How_to_
standing and match themselves up in small groups of opposing beliefs.
Students discuss and defend their positions. After some students Get_Students_Talking.pdf
defend their answers, the teacher can ask if others have been swayed.
If so, stand up. If not, what are your thoughts? Why did you disagree? There 20 cups in a gallon. Agree or disagree?
After hearing those who disagree, does anyone who has agreed want to
change their minds? This should be used when students have had 2/3 equivalent to 4/6. Agree or disagree?
some exposure to the content.
A square is a rectangle. Agree or disagree?
Annotated Student Drawings are student-made, labeled illustrations that
visually represent and describe students’ thinking about mathematical Additional Questioning:
concepts. Younger students may verbally describe and name parts of Has anyone been swayed into new thinking?
their drawings while the teacher annotates it for them. What is your new thinking?
Why do you disagree with what you have heard?
Does anyone want to change their mind?
What convinced you to change your mind?
Use when students have had sufficient exposure to
content.

http://formativeassessment.barrow.wikispaces.net/A
greement+Circles
Represent 747 by drawing rods and cubes.
Represent 3x2=2x3 by drawing arrays.
Describe the meaning of 5.60.

64 Volusia County Schools http://formativeassessment.barrow.wikispaces.net/A
nnotated+Student+Drawings
Mathematics Department
Grade 4 Math Curriculum Map
May 2016

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Card Sorts
Card Sorts is a sorting activity in which students group a set of cards
with pictures or words according to certain characteristics or category.
Students sort the cards based on their preexisting ideas about the
concepts, objects, or processes on the cards. As students sort the
cards, they discuss their reasons for placing each card into a designated
group. This activity promotes discussion and active thinking.

Commit and Toss Commit and Toss is a technique used to anonymously and quickly http://teachingmathrocks.blogspot.com/2012/09/voc
assess student understanding on a topic. Students are given a abulary-card-sort.html
Concept Card question. They are asked to answer it and explain their thinking. They
Mapping write this on a piece of paper. The paper is crumpled into a ball. Once Stephanie eats 5 apple slices during lunch. When
the teacher gives the signal, they toss, pass, or place the ball in a she gets home from school she eats more. Which
basket. Students take turns reading their "caught" response. statement(s) below indicates the number of apple
Once all ideas have been made public and discussed, engage students slices Stephanie may have eaten during the day?
in a class discussion to decide which ideas they believe are the most
plausible and to provide justification for the thinking. a. She eats 5 apple slices.
b. She eats 5 apple slices at least.
Concept Card Mapping is a variation on concept mapping. Students are c. She eats more than 5 apple slices.
given cards with the concepts written on them. They move the cards d. She eats no more than 5 apple slices.
around and arrange them as a connected web of knowledge. This e. I cannot tell how many apple slices were eaten.
strategy visually displays relationships between concepts.
Explain your thinking. Describe the reason for
the answer(s) you selected.

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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Concept Cartoons
Concept Cartoons are cartoon drawings that visually depict children or  www.pixton.com (comic strip maker)
Four corners adults sharing their ideas about common everyday mathematics. A decimal is a fraction.
Students decide which character in the cartoon they agree with most
and why. This formative is designed to engage and motivate students to Agree Strongly
uncover their own ideas and encourage mathematical argumentation. Agree
Concept Cartoons are most often used at the beginning of a new
concept or skill. These are designed to probe students’ thinking about Strongly Disagree
everyday situations they encounter that involve the use of math. Disagree
Not all cartoons have one “right answer.” Students should be given
ample time for ideas to simmer and stew to increase cognitive
engagement.

Four Corners is a kinesthetic strategy. The four corners of the
classroom are labeled: Strongly Agree, Agree, Disagree and Strongly
Disagree. Initially, the teacher presents a math-focused statement to
students and asks them to go to the corner that best aligns with their
thinking. Students then pair up to defend their thinking with evidence.
The teacher circulates and records student comments. Next, the
teacher facilitates a whole group discussion. Students defend their
thinking and listen to others’ thinking before returning to their desks to
record their new understanding.

http://debbiedespirt.suite101.com/four-corners-
activities-a170020

http://wvde.state.wv.us/teach21/FourCorners.html

Frayer Model graphically organizes prior knowledge about a concept Frayer Model
into an operational definition, characteristics, examples, and non-
examples. It provides students with the opportunity to clarify a concept Definition in your own words Facts/characteristics
or mathematical term and communicate their understanding.
For formative assessment purposes, they can be used to determine A quadrilateral is a shape •4 sides
students’ prior knowledge about a concept or mathematical term before with 4 sides. • may or may not be of equal
planning the lesson. Barriers that can hinder learning may be uncovered
Frayer Model with this assessment. This will then in turn help guide the teacher for length
beneficial instruction. • sides may or may not be

parallel

Examples Quadrilateral Nonexamples

• square • circle
• rectangle • triangle
• trapezoid • pentagon
• rhombus • dodecahedron

66 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Friendly Talk Probes
Friendly Talk Probes is a strategy that involves a selected response
section followed by justification. The probe is set in a real-life scenario in
which friends talk about a math-related concept or phenomenon.
Students are asked to pick the person they most agree with and explain
why. This can be used to engage students at any point during a unit. It
can be used to access prior knowledge before the unit begins, or assess
learning throughout and at the close of a unit.

Human Scatterplots Human Scatterplot is a quick, visual way for teacher and students to get http://www.sagepub.com/upm-
an immediate classroom snapshot of students’ thinking and the level of data/37758_chap_1_tobey.pdf
I Used to Think… confidence students have in their ideas. Teachers develop a selective
But Now I Know… response question with up to four answer choices. Label one side of the I USED TO THINK… BUT NOW I KNOW…
room with the answer choices. Label the adjacent wall with a range of AND THIS IS HOW I LEARNED IT
low confidence to high confidence. Students read the question and
position themselves in the room according to their answer choice and
degree of confidence in their answer.

I Used to Think…But Now I Know is a self-assessment and reflection
exercise that helps students recognize if and how their thinking has
changed at the end of a sequence of instruction. An additional column
can be added to include…And This Is How I Learned It to help students
reflect on what part of their learning experiences helped them change or
further develop their ideas.

67 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Justified List
Justified List begins with a statement about an object, process, concept Example 1
or skill. Examples and non-examples for the statement are listed. Put an X next to the examples that represent 734.
Students check off the items on the list that are examples of the
statement and provide a justification explaining the rule or reasons for ___700+30+4 ___7 tens 3 hundreds 4 ones
their selections. This can be done individually or in small group. Small
groups can share their lists with the whole class for discussion and ___730 tens 4 ones ___7 hundreds 3 tens 4ones
feedback. Pictures or manipulatives can be used for English-language
learners. ___734 ones ___seven hundred thirty-four

___seventy-four ___ 400+70+3

Explain your thinking. What “rule” or reasoning did
you use to decide which objects digit is another
way to state that number.

Example 2

K-W-L is a general technique in which students describe what they K-This what I W-This is what I L-This is what I
Know about a topic, what they Want to know about a topic, and what already KNOW WANT to find out LEARNED
they have Learned about the topic. It provides an opportunity for
K-W-L Variations students to become engaged with a topic, particularly when asked what What do you think the learning goal is about?
they want to know. K-W-L provides a self-assessment and reflection at
Learning Goals the end, when students are asked to think about what they have List any concepts or ideas you are familiar with related
Inventory (LGI) learned. The three phrases of K-W-L help students see the connections to this learning goal.
between what they already know, what they would like to find out, and List any terminology you know of that relates to this
what they learned as a result. goal.
List any experiences you have had that may have
Learning Goals Inventory (LGI) is a set of questions that relate to an helped you learn about the ideas in this learning goal.
identified learning goal in a unit of instruction. Students are asked to
“inventory” the learning goal by accessing prior knowledge. This
requires them to think about what they already know in relation to the
learning goal statement as well as when and how they may have
learned about it. The LGI can be given back to students at the end of
the instructional unit as a self-assessment and reflection of their
learning.

68 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Look Back
Muddiest Point Look Back is a recount of what students learned over a given What I Learned How I Learned it
instructional period of time. It provides students with an opportunity to
Odd One Out look back and summarize their learning. Asking the students “how they Scenario: Students have been learning about the
learned it” helps them think about their own learning. The information
Partner Speaks can be used to differentiate instruction for individual learners, based on attributes of three-dimensional shapes.
their descriptions of what helped them learn. Teacher states, “I want you to think about the
Muddiest Point is a quick-monitoring technique in which students are muddiest point for you so far when it comes to
asked to take a few minutes to jot down what the most difficult or three-dimensional shapes. Jot it down on this
confusing part of a lesson was for them. The information gathered is HINTcard. I will use the information you give to me
then to be used for instructional feedback to address student difficulties. to think about ways to help you better understand
three-dimensional shapes in tomorrow’s lesson.”
Odd One Out combines similar items/terminology and challenges
students to choose which item/term in the group does not belong. Show students three figures and ask:
Students are asked to justify their reasoning for selecting the item that Which is the odd one out?
does not fit with the others. Odd One Out provides an opportunity for Explain your thinking.
students to access scientific knowledge while analyzing relationships
between items in a group.

Partner Speaks provides students with an opportunity to talk through an Ask students to choose a different odd one out and
explain their thinking.
idea or question with another student before sharing with a larger group. Today we are going to explore different ways to
add three-digit numbers together.
When ideas are shared with the larger group, pairs speak from the
perspective of their partner’s ideas. This encourages careful listening What different kinds of strategies
and consideration of another’s ideas. can you use to add 395+525?

Turn to your partner and take turns discussing
your strategies. Listen carefully and be prepared
to share your partner’s ideas.

69 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
A Picture Tells a
Thousand Words A Picture Tells a Thousand Words, students are digitally photographed Question Generating Stems:
during a mathematical investigation using manipulatives or other
Question Generating materials. They are given the photograph and asked to describe what  Why does___?
they were doing and learning in the photo. Students write their  Why do you think___?
description under the photograph. The images can be used to spark  Does anyone have a different way to
student discussions, explore new directions in inquiry, and probe their
thinking as it relates to the moment the photograph was snapped. By explain___?
asking students to annotate a photo that shows the engaged in a  How can you prove___?
mathematics activity or investigation helps them activate their thinking  What would happen if___?
about the mathematics, connect important concepts and procedures to  Is___always true?
the experience shown in the picture and reflect on their learning.  How can we find out if___?
Teachers can better understand what students are gaining from the
learning experience and adjust as needed.

Question Generating is a technique that switches roles from the teacher
as the question generator to the student as the question generator. The
ability to formulate good questions about a topic can indicate the extent
to which a student understands ideas that underlie the topic. This
technique can be used any time during instruction. Students can
exchange or answer their own questions, revealing further information
about the students’ ideas related to the topic.

Sticky Bars Sticky Bars is a technique that helps students recognize the range of
ideas that students have about a topic. Students are presented with a
short answer or multiple-choice question. The answer is anonymously
recorded on a Post-it HINT and given to the teacher. The HINTs are
arranged on the wall or whiteboard as a bar graph representing the
different student responses. Students then discuss the data and what
they think the class needs to do in order to come to a common
understanding.

70 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Thinking Log
Thinking Logs is a strategy that informs the teacher of the learning  I was successful in…
Think-Pair-Share successes and challenges of individual students. Students choose the  I got stuck…
thinking stem that would best describe their thinking at that moment.  I figured out…
Three-Minute Pause Provide a few minutes for students to write down their thoughts using  I got confused when…so I…
the stem. The information can be used to provide interventions for  I think I need to redo…
Traffic Light individuals or groups of students as well as match students with peers  I need to rethink…
Cards/Cups/Dots who may be able to provide learning support.  I first thought…but now I realize…
 I will understand this better if I…
Think-Pair-Share is a technique that combines thinking with  The hardest part of this was…
communication. The teacher poses a question and gives individual  I figured it out because…
students time to think about the question. Students then pair up with a
partner to discuss their ideas. After pairs discuss, students share their  I really feel good about the way…
ideas in a small-group or whole-class discussion. (Kagan)

HINT: Varying student pairs ensures diverse peer interactions.
Three-Minute Pause provides a break during a block of instruction in
order to provide time for students to summarize, clarify, and reflect on
their understanding through discussion with a partner or small group.
When three minutes are up, students stop talking and direct their
attention once again to the teacher, video, lesson, or reading they are
engaged in, and the lesson resumes. Anything left unresolved is
recorded after the time runs out and saved for the final three-minute
pause at the end.
Traffic Light Cards/Cups/Dots is a monitoring strategy that can be used
at any time during instruction to help teachers gauge student
understanding. The colors indicate whether students have full, partial,
or minimal understanding. Students are given three different-colored
cards, cups, or dots to display as a form of self-assessment revealing
their level of understanding about the concept or skill they are learning.

71 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
Two-Minute Paper
Two-Minute Paper is a quick way to collect feedback from students  What was the most important thing you learned
Two Stars and a Wish about their learning at the end of an activity, field trip, lecture, video, or today?
other type of learning experience. Teacher writes two questions on the
board or on a chart to which students respond in two minutes.  What did you learn today that you didn’t know
Responses are analyzed and results are shared with students the before?
following day.
 What important question remains unanswered
Two Stars and a Wish is a way to balance positive and corrective for you?
feedback. The first sentence describes two positive commendations for
the student’s work. The second sentence provides one  What would help you learn better tomorrow?
recommendation for revision. This strategy could be used teacher-to-
student or student-to-student.

Two-Thirds Testing provides an opportunity for students to take an
ungraded “practice test” two thirds of the way through a unit. It helps to
identify areas of difficulty or misunderstanding through an instructional
unit so that interventions and support can be provided to help them learn
and be prepared for a final summative assessment. Working on the test
through discussions with a partner or in a small group further develops
and solidifies conceptual understanding.

Two-Thirds Testing

72 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

Formative Assessment Strategies/Mathematics K-5 (continued)

Name Description Additional Information
What Are You Doing
What Are You Doing and Why? is a short, simple monitoring strategy to Scenario: Students are decomposing a fraction
and Why? determine if students understand the purpose of the activity or how it will into the sum of two or more of its parts.
help them learn. At any point in an activity the teacher gets the students’
Whiteboarding attention and asks “What are you doing and why are you doing it?” 3 = 1 + 1 + 1 3 = 2 + 1 3 = 3 + 0
Responses can be shared with the class, discussed between partners, 8 8 8 8 8 8 8 8 8 8
or recorded in writing as a One-Minute Paper to be passed in to the
teacher. The data are analyzed by the teacher to determine if the class Teacher stops students in their tracks and asks,
understands the purpose of the activity they are involved in. “What are you do and why are you doing it?”

Whiteboarding is a technique used in small groups to encourage
students to pool their individual thinking and come to a group consensus
on an idea that is shared with the teacher and the whole class. Students
work collaboratively around the whiteboard during class discussion to
communicate their ideas to their peers and the teacher.

http://www.educationworld.com/a_lesson/02/lp
251-01.shtml

3-2-1 is a technique that provides a structured way for students to reflect Sample 1
upon their learning. Students respond in writing to three reflective  3 – Three key ideas I will remember
prompts. This technique allows students to identify and share their  2 – Two things I am still struggling with
successes, challenges, and questions for future learning. Teachers
have the flexibility to select reflective prompts that will provide them with  1 – One thing that will help me tomorrow
the most relevant information for data-driven decision making.
Sample 2

3-2-1

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Mathematics Department

Intervention/Remediation Guide

Resource Location Description

Intervention Lessons Math Diagnosis and Use for pre-requisite skills or remediation. For grades K-2, the
lessons consist of a teacher-directed activity followed by
(Student and Teacher pages) Intervention System problems. In grades 3-5, the student will first answer a series of
questions that guide him or her to the correct answer of a given
problem, followed by additional, but similar problems.

Meeting Individual Needs Planning section of each Provides topic-specific considerations and activities for

Topic in the enVision Math differentiated instruction of ELL, ESE, Below-Level and

Teacher’s Edition Advanced students.

Differentiated Instruction Close/Assess and Provides lesson-specific activities for differentiated instruction for
Intervention, On-Level and Advanced levels.
Differentiate step of each

Lesson in the enVision
Math Teacher’s Edition

Error Intervention Guided Practice step of Provides on-the-spot suggestions for corrective instruction.

each Lesson in the
enVision Math Teacher’s

Edition

ELL Companion Lesson Florida Interactive Lesson Includes short hands-on lessons designed to provide support for

Support for English teachers and their ELL students, useful for struggling students

Language Learners as well

74 Volusia County Schools Grade 4 Math Curriculum Map
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Mathematics Department

GLOSSARY OF TERMS

Definitions for the framework of the curriculum map components are defined below.

Pacing: the recommended timeline determined by teacher committee for initial delivery of instruction in preparation for state assessments

Domain: the broadest organizational structure used to group content and concepts within the curriculum map

Cluster: a substructure of related standards; standards from different Clusters may sometimes be closely related because mathematics is a
connected subject

Standard: what students should understand and be able to do

Learning Targets: the content knowledge, processes, and behaviors students should exhibit for mastery of the standards

Hints: additional information that serves to further clarify the expectations of the Learning Targets to assist with instructional decision-making
processes

Vocabulary: the content vocabulary and other key terms and phrases that support mastery of the learning targets and skills; for teacher and
student use alike

Standards for Mathematical Practice: processes and proficiencies that teachers should seek to purposefully develop in students

Resource Alignment: a listing of available, high quality and appropriate materials, strategies, lessons, textbooks, videos and other media
sources that are aligned with the learning targets and skills; recommendations are not intended to limit lesson development

Common Addition and Subtraction/Multiplication and Division Situations: a comprehensive display of possible addition, subtraction,
multiplication and division problem solving situations that involve an unknown number in varied locations within an equation

Formative Assessment Strategies: a collection of assessment strategies/techniques to help teachers discover student thinking, determine
student understanding, and design learning opportunities that will deepen student mastery of standards

Intervention/Remediation Guide: a description of resources available within the adopted mathematics textbook resource (enVisionMATH) that
provides differentiated support for struggling learners—ESE, ELL, and General Education students alike

75 Volusia County Schools Grade 4 Math Curriculum Map
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