CCSS Mathematical Practices

CCSS Mathematical Practices

First Grade

Make sense of problems and persevere in solving them.

Students should recognize problem solving words such as “in all”, “how many more”, “all together”, etc.

They should then have multiple strategies they could use to solve the problem. Such as, number line, pencil paper, virtual manipulatives, concrete models, counting on, verbal descriptions. Students should continually be asking themselves, “Does this make sense?” Students should compare and contrast solids.

Reason abstractly and quantitatively.

Students should be encouraged to use a variety of strategies when problem solving. These strategies could include acting it out, moving things around, determining a number sentence that could represent the problem. Students should then be expected to return to the problem to see if a solution makes sense.

Construct viable arguments and critique the reasoning of others.

Students should build a mathematical vocabulary and be encouraged to use it. They need to use prior knowledge and definitions in logical steps to justify an answer. Discussion should be encouraged when solving problems. Students should be comfortable asking questions to each other to help understand how to solve a problem.

“Explain you thinking” “Why is that true”

Model with mathematics.

Students should be encouraged to use different strategies to solve the same problem using number sentences – ex. 4+4+4 or 4 x 3. As students learn more mathematics, the ways they model situations with mathematics should become more efficient. They may use numbers, words, drawing pictures, using objects, acting out, making a chart or list, creating equations, etc.

Use appropriate tools strategically.

Students should have a toolbox of tools, manipulatives, used to solve problems and know when to use what tools. For instance, first graders decide it might be best to use colored chips to model an addition problem.

Attend to precision.

Students should use clear and precise language in their discussions with others and when they explain their own reasoning.

Look for and make use of structure.

Part to Whole

First graders begin to discern a pattern or structure. For instance, if students recognize 12+3=15, then they also know 3+12=15.

Look for and express regularity in repeated reasoning.

Students notice patterns in shapes, counting, and computation that help them gain a better conceptual understanding.

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First Grade

2011-2012

Mathematics Course of Study

1.OA 1Operations and Algebraic Thinking

Represent and solve problems involving addition and subtraction

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

VOCABULARY / EXAMPLES

Symbols

Explain

Combining

Part

Whole

Counting On
Addition

Subtraction

Separate

Number Sentence

Whole Numbers

Addend

Use objects, drawings, and equations with a symbol for the unknown number to represent the problem.

9-3 = ? or 3+? = 9

First Grade

2011-2012

Mathematics Course of Study

1.OA 1Operations and Algebraic Thinking (cont’d)

VOCABULARY / EXAMPLES

Difference

Plus

Minus

Equals

First Grade

2011-2012

Mathematics Course of Study

1.OA 2Operations and Algebraic Thinking

Represent and solve problems involving addition and subtraction
2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.
VOCABULARY / EXAMPLES

Symbols

Explain

Combining

Part

Whole

Counting On

Addition

Number Sentence

Whole Numbers

Addend

Plus

Equals

Use objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Counting on and counting on again (to add 3+2+4 a student write 3+2+4=? And thinks, “3,4,5 that’s 2 more, 6,7,8,9 that’s 4 more so 3+2+4=9”)

Making tens (4+8+6=4+6+8=10+8=18)

Using “Plus10, minus 1” (instead of thinking 9 +4 think 10+4 – 1)

Using doubles

First Grade

2011-2012

Mathematics Course of Study

1.OA 3Operations and Algebraic Thinking

Understand and apply properties of operations and the relationship between addition and subtraction
3.Apply properties of operations as strategies to add and subtract.
VOCABULARY / EXAMPLE

Turn Around Facts

Related Facts

Zero

Making a Ten

If 8 + 3 = 11 is known, then 3 + 8 + 11 is also known. (commutative property of addition)

To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (associative property)

Adding and subtracting zero does not change original number (identity property)

First Grade

2011-2012

Mathematics Course of Study

1.OA 4Operations and Algebraic Thinking

Understand and apply properties of operations and the relationship between addition and subtraction
4.Understand subtraction as an unknown-addend problem.
VOCABULARY / EXAMPLES

Addend

Related Facts

Tens Frames

Counting On

Number Lines

Subtract 10 – 8 by finding the number that makes 10 when added to 8.

Some strategies they may use are counting objects, creating drawings, counting up, using number lines or 10 frames to determine an answer.

First Grade

2011-2012

Mathematics Course of Study

1.OA 5Operations and Algebraic Thinking

Add and subtract within 20
5.Relate counting to addition and subtraction.
VOCABULARY / EXAMPLES

Skip Counting

Counting on

Counting Back

Counting on 2 to add 2

Skip Counting by 2’s, 5’s, 10’s

Remind students that when adding, the largest number comes last. When subtracting, the largest number comes first.

First Grade

2011-2012

Mathematics Course of Study

1.OA 6Operations and Algebraic Thinking

Add and subtract within 20
6.Addition and subtraction strategies within 20, demonstrating fluency for addition and subtraction within 10.
VOCABULARY / EXAMPLES

Counting on

Making ten

Related facts

Equivalent sums

Number line

Counting On (With and without using a number line)

Making ten (8+6=8+2+4=10+4=14)

Decomposing a number leading to a ten (13-4=13-3-1=10-1=9)

Related Facts (8+4=12, should also know 12 – 8 +4)

Creating equivalent but easier or known sums (adding 6+7 by creating the known equivalent 6+6+1=12+1=13)

Use hands on manipulatives – such as cubes – to assist

First Grade

2011-2012

Mathematics Course of Study

1.OA 7Operations and Algebraic Thinking

Work with addition and subtraction equations
7.Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
VOCABULARY / EXAMPLES

Equal (meaning the same quantity as)

Equal Sign

True

False

Same As

Not the Same As

Which of the following equations are true and which are false?

6=6
7=8-1
5+2=2+5
4+1=5+2

First Grade

2011-2012

Mathematics Course of Study

1.OA 8Operations and Algebraic Thinking

Work with addition and subtraction equations
8.Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.
VOCABULARY / EXAMPLES

Number Sentence

Equation

Determine the unknown number that makes the equation true in each of the equations

8+?=11
5 = ? – 3
6+6=?

Possible Student “think-throughs”

8+?=11: “8 and some number is the same as 11. 8 and 2 is 10 and 1 more makes 11. So the answer is 3”
5=? – 3: “This equation means I had some cookies and I ate 3 of them. Now I have 5. How many cookies did I have to start with? Since I have 5 left and I ate 3, I know I started with 8 because I count on from 5,6,7,8”

First Grade

2011-2012

Mathematics Course of Study

1.NTB 1Number and Operations in Base Ten

Extend the counting sequence
1.Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
VOCABULARY / EXAMPLES

Count

Count On

Count Forwards

To extend students’ understanding of counting, they should be given opportunities to count backwards by ones and tens.

First Grade

2011-2012

Mathematics Course of Study

1.NTB 2Number and Operations in Base Ten

Understand Place Value

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

10 can be thought of as a bundle of ten ones – called a “ten”.
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

VOCABULARY / EXAMPLES

Tens

Ones

Leftovers

Bundle

Provide multiple opportunities to count any number of objects up to 99 making bundles of tens with or without leftovers.

53 can be 53 ones; 5 groups of 10 with 3 ones leftover; 5 tens and 3 ones; fifty-three

First Grade

2011-2012

Mathematics Course of Study

1.NTB 3Number and Operations in Base Ten

Understand Place Value

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

VOCABULARY / EXAMPLES

More Than

Less Than

Greater Than

Most

Least

Greatest

Same As

Equal To

Not Equal To

Students use models that represent two sets of numbers. To compare, students first attend to the number of tens, then, if necessary, to the number of ones.

Students may also use pictures, number lines, and spoken or written words to compare two numbers.

First Grade

2011-2012

Mathematics Course of Study

1.NTB 4Number and Operations in Base Ten

Use place value understanding and properties of operations to add and subtract

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

VOCABULARY / EXAMPLES

Tens

Ones

Place Value

Two-Digit Numbers

One Digit Number

Insert examples from ARIZONA STANDARDS PAGES 11 AND 12

First Grade

2011-2012

Mathematics Course of Study

1.NTB 5Number and Operations in Base Ten

Use place value understanding and properties of operations to add and subtract

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

VOCABULARY / EXAMPLES

10 more

10 less

10 more than 43 is 53 because 53 is on more 10 than 43

10 less than 43 is 33 because 33 is one 10 less than 43

First Grade

2011-2012

Mathematics Course of Study

1.NTB 6Number and Operations in Base Ten

Use place value understanding and properties of operations to add and subtract

Subtract multiples of 10 in the range 10 -90 from multiples of 10 in the range 10 -90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

VOCABULARY / EXAMPLES

Counting Back

Counting by 10’s

Students may use base ten blocks, 100s charts, number lines, etc. to demonstrate and justify their thinking.

70-30: Seven 10s take away three 10s is four 10s

80-50: 80, 70 (one 10), 60 (two 10s), 50 (three 10s), 40 (four 10s), 30 (five 10s)

60 – 40: I know that 4 + 2 is 6 so four 10s + two 10s is six 10s so 60 -40 is 20

First Grade

2011-2012

Mathematics Course of Study

1.MD 1Measurement and Data

Measure lengths indirectly and by iterating length units

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

VOCABULARY / EXAMPLES

Measure

Lengths

Non-Standard Unit

Longer

Shorter

Taller

Higher

Examples for ordering

Order three students by height
Order pencils, crayons, markers by length
Build three towers and order them from shortest to tallest
Three students each draw one line, then order the lines from longest to shortest

Examples for comparing indirectly

Two students each have a length of string. Given a tower of cubes, each student compares his/her string to the tower. Then students make statements such as, “My string is longer than the cube tower and your string is shorter than the cube tower. So, my string is longer than your string.”

First Grade

2011-2012

Mathematics Course of Study

1.MD 2Measurement and Data

Measure lengths indirectly and by iterating length units

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end: understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

VOCABULARY / EXAMPLES

Non-standard unit

Measure

Length

Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

How many paperclips long is a pencil?

First Grade

2011-2012

Mathematics Course of Study

1.MD 3Measurement and Data

Tell and write time

Tell and write time in hours and half-hours using analog and digital clocks.

VOCABULARY / EXAMPLES

Time

Hour

Half Hour

Digital

Analog

Hour Hand

Minute Hand

Minutes

There are 60 minutes in one hour; so halfway between an hour, 30 minutes have passed.

Time on the hour is written in the same manner as it appears on a digital clock.

Half hour is written with “30” after the colon.

The idea of 30 being “halfway” is difficult for students to grasp. Students can write the numbers form 0 – 60 counting by tens on a sentence strip. Fold the paper in half and determine that halfway between 0 and 60 is 30. A number line on an interactive whiteboard may also be used to demonstrate this.

First Grade

2011-2012

Mathematics Course of Study

1.MD 4Measurement and Data

Represent and interpret data

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.

VOCABULARY / EXAMPLES

Tally charts

Picture graph

Bar graph

Table graph

Chart

Data

Compare

Altogether

Fewer

Units

Students create object graphs and tally charts using data relevant to their lives (e.g., favorite ice cream, eye color, pets, etc). Graphs may be constructed by groups of students as well as by individual students.

Counting objects should be reinforced when collecting, representing, and interpreting data. Students describe the object graphs and tally charts they create. They should also ask and answer questions based on these charts or graphs that reinforce other mathematics concepts such as sorting and comparing. The data chosen or questions asked give students opportunities to reinforce their understanding of place value, identifying ten more and ten less, relating counting to addition and subtraction and using comparative language and symbols.

First Grade

2011-2012

Mathematics Course of Study

1.G 1Geometry

Reason with shapes and their attributes

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size): build and draw shapes to possess defining attributes.

VOCABULARY / EXAMPLES

Attributes

Sides

Vertices

Closed

Faces

Edges

Irregular Shapes

Students use attribute language to describe given two-dimensional/ three-dimensional shapes: number of sides, numbers of vertices, straight sides, faces, closed. “A triangle has three straight sides and is closed.” Be sure to include irregular shapes.

Students would use above attributes to compare and contrast figures.

List two things that are the same and two things that are different between a triangle and a cube.

Given a circle and a sphere, students identify the sphere as being three-dimensional but both are round.

Given a trapezoid, find another two-dimensional shape that has two things that are the same.

First Grade

2011-2012

Mathematics Course of Study

1.G 2Geometry

Reason with shapes and their attributes
2.Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
VOCABULARY / EXAMPLES

Rectangle

Square

Trapezoid

Triangle

Half-circle

Quarter-circle

Cubes

Right rectangular prisms

Right circular cones

Right circular cylinders

Students may use pattern blocks, plastic shapes, tangrams, virtual manipulatives to make new shapes. The teacher can provide students with cutouts of shapes and ask them to combine them to make a particular shape.

Ex. What shapes can be made from four squares?

Students can make three-dimensional shapes with clay or dough, slice into two pieces and describe the resulting shapes.

First Grade

2011-2012

Mathematics Course of Study

1.G 3Geometry

Reason with shapes and their attributes
3.Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
VOCABULARY / EXAMPLES

Fractions

Halves

Thirds

Fourths

Quarters

Physical Models

Equal

Parts

Whole

Decompose

Be sure to use different sized circles and rectangles. Have students cut things into two equal pieces.

Students should recognize that halves of two different wholes are not necessarily the same size.

Students should reason that decomposing equal shares into more equal shares results in smaller equal shares.

Practice breaking candy bars equally with one friend, three friends, etc. Student should be able to compare pieces by placing them side by side and notice that one fourth is smaller than one half.