The Common Core Standards for Grade 8 (8.EE) highlight the importance of using scientific notation, integer exponents, and properties of exponents to build s

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8.EE.A.1:Knowandapplytheproperties,ofintegerexponentstogenerateequivalentnumerical expressions. For example,32 x 3a = 3-3 — ; = h

g.EE.A.2: Use square root and cube root symbols to represent.solutions to equations of the form

yz = .p andx3 = p, where ;;; ;.rtai* rational number.'Evaluate square roots of small perfect

squares and cube ,oot, o[ '*'fi p"*ttt cubes' Know that {2 is irrational'

8.EE.A.3: Use numbers expressed in the form of a single digit times an integer power of l0 to

estimate very large or r.ry ,i-rrll quantities, ,nd to "*p.E$

hi, *any times as much one is than the

other. For exa*ptu, n ti,i,Li)';i;;;;i:;i;;:i,h;;;;i*d"siit ' i' 3 times 108 and the poputation of

the world as 7 times lor,ina ininirine that tt u *ortd population is more than 20 times larget'

g.EE.A.4: Perform operations with numbers expressed in scientific notation' including problems

where both decimal ,.ri r.i..rtlfi. notation "r"

,s.J ur" scientific notation and choose units ot

appropriate size for *;;;;;; orut'y l"g" o' *'y small quantities (e'e'' use millimeters per

vear for seafloo*pr.ro,;;;. ili"rprli ,.#*in1 ""trilJ" if,rt has been genJrated bv technology'

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oMaior cluster

Cluster A: Work with radicals and integer exponents'

Grade 8 Overview

In this cruster students learn how to compute with integer exponents. students build on what they have learned about square

roots to solve equation, #h;';;;"f *)rL p ^nar;:';."il,Ji'i, , poriti"" ,"ii""ri"r*u"r, evaiuating perfect square and

oerfect cube roots. Students learn how to express r;iil;;.";;'d u.ry r*.ll ""*u." in scientific notation and express how manr

ii*., rrrg", or smaller ";;;;.; *ritten in ,.t.;ik;?;ii;" i;ih; r";tt .i. stra""tr use the properties o[ integer exponents

to perform operations *nh ;;;;r, written i" ,"i;;i'fi.-;;trtio.,. st'rd..,Ji't.rpr.t numbers written in scientific notation usrng

technologY.

Standards for Mathematical Practice

SFMP 2. Reason abstractly and quantitatively'

Students use reasoning to express how many times larger (or smaller) one number is than another when both are expressed in

scientific notation.

SFMP 5. Use appropriate tools strategically'

Students learn to read scientific notation as expressed by technology'

SFMP 5. Attend to Precision'

Studentscomputewithintegerexponentsandnumbersinscientificnotationaccurately.

116 The common core Mathematics companion: The standards Decoded' Grades 5-8

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Students learn how to compute using integer exponents building on their earlier experiences with adding and subtracting

integers. For any non-zero i.al .,u*6ers alnd i and integers n and m, the properties of integer exponents are as follows:

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Introduce the laws of integer exPonents one at time. Use

a conceptual approach as opposed to asking students to

memorize the rules.

Provide examples of the processes that lead to the rules for

each law, such as 42 x 43 – (4 " 4) " (4 " I x 4) : I x 4 x 4 x 4 x 4 – +5 . Allow students to try a few similar exPressions

to see if they can find the solution and posit a rule or

property they may discover.

Provide examples: 6272 – (6 t 6) (7 " 7) : (6 * 7) (6 x 7) – (6 * 7)2. Ask students to try several of their own to see if they

can discover a rule or property.

Provide examples: (sv)4 – (5 .5 .5) r (5. 5 . 5) * (5 . 5'5) * (5 . 5 .5) : -53×4

-5tz and (3 x7)a – (V .7) " (3 .7) x

(3 .7) " (3 .7): (3 '3 ') '3) , (7 '7 '7 '7): 3a x 74. Ask

students to try several of their own to see if they can discover

a rule or property.

Have students practice the properties by generating equivalent expressions and writing them in simplest form suchas3: *i-r:)-)- + = +

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Provide examples to lead students to discover how a

negative exponent translates to a positive exponent in the

denominator of a fraction, such as how 3-7 – + Assign a project for students to design and .r.'rt. posters

summa ilznrgthe rules they discovered to hang in the

classroom of the properties of exponents with integers.

Discover the properties of integer exPonents by.making sense

of the examples presented. For example, (3 x7)+ – () '7) " O .7)

" (3 .7) * (3 .7) -(3 .3 .3. 3) , (7 7 '7 '7) – 3a x 74 with the rule discovered as (ob)" – dnb'.

Generate equivalent expressions in simplest form for

products and quotients of numbers with integer exponents

having the same bases.

Students often confuse the rules. This occurs primarily when students are taught to memorize the rules rather than understand

what is happening in the properties by working with numerical expressions as in the suggestions above. It is important to present

examples ,.rd t"titrd".,t dir.ou.t what the ,i., ,r.. Then students should be encouraged to write their reasoning so thev can

clarify the explanations for themselves.

118 The Common Core Mathematics Companion: The Standards Decoded, Grades 6-8

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Stu.dents-learn that squaring-and cubing numbers are the inverse operations to finding square and.bube roots. This standard works with perfect squares and perfect cubes, and students will begin to recognize those nuirbers. Equations should include rational

numbers such as *2 = l- and *' = L and fractions where both the numerator and denominator are perfect squares or cubes:464

Square roots can be positive or negative becauseZ x2=4 and -Z x -2=4.

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Introduce squaring a number and taking the square root as inverse operations, providing students opportunities to practice squaring and taking roots.

Repeat the previous instruction for cubes and cube roots, also including fractions where the numerator and denominator are both perfect cubes.

Relate perfect square numbers and perfect cubes to geometric squares and cubes using square tiles and square cubes to build the numbers. Asquare root is the length of the side of a

square, and a cube root is the length of the side of a cube.

Encourage students to find patterns within the list of square numbers and then with cube numbers.

Facilitate a ciass discussion around the question, "fn the equation *2 = p, when cdn p be a negatiye number?" Students should come to the conclusion that it is not possible.

Discuss non-perfect squares and non-perfect cubes as

irrational numbers such as {2.

Recognize perfect squares and perfect cubes.

Solve equations containing cube and square roots.

Discover and explain the relationship between square and cube roots and the sides of a square and the edges of a cube, respectively, by using hands-on materials.

Reason that non-perfect squares and non-perfect cubes are irrational, including the square root of 2.

It is important for students to have multiple opportunities and exposures with perfect cubes. This is a new concept in the curriculum and many students struggle with finding cube roots. A common misconception for cube roots is that any number times 3 is a perfect cube. Building larger cubes from smaller ones gives students a visual that they can rely on.

Part 3 Expressions and Equations 119

This standard emphasizes scientific notation. Students write very large and very small numbers in scientific notation using positive and negative exponents. For eiample, 123,000 written in scientific notation is I.2l x I0', and 0.0008 written in scientific notation is 8 x 10+ . When mastered, students use the skill to determine how many times larger (or smaller) one number written in scientific notation is than another. To compare, if the exponent increases by l, the value increases l0 times. In the example

of the U.S. and world populations, the exponent increased by l, and the 7 is a little more than 2 times 3. So 2 x I0 makes for 20

times larger.

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lntroduce examples of very large and very small numbers in contexts. Contexts can be found in sources such as

government statistics websites, population sizes, land mass

in area, and science. Ask students why writing very large and very small numbers in scientific notation would be

beneficial. Who would use it?

Provide students with the opportunity to research very Iarge and very small numbers and present them written in scientific notation, along with the contexts, to the class.

Discuss why these numbers are considered estimates. Keep

a bank of these numbers and their contexts for students to use at a later time to create real-world problems.

Provide contextual problems for students to compare numbers written in scientific notation.

Use the bank of numbers created by the students to ask

about how some of the numbers are related to one another by comparing similar contexts.

o ljnderstand the benefits of using scientific notation.

3 Research to find examples of very large and very small numbers.

o Write very large and very small numbers in scientific notation.

c ljnderstand that some numbers written in scientific notation are estimates. Explain why that is true.

. Compare numbers written in scientific notation to determine how many times larger (or smaller) one number written in scientific notation is than another.

Students often confuse a very large number for a small number when written in scientific notation such as 4,000,000 for 4 x l0- This usually is a result of students tuying to memorize a rule about moving a decimal point to the left or the right. Instead of teaching

a rule, rely on students' background knowledge of negative exponents. Before rewriting a number in standard form, look to the

exponent to determine whether it is a small or large number. This can also be used as a check.

Students who do not understand the properties of exponents also make errors in computation with scientific notation. Teachers

may need to review these properties.

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12O The Common Core Mathematics Companion: The Standards Decoded, Grades 6-8

This standard builds on previous standards as now students use what they know about scientific notation and properties ofinteger exponents to solve problems. Quantities in the problems can be expressed in scientific notation and decimal form. Shrdents focus on the size of the measurement to determine which units are appropriate for the context such as millimeters for very small quantities. This standard also calls for students to use technology and be able to interpret the scientific notation used. The teacher needs to check the class calculators to be familiar with the notation used by those particular calculators as the notation used by calculators to express scientific notation is not standard.

Pose problems that require students to perform operations with numbers written in scientific notation.

Present problem-solving opportunities for students to choose correct units of measurement when working with very large and very small numbers including making conversions between units such as in the following problem : "An ayerdge ant is l}-t centimeters long. If you laid dnts end to end, how mdny would it take to make a line from I ew York City to Disney World?" The distance to Disney World from MC is 1,513 kilometers.

Provide students with calculators. Give them a calculation to perform that results in a number displayed in scientific notation. Facilitate a large group discussion about what the notation in the display means and how it is a form of scientific notation. Provide other opportunities for students to interpret scientific notation expressed with technology. Ask, "Do all calculators use the sdme display for scientific notation?"

Perform operations with numbers written in scientific notation. Solve both mathematical and real-world problems.

Choose correct units for very large and very small numbers when solving problems.

Discover and interpret the rules for scientific notation displayed on a given calculator.

When performing operations with numbers in scientific notation, such as (7 x 105) x (18 x 10e), some students will be overwhelmed with keeping track of what they should do. Encourage these students to color code the numbers such as

highlighting the numbers in exponential form in the given example so students remember to work them together.

Part 3 Expressions and Equations 121

8.EE.B*

8.EE.B,5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compore a distance-time graph to a distance-time equation to determine which of two mofing obiects has greater speed.

8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation I = mx for a line through the origin and the equation I = mx + b for a line intercepting the vertical axis at b.

oMajor cluster

Cluster B: Understand the connections between proportional relationships, lines, and Iinear equations. Grade 8 Overview

In this cluster students connect proportional relationships, lines, and linear equations. First, students compare proportional relationships represented in different ways such as graphs, tables, and linear equations. Unit rate is interpreted as the slope of a

line, and students learn that the slope is the same between any two points on a line by using similar triangles. Then the general equations for a line (y = ** + b and f = mx) are derived.

Standards for Mathematical Practice

SFMP 2. Reason abstractly and quantitatively.

Students compare two proportional relationships represented in different forms.

SFMP 6. Attend to precision.

Students give explanations that are precise and use appropriate vocabulary.

SFMP 7. Look for and make use of structure.

Students see a pattern that results in the general form ofa linear equation.

Related Content Standards

6.RP.A.3.b 7.8F,.8.4.a 8.G.A.4 B.F.A.2 B.F.A.3

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122 The Common Core Mathematics Companion: The Standards Decoded, Grades 5-8

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students build on their work from Grade 6 with unit rates and their work with prop_ortional relationships in Grade 7 to comparegraphs' tables' and equations.of li"..t 1p.opoiiio"al ,.trtionGr. sili."ts identrty tt. ,rii.rt" ,s slope i., graphs, tables, andequations to compare proportional tti'tionrhip, presented ,rr1s air.*rJrepresentations. For example, compare the unit rate in ixi"':'.:T.xT:fJ#:;t:l'.'#[li:X':r'#i:i""',"ffii::;il"";i,,;"aphoneu,ii,.-'"iir.,.;i:;;;;;;*i"?.,r..fl

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o Present a.1ingle, graphe,d, proportional relationship to the class. Facilitate a .trir discussion about the unit rate, using students' background knowledg., and interpret the unit rate as the slope of the line.

c Present a second, related, proportional relationship, written in a different formiuch as a table or equation. Facilitate a class discussion about how to .o-prr. the two situations. [Jse questions such as, "what is hafipuning in each situation? How are they the same? Different? Irow can we tell? What about the stoles? What do ti'uy tell us? What can we do n hglp us campare the slopes (gra'ph the second relationship)?"

o Provide opportunities for students to compare proportional relationships and write their conclusions ,rlrrg precise mathematical vocabulary.

Make use of the structureof a representation to compare {iffe19nt representations. For erample, students will find lhit if a problem asks them to compnr. the unit rates for information

-presented in a table ,"a information presented

in equation form, graphing both of them will make the comparison easier.

9o*plre ProPortional relationships presented in different forls. (graphs, tables,. equations, u.rurt descriptions) and explain comparisons in writing using clear r;; pr;cise mathematicil languag.._ !r*[rrirri, will include slope interpreted in contexiof the relationships.

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Errors occur when students are overwhelmed by being presented with too much information at a time. Encourage studentshaving difficultv making the compariso.r t" *""r, ,"itri;;;;;lrtt";;i;ip "i

, ,r*.. Graphing may be a difficurt sk,l for somestudents' use graph paplr larger trrr" r "rn

rorl-hese students so they can see the unit rate easier. students who are overwhelmed can also be helped by using graphs of experiences that are familiar to them. This makes theinformation more accessible so students ;il't;;;d.,*?fiffi t"".ir;o-oortionar relationships.

Part 3 Expressions and Equations 1zg

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Students gain additional knowledge about slope in this standard as they use similar triangles to explhin how the slope m of a line is the same between any two points on a given non-vertical line. Students understand positive/negative slopes, 0 slope, and

defined slopes. Through the use oJsimilar triangles, teachers lead students to derive the general equation (y = ** + b) of a line and$scover that m $ the slope and b is the y-intercept.

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6ilitate a class discussion about two similar triangles as in Explain, orally and/or in writing, using similar triangles,

IH'tr-'"'J;:ii,i::,l,yi::ff":?.'.:LTlffi #:ints Discover that b is the y-intercept and m is the slope in the general equation for a line, T = mx + b.

Determine the slope of a line from a graph, table, or linear equation.

Explain orally and/or in writing how proportional relationships, lines, and linear equations are related.

the example below:

Explain slope as rise over run (rise is the vertical distance and run is the horizontal distance) by having students see

that 7b (rise) is/ units and b, (run) is 3 units for a ratio

of 2 to 3 or written as 3 . O" similar triangle CED, A is

4 units and d" l, 6 ,rnits for a ratio of 4to 6, which is the same as Z to 3. Have students create other pairs of similar triangles to convince themselves that the slope of a line is the same between any two points on a non-vertical line.

Challenge students to find the slope of a horizontal line (0)

and a vertical line (undefined) and explain their reasoning.

Demonstrate how to find a slope using the formula fr-Tz xr-xZ

Lead students to discover the equation y – mx for a line that goes through the origin and T : mx + b for a line that goes through point b. Note that when b : 0, T : mx. The y-intercept is b.

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124 The Common Core Mathematics Companion: The Standards Decoded, Grades 5-8

Addressing Student Misconceptions and Common Errors

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Part 3 Expressions and Equations 125

8.EE.C*

8.EE.C.7:.Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively hansforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

8.EE.C.8: Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables correspond to points ofintersection oftheir graphs, because points ofintersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 haye no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pdirs of points, determine whether the line through the

first pair of points intersects the line through the second pair.

oMajor cluster

Cluster C: Analyze and solve linear equations and pairs of simultaneous linear equations. Grade 8 Overview

Students analyze and solve one variable linear equations for one, zero, ot infinitely many solutions, simplifuing the equations until they reach x = a, e. = a, or o = b (where a and b are different numbers). Students then apply that knowledge to analyzing and solving pairs of simultaneous linear equations also known as systems of linear equations in two variables.

Standards for Mathematical Practice

SFMP 1. Make sense of problems and persevere in solving them.

Students solve problems with systems of linear equations.

SFMP 2. Reason abstractly and quantitatively.

Shrdents analyze linear equations and systems oflinear equations.

SFMP 4. Model with mathematics.

Students model real-world problems with systems of equations.

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126 The Common Core Mathematics Companion: The Standards Decoded, Grades 6-8

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This standard has students solving linear equations. It is explained by g.EE.c.7.a and b. It is best to teach a and b together so that

};X':*,t'|]]::fi51l:'lln for au,future workwith line-ar equations' Students-solve equations thathave one"zeto' or-

infinitely many sorution, "rJr"rr," thos" solutionx;H#il. If ,h; l"i;;i;; i, in tr," 5or* r:o, there is onry one solution' If

c = a, there ar" infinit"ly l,a.,y ,ot,rtior,r. lf a = b;;il (;;;;'i ^na

n";;ff;;i ;mbers)' there are no solutions'

Linear equations can have ftactions and decimals as coefficients and can be sorved by expanding expressions with the distributive

ptop.rty and/or collecting like terms'

solve and anaryzeone variabre linear equations and explain

whether the ,oirtion has one, zero,or infinitely many

solutions.

Solve linear equations with rational coefficients' Use the

distributive d;;t*f't" 'Ppt-oPliate and combine like

terms when the equation calls for it'

A common error students make invorves apprying the distributive property w}en negative integers are involved' such as -7(-x – i'

The error occurs when *r,.v lry a *rrtipry ir,L -2 ;;;; :4 Stude;ts need repeatEd exposure to equations of this type' Prompting

students to consider.*ir,rl +j ,, "pt,-,, negative +,iir]rp, ."*".itr," *ir*r,i:;1#:ir#"g1"a d'iscussing tasks that involve

students analyzing t"o" ilttp' 'i"dt"t' "if-tottect many misconcepttons'

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. Provide pairs of students with three one-variable linear

equatio,*o,.eequationhasonesolution,onehasno solutionr, *e on; has an infinite number of solutions'

After students have , .ttr".e to solve, facilitate a discussion

on the results studentt f"""a' Demonstrate how when the

result is x = a, there is only one solution that will make the

equation true. Use ,.rUrtitltion when student results are

a -b(a); ask the stud;;ir to make sense of the results' Ask

questio,, ,".r, aS,,,Cdn a =b?,, An a|yze the equation to see

whytherearenopossiblesolutions.Forresults a = d.)use substitution to demonstrate how there are an

infinite number of solutions'

o Present students many opportunities to solve linear

equatio;;;-.l.rdi,g iholJ *i,

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