\n
![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-7th-Grade-Module-3-Lesson-4-Example-Answer-Key-1.png\")
\nAnswer:
\n
![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-7th-Grade-Module-3-Lesson-4-Example-Answer-Key-2.png\")
<\/p>\nExample 1.
\n
\nAnswer:
\n<\/p>\n
\u2192 What is happening when you factor and write equivalent expressions for parts (e), (f), (g), and (h)?
\n\u2192 In the same way that dividing is the opposite or inverse operation of multiplying, factoring is the opposite of expanding.
\n\u2192 What are the terms being divided by?
\n\u2192 They are being divided by a common factor.
\nHave students write an expression that is equivalent to 8x+4.
\n\u2192 Would it be incorrect to factor out a 2 instead of a 4?
\n![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-7th-Grade-Module-3-Lesson-4-Example-Answer-Key-3.png\")
<\/p>\n
Example 2
\nLet the variables x and y stand for positive integers, and let 2x, 12y, and 8 represent the area of three regions in the array. Determine the length and width of each rectangle if the width is the same for each rectangle.
\n![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-7th-Grade-Module-3-Lesson-4-Example-Answer-Key-4.png\")
\nAnswer:
\n\u2192 What does 2x represent in the first region of the array?
\n\u2192 The region has an area of 2x or can be covered by 2x square units.
\n\u2192 What does the entire array represent?
\n\u2192 The entire array represents 2x+12y+8 square units.
\n\u2192 What is the common factor of 2x, 12y, and 8?
\n\u2192 The common factor of 2x, 12y, and 8 is 2.
\n\u2192 What are the missing values, and how do you know?
\n\u2192 The missing values are x, 6y, and 4. If the products are given in the area of the regions, divide the regions by 2 to determine the missing values.
\n![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-7th-Grade-Module-3-Lesson-4-Example-Answer-Key-5.png\")
<\/p>\n
Example 3.
\nA new miniature golf and arcade opened up in town. For convenient ordering, a play package is available to purchase. It includes two rounds of golf and 20 arcade tokens, plus $3.00 off the regular price. There is a group of six friends purchasing this package. Let g represent the cost of a round of golf, and let t represent the cost of a token. Write two different expressions that represent the total amount this group spent. Explain how each expression describes the situation in a different way.
\nAnswer:
\n\u2192 What two equivalent expressions could be used to represent the situation?
\n\u2192 6(2g+20t-3)
\nEach person will pay for two rounds of golf and 20 tokens and will be discounted $3.00. This expression is six times the quantity of each friend\u2019s cost.
\n\u2192 12g+120t-18
\nThe total cost is equal to 12 games of golf plus 120 tokens, minus $18.00 off the entire bill.<\/p>\n
\u2192 What does it mean to take the opposite of a number?
\n\u2192 You can determine the additive inverse of a number or a multiplicative inverse.
\n\u2192 What is the opposite of 2?
\n\u2192 -2
\n\u2192 What is (-1)(2)?
\n\u2192 -2
\n\u2192 What is (-1)(n)?
\n\u2192 -n
\n\u2192 What are two mathematical expressions that represent the opposite of (2a+3b)?
\n\u2192 (-1)(2a+3b) or -(2a+3b)
\n\u2192 Use the distributive property to write (-1)(2a+3b) as an equivalent expression.
\n\u2192 -2a-3b or -2a+(-3b)
\n\u2192 To go from -2a-3b to -(2a+3b), what process occurs?
\n\u2192 The terms -2a and -3b are written as (-1)(2a) and (-1)(3b), and the -1 is factored out of their sum.<\/p>\n
Example 4.
\n\u2192 What does it mean to take the opposite of a number?
\n\u2192 You can determine the additive inverse of a number or a multiplicative inverse.
\n\u2192 What is the opposite of 2?
\n\u2192 -2
\n\u2192 What is (-1)(2)?
\n\u2192 -2
\n\u2192 What is (-1)(n)?
\n\u2192 -n
\n\u2192 What are two mathematical expressions that represent the opposite of (2a+3b)?
\n\u2192 (-1)(2a+3b) or -(2a+3b)
\n\u2192 Use the distributive property to write (-1)(2a+3b) as an equivalent expression.
\n\u2192 -2a-3b or -2a+(-3b)
\n\u2192 To go from -2a-3b to -(2a+3b), what process occurs?
\n\u2192 The terms -2a and -3b are written as (-1)(2a) and (-1)(3b), and the -1 is factored out of their sum.<\/p>\n
Example 5
\nRewrite 5a-(a-3b) in standard form. Justify each step, applying the rules for subtracting and the distributive property.
\nAnswer:
\n5a+(-(a+-3b)) Subtraction as adding the inverse
\n5a+(-1)(a+-3b) The opposite of a number is the same as multiplying by \u20131.
\n5a+(-1)(a)+(-1)(-3b) Distributive property
\n5a+-a+3b Multiplying by -1 is the same as the opposite of the number.
\n4a+3b Collect like terms<\/p>\n
Exercise 1.
\nRewrite the expressions as a product of two factors.
\na. 72t+8
\nAnswer:
\n8(9t+1)<\/p>\n
b. 55a+11
\nAnswer:
\n11(5a+1)<\/p>\n
c. 36z+72
\nAnswer:
\n36(z+2)<\/p>\n
d. 144q-15
\nAnswer:
\n3(48q-5)<\/p>\n
e. 3r+3s
\nAnswer:
\n3(r+s)<\/p>\n
Exercise 2.
\nWrite the product and sum of the expressions being represented in the rectangular array.
\nAnswer:
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Eureka-Math-Grade-7-Module-3-Lesson-4-Exercise-Answer-Key-20.png\")
\n2(12d + 4e + 3); 24d + 8e + 6<\/p>\n
b. Factor 48j+60k+24 by finding the greatest common factor of the terms.
\nAnswer:
\n12(4j+5k+2)<\/p>\n
Exercise 3.
\nFor each expression, write each sum as a product of two factors. Emphasize the importance of the distributive property. Use various equivalent expressions to justify equivalency.
\na. 2\u22193+5\u22193
\nBoth have a common factor of 3, so the two factors would be 3(2+5). Demonstrate that 3(7) is equivalent to 6+15, or 21.<\/p>\n
b. (2+5)+(2+5)+(2+5)
\nAnswer:
\nThis expression is 3 groups of (2+5) or 3(2)+3(5), which is 3(2+5).<\/p>\n
c. 2\u22192+(5+2)+(5\u22192)
\nAnswer:
\nRewrite the expression as
\n2\u22192+(5\u22192)+(2+5), so 2(2+5)+(2+5), which equals 3(2+5).<\/p>\n
d. x\u22193+5\u22193
\nAnswer:
\nThe greatest common factor is 3, so factor out the 3:<\/p>\n
e. 3(x+5). (x+5)+(x+5)+(x+5)
\nAnswer:
\nSimilar to part (b), this is 3 groups of (x+5), so<\/p>\n
f. 3(x+5). 2x+(5+x)+5\u22192
\nAnswer:
\nCombine like terms and then identify the common factor. 3x+15, where 3 is the common factor: 3(x+5). Or,
\n2x+2\u22195+(x+5), so that 2(x+5)+(x+5)=3(x+5). Or, use the associative property and write:
\n2x+(5\u22192)+(5+x)
\n2(x+5)+(5+x)
\n3(x+5).<\/p>\n
g. x\u22193+y\u22193
\nAnswer:
\nThe greatest common factor is 3, so 3(x+y). (x+y)+(x+y)+(x+y)
\nThere are 3 groups of<\/p>\n
h. (x + y) + (x+ y) + (x+y)
\nAnswer:
\nThere are 3 groups of (x+y), so 3(x+y)<\/p>\n
i. 2x + (y + x) + 2y
\nAnswer:
\nCombine like terms and then identify the common factor. 3x+3y, where 3 is the common factor. 3(x+y). Or,
\n2x+2y+(x+y), so that 2(x+y)+(x+y) is equivalent to 3(x+y). Or, use the associative property, and write:
\n2x+2y+(y+x)
\n2(x+y)+(x+y)
\n3(x+y).<\/p>\n
Exercise 4.
\na. What is the opposite of (-6v+1)?
\nAnswer:
\n-(-6v+1)<\/p>\n
b. Using the distributive property, write an equivalent expression for part (a).
\nAnswer:
\n6v-1<\/p>\n
Exercise 5.
\nExpand each expression and collect like terms.
\na. -3(2p-3q)
\nAnswer:
\n-3(2p+(-3q)) Subtraction as adding the inverse
\n-3\u22192p+(-3)\u2219(-3q) Distributive property
\n-6p+9q Apply integer rules<\/p>\n
b. -a-(a-b)
\nAnswer:
\n-a+(-(a+-b)) Subtraction as adding the inverse
\n-1a+(-1(a+-1b)) The opposite of a number is the same as multiplying by \u20131.
\n-1a+(-1a)+1b Distributive property
\n-2a+b Apply integer addition rules<\/p>\n
Question 1.
\nWrite each expression as the product of two factors.
\na. 1\u22193+7\u22193
\nAnswer:
\n3(1+7)<\/p>\n
b. (1+7)+(1+7)+(1+7)
\nAnswer:
\n3(1+7)<\/p>\n
c. 2\u22191+(1+7)+(7\u22192)
\nAnswer:
\n3(1+7)<\/p>\n
d. h\u22193+6\u22193
\nAnswer:
\n3(h+6)<\/p>\n
e. (h+6)+(h+6)+(h+6)
\nAnswer:
\n3(h+6)<\/p>\n
f. 2h+(6+h)+6\u22192
\nAnswer:
\n3(h+6)<\/p>\n
g. j\u22193+k\u22193
\nAnswer:
\n3(j+k)<\/p>\n
h. (j+k)+(j+k)+(j+k)
\nAnswer:
\n3(j+k)<\/p>\n
i. 3(j+k) 2j+(k+j)+2k
\nAnswer:
\n3(j+k)<\/p>\n
Question 2.
\nWrite each sum as a product of two factors.
\na. 6\u22197+3\u22197
\nAnswer:
\n7(6+3)<\/p>\n
b. (8+9)+(8+9)+(8+9)
\nAnswer:
\n3(8+9)<\/p>\n
c. 4+(12+4)+(5\u22194)
\nAnswer:
\n4(1+4+5)<\/p>\n
d. 2y\u22193+4\u22193
\nAnswer:
\n3(2y+4)<\/p>\n
e. (x+5)+(x+5)
\nAnswer:
\n2(x+5)<\/p>\n
f. 3x+(2+x)+5\u22192
\nAnswer:
\n4(x+3)<\/p>\n
g. f\u22196+g\u22196
\nAnswer:
\n6(f+g)<\/p>\n
h. (c+d)+(c+d)+(c+d)+(c+d)
\nAnswer:
\n4(c+d)<\/p>\n
i. 2r+r+s+2s
\nAnswer:
\n3(r+s)<\/p>\n
Question 3.
\nUse the following rectangular array to answer the questions below.
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Eureka-Math-Grade-7-Module-3-Lesson-4-Problem-Set-Answer-Key-40.png\")
\na. Fill in the missing information.
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/03\/Eureka-Math-Grade-7-Module-3-Lesson-4-Problem-Set-Answer-Key-41.png\")
\nb. Write the sum represented in the rectangular array.
\nAnswer:
\n15f + 5g + 45<\/p>\n
c. Use the missing information from part (a) to write the sum from part (b) as a product of two factors.
\nAnswer:
\n5(3f+g+9)<\/p>\n
Question 4.
\nWrite the sum as a product of two factors.
\na. 81w+48
\nAnswer:
\n3(27w+16)<\/p>\n
b. 10-25t
\nAnswer:
\n5(2-5t)<\/p>\n
c. 12a+16b+8
\nAnswer:
\n4(3a+4b+2)<\/p>\n
Question 5.
\nXander goes to the movies with his family. Each family member buys a ticket and two boxes of popcorn. If there are five members of his family, let t represent the cost of a ticket and p represent the cost of a box of popcorn. Write two different expressions that represent the total amount his family spent. Explain how each expression describes the situation in a different way.
\nAnswer:
\n5(t+2p)
\nFive people each buy a ticket and two boxes of popcorn, so the cost is five times the quantity of a ticket and two boxes of popcorn.
\n5t+10p
\nThere are five tickets and 10 boxes of popcorn total. The total cost will be five times the cost of the tickets, plus 10 times the cost of the popcorn.<\/p>\n
Question 6.
\nWrite each expression in standard form.
\na. -3(1-8m-2n)
\nAnswer:
\n-3(1+(-8m)+(-2n))
\n-3+24m+6n<\/p>\n
b. 5-7(-4q+5)
\nAnswer:
\n5+-7(-4q+5)
\n5+28q+(-35)
\n28q-35+5
\n28q-30<\/p>\n
c. -(2h-9)-4h
\nAnswer:
\n-(2h-9)-4h
\n-(2h+(-9))+(-4h)
\n-2h+9+(-4h)
\n-6h+9<\/p>\n
d. 6(-5r-4)-2(r-7s-3)
\nAnswer:
\n6(-5r-4)-2(r-7s-3)
\n6(-5r+-4)+-2(r-7s+-3)
\n-30r+-24+-2r+14s+6
\n-30r+-2r+14s+-24+6
\n-32r+14s-18<\/p>\n
Question 7.
\nCombine like terms to write each expression in standard form.
\na. (r-s)+(s-r)
\nAnswer:
\n0<\/p>\n
b. (-r+s)+(s-r)
\nAnswer:
\n-2r+2s<\/p>\n
c. (-r-s)-(-s-r)
\nAnswer:
\n0<\/p>\n
d. (r-s)+(s-t)+(t-r)
\nAnswer:
\n0<\/p>\n
e. (r-s)-(s-t)-(t-r)
\nAnswer:
\n2r-2s<\/p>\n
Question 1.
\nWrite the expression below in standard form.
\n3h-2(1+4h)
\nAnswer:
\n3h-2(1+4h)
\n3h+(-2(1+4h)) Subtraction as adding the inverse
\n3h+(-2\u22191)+(-2h\u22194) Distributive property
\n3h+(-2)+(-8h) Apply integer rules
\n-5h-2 Collect like terms<\/p>\n
Question 2.
\nWrite the expression below as a product of two factors.
\n6m+8n+4
\nAnswer:
\nThe GCF for the terms is 2. Therefore, the factors are 2(3m+4n+2).<\/p>\n","protected":false},"excerpt":{"rendered":"
Engage NY Eureka Math 7th Grade Module 3 Lesson 4 Answer Key Eureka Math Grade 7 Module 3 Lesson 4 Example Answer Key Example 1. Answer: \u2192 What is happening when you factor and write equivalent expressions for parts (e), (f), (g), and (h)? \u2192 In the same way that dividing is the opposite or … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/77984"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=77984"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/77984\/revisions"}],"predecessor-version":[{"id":137768,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/77984\/revisions\/137768"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=77984"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=77984"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=77984"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}