Equal and Equivalent: IM 6.6.8
[url=https://im.kendallhunt.com/MS/teachers/1/6/8/preparation.html]“Equal and Equivalent”[/url] from IM Grade 6 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 6.6.8
- IM 6.6.8 Lesson: Equal and Equivalent
- Practice 6.6.8
- IM 6.6.8 Practice: Equal and Equivalent
IM 6.6.8 Lesson: Equal and Equivalent
Find a solution to each equation mentally.
[math]3+x=8[/math]
[math]10=12-x[/math]
[math]x^2=49[/math]
[math]\frac{1}{3}x=6[/math]
[size=150]Here is a diagram of [math]x+2[/math] and [math]3x[/math] when [math]x[/math] is 4. Notice that the two diagrams are lined up on their left sides.[br][br][img]data:image/png;base64,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[/img][br][br]In each of your drawings below, line up the diagrams on one side.[/size]
Draw a diagram of x+2, and a separate diagram of 3x, when x is 3.
Draw a diagram of x+2, and a separate diagram of 3x, when x is 2.
Draw a diagram of x+2, and a separate diagram of 3x, when x is 1.
Draw a diagram of x+2, and a separate diagram of 3x, when x is 0.
When are [math]x+2[/math] and [math]3x[/math] equal? When are they not equal? Use your diagrams to explain.
Draw a diagram of x+3, and a separate diagram of 3+x.
When are [math]x+3[/math] and [math]3+x[/math] equal? When are they not equal? Use your diagrams to explain.
Here is a list of expressions. Find any pairs of expressions that are equivalent. If you get stuck, try reasoning with diagrams.
Below are four questions about equivalent expressions. For each one:
[size=150][list][*]Decide whether you think the expressions are equivalent.[/*][*]Test your guess by choosing numbers for [math]x[/math] (and[math]y[/math], if needed).[/*][/list][/size][br]Are [math]\frac{x\cdot x\cdot x\cdot x}{x}[/math]and [math]x\cdot x\cdot x[/math] equivalent expressions?
Are [math]\frac{x+x+x+x}{x}[/math] and [math]x+x+x[/math] equivalent expressions?
Are [math]2(x+y)[/math] and [math]2x+2y[/math] equivalent expressions?
Are [math]2xy[/math] and [math]2x\cdot2y[/math] equivalent expressions?
IM 6.6.8 Practice: Equal and Equivalent
Draw a diagram of x+3 and a diagram of 2x when x is 1.
Draw a diagram of x+3 and a diagram of 2x when x is 2.
Draw a diagram of x+3 and a diagram of 2x when x is 3.
Draw a diagram of x+3 and a diagram of 2x when x is 4.
When are [math]x+3[/math] and [math]2x[/math] equal? When are they not equal? Use your diagrams to explain.
Do [math]4x[/math] and [math]15+x[/math] have the same value when [math]x[/math] is 5?
Are [math]4x[/math] and [math]15+x[/math] equivalent expressions? Explain your reasoning.
Check that [math]2b+b[/math] and [math]3b[/math] have the same value when [math]b[/math] is 1.
Check that [math]2b+b[/math] and [math]3b[/math] have the same value when [math]b[/math] is 2.
Check that [math]2b+b[/math] and [math]3b[/math] have the same value when [math]b[/math] is 3.
Do [math]2b+b[/math] and [math]3b[/math] have the same value for all values of [math]b[/math]? Explain your reasoning.
Are [math]2b+b[/math] and [math]3b[/math] equivalent expressions?
80% of x is equal to 100.
Write an equation that shows the relationship of 80%, [math]x[/math], and 100.
Use your equation to find [math]x[/math].[br][br]
For each story problem, write an equation to represent the problem and then solve the equation. Be sure to explain the meaning of any variables you use.
Jada’s dog was [math]5\frac{1}{2}[/math] inches tall when it was a puppy. Now her dog is [math]14\frac{1}{2}[/math] inches taller than that. How tall is Jada’s dog now?
Lin picked [math]9\frac{3}{4}[/math] pounds of apples, which was 3 times the weight of the apples Andre picked. How many pounds of apples did Andre pick?
Find these products.
[math](2.3)\cdot(1.4)[/math]
[math](1.72)\cdot(2.6)[/math]
[math](18.2)\cdot(0.2)[/math]
[math]15\cdot(1.2)[/math]
Calculate [math]141.75\div2.5[/math] using a method of your choice. Show or explain your reasoning.