1

Eureka Math™ Tips for Parents

Grade 8 Module 1

Prepared by The Eureka Math 6-8 Writing Team

What Came Before this Module: Students used whole number exponents to denote powers of ten, expanded the use of exponents to include bases other than ten and evaluated expressions limited to whole-number exponents. Students also learned how to apply exponents in various formulas.

What Comes After this Module: Students will learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence.

We use the definition above to prove that the following statement is true for all integer exponents b.

𝑥!! = !!!  

+ • Ask your child what they

learned in school today and ask them to show you an example.

• Complete the Sprint activity on the next page with your child.

• Ask your child to show you why the equation below is true.

𝑥!! ∙ 𝑥!! = 𝑥!!"

• Ask your child to determine the value of n and explain to you why they think their solution is correct.

2! ∙ 4! = 2! ∙ 2! = 2!

How Can You Help At Home?

Key Words Scientific Notation: The scientific notation is the representation of a number as the product of a finite decimal ,d,and a power of 10. The decimal d must be greater than or equal to one and less than 10. The exponent of the power of 10 must be an integer. For example, the scientific notation for 192.7 is 1.927  ×10!. An example of a number that is not written in scientific notation is 0.234567  ×10!. because 0.234567 is not greater than or equal to 1 and less than 10.

Order of Magnitude: The order of magnitude of a finite decimal is the exponent in the power of 10 when that decimal is expressed in scientific notation. For example, the order of magnitude of 192.7 is 2 because when 192.7  is expressed in scientific notation as 1.927  ×10!, 2 is the exponent of 10!.

Key Common Core Standards:

• Work with radicals and integer exponents. o Know and apply the properties of integer exponents to general

equivalent numerical expressions. o Use numbers expressed in the form of a single digit times an

integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

o Use, interpret and perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

In general, if x is any number and m, n are positive integers, then 𝑥! ∙ 𝑥! = 𝑥!!!

In general, if x is nonzero and m, n are positive integers, then !!

!!= 𝑥!!! , 𝑖𝑓  𝑚 > 𝑛

𝑥! ∙ 𝑥! = 𝑥!!!  

(𝑥!)! = 𝑥!"

(𝑥𝑦)! = 𝑥!𝑦!  

The Law of Exponents

For x, y > 0, and all integers a, b, the following holds:

For any positive number x and for any positive integer n, we define

x!! is just the reciprocal,!!, of x

𝑥!! = !!!    

Eureka Math, A Story of Ratios Grade 8

Module 1

For more information visit commoncore.org

Sprints help develop fluency, build excitement towards mathematics and encourage students to do their personal best! They are not necessarily a competition among classmates, but a quest to improve upon a student’s previous time, ultimately helping them achieve the desired fluency when they’re working with numbers as well as provide a feeling of achievement when their second sprint shows improvement. During the Sprint activity, your role as the parent will be the same as the role of the teacher when the class is completing this activity. You will keep track of the time as well as be an exciting and encouraging coach for your child. You will give your child the following: a copy of Sprint A and Sprint B. You can make a copy of this newsletter or use the original and fold the newsletter in half so your child only sees one Sprint at a time. You can use a stopwatch to record the time. For these modified sprints, please give your child 15 seconds to complete the 11 problems. Have fun!

Read on to learn a little bit about Eureka Math, the creators of A Story of Ratios:

Eureka Math is a complete, PreK–12 curriculum and professional development platform. It follows the focus and coherence of the Common Core State Standards (CCSS) and carefully sequences the progression of mathematical ideas into expertly crafted instructional modules.

This curriculum is distinguished not only by its adherence to the CCSS; it is also based on a theory of teaching math that is proven to work. That theory posits that mathematical knowledge is conveyed most effectively when it is taught in a sequence that follows the “story” of mathematics itself. This is why we call the middle school portion of Eureka Math "A Story of Ratios." The sequencing has been joined with methods of instruction that have been proven to work, in this nation and abroad. These methods drive student understanding beyond process, to deep mastery of mathematical concepts.

The goal of Eureka Math is to produce students who are not merely literate, but fluent, in mathematics. Your student has an exciting year of discovering the story of mathematics ahead!

Sample Problem from Module 1: (Example taken from Module 1, Lesson 13) Question: Compare 2.01  ×  10!" and 2.8  ×  10!". Which number is larger?

1. 2!   ∙ 2! 2. 2!   ∙ 2! 3. 2!   ∙ 2!

4. 99! ∙ 99! 5. 99! ∙ 99! 6. 99! ∙ 99! 7. 𝑟! ∙ 𝑟! 8. 𝑠! ∙ 𝑠! 9. 𝑥! ∙ 𝑥! 10. 5! ∙ 125 11. 8 ∙ 2!

1. 5!   ∙ 5! 2. 5!   ∙ 5!

3. 5!   ∙ 5!

4. 11!" ∙ 11! 5. 11!" ∙ 11!

6. 11!" ∙ 11! 7. 𝑥! ∙ 𝑥!

8. 𝑦! ∙ 𝑦!

9. 𝑧! ∙ 𝑧! 10. 2!! ∙ 4

11. 2!! ∙ 16

Timed Challenge: The SPRINT! Can you beat your personal best?

Directions: Rewrite each item as an equivalent expression in exponential notation. All letters denote numbers. Sprint A Sprint B

Answer: 2.01×10!" = 2.01×10!×10!" = 201×10!" Since 201 > 2.8, we have 201×10!" > 2.8×10!", and since 201×10!" = 2.01×10!", we conclude 2.01×10!" > 2.8×10!".

Sprint A 1. 2!   2. 2!   3. 2!   4. 99!   5. 99! 6. 99!!   7. 𝑟!"   8. 𝑠!"   9. 𝑥!   10. 5!   11. 2!"  

Sprint B 1. 5!   2. 5!   3. 5!   4. 11!"   5. 11!" 6. 11!"   7. 𝑥!"   8. 𝑦!"   9. 𝑧!"   10. 2!"   11. 2!"  

Anwers to the Sprints

1 Prepared by The Eureka Math 6-8 Writing Team

y Erin Schweng, Math Coach

The Concept of Congruence

In this -lesson module, students learn about translations, reflections, and rotations in the plane and how to use them to precisely define the concept of congruence. Up to this point, “congruence” has been taken to mean, intuitively, “same size and same shape.” Because this module begins a serious study of geometry, this intuitive definition must be replaced by a precise definition. This module is a first step; its goal is to provide the needed intuitive background for the precise definitons that are introduced in this module for the first time. Students are also introduced to the Pythagorean Theorem.

Key Words Transformation: A rule, to be denoted by , that assigns each point 𝑃 of the plane a unique point which is denoted by (𝑃). Basic Rigid Motion: A basic rigid motion is a rotation, reflection, or translation of the plane. Translation: A basic rigid motion that moves a figure along a given vector. Rotation: A basic rigid motion that moves a figure around a point, 𝑑 degrees. Reflection: A basic rigid motion that moves a figure across a line. Image of a point, image of a figure: Image refers to the location of a point, or figure, after it has been transformed. Sequence (Composition) of Transformations: More than one transformation.

Vector : A Euclidean vector (or directed segment) 𝐴𝐵 is the line segment 𝐴𝐵 together with a direction given by connecting an initial point 𝐴 to a terminal point 𝐵. Congruence: A congruence is a sequence of basic rigid motions (rotations, reflections, translations) of the plane. Transversal: Given a pair of lines 𝐿 and 𝑀

in a plane, a third line 𝑇 is a transversal if it

intersects 𝐿 at a single point and intersects

𝑀 at a single but different point.

Grade

Module

Eureka Math™ Tips for Parents

What Came Before this Module: Students used their knowledge of operations on numbers to include integer exponents and transformed expressions. Students made conjectures about how zero and negative exponents of a number should be defined and proved the properties of integer exponents. What Comes After this Module: In Module , students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles. Students learn the definition of a dilation, its properties and how to compose them. One overarching goal of this module is to replace the common idea of “same shape, different sizes” with a definition of similarity that can be applied to shapes that are not

polygons, such as ellipses and circles.

Describe, intuitively, what kind of transformation will be required to move Figure A on the left to its image on the right.

+ How Can You Help At

Home?

Every day, ask your child what they learned in school and ask them to show you an example.

Be excited to learn new ideas in math! Show your child that math is fun, even when it might be new and challenging!

Ask your child to describe a translation, reflection and rotation in his/her own words.

Watch the video posted below with your child and discuss each

of the transformations.http://www.youtube.com/watch?

v=O2XPy3ZLU7Y

Key Common Core Standards:

Understand congruence and similarity using physical models, transparencies, or geometry software.

o Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure. Parallel lines are taken to parallel lines.

o Understand that a two-dimensional figure is congruent to another if the

second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

o Use informal arguments to establish facts about the angle sum and

exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Understand and apply the Pythagorean Theorem. o Explain a proof of the Pythagorean Theorem and its converse.

o Apply the Pythagorean Theorem to determine unknown side lengths in

right triangles in real-world and mathematical problems in two and

three dimensions.

Transformations in this module

Translation

Rotation

Reflection

Eureka Math, A Story of Ratios

For more information visit commoncore.org

Grade

Module

Below are two examples of the type of problems involving reflections your child will see in this unit.

Spotlight on a model

frequently used in this

module:

Reflections

Sample Problem from Module 2: (Example taken from Module 2, Lesson 6)

Here is an example of a problem using rotations:

Use the diagram to the right for problems 1-5. Use your transparency as needed.

1. Looking only at segment BC, is it possible that a 180° rotation would map BC onto B’C’? Why

or why not?

It is possible because the segments are parallel.

2. Looking only at segment AB, is it possible that a 180° rotation would map AB onto A’B’? Why

or why not?

It is possible because the segments are parallel.

3. Looking only at segment AC, is it possible that a 180° rotation would map AC onto A’C’? Why

or why not?

It is possible because the segments are parallel.

4. Connect point B to point B’, point C to point C’ and point A to point A’. What do you notice?

What do you think that point is?

All of the lines intersect at one point. The point is the center of rotation. I checked by using my transparency.

5. Would a rotation map triangle ABC onto triangle A’B’C’? If so, define the rotation (i.e. degree and center). If not

explain why not.

Let there be a rotation 180° around point (0,-1). Then Rotation (∆𝐴𝐵𝐶) = ∆𝐴′𝐵′𝐶′.

Problem: Given a right triangle with a hypotenuse with length 13 units and a leg with length 5 units, as shown, determine the length of the other leg.

=

=

=

=

=

=

=

Solution: Pythagorean Theorem

The length of the leg is

12 units.

A positive attitude towards math is very important in helping your child succeed in school. In the ever-

changing world we live in, a strong foundation in math, paired with excellent problem

solving skills may open many doors for your child in the years to

come!

ABC was reflected across line L. Notice how Point B and line L did not move in this transformation. Also, notice how the image was labeled and how this is different from the original figure.

Figure D was also reflected across line L.