Sample Checkpoint

Checkpoint 6A
Problem 6-29

Rewriting Equations With More Than One Variable

Answers to problem 6-29:  a: y=3x−105=35x−2, b: x=y−bm, c: r2=Aπ

Rewriting equations with more than one variable may be done in a variety of ways but normally you follow the same steps as you would for solving an equation with a single variable. Commonly, the first step is to multiply all the terms by a common denominator to remove all of the fractions. Then solve in the usual way. Collect like terms. Isolate the specified variable terms on one side and everything else on the other. Finally, divide or undo the exponent so that the variable is alone. The final answer will be an equation that involves variables and possibly numbers.

Example 1:  Solve for y:  2x+3y−9=0

Solution: 2x + 3y − 9 = 0



subtract 2x, add 9 on each side


3 divide by 3



Example 2:  Solve for r:  V = 43πr3

Solution: V = 43πr3


3V = 4πr3

multiply by 3 on each side


divide by 


cube root

Now we can go back and solve the original problems.

  1. −3x+5y=−105y=3x−10 y=3x−105 y=x−2

  1. y=mx+by−b=mx+by−bm=x

  1. A=πr2Aπ=r2

Here are some more to try.  Solve each equation for the specified variable.

  1. 2x−3y=9(for x)

  1. 2x−3y=9(for y)

  1. 5x+3y=15(for y)

  1. 4n=3m−1(for m)

  1. 2w+2l=P(for w)

  1. 2a+b=c (for a)

  1. I=ER(for R)

  1. y=14x+1(for x)

  1. c2=a2+b2(for b2)

  1. V=s3(for s)

  1. S=4πr2(for r2)

  1. m=a+b+c3(for a)

  1. a3+b=c2(for a)

  1. Fa=m(for a)

  1. A=12h(b1+b2)(for b1)

  1. V=13πr2h(for h)


  1. x=3y+92

  1. y=2x−93

  1. y=15−5×3

  1. m=4n+13

  1. w=P−2l2

  1. a=c−b2

  1. R=EI

  1. x=4(y−1)

  1. b2=c2−a2

  1. s=V3

  1. r2=S4π

  1. a=3m−b−c

  1. a=c2−b3

  1. a=Fm

  1. b1=2Ah−b2

  1. h=3Vπr2

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Algebra Tiles Blue Icon

Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
  • Solving equations
  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
  • Complete the square.
  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.