# Lesson 5

Reasoning about Equations and Tape Diagrams (Part 2)

### Lesson Narrative

This lesson parallels the previous one, except the focus is on situations that lead to equations of the form $$p(x+q)=r.$$ Tape diagrams are used to help students understand why these situations can be represented with equations of this form, and to help them reason about solving equations of this form. Students also attend to the meaning of the equation’s solution in the context (MP2). Note that we are not generalizing solution methods yet; just using diagrams as a tool to reason about solving equations.

### Learning Goals

Teacher Facing

• Coordinate tape diagrams, equations of the form $p(x+q)=r$, and verbal descriptions of the situations.
• Explain (orally and in writing) how to use a tape diagram to determine the value of an unknown quantity in an equation of the form $p(x+q)=r$.
• Interpret (in writing) the solution to an equation in the context of the situation it represents.

### Student Facing

Let’s use tape diagrams to help answer questions about situations where the equation has parentheses.

### Student Facing

• I can draw a tape diagram to represent a situation where there is more than one copy of the same sum and explain what the parts of the diagram represent.
• I can find a solution to an equation by reasoning about a tape diagram or about what value would make the equation true.

Building Towards

### Glossary Entries

• equivalent expressions

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, $$3x+4x$$ is equivalent to $$5x+2x$$. No matter what value we use for $$x$$, these expressions are always equal. When $$x$$ is 3, both expressions equal 21. When $$x$$ is 10, both expressions equal 70.

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