options: Understand properties of multiplication and the relationship between multiplication and division.

Use properties of operations to generate equivalent expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Interpret the structure of expressions.

Represent and solve problems involving multiplication and division.

Understand solving equations as a process of reasoning and explain the reasoning.

Multiply and divide within 100.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Write and interpret numerical expressions.

Add and subtract within 20.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand and apply properties of operations and the relationship between addition and subtraction.

Work with addition and subtraction equations.

Use the four operations with whole numbers to solve problems.

Solve equations and inequalities in one variable.

Reason about and solve one-variable equations and inequalities.

Use polynomial identities to solve problems.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Solve systems of equations.

Create equations that describe numbers or relationships

Rewrite rational expressions.

Generate and analyze patterns.

Represent and solve problems involving addition and subtraction.

Gain familiarity with factors and multiples.

Work with equal groups of objects to gain foundations for multiplication.

Work with radicals and integer exponents.

Represent and solve equations and inequalities graphically.

Analyze patterns and relationships.

Write expressions in equivalent forms to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.
problem_activity: Activity
This task culminates in writing a formula for the area of triangles. By now students are likely to have developed the intuition that the area of a triangle is half of that of a parallelogram with the same base and height. This activity encapsulates that work in an algebraic expression.
Students first find the areas of several triangles given base and height measurements. They then generalize the numerical work to arrive at an expression for finding the area of any triangle (MP8).
If needed, remind students how they reasoned about the area of triangles in the previous lesson (i.e. by composing a parallelogram, enclosing with one or more rectangles, etc.). Encourage them to refer to their previous work and use tracing paper as needed. Students might write
\(b \boldcdot h \div 2\)
or
\(b \boldcdot h \boldcdot \frac12\)
as the expression for the area of any triangle. Any equivalent expression should be celebrated.
At the end of the activity, consider giving students a chance to reason more abstractly and deductively, i.e., to think about why the expression
\(b \boldcdot h \div 2\)
would hold true for all triangles. See Activity Synthesis for prompts and diagrams that support such reasoning.
Launch
Arrange students in groups of 2–3. Explain that they will now find the area of some triangles using what they know about base-height pairs in triangles and the relationships between triangles and parallelograms.
Give students 5–6 minutes to complete the activity and access to geometry toolkits, especially tracing paper. Ask them to find the area at least a couple of triangles independently before discussing with their partner(s).
Student Facing
For each triangle:
Identify a base and a corresponding height, and record their lengths in the table.
Find the area of the triangle and record it in the last column of the table.
[IMG]
|              triangle               | base (units) | height (units) | area (square units) |
|-------------------------------------|--------------|----------------|---------------------|
|      **          A         **       |              |                |                     |
|      **          B         **       |              |                |                     |
|      **          C         **       |              |                |                     |
|      **          D         **       |              |                |                     |
| **          any triangle         ** |    \(b\)     |     \(h\)      |                     |
In the last row, write an expression for the area of any triangle, using
\(b\)
and
\(h\)
.
Student Response
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click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may not be inclined to write an expression using the variables
\(b\)
and
\(h\)
and instead replace the variables with numbers of their choice. Ask them to reflect on what they did with the numbers for the first four triangles. Then, encourage them to write the same operations but using the letters
\(b\)
and
\(h\)
rather than numbers.
Activity Synthesis
Select a few students to share their expression for finding the area of any triangle. Record each expression for all to see.
To give students a chance to reason logically and deductively about their expression, ask, “Can you explain why this expression is true for
any
triangle?”
Display the following diagrams for all to see. Give students a minute to observe the diagrams. Ask them to choose one that makes sense to them and use that diagram to explain or show in writing that the expression
\(b \boldcdot h \div 2\)
works for finding the area of any triangle. (Consider giving each student an index card or a sheet of paper on which to write their reasoning so that their responses could be collected, if desired.)
[IMG]
When dealing only with the variables
\(b\)
and
\(h\)
and no numbers, students are likely to find Jada's and Lin's diagrams more intuitive to explain. Those choosing to use Elena's diagram are likely to suggest moving one of the extra triangles and joining it with the other to form a non-rectangular parallelogram with an area of
\(b \boldcdot h\)
. Expect students be less comfortable reasoning in abstract terms than in concrete terms. Prepare to support them in piecing together a logical argument using only variables.
If time permits, select students who used different diagrams to share their explanation, starting with the most commonly used diagram (most likely Jada's). Ask other students to support, refine, or disagree with their arguments. It time is limited, consider collecting students' written responses now and discussing them in an upcoming lesson.
Representation: Develop Language and Symbols.
Create a display of important terms and concepts. Invite students to suggest language or diagrams to include that will support their understanding. Include the following terms and maintain the display for reference throughout the unit: opposite vertex, base (of a triangle), height (of a triangle), and formula for finding area.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion when students share their expressions for finding the area of any triangle. Provide students with 1–2 minutes of quiet think time to begin to consider why their expression is true for any triangle, before they continue work with a partner to complete their response. Select 1 or 2 pairs of students to share with the class, then call on students to restate their peers' reasoning. This will give more students the chance to use language to interpret and describe expressions for the area of triangles.
Design Principle(s): Support sense-making; Maximize meta-awareness


correct_option_index: 3