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

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Create equations that describe numbers or relationships

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

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

Use polynomial identities to solve problems.

Understand the relationship between zeros and factors of polynomials.

Rewrite rational expressions.

Work with addition and subtraction equations.

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

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

Represent and solve problems involving addition and subtraction.

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

Add and subtract within 20.

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

Gain familiarity with factors and multiples.

Perform arithmetic operations on polynomials.

Interpret the structure of expressions.

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

Represent and solve equations and inequalities graphically.

Reason about and solve one-variable equations and inequalities.

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

Use the four operations with whole numbers to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write and interpret numerical expressions.

Solve systems of equations.

Represent and solve problems involving multiplication and division.

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

Multiply and divide within 100.

Generate and analyze patterns.

Write expressions in equivalent forms to solve problems.
problem_activity: Optional activity
This activity examines how measurement errors behave when they are added together. In other words, if I have a measurement
\(m\)
with a maximum error of 1% and a measurement
\(n\)
with a maximum error of 1%, what percent error can
\(m + n\)
have? In addition to examining accuracy of measurements carefully (MP6), students work through examples and look for patterns (MP8) in order to hypothesize, and eventually show, how percent error behaves when measurements with error are added to one another.
Monitor for students who look for patterns, recognize the usefulness of the distributive property, or formulate the problem abstractly with variables.
Launch
Read the problem out loud and ask students what information they would need to know to be able to solve the problem. Students may say that they need to know what length the boards are supposed to be, because it is likely that they haven't realized that they can solve the problem without this information. Explain that floor boards come in many possible lengths, that 18-inch and 36-inch lengths are both common, but the boards can be anywhere between 12 and 84 inches. Ask students to pick values for two actual lengths and figure out the error in that case. Then they can pick two different examples, make the calculations again, and look for patterns.
Provide access to calculators.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams. If students are unsure where to begin, suggest that they draw a diagram to help illustrate the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
A wood floor is made by laying multiple boards end to end. Each board is measured with a maximum percent error of 5%. What is the maximum percent error for the total length of the floor?
Student Response
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Anticipated Misconceptions
Some students may pick some example lengths but then struggle with knowing what to do with them. Ask them “what would be the maximum measured lengths? The minimum? What would be the error if both measurements were maximum? What if they were both minimum?”
Some students may pick numbers that make the calculations more complicated, leading to arithmetic errors. Suggest that they choose simple, round numbers for lengths, like 50 inches or 100 centimeters.
Activity Synthesis
The goal of this discussion is for students to generalize from their specific examples of measurements to understand the general pattern and express it algebraically.
Poll the class on the measurements they tried and the maximum percent error they calculated. Invite students to share any patterns they noticed, especially students who recognized the usefulness of the distributive property for making sense of the general pattern.
Guide students to use variables to talk about the patterns more generally.
If a board is supposed to have length
\(x\)
with a maximum percent error of 5%, then the shortest it could be is
\(0.95x\)
and the longest it could be is
\(1.05x\)
.
If another board is supposed to have length
\(y\)
, it could be between
\(0.95y\)
and
\(1.05y\)
.
When the boards are laid end-to-end, the shortest the total length could be is
\(0.95x + 0.95y\)
, which is equivalent to
\(0.95(x + y)\)
.
The longest the total length could be is
\(1.05x + 1.05y\)
, or
\(1.05(x + y)\)
.
Because of the distributive property, we can see that the maximum percent error is still 5% after the board lengths are added together.
One interesting point to make, if students have also done the previous activity about measuring pencils, is that you could
measure
the sum of the board lengths with a lower percent error than you could measure each individual board (assuming your tape measure is long enough), just like an error of 1 mm was a smaller percentage of the length of the longer pencil.


correct_option_index: 11