options: Write and interpret numerical expressions.

Represent and solve problems involving multiplication and division.

Generate and analyze patterns.

Rewrite rational expressions.

Work with radicals and integer exponents.

Represent and solve equations and inequalities graphically.

Multiply and divide within 100.

Solve systems of equations.

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

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

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

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

Understand the relationship between zeros and factors of polynomials.

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

Analyze patterns and relationships.

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

Reason about and solve one-variable equations and inequalities.

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

Solve equations and inequalities in one variable.

Perform arithmetic operations on polynomials.

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

Add and subtract within 20.

Write expressions in equivalent forms to solve problems.

Use the four operations with whole numbers to solve problems.

Represent and solve problems involving addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

Interpret the structure of expressions.

Create equations that describe numbers or relationships

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

Use polynomial identities to solve problems.

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

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.
problem_activity: Activity
Students have just learned the meaning of the
\(y\)
-intercept for a line and have been interpreting slope in context. In this activity, they investigate the
\(y\)
-intercept and slope together and investigate what happens when their values are switched.
In contexts like this one, the
\(y\)
-intercept and the slope come with natural units and understanding this can help graph accurately. Specifically, the
\(y\)
-intercept is the number of pages Lin read before she starts gathering data for the graph. The slope, on the other hand, is a rate: it’s the number of pages Lin reads
per day
.
Watch for students who understand the source of Lin’s error (confusing the
\(y\)
-intercept with the slope) and invite them to share this observation during the discussion.
Launch
Work time followed by whole-class discussion.
Representation: Internalize Comprehension.
Use MLR6 Three Reads to supporting reading comprehension of the word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (Lin’s reading assignment). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for and amplify the important quantities that vary in relation to each other in this situation: first 30 pages and 40 pages each day. After the third read, ask students to brainstorm possible strategies to answer the question, “What does the vertical intercept mean in this situation?”
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports.
Use this to amplify mathematical uses of language to communicate about vertical intercepts, slope, and constant rate. Invite students to use these words when describing their ideas. Ask students to chorally repeat phrases that include these words in context.
Design Principle(s): Support sense-making, Optimize output (for explanation)
Student Facing
Lin has a summer reading assignment. After reading the first 30 pages of the book, she plans to read 40 pages each day until she finishes. Lin makes the graph shown here to track how many total pages she'll read over the next few days.
After day 1, Lin reaches page 70, which matches the point
\((1,70)\)
she made on her graph. After day 4, Lin reaches page 190, which does not match the point
\((4,160)\)
she made on her graph. Lin is not sure what went wrong since she knows she followed her reading plan.
[IMG]
Sketch a line showing Lin's original plan on the axes.
What does the
vertical intercept
mean in this situation? How do the vertical intercepts of the two lines compare?
What does the slope mean in this situation? How do the slopes of the two lines compare?
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Jada's grandparents started a savings account for her in 2010. The table shows the amount in the account each year.
If this relationship is graphed with the year on the horizontal axis and the amount in dollars on the vertical axis, what is the vertical intercept? What does it mean in this context?
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "year"}, {"_attributes": {"scope": "col"}, "_value": "amount in dollars"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "2010"}, {"_value": "600"}]}, {"td": [{"_value": "2012"}, {"_value": "750"}]}, {"td": [{"_value": "2014"}, {"_value": "900"}]}, {"td": [{"_value": "2016"}, {"_value": "1050"}]}]}]}]}
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask students:
“How did your graph compare to Lin’s?” (It is steeper but starts off at 30 instead of 40 pages.)
“Which point does your graph have in common with Lin’s?” (The point
\((1,70)\)
, 70 pages read after one day.)
Invite selected students to share what’s the likely source of Lin’s error (confusing the
\(y\)
-intercept with the slope). In this context, the
\(y\)
-intercept is the number of pages Lin read before she starts counting the days (30), and the slope is the number of pages Lin reads per day (40).
Emphasize how the
\(y\)
-intercept and slope influence the graph of a line.
The
\(y\)
-intercept indicates where the line touches or crosses the
\(y\)
-axis.
The slope indicates how steep the line is.


correct_option_index: 10