All of the following are examples of algebraic thinking a young student would demonstrate in kindergarten EXCEPT:
Acting out a situation.
Recognizing patterns in sounds (clapping).
Applying properties of addition.
Adding and subtracting with fingers.
Three of these are the strands of algebraic thinking described by Blanton and Kaput. Which one is not considered a strand by itself?
Structures in the number system.
Meaningful use of symbols.
Mathematical modeling.
Patterns, relations and functions.
A tool called __________________, is normally thought of as teaching numeration but can help students to connect place value and algebraic thinking.
Open number line.
Grid paper.
Calculator.
Hundreds chart.
Making sense of properties of the operations is a part of learning about generalizations. Identify the statement below that a student might use to explain the associative property of addition.
“ When you add three number you can add the first two and then add the third or add the second and third and then the first. Either way you get the same answer”.
“ When you add two number in any order you will get the same answer”.
“ When you have a subtraction problem you can think addition by using the inverse”.
“When you add zero to any number you get the number you started with”.
What is one method that students can use to show that they are generalizing properties?
Symbols.
Written examples.
Equations with numbers.
Model with manipulatives.
The ________ property is central to learning multiplication basic facts and the algorithms for the operation.
Associative.
Multiplicative identity.
Distributive
Inverse relationship of addition and subtraction.
Patterns are found in all areas of mathematics. Below are examples of repeating patterns EXCEPT:
Patterns that have core the repeats.
Patterns in number i.e. place value.
Patterns in seasons, days, music.
Patterns in skip counting.
These patterns are technically referred to as sequences and they involve a step-to-step progression.
Recursive.
Covariational.
Correspondence.
Linear.
This method of recording can help students think about how two quantities vary from step to step.
Table.
Growing patterns can be represented in multiple ways. Identify the representation below that actually illustrates covariation.
A table.
Physical model.
Graph.
Students need to be familiar and use the language to describe functions of graphs. All of vocabulary below will support the knowledge of functions EXCEPT:
Discrete are isolated or selected values.
Covariational is the input generated by the output
Range is the corresponding possible values for the dependent variable.
Domain is the possible values of the independent variable.
All of the statements below relate students’ understanding of the equal sign EXCEPT:
Understanding or confusion with the equal sign does not usually cause difficulties understanding the process of solving equations.
Because of their early experiences, many students tend to believe the equal sign represents “and the answer is”.
The equal sign is one of the principle methods of representing important relationships within the number system.
The equal function can be represented concretely by a number balance scale, which can lead to deeper conceptual understanding.
Complete this statement, “The use of a two-pan balance scale or semi-concrete drawings of a balance help develop a strong understanding of..”.
Pattern identification
Function patterns.
Abstract concept of equality.
Conjecture.
The statements below are students’ views of equations EXCEPT.
Relational-structural view
Relational-computational view
Correspondence-relational view
Operational view
What is a reason for students to create graphs of functions?
They are representing them in the manner that makes it the hardest to visualize relationships between patterns.
They should be provided to them with examples within a real-life context
They should place the independent variable (step number) along the vertical axis.
They should always be given specific data, equations, or numbers.
Identify the true statement for all proportional relationships.
They can only be represented accurately with an equation.
They will always show in a graph as a straight line that passes through the origin.
They will always have a positive slope.
They are more challenging for students to generalize than a non-proportional one.
What is an early misconception about variables?
A constant value.
A symbol of relationships.
A placeholder for one exact number.
A quantity that varies.
Using expressions and variables in elementary classrooms should be evident with all of the following EXCEPT:
Involve situation with a specific unknown.
Express it in symbols.
Use letters in place of an open box.
Use specific data, numbers and equations.
Mathematical modeling is one of the eight Standards for Mathematical Practice. Three of the statements reference the true meaning of mathematical modeling. Identify the one that is often mistaken for modeling
Links classroom mathematics to everyday life.
Process of choosing appropriate mathematics for situations.
Visual models, such as manipulatives and drawings of pattern.
Analyzing empirical situations to better understand.
The term algebraic thinking is used instead of the term algebra because algebraic thinking goes beyond the topics that are typically found in an algebra course. All of the ideas below could be used as “algebraified” activity EXCEPT:
Familiar formulas for measuring a geometric shape.
Data from census reports and survey.
Experiments that look for functional relations
Strategies for model-based problems.
The part-whole construct is the concept most associated with fractions, but other important constructs they represent include all of the following EXCEPT:
Measure.
Reciprocity
Division.
Ratio.
All of the following are fraction constructs EXCEPT:
Part-whole
Measurement.
Iteration
Division
Fraction misconceptions come about for all of the following reasons. The statements below can be fraction misconceptions EXCEPT.
Many meanings of fractions.
Fractions written in a unique way.
Students overgeneralize their whole-number knowledge
Teachers present fractions late in the school year.
Models provide an effective visual for students and help them explore fractions. Identify the statement that is the definition of the length model.
Location of a point in relation to 0 and other values.
Part of area covered as it relates to the whole unit.
Count of objects in the subset as it relates to defined whole.
A unit or length involving fractional amounts.
The following visuals/manipulatives support the development of fractions using the area model EXCEPT:
Pattern blocks.
Tangrams
Cuisenaire rods.
Geoboards.
A _______ is a significantly more sophisticated length model than other models.
Number line.
Cuisenaire rods
Measurement tools.
Folded paper strips.
What is a common misconception with fraction set models?
There are not many real-world uses.
Knowing the size of the subset rather than the number of equal sets
Knowing the number of equal sets rather than the size of subsets
There are not many manipulatives to model the collections.
Complete this statement, “Comparing two fractions with any representation can be made only if you know the..”.
Size of the whole
Parts all the same size.
Fractional parts are parts of the same size whole.
Relationship between part and whole
What is the definition of the process of partitioning?
Equal shares
Equal-sized parts
Equivalent fractions
Subset of the whole.
Locating a fractional value on a number line can be challenging but is important for students to do. All of the statements below are common errors that students make when working with the number line EXCEPT:
Use incorrect notation.
Change the unit.
Use incorrect subsets.
Count the tick marks rather than the space.
Counting precedes whole-number learning of addition and subtraction. What is another term for counting fraction parts?
Equalizing.
Iterating
Partitioning.
Sectioning.
The term improper fraction is used to describe fractions greater than one. Identify the statement that is true about the term improper fraction.
Is a clear term, as it helps students realize that there is something unacceptable about the format.
Should be taught separately from proper fractions.
Are best connected to mixed numbers through the standard algorithm.
Should be introduced to students in a relevant context.
What does a strong understanding of fractional computation relies on?
Estimating with fractions.
Iteration skills.
Whole number knowledge.
Fraction equivalence.
All of the models listed below support the understanding of fraction equivalence EXCEPT:
Graph of slope
Shapes created on dot paper
Plastic, circular area models.
Clock faces
The way we write fractions is a convention with a top and bottom number with a bar in between. Posing questions can help students make sense of the symbols. All of the questions would support that sense making EXCEPT:
What does the denominator in a fraction tell us?
What does the equal symbol mean with fractions?
What might a fraction equal to one look like?
How do know if a fraction is greater than, less than 1?
How do you know that 4/6 = 2/3 ? Identify the statement below that demonstrates a conceptual understanding.
They are the same because you can simplify 4/6 and get 2/3.
Start with 2/3 and multiply the top and bottom by 2 and you get 4/6.
If you have 6 items and you take 4 that would be 4/6. You can make 6 groups into 3 groups and 4 into 2 groups and that would be 2/3.
If you multiply 4 x 3 and 6 x 2 they’re both 12.
What does it mean to write fractions in simplest term?
Finding equivalent numerators.
Finding equivalent denominators.
Finding multipliers and divisors.
Finding equivalent fractions with no common whole number factors.
Comparing fractions involves the knowledge of the inverse relationship between number of parts and size of parts. The following activities support the relationship EXCEPT:
Iterating.
Equivalent fraction algorithm.
Estimating.
Estimating with fractions means that students have number sense about the relative size of fractions. All of the activities below would guide this number sense EXCEPT:
Comparing fractions to benchmark numbers.
Find out the fractional part of the class are wearing glasses.
Collect survey data and find out what fractions of the class choose each item.
Use paper folding to identify equivalence.
Teaching considerations for fraction concepts include all of the following EXCEPT:
Iterating and partitioning.
Procedural algorithm for equivalence
Emphasis on number sense and fractional meaning.
Link fractions to key benchmarks.
To guide students to develop a problem-based number sense approach for operations with fractions all of the following are recommended EXCEPT:
Address common misconceptions regarding computational procedures.
Estimating and invented methods play a big role in the development.
Explore each operation with a single model.
Use contextual tasks.
Identify the problem that solving with a linear model would not be the best method.
Half a pizza is left from the 2 pizzas Molly ordered. How much pizza was eaten?
Mary needs 3 1/3 feet of wood to build her fence. She only has 2 3/4 feet. How much more wood does she need?
Millie is at mile marker 2 1/2. Rob is at mile marker 1. How far behind if Rob?
What is the total length of these two Cuisenaire rods placed end to end?
Adding and subtraction fractions should begin with students using prior knowledge of equivalent fractions. Identify the problem that may be more challenging to solve mentally.
Luke ordered 3 pizzas. But before his guests arrive he got hungry and ate 3/8 of one pizza. What was left for the party?
Linda ran 1 1/2 miles on Friday. Saturday she ran 2 1/8 miles and Sunday 2 3/4. How many miles did she run over the weekend?
Lois gathered 3/4 pounds of walnuts and Charles gathered 7/8 pounds. Who gathered the most? How much more?
Estimate the answer to 12/13 + 7/8.
Different models are used to help illustrate fractions. Identify the model that can be confusing when you are learning to add fractions.
Area.
Set.
Length.
Linear models are best represented by what manipulative?
Pattern Blocks
Circular pieces.
Ruler.
Identify the manipulative used with linear models that you can decide what to use as the “whole”.
Circular pieces
Number Line.
Cuisenaire Rods
All of the statements below are examples of estimation or invented strategies for adding and subtracting fractions EXCEPT:
Decide whether fractions are closest to 0, 1/2, or 1.
Look for ways different fraction parts are related.
Decide how big the fraction is based on the unit.
Look for the size of the denominator
Complete the statement, “Developing the algorithm for adding and subtracting fractions should..”.
Be done side by side with visuals and situations.
Be done with specific procedures
Be done with units that are challenging to combine.
Be done mentally without paper and pencil.
What statement is true about adding and subtracting with unlike denominators?
Should be introduced at first with tasks that require both fractions to be changed.
Is sometimes possible for students, especially if they have a good conceptual understanding of the relationships between certain fractional parts and a visual tool, such as a number line.
Is a concept understood especially well by students if the teacher compares different denominators to “apples and oranges.”
Should initially be introduced without a model or drawing.
Students are able to solve adding and subtracting fractions without finding a common denominator using invented strategies. The problems below would work with the invented strategies EXCEPT:
3/4 + 1/8
1/2 - 1/8
5/6 - 1/7
2/3 + 1/2
What is helpful when subtracting mixed number fractions?
Deal with the whole numbers first and then work with the fractions.
Always trade one of the whole number parts into equivalent parts.
Avoid this method until the student fully understands subtraction of numbers less than one.
Teach only the algorithm that keeps the whole number separate from the fractional part.
Common misconceptions occur because students tend to overgeneralize what they know about whole number operations. Identify the misconception that is not relative to fraction operations.
Adding both numerator and denominator.
Not identifying the common denominator.
Difficulty with common multiples.
Use of invert and multiply.
All of the activities below guide students to understand the algorithm for fraction multiplication EXCEPT:
Multiply a fraction by a whole number.
Multiply a whole number by a fraction.
Subdividing the whole number.
Fraction of a fraction- no subdivisions.
This model is exceptionally good at modeling fraction multiplication. It works when partitioning is challenging and provides a visual of the size of the result.
Area model.
Linear model.
Set model.
Circular model.
What is one of the methods for finding the product of fractional problems when one of the numbers is mixed number?
Change to improper fraction.
Compute partial products
Linear modeling
Associative property
Each the statements below are examples of misconceptions students have when learning to multiply fractions EXCEPT:
Treating denominators the same as addition and subtraction.
Matching multiplication situations with multiplication situations.
Estimating the size of the answer incorrectly
Multiplying the denominator and not numerator.
It is recommended that division of fractions be taught with a developmental progression that focuses on four types of problems. Which statement below is not part of the progression?
A fraction divided by a fraction.
A whole number divided by a fraction.
A whole number divided by a mixed number.
A whole number divided by a whole number.
A ______ interpretation is a good method to explore division because students can draw illustrations to show the model.
Estimation and invented strategies are important with division of fractions. If you posed the problem 1/6 ÷ 4 you would ask all of the questions EXCEPT:
Will the answer be greater than 4?
Will the answer be greater than one?
Will the answer be greater than 1/2?
Will the answer be greater than 1/6?
Based on students experience with whole number division they think that when dividing by a fraction the answer should be smaller. This would be true for all of the following problems EXCEPT:
1/6 ÷ 3
5/6 ÷ 3
3/6 ÷ 3
3 ÷ 5/6
