What is it advisable to do when you are exploring decimal numbers?
10 to one multiplicative relationship.
Rules for placement of the decimal.
Role of the decimal point.
How to read a decimal fraction.
What is an early method to use to help students see the connection between fractions and decimals fractions?
Show them how to use a calculator to divide the fraction numerator by the denominator to find the decimal.
Be sure to use precise language when speaking about decimals, such as “point seven two.”
Show them how to round decimal numbers to the closest whole number.
Show them how to use base-ten models to build models of base-ten fractions.
The 10-to-1 relationship extends in two directions. There is never a smallest piece or a largest piece. Complete the statement, “The symmetry is around..”.
The decimal point.
The ones place.
The operation being conducted.
The relationship between the adjacent pieces.
The following decimals are equivalent 0.06 and 0.060. What does one of them show that the other does not show?
More place value.
More hundreds.
More level of precision.
Closer to one.
Using precise language can support students’ understanding of the relationship between fractions and decimal fractions. All of the following are true statements EXCEPT:
0.75 = 3/4
Five and two-tenths is the same as five point two.
Six and three-tenths = 6 3/10
7. 03 = 7 30/100
What is the most common model used for decimal fractions?
Rational number wheel.
Base ten strips and squares.
10 x 10 grids.
Number line.
A common set model for decimal fraction is money. Identify the true statement below.
Money is a two-place system.
One-tenth a dime proportionately compares to a dollar.
Money should be an initial model for decimal fractions.
Money is a proportional model.
All of the statements below are true of this decimal fraction 5.13 EXCEPT:
5 + 1/10 + 3100
Five and thirteen-hundredths.
513/100
Five wholes, 3 tenths and 1 hundredth.
Approximation with compatible fractions is one method to help students with number sense with decimal fractions. All of the statements are true of 7.3962 EXCEPT:
Closer to 7 than 8.
Closer to 7 3/4 than 7 1/2
Closer to 7.3 than 7 1/5
Closer to 7.4 than 7.5
There are several errors and misconceptions associated with comparing and ordering decimals. Identify the statement below that represents the error with internal zero.
Students say 0.375 is greater than 0.97.
Students see 0.58 less than 0.078.
Students select 0 as larger than 0.36
Students see 0.4 as not close to 0.375
Understanding that when decimals are rounded to two places (2.30 and 2.32) there is always another number in between. What is the place in between called?
Place value.
Density.
Relationships.
Equality
Instruction on decimal computation has been dominated by rules. Identify the statement that is not rule based.
Line up the decimal points.
Count the decimal places.
Shift the decimal point in the divisor.
Apply decimal notation to properties of operations.
Decimal multiplication tends to be poorly understood. What is it that students need to be able to do?
Discover the method by being given a series of multiplication problems with factors that have the same digits, but decimals in different places.
Discover it on their own with models, drawings and strategies.
Be shown how to estimate after they are shown the algorithm.
Use the repeated addition strategy that works for whole number.
The estimation questions below would help solve this problem EXCEPT: - A farmer fills each jug with 3.7 liters of cider. If you buy 4 jugs, how many liters of cider is that?
Is it more than 12 liters?
What is the most it could be?
What is double 3.7 liters?
Is it more than 7 x 4?
Understanding where to put the decimal is an issue with multiplication and division of decimals. What method below supports a fuller understanding?
Rewrite decimals in their fractional equivalents.
Rewrite decimals as whole numbers, compute and count place value.
Rewrite decimals to the nearest tenths or hundredths.
Rewrite decimals on 10 by 10 grids.
What is a method teachers might use to assess the level of their students understanding of the decimal point placement?
Ask them to show all computations.
Ask them to show a model or drawing.
Ask them to explain or write a rationale.
Ask them to use a calculator to show the computation.
What is it that students can understand if they can express fractions and decimals to the hundredths place?
Place value
Computation of decimals.
Percents.
Density of decimals.
The main link between fractions, decimals and percents are _______________.
Expanded notation.
Terminology.
Equivalency.
Physical models.
The following are guidelines for instruction on percents EXCEPT:
Use terms part, whole and percent.
Use models, drawings and contexts to explain their solutions.
Use calculators
Use mental computation.
Estimation of many percent problems can be done with familiar numbers. Identify the idea that would not support estimation.
Substitute a close percent that is easy to work with.
Use a calculator to get an exact answer.
Select numbers that are compatible with the percent to work with
Convert the problem to one that is simpler
Complete this statement, “A ratio is a number that relates two quantities or measures within a given situation in a..”.
Multiplicative relationship.
Difference relationship.
Additive relationship.
Multiplicative comparison.
What is the type of ratio that would compare the number of girls in a class to the number of students in a class?
Ratio as rates.
Ratio as quotients.
Ratio as part-whole.
Ratio as part-to-part
What should you keep in mind when comparing ratios to fractions?
Conceptually, they are exactly the same thing
They have the same meaning when a ratio is of the part-to-whole type.
They both have a fraction bar that causes students to mistakenly think they are related in some way.
Operations can be done with fractions while they can’t be done with ratios.
In the scenario “Billy’s dog weighs 10 pounds while Sarah’s dog weighs 8 pounds", the ratio 10/8 can be interpreted in the following ways EXCEPT:
For every 5 pounds of weight Billy’s dog has, Sarah’s dog has 4 pounds.
Billy’s dog weighs 1 1/4 times what Sarah’s dog does.
Sarah’s dog weighs 8 out of a total of 10 dog pounds.
Billy’s dog makes up 5/9 of the total dog weight.
A _____________ refers to thinking about a ratio as one unit.
Ratio as a rate.
Cognitive task.
Composed unit.
The following statement are ways to define proportional reasoning EXCEPT:
Ratios as distinct entities.
Develop a specialized procedure for solving proportions.
Sense of covariation.
Recognize proportional relationships distinct from nonproportional relationships.
Identify the problem below that is a constant relationship.
Janet and Jean were walking to the park, each walking at the same rate. Jean started first. When Jean has walked 6 blocks, Janet has walked 2 blocks. How far will Janet be when Jean is at 12 blocks?
Kendra and Kevin are baking muffins using the same recipe. Kendra makes 6 dozen and Kevin makes 3 dozen. If Kevin is using 6 ounces of chocolate chips, how many ounces will Kendra need?
Lisa and Linda are planting peas on the same farm. Linda plants 4 rows and Lisa plants 6 rows. If Linda’s peas are ready to pick in 8 weeks, how many weeks will it take for Lisa’s peas to be ready?
Two weeks ago, two flowers were measured at 8 inches and 12 inches, respectively. Today they are 11 inches and 15 inches tall. Did the 8-inch or 12- inch flower grow more?
Covariation means that two different quantities vary together. Identify the problem that is about a covariation between ratio.
Apples are 4 for $2.00.
2 apples for $1.00 and 1 for $0.50.
Apples at Meyers were 4 for $2.00 and at HyVee 5 for $3.00.
Apples sold 4 out of 5 over oranges.
Using proportional reasoning with measurement helps students with options for finding what?
Conversions.
Similarities.
Differences.
Rates.
What is one method for students to see the connection between multiplicative reasoning and proportional reasoning?
Solving problems with rates.
Solving problems with scale drawings.
Solve problems with between ratios.
Solving problems with costs.
The following are examples of connections between proportional reasoning and another mathematical strand EXCEPT:
The area of a rectangle is 8 square units and the length is four units long. How long is the width?
The negative slope of the line on the graph represents the fact that, for every 30 miles the car travels, it burns one gallon of gas.
The triangle has been enlarged by a scale factor of 2. How wide is the new triangle if its original width is 4 inches?
Sandy ate 1/4 of her Halloween candy and her sister also ate 1/2 of Sandy’s candy. What fraction of Sandy’s candy was left?
Which of the following is an example of using unit rate method of solving proportions?
If 2/3 = x/15, find the cross products, 30 = 3x, and then solve for x. x = 10.
Allison bought 3 pairs of socks for $12. To find out how much 10 pairs cost, find that $12 divided by 3 is $4 a pair, and multiply $4 by 10 for a total of $40.
A square with a length of 2 inches was enlarged by a scale factor of 4 and is now 8 inches long.
If 5 candy bars cost $4.50, then 10 would cost $9. (Because 5 × 2 = 10, multiply $4.50 by 2).
Which of the following is an example of using a buildup strategy method of solving proportions?
Allison bought 3 pairs of socks for $12. To find out how much ten pairs cost, find that $12 divided by 3 is $4 a pair, and multiply $4 by 10 for a total of $40.
A variety of methods will help students develop their proportional thinking ability. All of the ideas below support this thinking EXCEPT:
Provide ratio and proportional tasks within many different contexts
Provide examples of proportional and non-proportional relationships to students and ask them to discuss the differences.
Relate proportional reasoning to their background knowledge and experiences.
Provide practice in cross-multiplication problems.
Creating ratio tables or charts helps students in all of the following ways EXCEPT:
Application of build up strategy
Organize information
Show nonproportional relationships.
Used to determine unit rate.
What statement below describes an advantage of using strip diagrams, bar models, fraction strips or length models to solve proportions?
A concrete strategy that can be done first and then connected to equations.
A strategy that connects ratio tables to graphs.
A common method to figure out how much goes in each equation
A strategy that helps set up linear relationships.
Posing problems for students to solve proportions situations with their own intuition and inventive method is preferred over what?
Scaling up and down
Ratio tables
Graphs.
Cross products
Graphing ratios can be challenging. Identify the statement that would NOT be a challenge.
Slope m is always one of the equivalent ratios.
Decide what points to graph
Which axes to use to measure
Sense making of the graphed points.
When a teacher assigns an object to be measured students have to make all of these decisions EXCEPT:
What attribute to measure?
What unit they can use to measure that attribute?
How to compare the unit to the attribute?
What formulas they should use to find the measurement?
Identify the statement that is NOT a part of the sequence of experiences for measurement instruction.
Using measurement formulas
Using physical models
Using measuring instruments.
Using comparisons of attributes
All of the ideas below support the reasoning behind starting measurement experiences with nonstandard units EXCEPT:
They focus directly on the attribute being measured.
Avoids conflicting objectives of the lesson on area or centimeters.
Provides good rational for using standard units.
Understanding of how measurement tools work.
When using a nonstandard unit to measure an object, what is it called when use many copies of the unit as needed to fill or match the attribute?
Iterating
Tiling
Comparing.
Matching.
There are three broad goals to teaching standard units of measure. Identify the one that is generally NOT a key goal.
Familiarity with the unit.
Knowledge of relationships between units
Estimation with standard and nonstandard units
Ability to select and appropriate unit.
The Common Core State Standards and the National Council of Teachers of Mathematics agree on the importance of what measurement topic?
Students focus on customary units of measurement.
Students focus on formulas versus actual measurements.
Students focus on conversions of standard to metric.
Students focus on metric unit of measurement as well as customary units.
All of these statements are true about reasons for including estimation in measurement activities EXCEPT:
Helps focus on the attribute being measured.
Helps provide an extrinsic motivation for measurement activities.
Helps develop familiarity with the unit.
Helps promote multiplicative reasoning.
Young learners do not immediately understand length measurement. Identify the statement below that would not be a misconception about measuring length.
Measuring attribute with the wrong measurement tool.
Using wrong end of the ruler
Counting hash marks rather than spaces.
Misaligning objects when comparing
The concept of conversion can be confusing for students. Identify the statement that is the primary reason for this confusion.
Basic idea if the measure is the same as the unit it is equal
Basic idea that if the measure is larger the unit is longer.
Basic idea that if the measure is larger the unit is shorter.
Basic idea that if the measure is shorter the unit is shorter.
Comparing area is more of a conceptual challenge for students than comparing length measures. Identify the statement that represents one reason for this confusion.
Area is a measure of two-dimensional space inside a region
Direct comparison of two areas is not always possible
Rearranging areas into different shapes does not affect the amount of area
Area and perimeter formulas are often used interchangeably.
As students move to thinking about formulas it supports their conceptual knowledge of how the perimeter of rectangles can be put into general form. What formula below displays a common student error for finding the perimeter?
P = l + w + l + w
P = l + w
P = 2l + 2w
P = 2(l + w)
What language supports the idea that the area of a rectangle is not just measuring sides?
Height and base.
Length and width
Width and Rows
Number of square units
Challenges with students’ use of rulers include all EXCEPT:
Deciding whether to measure an item beginning with the end of the ruler
Deciding how to measure an object that is longer than the ruler
Properly using fractional parts of inches and centimeters
Converting between metric and customary units
Volume and capacity are both terms for measures of the “size” of three-dimensional regions. What statement is true of volume but not of capacity?
Refers to the amount a container will hold
Refers to the amount of space of occupied by three-dimensional region
Refers to the measure of only liquids
Refers to the measure of surface area
The statements below represent illustrations of various relationships between the area formulas? Identify the one that is NOT represented correctly
A rectangle can be cut along a diagonal line and rearranged to form a nonrectangular parallelogram. Therefore the two shapes have the same formula.
A rectangle can be cut in half to produce two congruent triangles. Therefore, the formula for a triangle is like that for a rectangle, but the product of the base length and height must be cut in half
The area of a shape made up of several polygons (a compound figure) is found by adding the sum of the areas of each polygon
Two congruent trapezoids placed together always form a parallelogram with the same height and a base that has a length equal to the sum of the trapezoid bases. Therefore, the area of a trapezoid is equal to half the area of that giant parallelogram, ½ h (b1 +b2).
What is the most conceptual method for comparing weights of two objects?
Place objects in two pans of a balance.
Place objects on a spring balance and compare
Place objects on extended arms and experience the pull on each.
Place objects on digital scale and compare.
Identify the attribute of an angle measurement
Base and height
Spread of angle rays
Unit angle
Degrees.
Steps for teaching students to understand and read analog clocks include all of the following EXCEPT:
Begin with a one-handed clock.
Discuss what happens with the big hand as the little hand goes from one hour to the next
Predict the reading on a digital clock when shown an analog clock.
Teach time after the hour in one-minute intervals
All of these are ideas and skills for money that students should be aware of in elementary grades EXCEPT:
Making change
Solving word problems involving money
Values of coins
Solving problems of primary interest
