Formative Assessment: Check Understanding
Math Expressions: Common Core
Grade 3
Unit 1: Multiplication and Division with 0-5, 9, and 10
Lesson 1: Multiply With 5
Ask students to explain how they can use multiplication to find the total of 5 + 5 + 5 + 5 + 5 + 5 + 5.
Students should give the equation 7 × 5 = 35.
Lesson 2: Multiplication as Equal Groups
Ask students to explain when an Equal Shares drawing is faster to make than an Equal Groups
drawing. Students should explain that an Equal Shares drawing is faster when there are many
things in each group.
Lesson 3: Multiplication and Arrays
Ask students to explain why switching the order of the factors in a multiplication equation does not
change the product. Students should use arrays to show that order of the factors does not change
the product.
Lesson 4: The Meaning of Division
Ask students what multiplication with an unknown number they can write to find 20 ÷ 5 = ⧠.
Students should write ⧠ × 5 = 20 and give the answer of 4.
Lesson 5: Multiply and Divide with 2
Ask students to describe any patterns found in the 2s count-bys and multiplications and to explain
how these patterns can help them when multiplying. Students may describe that the count-bys skip
a number between and that all products of 2s are even. So if a product of 2 is odd, they know they
need to find their error and correct it.
Lesson 6: Building Fluency with 2s and 5s
Ask students to name the other multiplications and divisions they know if they know that 7 × 2= 14.
Students should name 2 × 7 = 14, 14 ÷ 7 = 2, and 14 ÷ 2 =7.
Lesson 7: Multiply and Divide with 10
Ask students to explain how they know a number is a 10s count-by and give an example. Students
should explain that any number that is a 10s count-by will end in 0. Example: 90 is a count-by
because it ends in 0.
Lesson 8: Multiply and Divide with 9
Ask students to explain how to find the product 4 × 9 using a 10s multiplication and using the Quick
9s method.
Lesson 9: Building Fluency with 2s, 5s, 9s, and 10s
Ask students to explain how a Fast Array drawing can be used to solve a division problem. Students
should explain how to use a related multiplication or division using the Fast Array drawing for the
numbers to solve a division.
Lesson 10: Multiply and Divide with 3
Ask students to explain the different strategies they can use to find 9 × 3. Some strategies students
may describe are skip counting, using fingers, and using the 5s shortcut.
Lesson 11: Multiplication and Area
Ask students to explain different ways they can find the area of a rectangle that is 6 rows by 4
columns. Students may describe counting square units in the rectangle, multiplying the side lengths,
or separating the rectangle into two smaller rectangles and adding the area of the smaller
rectangles.
Lesson 12: Multiply and Divide with 4
Ask students to explain how they can use the answers to 2 × 4 and 6 × 4 to find the answer to 8 × 4.
Students should make a 8 × 4 rectangle and decompose it to explain why this can be done.
Lesson 13: Use the Strategy Cards
Ask students to explain how they can use relationships in a Fast Array to find divisions they don’t
know. Students should describe how multiplication can be used to find a division using the
numbers in the Fast Array.
Lesson 14: Building Fluency with 2s, 3s, 4s, 5s, 9s, and 10s
Ask students to explain how they can use a multiplication strategy to find the answer to 7 × 4.
Students may explain finding the area of a rectangle to finding the product or they may explain how
to find it using the Distributive Property or the 5s shortcut.
Lesson 15: Multiply and Divide with 1 and 0
Ask students to describe any patterns found in the 1s and 0s multiplications and divisions and to
explain how these patterns can help them when multiplying by 1 or 0. Students should explain that
knowing these patterns or rules helps them to know the answer as soon as they see 0 or 1 as a
factor.
Lesson 16: Solve and Create Word Problems
Ask students to describe the difference between an array and an equal groups multiplication
problem. Students should explain that in an array problem, rows and column are usually
mentioned. In an Equal Groups problem, equal groups are described.
Lesson 17: Play Multiplication and Division Games
Ask students to explain how you can use 8 ÷ 2 =4 to help you find 8 ÷ 4. Students should explain
that for a set of 3 numbers such as 8, 2, and 4 there are two divisions and two related
multiplications. For example, if you know 8 ÷ 2 = 4, then you know 8 ÷ 4 = 2 and you also know 2 ×
4 = 8 and 4 × 2 = 8.
Lesson 18: Building Fluency with 0s, 1s, 2s, 3s, 4s, 5s, 9s, and 10s
Ask students to explain why they can write a multiplication equation to solve a division problem.
Students should explain that multiplication and division undo each other.
Lesson 19: Focus on Mathematical Practices
Unit 2: Multiplication and Division with 6s, 7s, 8s and Multiply with Multiples of 10
Lesson 1: Multiply and Divide with 6
Ask students to describe a strategy they can use to find a 6s multiplication they don’t know.
Students may describe starting with 5 × 6 and counting on or adding on to the fact before, doubling
a 3s multiplication, or combining two multiplications they know.
Lesson 2: Solve Area Word Problems
Ask students to describe two ways to find the area of a rectangle. Students should describe tiling
the rectangle with square units or multiplying the length and width as ways to find the area of a
rectangle.
Lesson 3: Multiply and Divide with 8
Ask students to describe a strategy they could use to find 9 × 8 if they do not recall it. Students
should describe starting with 5 × 8 and then counting by 8s from 40, combining two facts they
know, doubling 4s multiplication, adding on to the fact behind, subtracting from the fact ahead or
using the Commutative Property of Multiplication to check if they know 8 × 9.
Lesson 4: Write Word Problems and Equations
Ask students to describe how they know multiplication is the operation to use to solve a problem.
Students should describe that when the problem involves finding the total of equal groups,
columns, rows, or area, you multiply.
Lesson 5: Multiply and Divide with 7
Ask students to describe how to find the answer to 42÷ 7. Students should describe how to find the
answer to 42 ÷ 7. Students should describe using a related Fast Array or related multiplication (⧠ ×
7 = 42, ⧠ = 6).
Lesson 6: Square Numbers
Ask students to give an example of a square number and tell why it is a square number. For
example, 16 is a square number because it has two factors that are the same: 4.
Lesson 7: Practice with 6s, 7s, and 8s
Ask students when they know a word problem can be solved by division. Students should describe
that when a word problem involves a total and one of the following needs to be found – the number
of groups, the size of the groups, the number of columns or rows, or the side length of a square –
then the problem can be solved by division.
Lesson 8: Building Fluency with 0s-10s
Ask students what strategy they use to memorize a multiplication they do not recall. Allow students
to share their methods.
Lesson 9: Equations and Word Problems
Ask students to describe how they would begin to write a word problem for the equation 42 ÷ 6 = n.
Students should describe a situation that has a total of 42 objects that can be divided into groups of
6 or into 6 groups.
Lesson 10: Write First Step Questions for Two Step Problems
Ask students to write a two-step problem and tell what question they would need to answer first.
Lesson 11: Make Sense of Two Step Word Problems
Ask students to describe the rules they need to follow when solving an equation for a two-step
problem. Students should describe the rules for the Order of Operations.
Lesson 12: Multiply with Multiples of 10
Ask students to explain how they can use a multiplication strategy to find the answer to 7 × 40.
Students may explain that they can use place value and properties to find the product or they may
explain how to find it using a basic multiplication and then multiplying by 10 mentally.
Lesson 13: Play Multiplication and Division Games
Ask students to choose one of the function tables on Student Book page 146 and to explain how
they determined the rule they used to complete the table.
Lesson 14: Building Fluency with 0s-10s
Ask students to write a basic multiplication fact and use their knowledge of patterns to explain how
they know their answer is correct.
Lesson 15: Focus on Mathematical Practices
Unit 3: Measurement, Time, and Graphs
Lesson 1: Customary Units of Length
Ask students to describe how to measure a line segment to the nearest ¼ inch. Students should
describe that they would put the ruler underneath the line segment and line up 0 on the ruler with
the left end of the line. Then count the repeated units to the other end of the line and decide which
¼ unit the line segment is closer to.
Lesson 2: Customary Units of Liquid Volume
Ask students to name the units of liquid volume they worked with in this lesson from largest to
smallest. Students should name gallon, quart, pint, cup, and fluid ounce.
Lesson3: Metric Units of Liquid Volume
Ask students to explain the relationship between a liter and a milliliter. Students should explain
that a liter contains 1,000 ml. A liter is much larger than a milliliter.
Lesson 4: Customary Units of Weight and Metric Units of Mass
Ask students to explain the strategies they use to estimate the weight or mass of an object. Students
should explain that they use a benchmark with a similar size. They should include an example of a
benchmark for ounce, pound, gram and kilogram in their explanation.
Lesson 5: Solve Word Problems Involving Liquid Volume and Mass
Ask students to explain how they know whether to add, subtract, multiply, or divide to solve a
problem. Students should explain that if the problem asks for a total they would use addition or
multiplication. If the groups are equal, it is easier to use multiplication to find the total. If the
problem asks for how much more or less a quantity is than another, then subtract. If the problem
asks about how many equal groups or how many in each group, dividing will solve the problem.
Lesson 6: Tell Time
Show 1:25 on an analog clock. Ask students to say the time in two ways. Then ask them to write the
time. Students should say the time as one twenty five and twenty-five minutes after one. Students
should write the time as 1:25.
Lesson 7: Before and After the Hour
Write the time 4:37 on the board. Ask students to show the time on a clock. Then have students
explain the two different ways the time can be expressed. Students should be able to explain that
the time can be expressed as a time after the hour and as a time before the hour; in this case, thirtyseven minutes after 4 or twenty- three minutes before 5.
Lesson 8: Elapsed Time
Ask students to tell how they would find the elapsed time between 3:15 and 4:45. Students should
be able to explain that they can find elapsed time by either counting the hours on a clock and adding
the minutes or by using a number line with appropriate time intervals for the problem marked off.
Lesson 9: Add and Subtract Time
Ask students to explain how they know to jump forward or backward when adding and subtracting
on the number line. Students should explain that if they know the start time and the elapsed time
and want to find the end time, they can find the start time and make forward jumps to add the
times. If they know the end time and the elapsed time and need to find the start time, they will start
with the end time and make backward jumps to subtract to find the start time.
Lesson 10: Solve Word Problems Involving Time
Ask Students to explain how they would use a clock to add units of time. Students should explain
that they start on the start time and add the times counting by hours or minutes around the face.
Then look at the time where they end up on the clock.
Lesson 11: Read and Create Pictographs and Bar Graphs
Ask students to explain how they determine what a bar on a graph represents and to give an
example using the graph about career choices on Student Book page 198. Students should explain
that you can tell what a bar represents by looking at the title and the labels. For example, the bar
next to Doctor represents the number of students who chose a doctor for their future career.
Looking at the end of the bar and following the line to the scale shows that 60 students chose
Doctor as their future career.
Lesson 12: Read and Create Bar Graphs with Multi-digit Numbers
Ask students to explain how they draw a bar graph. Students should include in their description
that they first decide what the bars on the graph represent and the determine how to label the scale
to best display the data.
Lesson 13: Represent and Organize Data
Ask students to name the different ways they have learned to represent and organize data to show
how often something occurs and how they are different. Students should be able to explain that
they can represent data in tally charts or frequency tables and in line plots. They should include in
their explanation that while the tables use tally marks or numbers to show how often something
occurs, the line plot uses dots placed above a number line.
Lesson 14: Use Graphs to Solve Time and Measurement Problems
Ask students to describe how they can use data in a bar graph or a line plot to solve word problems.
Students should be able to explain that they have to understand what the problem is asking them to
do. Then they have to find the correct data in the bar graph or line plot and determine what
operation to use. After solving the problem, they need to check their work for reasonableness of the
answer and that they have answered the question asked.
Lesson 15: Focus on Mathematical Practices
Unit 4: Multi-digit Addition and Subtraction
Lesson 1: Make Place Value Drawings
Ask students to explain how the number 251 is different from the number 521. Students should
explain that the values are different because a 5 in the hundreds place has a value greater than a 2
in the hundreds place. Students may use place value drawings with or without dots to show how
the numbers are different.
Lesson 2: Build Numbers
Ask students to give the value of the 8 in 384 and use the Secret Code Cards to show they are
correct. Students should explain that the 8 is in the tens place and has a value of 80. Students should
expand the Secret Code Cards to show 384 and to show that the 8 has a value of 80.
Lesson 3: Place Value in Word Problems
Ask students to explain how they can find how many baskets of 100 peaches can be made with 742
peaches and how many will be left over. Students should explain that the hundreds place tells them
how many baskets of 100 can be made: 7. The tens and ones tells how many will be left over: 42.
Lesson 4: Practice with Place Value
Ask students to explain how to use place value to add 700 + 500. Students should explain that 7
hundreds plus 5 hundreds is 12 hundreds or 1 thousand 2 hundred.
Lesson 5: Round to the Nearest Hundred
Ask students to explain how to round a number with a 7 in the tens place to the nearest hundred.
Students should explain that a number with a 7 in the tens place is closer to the next hundred, so
the number should be rounded up to the next hundred.
Lesson 6: Round to the Nearest Ten
Ask students how to estimate a sum or difference by rounding and give an example. Students
should explain using their example to first round the numbers to the nearest ten or nearest
hundred. Then add or subtract the rounded numbers.
Lesson 7: Explore Multi-digit Addition
Ask students to explain how to add 614 and 293 numerically and with a proof drawing. Students
should explain how to add using the New Groups Above method, the New Groups Below method or
the Show all Totals method for addition, and make a proof drawing.
Lesson 8: Discuss Addition Methods
Ask students how to subtract 150 – 70 by counting on by tens. Students should explain one of the
three methods. Count on by tens; start with 70 and count on to 150. Place value: 7 tens + 8 tens = 15
tens. Or show the counting on with a drawing.
Lesson 9: The Grouping Concept in Addition
Ask students to explain when they need to group and how to group when adding. Students should
explain that they need to group when the ones or tens are more than 9. They should also explain
where to write the new group number for the method they are using.
Lesson 10: Practice Addition
Ask students to discuss examples of common errors they identified. Students should be able to
explain that some common errors they found included forgetting to make a new hundred, writing
the ones above the tens column and the new 1 ten in the ones column, forgetting to make a new ten,
and forgetting to make a new ten and a new hundred.
Lesson 11: Ungroup to Subtract
Ask students to explain a method they learned for subtracting multi-digit numbers. Students should
explain the Expanded method, the Ungroup First method, or the Common U.S. method for
subtracting.
Lesson12: Subtract Across Zeros
Ask students to explain a method to subtract across zeros in the top number. Students should be
able to explain the process of ungrouping to subtract.
Lesson 13: Discuss Methods of Subtracting
Ask students to explain two subtraction methods – ungrouping from the left and ungrouping from
the right. Students should be able to explain the process of ungrouping to subtract.
Lesson 14: Relate Addition and Subtraction
Ask students to explain how addition and subtraction are related. Students should explain that one
undoes the other. If the addend in an addition is subtracted from the sum, the result will be the
other addend. The numbers in an addition and its related subtraction are the same. The proof
drawings show the same numbers and the after grouping matches the before ungrouping and the
before grouping matches the after ungrouping.
Lesson 15: Subtraction Practice
Ask students to explain how they decide when to ungroup in a subtraction problem, and to give an
example ofa subtraction where all three places need to be ungrouped. Students should explain
that when the number in the place they are subtracting is larger than the one they are subtracting
from, they need to ungroup a larger place. An example where all three places need to be ungrouped
is 345 – 299.
Lesson 16: Addition and Subtraction Practice
Ask students how they could subtract 398 from 600 mentally. Students should suggest subtracting
400 from 600 and adding 2 to 200 for an answer of 202.
Lesson 17: Solve Word Problems
Write this problem on the board. Yvette had 18 mysteries and 15 biographies. Then she bought a
group of 12 science fiction books. How many books does Yvette have now? Ask students to describe
a strategy they would use to solve the problem. Students should be able to explain they would write
the equation 18 + 15 + 12 = n. Nest, use the Commutative Property to switch the order of addends:
18 + 12 +15 = n. Then use the Associative Property to group the numbers to make them easier to
add.
Lesson 18: Focus on Mathematical Practices
Unit 5: Write Equations to Solve Word Problems
Lesson 1: Addition and Subtraction Situations
Ask students to summarize what methods they learned for solving addition and subtraction word
problems. Present the following problem, questions, and instructions to students: Cory had 12
grapes. He ate 4 of them. How many grapes does he have now?
 Is the problem an Add To or a Take from problem?
Then have students:
 Draw a Math Mountain for the problem.
 Write an equation for the problem.
 Solve the problem.
Lesson 2: Word Problems with Unknown Addends or Unknown Factors
Have volunteers give a word problem with an unknown addend and another with an unknown
factor. Then ask students to write an equation to solve each problem.
Lesson 3: Word Problems with Unknown Starts
Ask students to explain what a situation equation and a solution equation are. Students should
explain that a situation equation shows the order of the information in the problem and a solution
equation shows the operation that can be used to solve the problem.
Lesson 4: Comparison Problems
Write a comparison problem with an unknown difference on the board. Students should draw
comparison bars and then write an equation to solve the problem.
Lesson 5: Comparison Problems with Misleading Language
Ask students to explain how they can make sense of a problem with misleading language. Students
should be able to explain that if they can’t use the problem as it is stated to determine who has
more or fewer, they can find the comparison statement in the problem and restate it in terms of the
other person in the problem.
Lesson 6: Word Problems with Extra, Hidden, or Not Enough Information
Ask students to write and solve one example each of problems with extra, hidden, or not enough
information.
Lesson 7: Write First Step Questions for Two Step Problems
Ask students how they know a problem is a two-step problem. Student’s explanations should
include that when they read the problem to make sense of it, they realize that a single number is
needed in order to do the second step, which will solve the problem and answer the questions
asked.
Lesson 8: Solve Two Step Word Problems
Ask students to write an equation involving two-steps. Then explain how they could assess the
reasonableness of the answer using a mental math strategy. Students should explain the strategy
they used for their numbers.
Lesson 9: Equations and Two Step Word Problems
Write this problem on the board. Jared has 2 boxes in his room. Each box contains a number of
trays, and each tray contains 2 miniature trucks. Jared has 16 trucks in all. Ask students to write an
equation with a variable to solve the problem and explain how they would solve it. Students should
write 2 × t × 2 = 16, t = 4.
Lesson 10: Practice with Two Step Word Problems
Ask students to choose one of the problems on Student Book page 290 and write a two-step
equation to represent the problem. Students should be able to explain how they decided which
operations to use in the equation and which number is the unknown number they are trying to find.
Lesson 11: Focus on Mathematical Practices
Unit 6: Polygons, Perimeter, and Area
Lesson 1: Triangles
Ask students what characteristics all triangles share and how they can be grouped into smaller
categories. Students should explain that all triangles have three sides and three angles. Triangles
can be grouped into smaller groups by the number of sides of equal length or by the types of angles.
Lesson 2: Parallelograms, Rectangles, Squares, and Rhombuses
Ask students to draw a quadrilateral and describe it. Students’ descriptions should include the sides
that are of equal length, the sides that are parallel, and if there are any right angles.
Lesson 3: Draw Quadrilaterals
Ask students to name a subgroup of quadrilaterals and give its attributes. For example, students
may name rectangles and give the attributes of opposite sides parallel and the same length and four
right angles.
Lesson 4: Classify Quadrilaterals
Ask students to name the subcategories of quadrilaterals. Students should name parallelograms,
rectangles, squares, rhombuses, and trapezoids.
Lesson 5: Perimeter and Area
Ask students to explain what the perimeter and area of a figure are and to tell how both attributes
can be measured.
Lesson 6: Side Lengths with Area and Perimeter
Ask students to describe how to find an unknown side length when the perimeter or the area of a
rectangle is known.
Lesson 7: Compare Areas and Perimeters
Ask students to describe if they were given a rectangle with a certain perimeter, how they would
draw it so that it has the greatest area. Suggest they use a perimeter of 18 inches to help them
explain.
Lesson 8: Area of Rectilinear Figures
Draw a rectilinear figure on the class Math Board or on Centimeter Dot Paper and ask students to
explain two different ways to decompose it to find the area.
Lesson 9: Solve Perimeter and Area Problems
Ask students to write examples of and explain how they would solve a real world area and a real
world perimeter problem.
Lesson 10: Tangram Shapes and Area
Ask students to explain how area and perimeter can be used to describe figures and to tell what
kinds of problems can be solved by finding the area and perimeter of rectangles.
Lesson11: Focus on Mathematical Practices
Unit 7: Explore Fractions
Lesson 1: Understand Fractions
Ask students to explain what unit fractions are and how they are used to build other fractions.
Lesson 2: Model Fractions
Ask students to use examples to explain how both fraction bars and number lines can be used to
show fractions.
Lesson 3: Locate Fractions on the Number Line
Ask students to use examples to explain how to locate fractions on the number line.
Lesson 4: Compare Unit Fractions
Ask students to describe the two methods they can use to compare unit fractions.
Lesson 5: Compare Fractions
Ask students to explain how they can compare fractions that have the same denominator and how
they can compare fractions that have the same numerator.
Lesson 6: Introduce Equivalence
Ask students how using fraction strips can help them to find equivalent fractions. Students should
explain that fraction strips show that two fractions with different numerators and denominators
can name the same amount.
Lesson 7: Equivalent Fractions
Ask students to use examples to explain how to use number lines to find equivalent fractions.
Lesson 8: Problem Solving With Fractions
Ask students to explain how they can use their understanding of unit fractions, comparing fractions,
and equivalent fractions to solve word problems. Students should include the following points in
their explanation: unit fractions are equal parts of the fraction and you can put them together to
make other fractions; when comparing fractions with the same denominators, the one with the
greater numerator is the greater fraction; when comparing fractions with the same numerators, the
one with the greater denominator is the lesser fraction; and equivalent fractions are fractions that
represent the same part of the whole.
Lesson 9: Focus on Mathematical Practices