5.NBT.7 - About the Math, Learning Targets, and Rigor

Grade 4 G/T Place Value and Decimals

5.NBT.7

About the Math

Full Standard

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

 

Learning Targets (I can)

  • Represent addition of decimals with base ten models and number lines.
  • Represent subtraction of decimals base ten models and number lines.
  • Explain how addition and subtraction of decimals are related.
  • Estimate sums and differences of decimals.
  • Add decimals using partial sums and an addition algorithm.
  • Subtract decimals using partial differences and a subtraction algorithm.
  • Represent multiplication of decimals with base ten models.
  • Estimate products of decimals to tell if a product is reasonable.
  • Multiply decimals using partial products and a multiplication algorithm.
  • Represent division with decimals using base ten models.
  • Estimate quotients of decimals to tell if a quotient is reasonable.
  • Divide decimals with partial quotients.
  • Solve problems with computation of decimals.

 

About the Math 

Computation with decimals is more than "lining up decimals" (+/-) or "moving the decimal point" (x/÷). Simply, our students must understand computation with decimals. They must work to estimate sums, differences, products, and quotients. They should transfer whole number computation strategies to decimals. 

  • Represent addition of decimals with base ten models and number lines.
  • Represent subtraction of decimals base ten models and number lines.

Using base ten blocks can be challenging for some students as the "flat" now has a value of 1. Using decimal grids can help. Regardless, students should have ample opportunities to add and subtract decimals on a number line. Students should make connections between decimals and related whole numbers. For example, 33 + 45 has some similarities to 0.33 + 0.44. The addends are referring to a difference place value and the sum will in turn refer to a different place value. Here, 33 ones + 45 ones equals 78 ones or 7 tens and 8 ones or simply 78. For 0.33 + 0.44, one is adding 33 hundredths + 44 hundredths which equals 78 hundredths which is also 7 tenths and 8 hundreds or simply 0.78.

  • Explain how addition and subtraction of decimals are related.

Students should be able to explain how addition and subtraction of decimals is related as they are related with whole numbers. Students should be able to use adding up to subtract on a number line. 

  • Estimate sums and differences of decimals.

When computing with decimals, it is important to ask the students to estimate the answers. Prior to computing ask the students to make estimates by rounding to nice whole numbers. Estimating helps students focus on the meaning of the numbers and operations and not on counting decimal places. Students will need a solid understanding of decimal place value prior to computing with decimals.

  • Add decimals using partial sums and an addition algorithm.
  • Subtract decimals using partial differences and a subtraction algorithm.

Often, partial sums and partial differences are more efficient than using a procedural algorithm. Decimals can be decomposed and added just as fractions and whole numbers can be.

  • Represent multiplication of decimals with base ten models.

Students should work with representations of whole numbers times decimals to first develop understanding of multiplication with decimals. From there, students can develop understanding of decimal times decimal. Students can then use these two ideas to apply partial products to decimal numbers.

When multiplying decimals, connect their understanding to fractions. So when multiplying .3 x .4, you are multiplying 3/10 x 4/10 which equals 12/100. This helps students make sense that .3 x .4 = .12. Students need to understand that when multiplying two numbers less than one, the product may be smaller than both the factors. We can think of this type of multiplication as finding part of, this helps them make sense that the product will be smaller. Often times students try to apply a misunderstanding that when you multiply, the answer "gets bigger". This is true in whole numbers but does not apply to fractions and decimals. We apply this same thinking to division of decimals. When dividing two decimals less than one (such as 0.5 ÷ 0.1) the quotient will be greater than either decimal. So the quotient will be 5. Students may be confused by this if they are under the misconception that when you divide the quotient will be smaller.

  • Estimate products of decimals to tell if a product is reasonable.

As noted above, students should estimate products of decimals before finding actual products. Doing so has many benefits. Most importantly, it helps develop students number and operational sense.  There are various ways to estimate products with decimals. We can always round. We can also think of a decimal as being between two whole numbers thus generating a range for a product. For example, 5.48 x 8 is between 5 x 8 and 6 x 8 as 5.48 is between 5 and 6. There are other strategies as well.

  • Multiply decimals using partial products and a multiplication algorithm.

Decimals can be decomposed as whole numbers are. We can apply the Distributive Property to multiplication with decimals. Consider 4.5 x 6. We can think of it as 4 x 6 + 0.5 x 6 or 24 + 3.

  • Represent division with decimals using base ten models.

Understanding of division with decimals must also be grounded in understanding. It is also important that student work with division of decimals is restricted to friendly computations during first exposure (if not all of grade 5 instruction). Without careful consideration, quotients can easily enter 4, 5, or more decimal places. 

  • Estimate quotients of decimals to tell if a quotient is reasonable.

Estimating quotients with decimals can be done in many ways. Here rounding may lose efficiency. As with other operations, it makes sense to reason about the decimal and think about friendly, compatible numbers. For example, 83.8 ÷ 21.5 might be best estimated as about 80 ÷ 20 or even 84 ÷ 20.

  • Divide decimals with partial quotients.

Note that students in grade 5 are not exposed to the division algorithm. They rely on partial quotients. The same is true for dividing decimals. Because of this, number selection is critical. 

 

Other considerations for computing with decimals:

  • +/- with decimals: Situations through hundredths. Develop through use models and tools (number lines). Use procedure when students are ready.
  • x with decimals: Situations should include whole x tenth, whole x hundredth, and tenth x tenth. Hundredth x hundredth is not required. Develop through use of models and tools. Use procedure when students are ready.
  • ÷ with decimals: Situations need to be thoughtful. It is grounded in meaning of the computation. It centers on context and number choice. It is necessary to avoid repeating decimals and quotients beyond hundredths. The examples below should be helpful.

 

Situations to consider for division:

Division Situations

Expression

Situation

12 ÷ .4

How many tenths are in a whole?

(Whole number divided by tenths, clearly divisible by the divisor)

2.7 ÷ 3

How can we partition a tenth?

(Tenths divided by whole number clearly divisible by divisor, quotient in tenths)

5.5 ÷ .5

How many tenths are in tenths?

(Tenths divided by tenths, clearly divisible by divisor)

3.68 ÷ 2

How can we partition a hundredth?

(Hundredth divided by whole) clearly divisible by divisor

3.15÷ .3

How many tenths are in a hundredth?

(Hundredths divided by tenth clearly divisible by divisor)

4 ÷ .25

How many hundredths are in a whole?

(Whole divided by hundredth clearly divisible by divisor. Numbers should be carefully selected.


Essential vocabulary for this standard includes decimal, tenths, hundredths, add, subtract, multiply, divide, addend, sum, difference, factor, product, quotient, and whole number.

 

 

Progression of Standard within Grade 4 G/T

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Progression Throughout Year
Quarter 1 Quarter 2 Quarter 3 Quarter 4

 

 
  • Represent addition of decimals with base ten models and number lines.
  • Represent subtraction of decimals base ten models and number lines.
  • Explain how addition and subtraction of decimals are related.
  • Estimate sums and differences of decimals.
  • Add decimals using partial sums and an addition algorithm.
  • Subtract decimals using partial sums and a subtraction algorithm.
  • Represent multiplication of decimals with base ten models.
  • Estimate products of decimals to tell if a product is reasonable.
  • Multiply decimals using partial products and a multiplication algorithm.
  • Represent division with decimals using base ten models.
  • Estimate quotients of decimals to tell if a quotient is reasonable.
  • Divide decimals with partial quotients.
  • Estimate sums, differences, products, and dividends with decimals.
  • Add, subtract, or multiply decimals with various strategies including standard algorithms.
  • Divide decimals with partial quotients.

 

 

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.

Progression Across Grades
Grade 4 Grade 6

Fluently add and subtract multi-digit whole numbers using the standard algorithm (4.NBT.4)

Fluently add, subtract, multiply, and divide decimals using the standard algorithm (6.NS.3)

 

 

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Instructional Tasks

TASKS

These tasks can be used with small group or whole group instruction.

  • How is adding and subtracting with decimals similar to adding and subtracting with whole numbers? How are they different?
  • Why does"lining up the decimal points” when you are adding and subtracting with decimals help when finding the answers? Why does this make sense mathematically?
  • How can estimation help when you multiply and divide with decimals? Use examples to support your answer.
  • Without doing the calculations, what can you conclude about the products of the following equations?
    • 3.4 x 6
    • 3.45 x 6
    • 3.452 x 6
  • How can you use 45 ÷ 3 = 15 to help you with 4.5 ÷ 3?
  • When Jamie added .4 + .7 he got .11. Explain to him what he did wrong.
  • Write two decimals when added together have a sum of 14.5.

 

SLIDE-BASED TASKS 

 These links are HCPSS created instructional tasks. These tasks are provided in Google slides. These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.

Module 7 • Operations with Decimals

Estimate Sums and Differences of Decimals:

Estimate Products and Quotients of Decimals:

 

 

Additional Tasks 

These links provide instructional ideas connected to this standard. 

CC Indicates colleague created HCPSS instructional tasks. These tasks are provided in Google slides. When selected, a copy of the file is added to your drive for classroom use.

 

Tasks From Print Resources

These publications have been provided for each school. They are typically stored in team closets or the media center. Check with your team leader if you cannot find them. 

Print Resources
Book Thumbnail Book Title Grade Pages
Teaching Student-Centered Mathematics 3-5 Exact Sums & Differences,  Activity 7.11, Page 198
Where Does the Decimal Go? Multiplication, Activity 7.12, Page 199 
Where Does the Decimal Go? Division, Activity 7.13, Page 200
Fundamentals 5 40-43 "Decimal Draw"
44-47 "X Zone"
Nimble with Numbers 6-7 66-67 Decimal Draw
Hands-On Standards 5 44-45 "Add and Subtract Decimals"
52-53 "Multiply and Divide Decimals"
Groundworks: Reasoning about Numbers 5 80-87 Grid Sums
Brain-Compatible Activities for Mathematics 4-5 61-62 Dollars and Sense"
77-79 "Calling All Construction Workers"
 Number SENSE 4-6 75-76 "About How Many Is It?"
196-200 "What Did I Buy?"
221-225 "Where's the Point?"
Problem-Driven Math 5 7, 47-49 "Un-Sound System"
9, 61-62 "Creative Costs"
Problem #s 16, 18, 19, and 30

Math In Practice

Teaching Fifth-Grade Math

 5 Module 5 & 6

 

 

 

 

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Independent Work

Centers

These print resources can be used during independent or center time. These resources could also be used as lesson seeds.

 

 

INDEPENDENT PRACTICE/HOMEWORK/ASSESSMENT

 These resource sheets can be used for independent practice, homework, or assessment. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.

 

 

Links to an external site.Wholes x Tenths: Represent Multiplication Using Base Ten Model

Links to an external site.

 

 

 

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Assessment

Learning Targets

rubric

  • Represent addition of decimals with base ten models and number lines.
  • Represent subtraction of decimals base ten models and number lines.
  • Explain how addition and subtraction of decimals are related.
  • Estimate sums and differences of decimals.
  • Add decimals using partial sums and an addition algorithm.
  • Subtract decimals using partial differences and a subtraction algorithm.
  • Represent multiplication of decimals with base ten models.
  • Estimate products of decimals to tell if a product is reasonable.
  • Multiply decimals using partial products and a multiplication algorithm.
  • Represent division with decimals using base ten models.
  • Estimate quotients of decimals to tell if a quotient is reasonable.
  • Divide decimals with partial quotients.

Learning targets identify what students should be able to do. This rubric can be applied to tasks and observations for assessment and/or grading. 

 

 

 

 

 

 

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 This course content is offered under a CC Attribution Non-Commercial Share Alike Links to an external site. license. Content in this course can be considered under this license unless otherwise noted.