Building Sense of “Ten”ness

When is this property introduced?

The foundations for place value are introduced when students begin regrouping ten ones into one ten and building an understanding of unitizing.

Place Value is a much more complex concept than we often give it credit for as it is typically taught at a very surface level. Misconceptions are easily and often developed, creating significant issues later. Explore possible misconceptions you may have by working through a “learning activity for teachers” and exploring the research around place value.

How might this property be introduced?

Many models can be utilized to represent place value. Below, you will find place value explored from Kindergarten to Grade 9.

Kindergarten

CURRENT Curricular Outcomes:

• Number 1 Say the number sequence 1 to 10 by 1s, starting anywhere from 1 to 10 and from 10 to 1.
• Number 2 Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots.
• Number 3 Relate a numeral, 1 to 10, to is respective quantity
• Number 4 Represent and describe numbers 2 to 10, concretely and pictorially
• Number 5 Compare quantities 1 to 10, using one-to-one correspondence
• Patterns and Relations 2 Sort a set of objects based on a single attribute and explain the sorting rule
• Shape and Space 1 Use direct comparison to compare two objects based on a single attribute, such as length (height), mass (weight) and volume (capacity)
• Shape and Space 2 Sort 3-D objects, using a single attribute

Process of Complexity:

Spend a significant amount of time on the number 5. Move to numbers more than 5 but less than 10. Move to 10.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Longer/shorter/same
• More/Less/same
• Count(ing)
• Match
• Total Number

The Process:

Strategy Part 1: Concrete

> Counting Blocks

• Provide students with 5 objects that are each a different color (eg. Unifix cubes, counting bears, etc.) and a tray or container.
• Have them count the 5 objects as they place them in the tray.
• Students empty trays.
• Bring a student to the front. Ask them to pick a color to start with, announce the color, and have the student count the objects in front of the class.
• Say, “There are 5 blocks altogether. Let’s count as we remove the blocks.”
• Have the student remove the last block he put into the tray.
• Say, “We had 5. Now how many are there?” (Student may need to recount – 4)
• Say, “We have 4. Let’s remove 1.” (Student removes one of his choice.)
• Ask, “How many are there now?” (3)
• Repeat until all blocks are out of the tray.
• Bring a different student to the front. Have them pick a different color to start with, announce the color, and have the student count the objects in front of the class.
• Ask, “How many blocks are there altogether?” (5)
• Ask, “What did you notice about the total number of blocks?” (same)
• Repeat process for counting backwards.
• Bring a different student to the front. Have them pick a different color to start with and announce the color. 
• Say, “Before he counts, in your head, figure out how many blocks there are. What number will he say? Will it be more, less, or the same? When you think you know, give me a thumbs up (or whatever -I am ready- strategy you use in class)” Give a bit of time for them to predict.
• Say, “Whisper to your partner – How many blocks do you think he will say there are altogether?” (5)
• Have the student count.
• Repeat, starting with different colors each time.
• Repeat the activity on different days, using different objects.
• Repeat the activity using a 5 frame.

Strategy Part 2: Concrete 

> Connecting counting to digits

• Provide students with 5 colored dots to put on their fingers (different color for each finger) and 5 objects that match the colors of the dots.
• Students count the objects as they touch a colored finger to the same colored object.
• You can repeat the process from Strategy Part 1.

> Counting digits

• Students count their fingers (that still have dots on them), starting on a different color each time. They should count forwards as they put their fingers down and backwards as they put their fingers up. (Or vice versa)

Strategy Part 3: Pictorial

> Exploring 5 frames

• Provide students with a blank 5 frame and bingo daubers, markers or pencil crayons. Have them color each square in a different color.
• Students count the squares, starting on a different color each time. 

Additional Strategy

> Exploring Tally Marks

• Provide students with popsicle sticks or other stick like options. Have them count them into piles of 5.
• Have them recount each pile, taking the 5th stick and laying across the top of the other 4 to “prove” that they double checked the count.
• Note: This isn’t about having them count higher than 10 or to skip count. Those outcomes are beyond Kindergarten. It is about seeing 5 and a way to introduce tally marks in a concrete way and a reason for placing the 5th mark across the previous 4. 

Formative / Summative Assessment:

• As students count, watch for one-to-one correspondence.
• Provide students with a set number of objects. Ask, “How many are there?” Ask, “will I have more, less or the same number if I start counting on a different object? How do you know?”

Moving Forward:

Provide students with objects where some of the colors are the same. Example, 2 red and 3 blue. Increase the numbers.

Grade 2

CURRENT Curricular Outcomes:

• Number 1 Say the number sequence 0 to 100 by
• 2s, 5s and 10s, forward and backward, using starting points that are multiples of 2, 5 and 10 respectively
• 10s, using starting points from 1 to 9
• 2s, starting from 1.
• Number 4 Represent and describe numbers to 100, concretely, pictorially and symbolically
• Number 5 Compare and order numbers up to 100
• Number 6 Estimate quantities to 100, using referents
• Number 7 Illustrate, concretely and pictorially, the meaning of place value for numerals to 100
• Number 8 Demonstrate and explain the effect of adding zero to, or subtracting zero from any number
• Number 9 Demonstrate an understanding of addition (limited to 1-and 2-digit numerals) with answers to 100 and the corresponding subtraction by:
• Using personal strategies for adding and subtracting with and without the support of manipulatives
• Creating and solving problems that involve addition and subtraction
• Using the commutative property of addition (the order in which numbers are added does not affect the sum)
• Using the associative property of addition (grouping a set of numbers in different ways does not affect the sum)
• Explaining that the order in which  numbers are subtracted may affect the difference
• Number 10 Apply mental mathematics strategies for basic addition facts and related subtraction facts to 18
• Patterns and Relations 2 Demonstrate an understanding of increasing patterns by describing, reproducing, extending, and creating numerical (numbers to 100) and non-numerical patterns using manipulatives, diagrams, sounds and actions

Process of Complexity:

• Focus on numbers to 20, especially on the numbers 11-19. These are often reversed when students write them down if these numbers are rushed. When students demonstrate understanding, move on to numbers to 99. When students demonstrate understanding, explore 100 without regrouping ten tens.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Digits
• Sets of ten
• Value
• Place
• Place Value, Base Value, Face Value
• Standard form/notation
• Non-standard form
• Expanded Notation
• Position 

Pre-Assessment:

> Administer MIPI+ Question #2

•  Find out more about administering and assessing the MIPI+ Question #2.

> Students who do not answer MIPI+ Question #2 correctly:

• Introduce and explore the Grade 2 high leverage strategy.

The Process:

Strategy Part 1: Concrete

> Connecting digits and fingers to the base 10 system

• Discuss with students: What is the difference between fingers and thumbs? Are thumbs fingers? (Yes). For the purpose of our learning, any time we talk about fingers, we are including thumbs. We don’t need to specifically use the word “thumbs”.
• Explain that another word for fingers is digits.
• Give students a few minutes to make any statements with a partner that they can think of regarding their digits that relate to numbers. Examples might include: “I have 10 digits”. :I have 5 digits on each hand.”, etc.
• Have students count out your digits with you. Hold up your fists so everyone can see. Put up one finger and say 1. Put up another finger and say 2. Keep going until all 10 fingers up and you say 10.
• Say, “I have one full set of fingers.”
• Bring up a student. Move over a little to the left (from the class’ perspective) and have the student stand on your left (from your perspective). 
• Say, “Let’s add one more finger.” Tell the student to put up one finger.
• Wiggle your full set of fingers (or raise them up high – whatever you want to bring students’ attention back to the full set) and say “One full set of fingers and…”
• Have the student hold up her finger (or wiggle it). Say “one extra.”
• Ask, “How many fingers altogether?” 11
• Formative Assessment opportunity: Watch for students who are counting the full set of fingers rather than seeing them as a “full set of 10.”
• Say, “We have one full set of ten fingers and one extra. Let’s add one more finger.” Have the student put up a second finger. Repeat the process “One full set of fingers and two extra. How many fingers in all?”
• Repeat for all numbers to 20. 
• Refer to sets of 10. Ie. 1 full set of 10 and 3 extra.
• Ask, “How many sets of 10?” (FACE VALUE: 1)
• Ask, “What’s the value of all of those 10s?” (PLACE VALUE: 10)
• Ask, “How many -fingers/dots/bears/ones- are there altogether? (BASE VALUE: 13)
• When you reach 20 fingers, say “Two full sets of fingers. How many extra ones?” None. Face Value of tens: 2; Place of tens: 20; Base value of ones (fingers): 20

> Repeat using Ten Frames

• Provide students with two ten frames and bingo chips (or other small manipulatives). Optional: Virtual Ten Frames for display.
• Repeat the process above, where students increase the value of the ten frames by 1 each time. They should always reference “one full set of dots and two extra is a total of 12 dots.”

> Repeat using other objects

• Use any other objects to represent groupings of ten OTHER THAN base ten blocks.
• Centers could contain baggies of pre-counted items. Students can use “buckets that only hold 10”, etc. 

Introduce base ten blocks. (ones and tens only). Give students lots of time to begin with exploring the relationship between the 1 and the 10. Do not rush this! They should be spending lots of time building numbers 1-20 using base ten blocks. It is very important that they understand than 10 ones is the same as 1 ten. “Life Size Printable base ten blocks”

Strategy Part 2: Pictorial 

> Connect the concrete representations to visuals.

• Work through the PICTORIAL section of “Introducing Place Value” slideshow with students. 
• Focus on the teen numbers. Students struggle with writing teen numbers.
• Repeat with other pictorial representations as needed and as available.

Strategy Part 3: Symbolic

> Connect the visuals to symbolic forms.

Work through the SYMBOLIC section of “Introducing Place Value” slideshow with students.

 Strategy Part 4: Moving to higher numbers

> Connect understanding to higher 2 digit numbers.

• Work through the PICTORIAL PUZZLES section of “Introducing Place Value” slideshow with students. This extends their understanding to numbers between 21 and 100.
• Do not rush to 100. Research shows that moving to 3 digit numbers is complex. The first time students see 100, don’t talk about regrouping the tens. It is just ten sets of ten with zero extra.

Use a “Bottoms Up” hundreds chart rather than a regular hundreds chart to look for patterns. This article explores the concept of a “Bottoms Up” hundreds chart.

Formative / Summative Assessment:

Conceptual Questions: 

• How does changing the order of the digits in a 2 digit number change the value of the 2 digit number?
• How many “ones” are there in the number 24? Convince me.
• How does knowing how the number 24 is built help you figure out if 26 is greater than or less than 24?

Moving Forward:

> Connect understanding to 3 digit numbers.

• Write the number 123 on the board. Ask students to think, pair, share ideas around using what they know about 2 digit numbers to help them figure out what is happening here. Current curriculum only requires students to work with numbers to 100. However, they should understand that you could say 12 tens and 3 ones.

Grade 3

 CURRENT Curricular Outcomes:

• Number 1 Say the number sequence 0 to 1000 forward and backward by
• 5s, 10s or 100s, using any starting point
• 3s, using starting points that are multiples of 3
• 4s, using starting points that are multiples of 4
• 25s, using starting points that are multiples of 25
• Number 2 Represent and describe numbers to 1000, concretely, pictorially and symbolically
• Number 3 Compare and order numbers to 1000
• Number 4 Estimate quantities less than 1000, using referents
• Number 5 Illustrate, concretely and pictorially the meaning of place value for numerals to 1000.
• Number 6 Describe and apply mental mathematics strategies for adding two 2-digit numerals
• Number 7 Describe and apply mental mathematics strategies for subtracting two 2-digit numerals
• Number 8 Apply estimation strategies to predict sums and differences of two 2-digit numerals in a problem-solving context
• Number 9 Demonstrate an understanding of addition and subtraction of numbers with answers to 1000 (limited to 1-, 2-and 3-digit numerals), concretely, pictorially and symbolically, by
• Using personal strategies for adding and subtracting with and without the support of manipulatives
• Creating and solving problems in context that involve addition and subtraction of numbers
• Patterns and Relations 4 Solve one-step addition and subtraction equations involving a symbol to represent an unknown number.

Process of Complexity:

If students’ understanding of place value is limited, begin with the Grade 2 strategy. Then, focus on counting by tens beyond one hundred before regrouping into hundreds. Once students demonstrate understanding, explore regrouping of 10 tens into 100. Explore numbers up to 1000. Explore 1000 as 10 hundreds. Explore 1000 as a regrouping of 10 hundreds.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Digits
• Sets of ten
• Sets of one hundred
• Value
• Place
• Place Value, Base Value, Face Value
• Standard form/notation
• Non-standard form
• Expanded Notation
• Position 

 Pre-Assessment:

> Administer MIPI+ Question #2

•  Find out more about administering and assessing the MIPI+ Question #2.

> Student who do not answer MIPI+ Question #2 correctly:

• Work through the Grade 2 high leverage strategy.

The Process:

Strategy Part 1/2: Concrete/Pictorial

You can separate the concrete from the pictorial or do them together.

> Building understanding of 100 as ten tens using Crayons.

• The first section of the Exploring Three Digit Numbers with Base Ten blocks google slide show “Working with Crayons” can be helpful for the next steps. The slideshow is PIctorial. It is strongly recommended that you have boxes of crayons (even homemade boxes would be fine) so that they can see 10 crayons fitting inside of one box and ten boxes fitting inside the “ultimate box”.
• Review grade 2 strategy of counting “how many full sets of crayons?”, “how many extra?”, and “how many crayons altogether?” using the examples of 2 digit numbers.
• “How many crayons is this?” 
• Skip count the boxes by saying  with students “1 set of 10, 2 sets of 10, 3 sets of 10,…9 sets of 10, 10 sets of 10, 11 sets of 10, 12 sets of 10”
• “Skip count to find out how many crayons there are”. 
• Skip count by 10’s by saying 10, 20, 30….90, 100, 110, 120”
• Draw their attention to the underlined numbers. 
• Ultimate Box of Crayons (10 packages of crayons): Students think pair share “How many crayons does the Ultimate Box hold?” Ensure that each student understands that it holds 10 boxes of crayons, and there are 10 crayons per box so there are 100 crayons altogether.
• How many…
• Boxes of crayons? Think pair share: Have students figure out how many boxes of crayons. Students need to understand that there are 10 boxes in the ultimate box and 2 boxes extra so there are 12 boxes of crayons
• Crayons? Think pair share: Students need to understand that you could describe the ultimate as 10 boxes of 10 OR 1 ultimate box 100; regardless there are 100 crayons in an ultimate box. There are also 2 boxes of 10 crayons which have 20 crayons altogether. 100 and 20 crayons is 120 crayons.
• How many…
• Ultimate boxes? 2 ultimate boxes
• Boxes of crayons? 10 boxes + 10 boxes + 1 box = 21 boxes with 6 extra
• Crayons? 2 ultimate boxes of 100 = 200 crayons, 1 box of 10 = 10 crayons, and 6 extra crayons = 216 crayons
• When you consider the 216: Students think pair share: Do they see the number of 
• ultimate boxes? Ultimate boxes is how many sets of 100 crayons we have. 2
• Boxes of crayons? How many sets of boxes of 10 we have: 21
• Crayons? How many crayons we have: 216
• Which would be easier to carry around? 1 ultimate box, 10 boxes of crayons or 100 crayons? 
• Think pair share. Discuss that usually, it will be better to carry around 1 ultimate box compared to the other two options. However, we often need those crayons in one of the other forms.
• Explore other numbers as examples. Always ask “How many ultimate boxes? Sets of boxes? crayons?

> Building understanding of 100 as ten tens with Base Ten Blocks.

• The second section of the Exploring Three Digit Numbers with Base Ten blocks google slide show “Working with Base Ten Blocks” can be helpful for the next steps. “Life Size Printable base ten blocks”
• What is this worth? Review base ten block values (ones and tens), if needed.
• Special way to skip count: 
• Skip count by tens, using the base ten blocks, by saying 1 ten, 2 tens, 3 tens, 4 tens, 5 tens, 6 tens, 7 tens, 8 tens, 9 tens, 10 tens, 11 tens, 12 tens…etc. 
• Skip count again but say 10, 20, 30,….80, 90, 100, 110, 120… Underline the tens as you go
• Skip count again but show the words. Underline the “number of tens” in the word. Ten, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety, One hundred
• What do you think the “ty” means? 
• Ask, “What do you think the -ty- means?” (tens ie. sixty is six-tens)

• What is the relationship between these two: 
• Provide students with a ten and a hundred base ten block.
• Give students a few minutes to find any and all relationships between the two. Do they notice:
• 10 tens fit in 1 hundred?
• The dimensions of the hundred are 10 by 10 (same lengths as the ten)
• The heights are the same (1 unit)
• I have…11 tens
• Students think pair share: If I replace 10 of the tens with a hundred, will by new value be the same as, larger than or smaller than my original value?
• I have…
• This is the visual representation for the answer. Say, “I can see the 11 tens.”
• Say, “Let’s replace 10 of the tens with a hundred.” <click>
• Say, “I can see the ten tens and the 1 ten.” <click>
• Say, “The ten tens is the same as 1 hundred because 10 x 10 is 100 and there is still the 1 extra 10.” <click>
• Say, “Ten tens + 1 Ten equals 110. 1 Hundred and 1 ten equals 110. So as long as I exchange ten tens for 1 hundred, which are the same value, my total value doesn’t change.”
• How many tens are there? (3 slides)
• Think pair share for each of these. 
• 11 tens⇒ 110 ⇒ 110
• 12 tens⇒ 120⇒ 120
• 13 tens⇒ 130⇒ 130
• How many ones are there?
• 100 in the hundred, 10, 10, 10 in the tens
• 130 ones
• 130
• How many tens?
• Have students skip count with you. 10 tens, 20 tens, 30 tens, 40 tens, 50 tens, 60 tens, 70 tens, 80 tens, 90 tens, 100 tens
• How much is 100 tens? 1000
• How many ones?
• Have students skip count with you. 100 ones, 200 ones, 300 ones, 400 ones, 500 ones….900 ones, 1000 ones.
• Ask, “Someone might say ten hundred ones. Is that correct or incorrect?” Allow time for discussion. It might not be the convention to say 10 hundred for 1000 but it’s not technically incorrect. Besides, we say 11 hundred, 12 hundred, etc. Why is 10 hundred so unusual?

Strategy Part 3: Symbolic

> Apply understanding of base ten to addition

• Everything you know about ones and tens for
• Question 1:
• 23: Students think pair share everything they know: 2 sets of tens; 3 ones; 23 ones. Say, “I know I could say 1 set of tens and 13 ones. For now, let’s leave those out and stick to using only the digits that are there in the number.”
• 41: Students think pair share everything they know: 4 sets of tens; 1 one, 41 ones
• +: Students think pair share everything they know: 6 full sets of tens, 4 ones, 64 ones
• Question 2: 
• Repeat for 58 + 24 (8 full sets of ten and 2 extra = 72)
• Question 3
• Repeat for 234+415; Students may say 23 sets of 10 and 4 extra plus 41 sets of ten and 5 extra OR they may say 2 sets of 100, 3 sets of 10 and 4 extra plus 4 sets of 100, 1 set of 10 and 5 extra (or ones). Ask which way is correct. (They both are).  234+415=649
• Question 4
• Repeat for 347+526 = 873
• Question 5
• Repeat for 279+453=732
• Question 6
• 637+784=1421

Formative / Summative Assessment:

• How many ones are there in 437?
• How many full sets of ten are there in 60? 75? 140?
• How many full sets of 100 are there in 300? 850? 473?
• How does knowing how the number 789 is built help you figure out if 780 is greater or less than 789? 800 is greater or less than 789?
• How does changing the order of the digits in a number change the value of the number?

 Moving Forward:

> Connect understanding to 4 digit numbers.

• Write the number 1234 on the board. Ask students to think, pair, share ideas around using what they know about 3 digit numbers to help them figure out what is happening here. Current curriculum only requires students to work with numbers to 1000. However, they should understand that you could say 12 hundreds, 3 tens and 4 ones.

Grade 4

CURRENT Curricular Outcomes:

• Number 1 Represent and describe whole numbers to 10 000, pictorially and symbolically
• Number 2 Compare and order numbers to 10 000
• Number 3 Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4- digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• Solving problems involving addition and subtraction.
• Number 9 Represent and describe decimals (tenths and hundredths), concretely, pictorially and symbolically
• Number 11 Demonstrate an understanding of addition and subtraction of decimals (limited to hundredths) by
• Using personal strategies to determine sums and differences
• Estimating sums and differences
• Using mental mathematics strategies

To solve problems

• Patterns and Relations 1 Identify and describe patterns found in tables and charts
• Patterns and Relations 6 Solve one-step equations involving a symbol to represent an unknown number

Process of Complexity:

Connect place value from 1000 (grade 3) to 10 000. Connect place value to tenths. Connect place value to hundredths. Estimating addition and subtraction answers. Solving addition and subtraction questions.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Digits
• Sets of ten, one hundred, one thousand
• Value
• Place
• Place Value, Base Value, Face Value
• Standard form/notation
• Non-standard form
• Expanded Notation
• Position 
• Decimals
• Tenths
• Hundredths 

Pre-Assessment:

> Administer MIPI+ Question #2

•  Find out more about administering and assessing the MIPI+ Question #2.

> Student who do not answer MIPI+ Question #2 correctly:

• Introduce and explore the Grade 2 and 3 high leverage strategy.

The Process:

This high leverage strategy will briefly address numbers within 10 000 by connecting to the grade 2 and 3 strategy. If students do not understand place value appropriately, go back to the grade 2 and 3 strategy and begin there.

This high leverage strategy will mainly focus on place value of decimals.

Strategy Part 0: Concrete 

> Connecting 4 digit numbers to 3 digit numbers

• Provide students with a single 1 of each of the following base ten blocks: 1, 10, 100. Have lots of hundreds ready to be used later. Have some (10 if possible) thousands, out of sight, ready to be used later. “Life Size Printable base ten blocks”
• Ask students “What relationships are there between the blocks? What patterns do we know?” Ensure that students state that there are 10 ones in 1 ten. 10 tens in 1 hundred. 100 ones in 1 hundred. 1 ten is 10 times as many as 1 one. Etc.
• Ask, “How much space would 100 hundreds take up on your desk? What does 100 hundreds look like?” Give students time to think about that. Think pair share.
• Ask, “What is another way to say 100 hundreds?” (10 000)
• Say, let’s layout 100 hundreds on the floor. It is helpful to lay them out as a 10 by 10 array, counting out 100 of them as you build. Then, skip counting by hundreds as you count them afterwards. 
• Ask, “How long would this be if we lined them up in a straight line if we started with the first one here…?” Let students line up according to where they think the 100th base ten block would go when you line them up in a row. (If you don’t have enough, discuss how you can lay down as many as possible and then borrow some from the middle to continue the line. Just don’t remove the first or last ones in the line.)
• Ask, “How many times bigger is 100 hundreds than 10 hundreds? (10 times) How do you know? Because 100 is ten times bigger than 10.
• Ask, “How many times bigger is 10 000 than 1000? (10 times).

> Connecting digits and fingers to decimals, limited to tenths, and the base 10 system

• Discuss with students: What is the difference between fingers and thumbs? Are thumbs fingers? (Yes). For the purpose of our learning, any time we talk about fingers, we are including thumbs. We don’t need to specifically use the word “thumbs”.
• Ask students if they know another word for fingers, other than thumbs. (digits)
• Say, “We are going to think about our fingers today as they relate to our hands.” Hold up both hands. “I have all of my fingers up. This is 1. 1 set of fingers”
• Have students count out your digits with you. Hold up your fists so everyone can see. Put up one finger and say “1 out of the 10 fingers are up”. Put up another finger and say “2 out of the ten fingers are up”. Keep going until all 10 fingers up and you say “10 out of the 10 fingers are up. I have 1 set of hands.”
• Bring up a student. Move over a little to the left (from the class’ perspective) and have the student stand on your left (from your perspective). 
• Say, “Let’s add one more finger.” Tell the student to put up one finger.
• Wiggle your set of hands (or raise them up high – whatever you want to bring students’ attention back to the full set) and say “One set of hands…”
• On the board write: 1
• Ask, “What does this 1 mean?” (1 set of hands)
• Have the student hold up her finger (or wiggle it). Say “and…one out of the ten I need for another full set.”
• On the board write: 1/10 beside the 1.
• Ask, “What does the 1/10 mean?” (You have 1 out of the 10 fingers needed to make a set of hands.
• Ask, “How do I say this? How many altogether?” 1 and one tenth.
• Say, “We have one set of hands and one of the ten I need to make another full set. Let’s add one more finger.” Have the student put up a second finger. Repeat the process “One set of hands and 2 out of the 10 I need to make a full set.”
• Say, “Each finger is worth 1 tenth. How many tenths do we have right now?” (12 tenths)
• Repeat for all numbers to 2. 
• When you reach 20 fingers, say “Two sets of hands. How many extra fingers?” (None.) You can write 0/10 “How many fingers…how many tenths?”) (20 tenths)
• Don’t focus on decimals yet. Just write everything out as fractions. 1 2/10

> Repeat using Ten Frames

• Provide students with two ten frames and bingo chips (or other small manipulatives). Optional: Virtual Ten Frames for display.
• Tell students that the whole ten frame is worth 1. Ask, what would a dot in this first box be worth? (1/10)
• Repeat the process above, where students increase the value of the ten frames by 1 each time. They should always reference “one full set of dots and two extra is 1 and 2/10 which is a total of 12 dots.”
• Don’t focus on decimals yet. Just write everything out as fractions. 1 3/10

> Repeat using other objects

• Use any other objects to represent groupings of ten tenths OTHER THAN base ten blocks.
• Centers could contain baggies of pre-counted items. Students can use “buckets that only hold 10 – the bucket would be worth 1” , etc.

• Connecting to Number lines: This Introducing the Place Value of Decimals: 1 hundred is 1 PART 4 slideshow may be helpful.
• Say, “We have a number line that shows 0 and 1, where 1 is our whole. 1 is always our whole. How many tenths do I need to make the whole?” (10)
• Ask, “Where would I place ten tenths on the number line?” (same spot as the 1)
• Ask, “How many tenths do I have at 0?” (0) “We can write that as 0 tenths”
• Ask, “Where does five tenths fit? How do I know?” (Half way between 0 tenths and 10 tenths)
• Ask, “What is five tenths as a decimal?” (0.5)
• Ask, “What other fractions do I know?” (1/10, 2/10, 3/10, 4/10, 6/10, 7/10, 8/10, 9/10)
• Ask, “Approximately where would eleven tenths go?” (click 4 times to go through the different representations.”

Strategy Part 4: Moving to hundredths

> Connect understanding to hundredths.

• This Introducing the Place Value of Decimals: 1 hundred is 1 PART 5 slideshow may be helpful. It walks you through the process of using the “original ones” in base ten blocks as “hundredths”. The process and questions are almost identical to the tenths versions.

Use a “Bottoms Up” hundredths chart rather than a regular hundreds chart to look for patterns. This article explores the concept of a “Bottoms Up” hundreds chart.

Other Resources:

> You Tube Videos

• Math Antics – Decimal Place Value

Formative / Summative Assessment:

• What is the relationship between hundredths, tenths and ones?
• What are the similarities and differences between tenths and tens?
• What are the similarities and differences between hundredths and hundreds?
• What patterns are there in how numbers are named and represented symbolically?
• How does changing the order of the digits in a number affect the value of the number?
• How are the 3’s in the number 3 333 related to each other?
• How is an understanding place value helpful when solving addition, subtraction, multiplication and/or division questions?

Moving Forward:

• Make connections to place value when working on 2/3 digit by 1 digit multiplication. An excellent high leverage strategy focuses on the area model as it connects from 2/3 digit x 1 digit multiplication to 2 digit x 2 digit to decimal multiplication to multiplication of binomials.

Grade 5

CURRENT Curricular Outcomes:

• Number 1 Represent and describe whole numbers to 1 000 000.
• Number 5 Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.
• Number 6: Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.
• Number 8 Describe and represent decimals (tenths, hundredths, thousandths), concretely, pictorially and symbolically
• Number 10 Compare and order decimals (to thousandths) by using
• Benchmark
• Place value
• Equivalent decimals
• Number 11 Demonstrate an understanding of addition and subtraction of decimals (limited to thousandths).
• Patterns and Relations 1 Determine the pattern rule to make predictions about subsequent elements
• Shape and Space Demonstrate an understanding of measuring length (mm) by
• Selecting and justifying referents for the unit mm
• Modelling and describing the relationship between mm and cm units, and between mm and m units.

Process of Complexity:

Connect place value from 10 000 (grade 4) to 100 000, then to 1 000 000. Connect place value to tenths. Connect place value to hundredths. Connect place value to thousandths.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Digits
• Sets of ten, one hundred, one thousand, ten thousand, one hundred thousand, one million
• Value
• Place
• Place Value, Base Value, Face Value
• Equivalent
• Referent
• Standard form/notation
• Non-standard form
• Expanded Notation
• Position 
• Decimals
• Tenths, Hundredths, Thousandths

Pre-Assessment:

> Administer MIPI+ Question #2

•  Find out more about administering and assessing the MIPI+ Question #2.

> Student who do not answer MIPI+ Question #2 correctly:

• Administer the Grade 4 MIPI+ question #2 in order to determine if the misconceptions are limited to decimals or if they also occur with whole numbers.
• If students struggle with both, Introduce and explore the Grade 2, 3 and 4 high leverage strategy.
• If students struggle with just the decimals, introduce and explore the Grade 4 high leverage strategy for decimals.

The Process:

This high leverage strategy will briefly address numbers within 1 000 000 by connecting to the grade 2, 3 and 4 strategy. If students do not understand place value appropriately, go back to the grade 2, 3 and 4 strategy and begin there.

This high leverage strategy will mainly focus on place value of decimals.

Strategy Part 0: Concrete 

> Connecting 5 digit numbers to 4 digit numbers

• Provide students with a single 1 of each of the following base ten blocks: 1, 10, 100. Have lots of hundreds ready to be used later. Have some (10 if possible) thousands, out of sight, ready to be used later. “Life Size Printable base ten blocks-ones, tens, hundreds” and Net of a 1000 cube
• Ask students “What relationships are there between the blocks? What patterns do we know?” Ensure that students state that there are 10 ones in 1 ten. 10 tens in 1 hundred. 100 ones in 1 hundred. 1 ten is 10 times as many as 1 one. Etc.
• Display a thousand block. Ask students to describe the number. (1000 – 10 hundreds, 100 tens, 1000 ones)
• Ask, “How much space would 100 thousands take up on your desk? What does 100 thousands look like?” Give students time to think about that. Think pair share.
• Ask, “What is another way to say 100 thousands?” (100 000)
• Say, let’s lay out 100 thousands on the floor. It is helpful to lay them out as a 10 by 10 array, counting out 1000 of them as you build. Then, skip counting by thousands as you count them afterwards. 
• Ask, “How long would this be if we lined them up in a straight line if we started with the first one here…?” Let students line up according to where they think the 100th base ten block would go when you line them up in a row. (If you don’t have enough, discuss how you can lay down as many as possible and then borrow some from the middle to continue the line. Just don’t remove the first or last ones in the line.)
• Ask, “How many times bigger is 100 thousands than 10 thousands? (10 times) How do you know? Because 100 is ten times bigger than 10.
• Ask, “How many times bigger is 100 000 than 10 000? (10 times). 
• Draw a number line on the board that is about 1 m long. Locate 0 and 1 billion on the number line. Ask students, “Where does 1 milion fit on this number line?” Give students time to think pair share. Discuss strategies.

• Possible strategies:
• 1 million is 1/1000 of 1 billion. Imagine cutting that number line between 0 and 1 billion into 1000 pieces. 1 million would be the first piece.
• 1 billion is 1000 million. Half way on the number line would be 500 million. Halfway between 0 and 500 million would be 250 million. Halfway between 0 and 250 million would be 125 million. Keep dividing in half until you get as close as you can to 1 million…or run out of room.

Strategy Part 1/2/3: Concrete/Pictorial/Symbolic

> Connect the concrete representations to visuals and symbolic. 

As students are building on their place value knowledge from Grade 4, you will be able to move away from concrete whenever students are ready. Regardless, start with the concrete. Give students the base ten blocks to help them visualize the size of numbers.

• Finding patterns in the place value chart to make predictions about thousandths.
• Use the Place Value of Decimals: 1 thousand is 1 PART 1 slideshow to discuss the questions below.
• What is the relationship between each column?  Give students time to think pair share. You can click through examples on the screen. It’s important that students understand the last statement “EVERYTHING is related to ones!” We are always representing as it connects to the total number of ones.
• What is the pattern as you move from the right to left? Students think pair share. There is a 3 in every position but every 3 has a different value based on its position. The value increases by 10 times as you increase the location to the left. Click through each example as you discuss. Again, reinforce the idea that “EVERYTHING is related to the number of ones.”
• What is the pattern as you move from the left to right? Students think pair share. There is a 3 in every position but every 3 has a different value based on its position. The value decreases by 10 times as you decrease the location to the right. Click through each example as you discuss. Again, reinforce the idea that “EVERYTHING is related to the number of ones.”
• Predict the number to the left of the 300 000: Students think pair share. Some may say, “Just add another 0.” If they do, ask, “what does it mean to add another 0? What’s happening to the value of the number?” Students need to understand that they are multiplying the previous value by 10 NOT adding a 0.
• Predict the number to the right of the 0.03: Students think pair share. Some students may want to add a 0 to the front. Ie. 0003. If they do, ask what this number really is. 3. Ask, “What is the pattern as we move from the left to the right?” (Divide by 10). Say, “Tell me in words how we use that to move from 300 to 30.” (300 divided by 10) “30 to 3” (30 divided by 10), Then we have 3 divided by 10 which is 3 tenths or 0.3. Then we have 0.3 divided by 10. We could also look at that as 3 divided by 100 which is 3 hundredths or 0.03. What will the next stage be? (3 divided by one thousand OR 0.03 divided by 10)  
• Ask, “How can we show 3 divided by 1000?” (3/1000). 
• Ask, “How can that be represented as a decimal? (0.003)
• Introduce the 1000 base ten blocks. This Place Value of Decimals: 1 thousand is 1 PART 2 slideshow may be helpful in the process for renaming the blocks. Students use base ten blocks while the teacher displays on the board. You might give 1 one thousand block to each pair of student.
• This is now 1 (The Whole): Say, “In the past, we have named this block as 1000. Why?” (Because there are 1000 of the small unit blocks in it). Say, “We chose to name it 1000 because it was helpful when we were talking about whole numbers. Today, we are going to change that and call this 1.”
• What number is this? (2)
• How many of these…do you need to make the whole? Ask students to answer think pair share. (10)
• There is 1 of these out of the 10 needed to make the whole. Ask, “How many do I have?” (1) 
• Say “I have 1 of these out of the 10 needed to make the whole. How else can I say this?” (one tenth)
• Say, “We would write this as 1 tenth.”
• Hold up three one-tenths: 
• Display the answer.
• Say, “we have 3 out of the 10 we need to make 1 whole.”
• Say, “that’s 3 out of”
• Say, “the 10 I need to make the whole”.
• Hold up five tenths: Click and read out loud once students respond.
• Show me 12 tenths: Click and read. Do NOT attempt to change 12/10 to 1 and 2 tenths on this screen.
• How many tenths to make a whole? Ask “How many tenths to make a whole?”. (10) 
• Say, “We know that 10 tenths = 1 whole.” 
• Say, “I have ten tenths and two tenths.”
• Say, “which I could write like this.”
• Say, “I could also say that I have 1 whole and 2 tenths.” 
• How many of these…do you need to make the whole? Ask students to answer think pair share. (100)
• There is 1 of these out of the 100 needed to make the whole. 
• Say “I have 1 out of the 100 needed to make the whole.”
• Ask, “How else can I say this?” (one hundredth)
• Hold up two one-hundredths: 
• Display the answer.
• Say, “we have 2 out of the 100 we need to make 1 whole.”
• Hold up six hundredths: Click and read out loud once students respond.
• How are the values of the five’s related? Students think pair share. (5 tenths is 10 times as many as 5 hundredths)
• Show me 12 hundredths: 
• How do these both represent twelve hundredths? Students think pair share. (1 tenth is the same as 10 hundredths)
• How many of these…do you need to make the whole? Ask students to answer think pair share. (1000)
• There is 1 of these out of the 1000 needed to make the whole. 
• Ask, “What might the fraction look like?” (1/1000)
• Say “I can write that as 1 out of the 1000 I need to make the whole. How else can I say this?” (one thousandth)
• Hold up five one-thousandths: 
• Display the answer.
• Say, “we have 5 out of the 1000 we need to make 1 whole.”
• Say, “that’s 5 thousandths”
• Hold up three thousandths: Click and read out loud once students respond.
• How are the values of the three’s related? 
• Show me 12 thousandths: Click and read. 
• Ask, “How many of you built it using just thousandths?”
• Ask, “How many of you built it using 1 hundredth and 2 thousandths?”
• Ask, “How does the information in the place value show both representations?”
• Show one hundred twenty-three thousandths. 
• Click through the different representations.
• Can you see…in this?
• Explore each representation on the following slides that students built and have them discuss how each decimal number represents what was built.
• Write the following as fractions and decimals.
• Have students work through and discuss each statement and write as a fraction and a decimal. Notice the different between the last two!
• How many thousandths/hundredths/tenths in 2.435? Students think pair share. (2435/243/24 tenths)
• If I build the number 0.132 as…Students think pair share. (same as the starting value) 
• Watch for students who state that the new value is larger because you have more blocks. These students don’t understand that 1 hundredth = 10 thousandths
• Watch for students who state that the new value is larger because thousands are bigger than hundreds OR thousandths are bigger than hundredths. These students are basing their understanding of place value of whole numbers where thousands are bigger than hundreds.
• Watch for students who say the new value is smaller because thousandths are smaller than hundredths. These students don’t understand that 1 hundredth = 10 thousandths
• What is the relationship between thousandths, hundredths, tenths and ones. Students think pair share or use as a formative assessment.
• Number lines: These questions have students think about where thousandths fit on a number line.
• More than 1, less than 1 or equal to 1? Students think pair share a response to each question. Do not solve!

Formative / Summative Assessment:

• What is the relationship between thousandths, hundredths, tenths and ones? 
• What are the similarities and differences between thousands and thousandths?
• What is the connection between the number line and a meter stick?
• What patterns are there in how numbers are named and represented symbolically?
• How does the pattern of the place value system, i.e., the repetition of ones, tens and hundreds within each period, make it possible to read and write numerals for numbers of any magnitude.
• How does changing the order of the digits in a number affect the value of the number?
• How are the 3’s in the number 333 333 related to each other?
• How is an understanding place value helpful when solving addition, subtraction, multiplication and/or division questions?

Moving Forward:

• Make connections to place value when working on 2 digit by 2 digit multiplication. An excellent high leverage strategy focuses on the area model as it connects from 2/3 digit x 1 digit multiplication to 2 digit x 2 digit to decimal multiplication to multiplication of binomials.
• Explore these same concepts when discussing the relationships between mm and cm units, and between mm and m units.

 Grade 6

CURRENT Curricular Outcomes:

• Number 1 Demonstrate an understanding of place value, including numbers that are
• Greater than one million
• Less than one thousandth
• Number 2 Solve problems involving whole numbers and decimal numbers
• Number 8 Demonstrate an understanding of multiplication and division of decimals (1-digit whole number multipliers and 1-digit natural number divisors)
• Patterns and Relations 1 Represent and describe patterns and relationships, using graphs and tables

Process of Complexity:

Students are building upon their understanding of place value explored during previous grades. This strategy explores place value in different bases in order to reinforce the concept of place value in base 10.  Implement James Tanton’s “Exploding Dots” to explore any of the curricular outcomes listed above to whatever extent meets the needs of your students.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Value
• Place
• Place Value
• Decimal
• Tenths
• Hundredths
• Thousandths

> Students who do not answer MIPI+ Question #2 correctly:

• Administer the Grade 4 MIPI+ question #2 in order to determine if the misconceptions are limited to decimals or if they also occur with whole numbers.
• If students struggle with both, Introduce and explore the Grade 2, 3, 4 and 5 high leverage strategy.
• If students struggle with just the decimals, introduce and explore the Grade 4 and 5 high leverage strategy for decimals.

The Process:

Strategy Part 1: Concrete

> Note

• At this stage, students should be able to extend their understanding of place value into numbers that are greater than one million and less than one thousandth. Concrete representations of place value should not be necessary if students, through MIPI+ question 2 or other assessments demonstrate understanding around place value. At any time, concrete representations can, and should, be explored with students.

Strategy Part 2: Pictorial

> Exploding Dots by James Tanton

Visit http://explodingdots.org to learn about Exploding Dots. It is completely free though you should complete the Teacher Registration. You can work through the “islands”, which include videos of James walking you through the learning, if you are looking for an online experience or you can access the “no tech” version.

Strategy Part 3: Symbolic

> Reinforce Place Value Concepts

• Whenever appropriate, reinforce the concept that each position in a number has a value of 10 times more than the position on the right and 10 times less than the position on the left. 

> Important!

• Never reference “moving the decimal” when multiplying or dividing by 10. “Moving the decimal” is not a mathematical concept.
• When multiplying decimals, it is not about removing the decimal, multiplying as if they are whole numbers, and then counting the total number of digits after the decimal places. This is not a mathematical concept. A high leverage strategy for exploring the concept of multiplication as it relates to decimals will be explored. Here is a quick activity you can do with students. “I solved 2.31 x 3 and ended up with 693 as the answer but I forgot to place my decimal. Where would the decimal go?” Knowing that 2 x 3 = 6 tells me my answer should be about 6. Therefore, the actual answer must be 6.93. This is a perfect opportunity for estimation to be practiced in an appropriate setting.

Formative / Summative Assessment:

• What patterns are there in how numbers are named and represented symbolically?
• How does the pattern of the place value system, i.e., the repetition of ones, tens and hundreds within each period, make it possible to read and write numerals for numbers of any magnitude.
• How does changing the order of the digits in a number affect the value of the number?
• How are the 3’s in the number 333 333 333 related to each other?
• How is an understanding place value helpful when solving addition, subtraction, multiplication and/or division questions?

Moving Forward:

• Make connections to place value when working on multiplication of decimals. An excellent high leverage strategy focuses on the area model as it connects from 2/3 digit x 1 digit multiplication to 2 digit x 2 digit to decimal multiplication to multiplication of binomials.

Grade 7

CURRENT Curricular Outcomes:

• Number 5 Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretey, pictorially and symbolically (limited to positive sums and differences).
• Patterns and Relations 3 Demonstrate an understanding of preservation of equality by
• Modeling preservation of equality, concretely, pictorially and symbolically
• Applying preservation of equality to solve equations

Process of Complexity:

Begin with like denominators of 10, 5 and then others. Continue with unlike denominators of 5 and 10, 2 and 10, and then others that are multiples of each other (ie. 3 and 6). Continue with numbers less than 8 that are not multiples of each other (ie. 3 and 4).

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Value
• Place
• Place Value
• Denominator
• Numerator
• Mixed Fraction
• Improper Fraction

> Student who do not answer MIPI+ Question #2 correctly:

• Administer the Grade 4 MIPI+ question #2 in order to determine if the misconceptions are limited to decimals or if they also occur with whole numbers.
• If students struggle with both, Introduce and explore the Grade 2, 3, 4 and 5 high leverage strategy.
• If students struggle with just the decimals, introduce and explore the Grade 4 and 5 high leverage strategy for decimals.

The Process:

Strategy Part 0: Symbolic for review

> Skip Counting by Fractions

• Write ¼ on the board.
• Say, “We are going to skip count by quarters. On your mini-whiteboards, write down the fractions you would say when skip counting by quarters. Keep going until I say stop.” Give students about 45 seconds to write as many as you can.
• Say, “Compare your list to your partner’s. How are they the same, similar, different?” Give students a short amount of time to compare and discuss. As a class, discuss. Did anyone write 4/4? How many wrote 1 instead? How are they the same, similar, different?
• Display the fractions bottoms up chart in improper form (page 1). Have students look for patterns. What do they notice?
• Repeat using mixed form (page 2).
• Ask, “How many quarters do we need to make one whole?” 4 quarters

   

Strategy Part 1/2/3: Concrete/Pictorial/Symbolic

> Making a whole / one full set

• Use the Fractions – Ten frames google slides document to 
• Explore naming and adding fractions with a denominator of 10.
• Explore renaming the ten frame as 1 instead of 10. 
• Play a simple Memory Match game that you can first introduce online and then students can play using the paper version. The paper version should be printed on either cardstock or paper that you have made dark on one side so that students can’t see through. It is recommended that you photocopy on several different colors of paper so that when pieces are found on the floor, you have a better chance of determining which collection they go to. Printing off one page will give you 3 sets of cards.
• Using the same ten frame papers, students create pairs that make exactly 1, less than 1, and more than 1. Discuss – How can you check they are exactly, less or more than 1 without counting each dot? Collate responses on the board into the three categories as students share answers. Write out the equation for each pairing, using the improper form to start. Students look for patterns within and across categories. For each equation, ask, “How many full sets? How many extra? How do we represent this?” Rewrite in mixed form.
• Continue the work with a box of 10 crayons representing the whole.
• Solve symbolic representations with denominators of 10. Explore both representations of improper and mixed fractions.
• Make connections to denominators of 5 and then denominators of student’s choice.
• Puzzler: Applying understanding to fractions with unlike denominators – 10 and 5
• Give students the puzzler and time to think pair share. What is the relationship between the two images? How is adding fractions with denominators of 10 and 5 the same as, similar to and different from adding fractions when both the denominators are 10.
• Use the other puzzlers to help students connect their understanding to adding fractions with unlike denominators
• 3 and 6
• 3 and 2

> Making Full Sets  

• The process above can be used to explore this concept using any types of manipulatives such as fraction circles, fraction tiles, pattern blocks (especially using pink double hexagons and black chevrons so you can work up to twelfths), etc.

Formative / Summative Assessment:

What is the relationship between the digits 1, 2 and 3 in the fraction 1⅔?

Grade 8

CURRENT Curricular Outcomes:

• Number 3 Demonstrate an understanding of percents greater than or equal to 0%, including greater than 100%
• Patterns and Relations 3 Demonstrate an understanding of preservation of equality by:
• Modelling preservation of equality, concretely, pictorially and symbolically
• Applying preservation of equality to solve equations

Process of Complexity:

Percents less than 100. Percentage off values (30% off of $50). Percents more than 100.

Key Vocabulary:

The following key vocabulary should be used with and by students. Words in italics are not expected at this grade level. They are included here as a support for the teacher. Key Vocabulary will be italicized every time it is used below in order to draw attention to its introduction and use.   

• Hundredths
• Per cent
• cent

Pre-Assessment:

> Administer MIPI+ Question #2

•  Find out more about administering and assessing the MIPI+ Question #2.

The Process:

This high leverage strategy will connect students’ previous work with base ten blocks and place value.

Strategy Part 1: Concrete 

> Connect concrete representations to previous learning.

• Note: If students are struggling to rename the base ten blocks, this process may be helpful: Introducing the Place Value of Decimals: 1 hundred is 1 PART 2 slideshow.
• Note: Depending on students’ level of understanding, this process may vary is length of time needed.
• Note: Once students have renamed the base ten blocks and can explain the connection to decimals, you can provide students with pictorial representations of the whole that students can write on and shade in. 
• “Introducing the Place Value of Percents”: Students use base ten blocks while the teacher displays on the board.
• Slide 3: 
• Provide students with 1 minute to find as many words as they can think of that have the word “cent” in them. As a class share the words.
• Ask, “What does “cent” mean?” Ensure that everyone understands that cent means hundred.
• Slide 4:
• Give students a moment to decide if they agree or disagree with the statement: “Percent” is another name for hundredths. Have them vote and then share their reasoning.
• Slide 5: 
• Ask, “What does 13% really mean?”
• 13 per hundred
• 13/100
• 0.13
• Discuss the idea that “per cent” means “per hundred” which is really all about hundredths.
• Slide 6:
• Explain that Base Ten Blocks are manipulatives and that we choose to assign them a value. They don’t intrinsically have a specific value. In the past, they may have named them as 1, 10, 100 or 1, 0.1, 0.01, etc. I could make the “100” worth 500, or 3, or 0.2. I choose its value. Hand out base ten blocks.
• Slide 7: 
• Say, “Today, we will now give this base ten block (hold up the “hundred”) a value of 100% – the whole.
• Slide 8: 
• Ask, “How many hundredths are there in this whole 100%?” Give students time to decide, discuss with a partner and explore as a class. (100 hundredths)
• Slide 9:
• Ask, “How else could you write 100%?”
• 100 per hundred
• 100/100
• 1.00 (and others as students share)
• Slide 10:
• Hold up 100% and say, “If this is 100%, what would 50% look like?” Give students time to discuss.
• Students may have covered up half of their hundred.
• Students may have pulled out 5 x 10% (the original tens blocks)
• Ask, “Which one is correct?” (They both are.)
• Ask, “What is the percent value for each of these? (Hold up the original tens blocks) (10% each)
• Slide 11:
• “How else could you write 50%?” 50 per hundred, 50/100, 0.50 etc.
• Slide 12:
• Say, “If this is 100%”, Hold up 100%, “and this is 10%”, Hold up the 10%, what does 1% look like? Why?”
• Slide 13:
• Ask, “How else could you write 1%?” 1 per hundred, 1/100, 0.01, etc.
• Slide 14:
• Have students “show you 3%” (They should hold up 3×1%)
• Slide 15: 
• Ask, “How could you represent 3%?” 3 per hundred, 3/300, 0.03, etc.
• Slide 16:
• Say, “Using your base ten blocks, show me 15%”
• Say, “Some of you showed it using 15 hundredths.”
• Say, “Some of you showed it using 1 tenth and 5 hundredths. Which way is correct?” (They both are)
• Slide 17:
• Ask, “How else might you write 15%?” 15 per hundred, 15/100, 0.15, etc.
• Slide 18:
• Say, “Choose an interesting percent value. Build it and write it as many different ways as you can.”
• Slide 19:
• Say, “This 100% represents some students. What do you notice? What do you wonder?” (Students may ask questions like – I wonder how many students, etc.) Discuss.  (Each square represents one student. 1% of 100 students = 1 student.)
• Slide 20:
• Say, “Now we know that 10% of the students have a dog. Show me 10%.” (Students may cover up all but 10 of the “hundredths”, or hold up a tenth)
• Slide 21:
• Say, “Now we know that this 100% actually represents 100 students. How many students have a dog? Convince me?” Discuss. (Questions like – what does each hundredth represent?) (10% of 100 students is 10 students.)
• Slide 22:
• Say, “Now the 100% represents 200 students. What do you notice? What do you wonder?” Discuss. (Each square represents 2 students. 1% of 200 students = 2 students)
• Slide 23: 
• Say, “Show me 10%. Has this changed from the previous question when it was only 100 students?” Discuss. 
• “It’s still 10% of the students who have a dog. How does that change our answer to the question – how many students have a dog? Convince me.”
• Solution: 20
• Slide 24:
• Say, “If this 100% represents money, what would 33% look like?” Discuss
• Slide 25:
• Say, “If this 100% represents $400, what do you notice? Wonder?” Discuss. (Each 1% has a value of $4)
• Slide 26:
• Ask, “How much is 33% of 400? Convince me.”
• Solution: 4 x 33 = 132
• Slide 27:
• Say, “If this 100% represents 450, what is 20%? Convince me.”
• Solution: 90
• 450 / 100 = 4.5; 4.5 x 20 = 90
• 450 / 10 = 45; 45 x 2 = 90
• Slide 28:
• Say, “If this 100% represents $25, what is 20% of $25?”
• Solution: $5
• 25 / 100 = 0.25; 0.25 * 20 = $5
• 25/5 = $5
• 25/10=2.5; 2.5*2 = $5
• Slide 29:
• Say, “These are all the equations we just explored. What do you notice? Wonder? Describe the pattern you see? Can you break the pattern? (ie. is there are situation where the pattern doesn’t work?”
• Slide 30:
• Ask, “How do you see solving 30% of 70? Which of these examples best matches what you see in your head?”
• Ask, “How are these strategies the same? Similar? Different?”
• Ask, “Why do these all give the same answer? Is this a special case or will it always work?”
• Slide 31:
• Say, “This is a puzzler. A puzzler is a bit more challenging. A puzzler is going to make you think. Because it’s a puzzler, you only have 2 minutes to see what you can figure out. What is 120% of 200?”
• Slide 32:
• This slide can be used during discussion to help explore the strategies used.
• Slide 33:
• Give students time to solve the question: “Pat sees a shirt for $70…but it is now on sale! It is now 20% off. How much does Pat pay for the shirt? (before tax)
• Slide 34:
• 10% + 5% = 15% Do you agree or disagree? 
• Note: students may
• Agree – if you are calculating something like 15% of total can be done as 10% of total + 5% of total.
• Disagree – if you are calculating specific types of discounts. Ie. 10% off and then 5% off the sale price.
• Slide 35:
• Provide students time to reason out: “If each square, within the 100% is worth 5, what is the total?”
• Solution: 5 x 100 = 500
• Slide 36:
• Provide students time to reason out: “If 4 squares are worth 8, what is the total?”
• Solution: 200; Did you figure out 
• How many groups of 4 squares are there? 25; 25 x value of 8 per group = 200
• What is 1 square worth? 2; 2 per square x 100 squares = 200
• Slide 37:
• Provide students time to reason out: “If 10% of the total is 30, what is the total?”
• Solution: 300; Did you figure out
• the value of 1 square? 30/10 = 3; multiply 3 x 100 square = 300
• how many groups of 10 there are? 10; Multiply 10 group x 30 in each group = 300
• Slide 38:
• Provide students time to reason out: “If 20% of the number i2 5, what is the number?”
• Solution: 25; Did you figure out
• the value of 1 square? 5/20 = 0.25; multiply 0.25 x 100 square = 25
• how many groups of 20 there are? 5; Multiply 5 groups x 5 in each group = 25

• Slide 39:
• Provide students time to reason out: “If 200% of the total is 30, what is the total?”
• Solution: 15; Did you figure out
• the value of 1 square? 30/200 = 0.15; multiply 0.15 x 100 squares = 15
• how many groups of 10 there are? 10; Multiply 10 group x 30 in each group = 300
• The value of  100%? 15; and then one of the above strategies
• Slide 40:
• Say, “These are all the equations we just explored. What do you notice? Wonder? Describe the pattern you see? Can you break the pattern? (ie. is there a situation where the pattern doesn’t work?” 

Strategy Part 2: Pictorial

> Connect to number lines.

• Use the “double clothesline” activity described in “Equality” to explore percents on a number line. (This section will be updated with a specific process)
• Simple example: 
• Create the two number lines. 
• One represents percents (0 ⇒ 1). You may choose to begin with 0% to 100% but you have to be careful about accidentally building the misconception that 50% is equivalent to 50.
• The other represents the main “number”. 
• Ask, “Where would I place 50%? (Halfway between 0 and 1)
• Write “200” on the non percent number line, matching to the 1.
• Say, “If 200 is the whole, what is 50%?” (100 – halfway)
• Ask, “Where would 25% go? (halfway between 0% and 50%)
• Ask, “What is 25% of 200?”)
• Ask, “What other percents might be easy to figure out?” (10%, 75%, 5%, 20%, etc.)
• As you identify each “easy” percent, find the equivalent value out of 200.
• Ask, “Where would 110% go on this number line? (after the 100%)
• Ask, “How could I use what I’ve already done to figure out 110%?” (Add 100% and 10%)
• Repeat for other numbers that are greater than 100 that
• Work out nicely (300, 400, etc.)
• Will create decimal answers (125, 350, etc)
• Repeat for other numbers that are less than 100 (ie. 50, 25, 10, 5, etc.).

Strategy Part 3: Symbolic

> Connect the concrete and visuals to symbolic forms.

• When using the base ten blocks to represent 35% of 300, what did you do first? (ie. find how much 1 block/1% is worth: 3). What did you do second? (ie. multiplied that by 35: 105; or multiplied that by 5: 15 and by 30: 90)
• When using the double clothesline/number line, what did you do first? Second? Etc.
• How can we use that to create an algorithm/process for solving 35% of 300 or any other percent by any number.

Formative / Summative Assessment:

• What is the relationship between percents and hundredths?
• How can you use what you know about multiplying whole numbers to help you with multiplying by percentages?
• When you have to multiply by a “trickier’ percent (ie. 37%), how can you use what you know about multiplication in order to make it easier to solve? (Students should connect to distributive property: 30% and 7%, 30% and 5% and 2%, etc.)

Grade 9