Lesson Map / Anticipated Student Responses / ü Evaluations
v Prompts/Probes
Students should be seated in 6 groups. Each student should have a white board and dry erase marker. Student will write answers on the white board. The manipulative is Cuisenaire Rods. (Cuisenaire Rods include unit pieces that are labeled “white” however, they may appear tan.)
Tell students:
You have been working so hard the last couple of weeks figuring out the mysteries in math. Since you are such good problem solvers, I have some more mysteries for you today!
Give each student a bag with 2 tan, 3 orange, 1 red and a light green.
Say:
In your bag, you will find some manipulatives. Work by yourself.
Pose and post the following:
What is the value of this amount? Write it on your white board and explain how you know. / § 37 (orange is ten, tan is one, red is 2, green is 3)
§ Build with rods various numbers (411, 114, 141, 41, 911, etc.
§ 3 tens (rods), 7 ones (units)
§ 1 and 1/4 (putting all orange together as the total)
§ 3.7
§ 3 7/10
§ 3 2/5
§ “What is one?”
§ 3.222 / v “What do you think?”
v “Whatever you think, as long as you can justify it.”
The teacher allows several students to present answers to the whole group and posts student solutions below the question. The following prompts might be used to clarify student thinking.
· How do you know?
· One what?
· Show me.
· Prove it.
· Why do you think that?
· Does this remind you of something you’ve seen before?
· How do you know this is 2 tens?
· What made you think of this quantity?
Begin with any student who built digits with rods.
Move onto whole number presentations and progress to fraction presentations.
If all students come up with the same answer (3.7), present them with another possibility (for example, 37). Ask students how this person came up with that number as an answer. / Students should justify their thinking. / This will be a short discussion. Prompt students to use precise vocabulary when presenting.
v Does this represent the value?
ü Do the students recognize the importance of defining one as the basic unit?
Show students the orange.
(If students came up with multiple answers ask the next sentence, if not, go to the 2nd sentence. If all students answered 3.7, skip to the next section.) You came up with multiple answers. I wonder if your thinking would change if I told you the value of this orange piece is one.
Allow silence for wondering.
Let’s try it. This is now one. (Hold up the orange rod.) Since the orange is one, write the value of the quantity in your bag and your reasoning for believing that. Post assignment and tell students to find the value and record on their whiteboard with their reasoning.
Allow students to share solutions and their thinking. As you observe and listen to student thinking, select a student to present the solution. Student will bring their white board to the front to post. / § Students may recount pieces.
§ “I don’t get it.”
§ “How can this be one?”
§ 37
§ 3.7 (because white is .1, green is .3, red is .2)
§ 3 7/10
§ green =¼, red=1/8, tan=2/16, and orange= 3
§ orange =3 ones, green=3/10, red=2/10 (or 1/5), tan=1/10, so 3 7/10
§ orange= 3 ones, green = 1/3, red = 1/4,tan= 1/8 / v Acknowledge earlier student thinking. 37 what ? . (Show pieces). Can this (the tan) be one if this is one (hold up orange)?
v Hold up orange and say, Can I build one with these pieces (tans)? How? How many did it take? (ten) What is the value of this piece (tan) now since this is one (orange)?
v Now that you have figured that out what is the value of this? (Hold up red.)
v Conversation can be repeated by holding up the rod if the student struggled to identify the value of the other colors.
v Encourage students to record on their white boards in decimals, not just fractions. Ask students who record as a fraction for another way to record this quantity.
Ask,
Did you change your mind? What made you change your mind about the value of the blocks? Why?
Ask,
How is this thinking different from the first solutions shared?
Through questioning or emphasizing student comments, the following key point should be made and highlighted during the discussion.
· It is important to identify one.
· We know 2 tans is 2 tenths because it takes 10 of these (tans) to make one.
· The red is 0.2 because it is equivalent to 2 tans (tenths).
· The green is 0.3 because it is equivalent to 3 tans (tenths).
§ The total amount is 3.7. / § Yes, because we knew what one was this time!
§ It is important to identify one.
§ We know 2 tans is 2 tenths because it takes 10 of these (tans) to make one.
§ The red is 0.2 because it is equivalent to 2 tans (tenths).
§ The green is 0.3 because it is equivalent to 3 tans (tenths).
§ The total amount is 3.7.
§ No, I was already thinking that.
§ In the beginning, we used different pieces as the one whole. In this challenge, we knew the orange was one.
§ Now we should have the same answer. / ü Teachers should listen for students who have misconceptions about the place value and identifying 1.
ü Do the student conversations justify student decisions?
ü Does this conversation compare the previous thinking to the solutions when the orange is 1?
Say,
That was too easy. Let’s try a new challenge.
Remember when you figured out the mystery amounts in the bags? I have more mystery pieces for you to figure out. I’m going to give you and your table mates a bag with a number on it. Using the orange pieces, you will build your number. You will also find a tan piece and mystery pieces all of the same color. Remember, the value of the orange at this time is one. Since this is the case, what is the value of the tan piece?
Post and tell students:
Working with your group, your challenge is to find the value of one mystery piece. You also need to find out how many mystery pieces it takes to make your assigned whole number written on the bag.
Challenge: Find the value of one mystery piece. Determine how many mystery pieces it takes to make the assigned whole number written on the bag.
Distribute one of the following bags to each group.
Dark Green bag: makes 6 and contains 15 green, 6 oranges, 1 tan
Yellow Bag: makes 5 and contains 15 yellow pieces, 5 oranges, 1 tan
Purple Bag: makes 4 and contains 15 purple pieces, 4 oranges, 1 tan
Black Bag: makes 7 and contains 15 black pieces, 7 oranges, 1 tan
Brown Bag: makes 8 and contains 15 brown pieces, 8 oranges, 1 tan
Blue Bag: makes 9 and contains 15 blue pieces,9 oranges and 1 tan / 1/10 or .1
§ Students may be confused on where to start.
§ Students may try to use all of the mystery pieces. / ü Do the students understand the assignment?
v What is one?
v What can you use to make your number?
v What is your number? Show me how you represent your number. Is your line of mystery pieces the same length as your number?
v How can lining the mystery pieces with the orange help you find their value? Is it more or less than one? One half? Why?
v Does the tan piece help you decide?
v Can you show me?
Students record in their journals [or whiteboards?] the value of their mystery piece and how many of these it took to make their number. / ü How do the student determine the value of the mystery piece? (measure with 0.1, estimate, ?)
ü What language do students use?
Show and Tell
Each group presents their results with magnetic pieces. They explain the value of the mystery piece, how they determined it, and how many mystery pieces it took to build the whole number on their bag (oranges).
Do not allow the groups to present in sequential order.
As students present their findings, the teacher records the results on a class chart.
After each mystery piece is presented, ask if this is the total, what is one mystery piece of this total? Why?
Total / Value of Mystery Piece / Number of Mystery Pieces Needed
8 / .8 / 10
6 / .6 / 10
5 / .5 / 10
4 / .4 / 10
9 / .9 / 10
7 / .7 / 10
What patterns do you notice on the chart? Allow students to share any patterns they see.
Beyond Show and Tell
Post a new chart. Tell students,
I saw some of the same patterns you did! Let’s see if we’re right. (Post the following patterns and discuss one at a time.)
Patterns / Proof
It takes 10 groups of the mystery piece to make the total.
Ask students, how can we prove this is true?
Allow students to talk to his/her partner about how they could prove this.
Ask a student for his/her idea.
Ask students, how could we record this?
Record in the chart 10 x .8 = 8
10 x .6 = 6, etc
Continue with the next pattern.
Patterns (cont’d) / Proof
1/10 of the total is the value of the mystery piece
Ask, how can we prove this pattern? Talk to your table mates and find out their ideas.
What part of the total is the mystery piece? How do you know?
Let’s record that.
The teacher records on the board 1/10 of 8 = .8
Repeat for each mystery piece.
I wonder if this is true for any amount of tenths and whole number less than ten? What haven’t we tried?
Let’s try one. Here is 3 (post magnetic strip on board), how many light green mystery pieces should it take if this is true?
Let’s see if it works (post magnetic strip of ten light green pieces).
Let’s look at a red mystery piece. What would be the total if I had ten red pieces?
Why?
(If needed, can be repeated with one.)
So, these patterns are true for any tenths and whole number less than ten. / Students aren’t sure about the value of their mystery piece.
One-tenth because it took ten mystery pieces to make the total. And we are talking about one of them.
§ The value of the mystery pieces is always a decimal or in the tenths place.
§ The number of mystery pieces is always 10.
§ The whole number and the mystery piece use the same digit.
§ The total is always a whole number in the ones place.
§ It takes 10 of the mystery pieces to make the total.
§ If I cut the total (divide) into 10 pieces, I get the value of the mystery pieces.
§ Use multiplication.
§ Ten groups of the mystery number equal the total.
§ I don’t know.
§ Each mystery piece is one-tenth of the whole because it took ten mystery pieces to make the whole.
§ Students seem stuck or aren’t sure.
§ I don’t know.
§ We didn’t try the light green, tan, or red.
§ We didn’t have a bag with 1, 2, or 3.
Ten.
Two.
§ We haven’t used two.
§ Ten .2 makes 2. We know it’s .2 because two tans (tenths) make the red. / v Can you guess and check?
ü Can students show the group of ten mystery pieces equaling the total?
§ How do we show “groups of” in math? How many groups do we have? What is the value of each group?
ü Do the students seem to understand the meaning of “one-tenth”?
Closure
You have solved a mystery today! We need to summarize our learning.
When the digits are the same, 10 of the amount in the tenths place is the same as the amount in the ones place.
When the digits are the same, 1/10 of the amount in the ones place is the same as the amount in the tenths place.
Please write a reflection in your journal.
Board Plan for Research Lesson