Math Matters of San Joaquin County

Lincoln Unified School District - Lodi Unified School District - Stockton Unified School District

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Sixth Grade Head Problem

Some hints:

The power of Math Matters techniques to help you create a safe environment for learning for your more mature students should not be underestimated. Sixth grade students, although they are less likely to show it, still benefit from the support and reassurance that all efforts are appreciated, all answers are respected, and the thinking of every individual is valued.

Since the body of accumulated mathematical understanding sixth grade students are expected to have is now quite large, head problems provide an excellent way to remind students of earlier learning. At the same time, the teacher is able to assess quickly what the class remembers.

The math word wall is as important in sixth grade as it is in second, and all the grades in between. If you have ever found that students don't remember today what was taught yesterday, the math word wall is for you!

Head problems can help you cover some sophisticated mathematical ideas on a daily basis. Use previews to get them started, and the word wall to keep them going. Not every concept will work in a head problem, but it is surprising how many will.

You might find some useful ideas for review on the fourth and fifth grade head problem pages. (Use the "Back" button on your browser to return to this page.)

Here are some examples just to get you started.

PREVIEW:

• Put 4x and 3n and 5w on the board and invite students to tell you what they mean. If they know it represents multiplication skip the next step.

• If no one is sure, review quickly that mathematicians had to invent different ways to express multiplication to avoid confusion. Give them the example of 3 times (x) x. Then review the use of the dot, the use of parentheses, and the placement of a number (coefficient) and a letter (variable) next to each other as alternative ways to show the operation of multiplication.

• On the board write: w = 3s and s = 11.

HP:

1. In your head, think of the number w will be if s equals 11. (33)

2. Subtract the third prime number. (33 - 5 = 28)

3. When I say go, show me two sevenths of your answer on your fingers. (8)

DEBRIEF:

1. I'll take a quiet hand for the first thing I asked in the head problem.

2. Signals for what ___ said.

3. Now a quiet hand for what that number is.

4. And I see agreement for that.

5. I'll take a quiet hand for the next step in the head problem.

6. Signals?

7. And everyone, when I say go, tell me what number we have now.

8. And with a quiet hand who would like to say what we did next.

9. Signals for that?

Continue to end of problem.

The above problem is only 3 steps, but reviews three distinct areas of previous math curriculum. By the time you have finished debriefing, students will have remembered as much as possible. You could even try the same head problem later in the day, going a little faster. The debriefing should go considerably more quickly the second time.

If the math word wall is started right as soon as possible, it can support the students' memory by giving them a resource to refer to.

PREVIEW:

• Draw a circle with a diameter of 20 and ask students what the radius is as you point to half of the diameter.

HP:

1. In your head, think of the number of degrees in a straight line. (180)

2. Divide it by the radius of this circle (point to the circle). Whisper that number to your neighbor. (18)

3. Add to it -10. (8)

4. Show me the answer on your fingers.

DEBRIEF:

As above.

Notice the added Mode of Response in step 2 above. If you think all students might not understand a step, but some will, whisper to your neighbor gives everyone the opportunity to stay involved in the head problem. The next time you ask for a radius more students will know what you mean. Soon you will be able to say, "Think of the radius of a circle with a diameter of..." and all students will be with you.

PREVIEW:

• Write the number 0.1 on the board. Ask someone to read it. (Most often students will say, "zero point one." Explain how that would be a good way to tell someone how to write the number, but does anyone know how to read it.)

• Ask how the number might be expressed as a fraction. (1/10)

• Repeat with the number 0.25. (Correctly read: twenty-five hundredths. Expressed as a fraction: 25/100.)

• On the board, write the numbers 0.09 and 0.5. Hold up a cube for step 3 and brush the faces if students need the review.

HP:

1. In your head, think of the number that represents the larger piece of one whole. (0.5)

2. Now, in your head, express that number as a fraction. (1/2)

3. Multiply that number by the number of faces on a cube. (3)

4. When I say go, clap your answer.

DEBRIEF:

As above.

The above head problem could be varied by using one fraction and one decimal to compare for step one, or using 3 different numbers. A percent might be included.

Step 3 might ask them to think of half that number (a fraction of a fraction). Be sure to stick with fractions it is reasonable for them to picture in their heads. Of course, then you will want to change your Mode of Response in step four. You might have them show you the denominator on their fingers or ask for a quiet hand to share an answer.

PREVIEW:

• Write 2 on the board and invite students to tell you some multiples of 2. Write them as given, but leave spaces if they are given out of order, so several multiples can be shown.

• Repeat with 3.

• Ask students to find the least multiple that is the same for both 2 and 3. (Be sure multiples are carried out to 12 so there are 2 possible common multiples.) If students choose 12, accept that they are multiples, but ask if they are the smallest multiples that are the same. If they choose 6, ask why they wouldn't choose 12.

• Introduce the term Least Common Multiple as the mathematical phrase used to refer to that number. Add Least Common Multiple with an example to your math word wall. Leave multiples on the board.

HP:

1. In your head, think of the least common multiple of 2 and 3. (6)

2. Multiply that by the product of 5 and 4. (120)

3. Now think of one third of that number. (40)

4. I'll take a quiet hand from someone who would like to share their answer.

DEBRIEF:

As above.

PREVIEW:

• On the board, write 4 + 3 - 6 + 1 and (4 + 3) - (6 + 1). Invite students to tell you how the two expressions are different. If no one brings up the significance of the parentheses, ask what they might mean. If no one is able to share with the class that the operations inside the parentheses must be done first, tell them that.

• Put an equal sign after the expression with parentheses and ask them to find an answer. (0)

(4+3) - (6 + 1) =

7 - 7 = 0

• Ask students how they what they would do first in the other problem. Be sure they understand that addition and subtraction are done from left to right regardless of which comes first.

• Have the find the answer to it. (2)

4 + 3 - 6 + 1 =

7 - 6 + 1 =

1 + 1 = 2

• Write on the board: (5+3) x (1 + 7) = ___

HP:

1. In your head, think of the answer to the problem on the board. (64)

2. Add those digits together. (10)

3. Square that number. (100)

4. Raise a quiet hand if you can tell me how many nickels would be a fair trade for that many pennies. (20)

DEBRIEF:

As above.

The above problem looks like it has a long preview. It might take a little longer than some, but you might only do one head problem a week that requires a little longer preview. If the same concept is included in head problems on several subsequent days, the new learning has a chance to become familiar and easy. A quick reminder is likely to be the only preview that is needed then.

Other mathematical ideas from the standards you might use in your head problems are below.

In your head, think of the...
... volume, area, surface area, or perimeter of ... (Draw and label a shape with its dimensions or give the shape and dimensions orally.)

... the sum of the angles of a quadrilateral, rectangle, square, or triangle.

... the number of sides, faces, vertices or edges on a 2 or 3 dimensional shape.

... the range, median, mode or mean from a graph.

... the number that comes next in a pattern like 2, 4, 1, 3, 0, 2, -1, ___

... the theoretical possibility of getting heads if you toss a regular penny one time.

... (using the equation 3x + 1 = y) the value of y if x = 4.

.... (using a proportion like n/9 = 2/3) the value of n.

... the largest or smallest factor of a number.

... the greatest common divisor (GCD) or greatest common factor (GCF) of two numbers.

All four operations, rounding, doubling, and tripling can all find their ways into your head problems. You will find ways to include fractions, decimals and percents in them too.

It might be helpful to look back at the standards for fourth and fifth grade to get ideas for how head problems might be used to review. And certainly the 6th grade standards are a source for new and continuing learning to include.

When challenging, standards-based head problems are done so that students feel safe and have fun, a surprising amount of math can be covered in less than 10 minutes a day. Your goal might be 5 minute head problems most days.

When your students hear you say, "In your head...," they will get quiet and be ready to listen. You've done it. They're hooked.

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